Thermomechanical Behavior of 

Amorphous Polymers During 

High-Speed Crack Propagation 

by Todd W. Bjerke 

ARL-TR-2793 August 2002

Approved for public release; distribution is unlimited.
NOTICES 

Disclaimers 

The findings in this report are not to be construed as an official Department of the Army 

position unless so designated by other authorized documents. 

Citation of manufacturer’s or trade names does not constitute an official endorsement or 

approval of the use thereof. 

Destroy this report when it is no longer needed. Do not return it to the originator. 
Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-2793 August 2002

Thermomechanical Behavior of 

Amorphous Polymers During 

High-Speed Crack Propagation 

Todd W. Bjerke 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
ii 

Abstract 

The dissipative processes that occur during opening and shear-dominated 

dynamic fracture of amorphous polymers were examined in a combined 

experimental, computational, and analytical investigation. Experiments were 

performed using two materials, nominally brittle polymethyl methacrylate and 

nominally ductile polycarbonate to quantify crack tip heating and identify 

dominant dissipative mechanisms. Shear-dominated dynamic fracture of 

polymethyl methacrylate was found to exhibit heating from craze formation and 

frictional sliding of the fracture surfaces aft of the propagating crack tip. Heating 

in polycarbonate during shear-dominated dynamic fracture was from two 

dissipative processes, the formation of an adiabatic shear band and plastic 

deformation surrounding the propagating crack. Plastic deformation heating 

was noted for opening mode fracture of polycarbonate. Finite-element 

simulations of dynamic crack growth in polycarbonate were performed to isolate 

the heating from thermoplasticity. The simulations indicated that although 

thermoplastic heating does occur, thermofracture heating may be significant. 

Heating from craze formation was observed during opening mode fracture of 

polymethyl methacrylate. A dissipative cohesive zone model was developed to 

predict heating from thermofracture mechanisms associated with polymer 

crazing. The model predictions were consistent with measurements of single 

craze heating during opening mode fracture of polymethyl methacrylate. 
iii

Acknowledgments 

The work reported within this report was performed while on a long term 

training assignment at the University of Delaware. I would like to express my 

gratitude to Professor Lambros, now at the University of Illinois at UrbanaChampagne, for his technical guidance during the course of this research effort. I 

would also like to thank Professor Geubelle from the University of Illinois at 

Urbana-Champagne for the insightful discussions regarding this project. The 

assistance of Dr. Kubair, also from the University of Illinois, is sincerely 

appreciated. There are many people at the U.S. Army Research Laboratory I 

would like to thank for their helpfulness. In particular, Dr. McKnight, Dr. Hsieh, 

and Mr. Dehmer for sharing their insights of polymer mechanics with me, and 

Mr. Hubbard, Mr. Patton, Mr. Moy, Mr. Napadensky, and Mr. Gniazdowski for 

assisting me with equipment and testing needs. 
iv 

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v

Contents 

Acknowledgments iii

List of Figures ix

List of Tables xix

1. Introduction 1

1.1 Background and Motivation...............................................................1

1.2 Literature Review .................................................................................3

1.2.1 Mechanics of Propagating Cracks..........................................3 

1.2.1.1 Cohesive Failure Model .............................................4

1.2.1.2 Shear-Dominated Dynamic Crack Growth .............6

1.2.2 Thermomechanical Coupling .................................................9 

1.2.3 Crack Tip Temperature Measurements ..............................12 

1.3 Research Objectives............................................................................13

2. Experimental Techniques 14

2.1 Material Selection ...............................................................................14

2.2 Experimental Configuration and Impact Event.............................15

2.3 Gas Gun ...............................................................................................19

2.4 Projectile Contact, Crack Initiation, and Crack-Speed 

Measurements.....................................................................................20

2.5 Dynamic Energy Release Rate Measurements...............................25

2.6 Thermal Measurements.....................................................................31

2.6.1 Infrared Detector System and Operation............................35 

2.6.2 Alignment and Focusing Procedure....................................37 

2.6.3 Analysis of Specimen Translation........................................39 

2.6.4 Calibration...............................................................................40 

2.6.5 Sources of Potential Error......................................................44 
vi 

2.6.5.1 Detector Cross Talk...................................................44 

2.6.5.2 Depth of Field ............................................................44 

2.6.5.3 Spherical Aberrations ...............................................44 

2.6.5.4 Surface vs. Internal Heating ....................................46 

2.6.5.5 Temporal Resolution ................................................46 

2.6.5.6 Infrared Emission from Heated Surroundings.....47 

2.6.5.7 Effect of Material Damage on Surface 

Emissivity ..................................................................47 

2.6.5.8 Adiabatic Conditions................................................47 

3. Opening Mode Experiment 48

3.1 Introduction.........................................................................................48

3.2 Crack Tip Speed Measurements.......................................................51

3.3 Dynamic Energy Release Rate Measurements...............................54

3.4 Temperature Field Measurements...................................................56

3.4.1 PC Experiments ......................................................................58 

3.4.2 PMMA Experiments...............................................................63 

3.4.3 Discussion of Temperature Field Features .........................68 

3.5 Thermal Dissipation of Mechanical Work ......................................70

3.6 Fracture Surface Micrography..........................................................74

3.7 Additional Comments on Opening Mode Crack Tip Behavior...77

4. Shear-Dominated Fracture Experiments 80

4.1 Introduction.........................................................................................80

4.2 Crack Tip Speed Measurements.......................................................82

4.3 Temperature Field Measurements...................................................83

4.3.1 PC Experiment........................................................................84 

4.3.2 PMMA Experiments...............................................................87 

4.4 Thermal Dissipation of Mechanical Work ......................................90

4.5 Model Friction Experiments .............................................................92

4.6 Fracture Surface Micrography..........................................................93
vii

4.7 Additional Comments on Shear-Dominated Crack Tip 

Behavior ........................................................................................................98

5. Effect of Plasticity in Thermomechanical Heating of Ductile 

Polymers During Opening-Mode Dynamic Fracture 100

5.1 Introduction.......................................................................................100

5.2 Finite-Element Model ......................................................................101

5.3 Material Model..................................................................................106

5.4 Simulation Results and Discussion................................................110

6. Dissipative Cohesive Zone Model 115

6.1 Introduction.......................................................................................115

6.2 Cohesive Zone Model ......................................................................119

6.2.1 Thermal Dissipation in a Cohesive Zone..........................123 

6.2.2 Cohesive Traction Laws and Stress Distributions ...........126 

6.3 Comparison of Dissipative Cohesive Zone Model With 

Experiments.......................................................................................137

6.3.1 Single Craze Failure Mode ..................................................137 

6.3.2 Distributed Damage Failure Mode ....................................142 

6.4 Further Analysis of Temperature Field Measurements..............147

6.4.1 Flanking Thermal Signal Decay .........................................148 

6.4.2 Cohesive Zone Length.........................................................155 

6.4.3 Effect of Temperature on μ and cp.....................................156 

7. Conclusions 156

7.1 Conclusions .......................................................................................156

7.2 Future Work ......................................................................................159

8. References 161

Appendix A. Closed Form Solution of Cohesive Zone Length 171
viii

Appendix B. Closed Form Solution of Cohesive Zone Opening 

Rate 173

Report Documentation Page 177
ix

List of Figures 

Figure 1. Three modes of crack loading............................................................4 

Figure 2. Schematic illustration of a cohesive zone idealization at the 

tip of a crack to model the inelastic material response. The applied 

cohesive stress, σ(δ), cancels the stress singularity at the crack tip.........5 

Figure 3. Crazing process at the tip of an advancing crack tip. Stresses 

ahead of the crack tip cause molecular chain alignment to occur. 

Further stretching of the material causes the aligned molecules to 

form large bundles called fibrils and the creation of small voids 

between fibrils. The location of molecular chain scission is the 

crack tip. Fibril recoil occurs as the crack faces separate. ........................7 

Figure 4. Asymmetric loading geometries used to generate an 

adiabatic shear band and a shear-dominated crack tip. The left is 

the method introduced by Kalthoff and Winkler (1987). The right is 

the configuration used by Rosakis et al. (1999) and Ravi-Chandar et 

al. (2000)............................................................................................................8 

Figure 5. Low-rate tensile stress-strain curves for high molecular 

weight PMMA showing rate sensitivity of Young’s modulus and 

ultimate strength. Two tests were performed for each strain rate. 

Brittle failure occurred during each test. ...................................................16 

Figure 6. Low-rate tensile stress-strain curves for PC showing strainrate sensitivity of yield strength. Young’s modulus was not rate 

sensitive. Two tests were performed for each strain rate.......................16 

Figure 7. Opening mode dynamic fracture experiment configuration 

also showing infrared detector coverage and crack propagation 

gage. Crack propagates from right to left.................................................18 

Figure 8. Shear-dominated dynamic fracture experiment configuration 

also showing infrared detector coverage and painted wires for 

coarse crack speed measurement. Crack propagates from left to 

right.................................................................................................................18 

Figure 9. Photograph of gas gun, impact chamber, and infrared 

detector system..............................................................................................20 

Figure 10. Projectile velocity measurement system using two infrared 

break-beams at the muzzle of the launch tube. Projectile interrupts 

infrared beam, causing infrared detectors to change response..............21
x 

Figure 11. Crack propagation gage constructed from thin lines of 

conducting silver paint. Top illustration shows gage configuration 

with the crack propagating from left to right, severing the painted 

lines sequentially. Bottom photograph is of a completed gage on a 

PC fracture specimen. Crack initiation gage is also shown...................23 

Figure 12. Schematic of the crack propagation speed measurement 

circuit using painted silver lines on the polymer test specimen. The 

propagating crack sequentially severs the painted lines, changing 

the overall resistance of the circuit. ............................................................24 

Figure 13. Strain gage coordinate system showing the gage orientation 

angle, α. Strain gage is fixed in space and the coordinate system 

origin is attached to the propagating crack tip that moves from left 

to right. ...........................................................................................................26 

Figure 14. Coordinate system used in the constant crack tip speed 

analysis for determining the dynamic energy release rate from a 

strain gage. Coordinate system origin is at the crack tip and 

propagates with the crack............................................................................28 

Figure 15. Effect of the strain gage orientation on the location of peak 

strain as the crack propagates past the gage at 400 m/s in PMMA. 

Crack tip is located at x = 0. Acute angle results in the peak strain 

occurring 5 mm prior to crack passage, whereas the obtuse angle 

gives a peak strain when the crack is directly beneath the gage. ..........29 

Figure 16. Sensitivity of strain gage response to crack tip speed for 

PMMA. Obtuse angle orientation is used. ...............................................30 

Figure 17. Position and orientation of strain gages used to measure the 

dynamic energy release rate during opening mode dynamic 

fracture. Angle shown is for PMMA (119° was used for PC). Crack 

propagates from left to right. ......................................................................32 

Figure 18. Planck distribution for a black body emitter at several 

temperatures showing the shift in emitted radiation wavelength as 

the body is heated. ........................................................................................34 

Figure 19. Spectral responsivity of several common photon detectors 

(Zehnder and Rosakis 1993). .......................................................................34 

Figure 20. Optical configuration used to measure the temperature field 

surrounding a propagating crack tip. Infrared detector array 

coverage is also shown.................................................................................36 

Figure 21. Hot-wire jig used for alignment and focusing the infrared 

detector system. The 76-μm-diameter chromel-A wire serves as an 

infrared radiation source by passing current through it.........................38
xi

Figure 22. Finite-element mesh and boundary conditions used for 

numerical simulation of the projectile impacting the polymer 

specimen. Steel projectile moves from left to right. ................................41 

Figure 23. Finite-element results showing the horizontal displacement 

of a point 10 mm ahead of the initial notch tip due to projectile 

impact at various speeds..............................................................................41 

Figure 24. Typical calibration for one channel of the infrared detector 

system showing results for PMMA and PC..............................................43 

Figure 25. Typical calibration for one channel of the infrared detector 

system showing results for PMMA and PC using a grooved 

specimen. Groove was polished with 2000-grit sandpaper to 

achieve smooth finish...................................................................................43 

Figure 26. Infrared detector element cross talk observed using a 76-μm 

heated wire. Cross talk of 7%–10% was recorded...................................45 

Figure 27. Detector output measured as the detector array was 

translated in and out of focus to obtain an estimate of the infrared 

measurement system depth of focus..........................................................45 

Figure 28. Temperature increase of several x locations below the 

surface of a semi-infinite PMMA solid with a suddenly applied 

surface temperature. Initial ambient temperature of solid was 293 

K and suddenly applied constant surface temperature was 393 K. ......49 

Figure 29. Typical crack propagation gage circuit output. The abrupt 

increases in voltage are generated by the painted wires being 

severed by the propagating crack...............................................................52 

Figure 30. Crack tip position history obtained from the crack 

propagation gage timing data for PMMA. Two different 

experiments are shown, corresponding to different projectileimpact speeds. ...............................................................................................52 

Figure 31. Crack tip propagation speed in PMMA for two experiments 

with different projectile-impact speeds. Propagation speed was 

determined from crack propagation gage timing data and the 

distance between the painted wires. Error bar shows typical 

measurement uncertainty for crack tip speed (estimated from 

Lambros and Rosakis [1995]). .....................................................................53 

Figure 32. Opening mode crack tip speeds for PC and PMMA as 

measured by the crack propagation gage for experiments with a 

projectile-impact speed of 40 m/s. .............................................................54 

Figure 33. Measured strain from the strain gage circuits for a crack 

propagating at 0.55 cR in PC. Distance between strain gages was 

8 mm. ..............................................................................................................55
xii 

Figure 34. Measured strain from the strain gage circuits for a crack 

propagating at 0.55 cR in PMMA. Separation between strain gages 

was 8 mm. ......................................................................................................55

Figure 35. Measured strain from the strain gage circuits for a crack 

propagating in PMMA. The crack accelerated to a terminal speed 

of ~450 m/s....................................................................................................56 

Figure 36. Dynamic energy release rate for various crack tip speeds in 

PC and PMMA. .............................................................................................57 

Figure 37. Thermal detector output (after calibration) for opening 

mode fracture of PC with a crack tip speed of 0.55 cR. ............................59 

Figure 38. Thermal detector output from two detector elements for an 

opening mode crack propagating in PC at a terminal speed of 0.55 

cR. Thermoelastic cooling is observed prior to dissipative crack tip 

heating. ...........................................................................................................59 

Figure 39. Thermal detector output (after calibration) for opening 

mode fracture of PC with a crack tip speed of 0.45 cR. ............................61 

Figure 40. Temperature contours for opening mode fracture of PC 

with a crack tip speed of 0.55 cR. The large lateral span on the heataffected region suggests plastic deformation is a significant heating 

mechanism near the propagating crack tip...............................................62 

Figure 41. Temperature contours for opening mode fracture of PC 

with a crack tip speed of 0.45 cR. The lateral span on the heataffected region is smaller than the higher speed crack tip 

experiment. ....................................................................................................62 

Figure 42. Thermal detector output (after calibration) for opening 

mode fracture of PMMA with a crack tip speed of 0.55 cR. ....................64 

Figure 43. Thermal detector output (after calibration) of opening mode 

fracture of PMMA with the crack tip propagating between two 

detector elements. .........................................................................................65 

Figure 44. Thermal detector output (after calibration) for opening 

mode fracture of PMMA with a crack tip speed of 0.24 cR. ....................65 

Figure 45. Thermal detector output (after calibration) for opening 

mode fracture of low molecular weight PMMA with a crack tip 

speed of 0.24 cR. .............................................................................................66 

Figure 46. Thermal detector output (after calibration) for opening 

mode fracture of PMMA with the detector array rotated to the 

horizontal position. Crack tip may not have passed directly 

through the region monitored by the infrared detector system. 

Crack tip propagation speed is 0.57 cR.......................................................67
xiii

Figure 47. Temperature contours for opening mode fracture of PMMA 

with a crack tip speed of 0.55 cR. The contours are highly localized 

compared to those of PC, suggesting that plastic deformation does 

not contribute to the crack tip temperature increase. ..............................68 

Figure 48. Rate of plastic work contours for opening mode fracture of 

PC with a crack tip speed of 0.55 cR. The heat source was assumed 

to be only ductile plastic deformation with a β of 0.5. ............................72 

Figure 49. Rate of plastic work contours for opening mode fracture of 

PC with a crack tip speed of 0.45 cR. The heat source was assumed 

to be only ductile plastic deformation with a β of 0.5. The span of 

the contours are smaller than for the higher crack tip speed 

experiment, illustrating the influence of crack propagation speed. ......72

Figure 50. Rate of internal heat generation contours for opening mode 

fracture of PMMA with a crack tip speed of 0.55 cR. Internal 

heating was assumed to be the only source of crack tip heating...........73 

Figure 51. Photograph of PC fracture surface showing unique regions 

of smooth and rough surface texture. ........................................................75 

Figure 52. SEM images of the smooth and rough regions of a PC 

fracture surface. The crack propagated from the bottom of the 

figure to the top.............................................................................................76 

Figure 53. Photograph of a PMMA fracture surface showing the 

transition of surface roughness and eventual branching........................77 

Figure 54. SEM image of the smooth and rough regions of a highspeed crack growth PMMA fracture surface. The crack propagated 

from the bottom of the figure to the top. The rough region has a 

surface roughness of about 100–200 μm....................................................78 

Figure 55. Photograph of a low-speed crack growth PMMA fracture 

surface showing a glassy smooth surface corresponding to the 

passage of a single craze. .............................................................................79 

Figure 56. Opening mode maximum crack tip temperature as 

dependent on crack tip speed. ....................................................................79 

Figure 57. Empirical relationship between maximum crack tip 

temperature and dynamic energy release rate for opening mode 

fracture of PMMA.........................................................................................81 

Figure 58. Typical output from the break-wire circuit used to 

determine the crack tip speed for shear-dominated dynamic 

fracture of PMMA. Output from the crack initiation and two crackspeed measurement circuits is shown. ......................................................83
xiv 

Figure 59. Typical thermal detector output (after calibration) for sheardominated fracture of PC with side grooves to suppress crack 

kinking. ∆T denotes temperature change from ambient, and t = 0 

corresponds to time of impact.....................................................................85

Figure 60. Temperature history of an adiabatic shear band and 

arrested crack in a PC specimen without side grooves. ∆T denotes 

temperature change from ambient, and t = 0 corresponds to time of 

impact. ............................................................................................................86

Figure 61. Temperature contours around crack tip region for sheardominated fracture of PC. Temperature increase from room 

temperature (K). Steady-state crack growth has been assumed. ..........87 

Figure 62. (a) Thermal detector output (after calibration) for sheardominated fracture of PMMA with side grooves to suppress crack 

kinking. ∆T denotes temperature change from ambient and t = 0 

corresponds to time of impact. (b) Initial portion of thermal signal 

from shear-dominated fracture of PMMA with side grooves. 

Temperature field surrounding crack tip is shown. ∆T denotes 

temperature change from ambient and t = 0 corresponds to time of 

impact.. ...........................................................................................................88 

Figure 63. Temperature contours at crack tip region for sheardominated fracture of PMMA. Temperature increase from room 

temperature (K). Steady-state crack growth has been assumed. ..........89 

Figure 64. Plastic work rate contours at crack tip region for sheardominated fracture of PC. β = 0.5 and steady-state crack growth have 

been assumed. .....................................................................................................90 

Figure 65. Rate of internal heat generation contours at crack tip region 

for shear-dominated fracture of PMMA. Negligible plastic work 

rate and steady-state crack growth have been assumed.........................91 

Figure 66. Configuration for high-speed friction experiments. Infrared 

detector monitored the sliding interface between the top and 

middle PMMA pieces...................................................................................94 

Figure 67. Typical thermal detector output (after calibration) for highspeed friction experiments. ∆T denotes temperature change from 

ambient and t = 0 corresponds to time of impact.....................................94 

Figure 68. Photograph of fracture surface from shear-dominated 

dynamic fracture of PC with side grooves to suppress crack 

kinking. Fracture surface ripples are visible on the crack face..............95 

Figure 69. Photograph of fracture surface from shear-dominated 

dynamic fracture of PMMA with side grooves to suppress crack 

kinking. White-colored wear layer is visible on the crack face.............96
xv

Figure 70. SEM image of fracture surface from shear-dominated 

dynamic fracture experiment using PC with side grooves. The 

crack propagated from left to right leaving surface ripples covered 

with a molten coating...................................................................................96 

Figure 71. SEM image of fracture surface from shear-dominated 

dynamic fracture using PMMA with side grooves. Left-side image 

was taken in the interior of the panel. Right-side image is near 

location where infrared detector monitored. The crack propagated 

from left to right. ...........................................................................................97 

Figure 72. SEM image of PMMA high-speed friction surface. The 

mating surface slid over the surface shown from left to right. 

Deposits are visible in the friction surface. The original surface 

condition prior to the experiment is shown on the right. .......................98 

Figure 73. Measured temperature field surrounding a crack tip 

propagating in PC at 550 m/s under mode I loading. ..........................102 

Figure 74. Finite-element mesh used for the numerical simulations. 

Element size in the crack tip region is 80 μm. The plane-stress model 

has 77,600° of freedom...............................................................................104 

Figure 75. Dynamic crack initiation time measured during the 

opening-mode fracture experiments using PC for various projectileimpact speed................................................................................................105 

Figure 76. Crack tip position history measured during dynamic 

opening-mode crack growth in PC. Position history was used for 

timing the sequential release of nodes along the plane of symmetry 

to simulate crack tip propagation in the numerical simulation...........105 

Figure 77. Nominal stress-strain data for dynamic compression of PC 

reported by Li and Lambros (2001). Results from three different 

strain rates are shown.................................................................................107 

Figure 78. Approximation of the plastic part of the nominal stressstrain curve for PC at a strain rate of 1200 s-1 as used in the 

numerical simulation material model......................................................109 

Figure 79. Approximation of the plastic part of the nominal stressstrain curve for PC at a strain rate of 1700 s-1 as used in the 

numerical simulation material model......................................................109 

Figure 80. Approximation of the plastic part of the nominal stressstrain curve for PC at a strain rate of 2200 s-1 as used in the 

numerical simulation material model......................................................109 

Figure 81. Cauchy (true) stress as a function of plastic logarithmic 

strain for three different strain rates of dynamic compression for 

PC. .................................................................................................................110 
xvi 

Figure 82. Temperature increase contours at the tip of a crack tip 

propagating in PC at a speed of 550 m/s predicted from the 

numerical simulation. Adiabatic heating was due solely to plastic 

work. .............................................................................................................111

Figure 83. Temperature contours surrounding a crack tip propagating 

in PC at 550 m/s under mode I loading. .................................................112 

Figure 84. Temperature increase of a vertical array of finite elements 

located 3.2 mm from the initial notch tip for comparison to infrared 

detector measurements. .............................................................................113 

Figure 85. The initiation, widening, and breakdown of a polymer 

craze during opening-mode fracture. Aligned molecular chains 

form thick fibrils that resist crack opening..............................................116 

Figure 86. Cohesive zone concept at the tip of a propagating crack. 

Crack face opening, δ, is resisted by a cohesive traction σ ...................118 

Figure 87. Far field and near tip contours used for calculating energy 

flux into a crack tip with a cohesive zone of length L (Freund 1990)..120 

Figure 88. Heat source smaller than the infrared detector element 

passing over the region monitored by the detector. The element 

responds to the average radiation incident on the entire area 

monitored.....................................................................................................125 

Figure 89. Illustration of intrinsic and extrinsic cohesive traction laws 

(Kubair and Geubelle 2001). ......................................................................127 

Figure 90. Form of constant and linear decay stress distributions used 

in the dissipative cohesive zone model. ..................................................128 

Figure 91. Temperature increase predictions made with the dissipative 

cohesive zone model using the constant stress and linear decay 

stress distributions. Baseline material properties for PMMA and a 

crack propagation speed of 300 m/s were used. Ambient 

temperature T0 is 295 K. .............................................................................129 

Figure 92. Calculated form of the constant stress and linear decay 

traction laws obtained from matching cohesive zone stress 

distributions.................................................................................................131 

Figure 93. Temperature increase predictions using a linear decay σ(ξ) 

relation from parametric study with E varied........................................132 

Figure 94. Internal heating rate predictions using a linear decay σ(ξ)

relation from parametric study with E varied........................................133 

Figure 95. Temperature increase predictions using a linear decay σ(ξ)

relation from parametric study with G varied........................................133 

Figure 96. Internal heating rate predictions using a linear decay σ(ξ) 

relation from parametric study with G varied........................................134 
xvii

Figure 97. Temperature increase predictions using a linear decay σ(ξ) 

relation from parametric study with σc varied. ............................................134

Figure 98. Internal heating rate predictions using a linear decay σ(ξ) 

relation from parametric study with σc varied.......................................135 

Figure 99. Temperature increase predictions using a linear decay σ(ξ)

relation from parametric study with a& varied. ......................................135 

Figure 100. Internal heating rate predictions using a linear decay σ(ξ)

relation from parametric study with a& varied. ......................................136 

Figure 101. Leading portion of experimentally measured temperature 

signal from two different PMMA opening-mode fracture 

experiments with crack tip speeds of 300 m/s. Inset at top left 

shows a longer time span of the signals. .................................................138 

Figure 102. Rate of internal heating derived from dynamic openingmode fracture of PMMA. The crack tip speeds were 300 m/s, and 

the experimental signals are from the peak detector.............................139 

Figure 103. Comparison of the temperature increase predicted by the 

dissipative cohesive zone model with experimental data for 

opening-mode fracture of PMMA with a crack speed 300 m/s. 

Constant stress and linear decay stress distributions were used in 

the model......................................................................................................140

Figure 104. Form of the quad-high stress distribution compared to the 

constant stress and linear decay stress distribution...............................141 

Figure 105. Calculated shape of the quad-high traction law. The 

constant stress and linear decay traction laws are shown for 

comparison. The baseline parameters for 300 m/s crack growth in 

PMMA were used. ......................................................................................141

Figure 106. Temperature increase predicted by the dissipative 

cohesive zone model compared to experimental data for 300 m/s 

crack growth in PMMA..............................................................................142 

Figure 107. Internal heating predicted by the dissipative cohesive zone 

model compared to experimental data for 300 m/s crack growth in 

PMMA. .........................................................................................................143

Figure 108. Leading portion of experimentally measured temperature 

signal from three different PMMA opening mode fracture 

experiments with crack tip speeds of 300, 400, and 600 m/s. Inset 

at top left shows a longer time span of the signals. Peak detector 

only shown for 400 and 600 m/s experiments. ......................................144 

Figure 109. Rate of internal heating derived from dynamic opening 

mode fracture of PMMA. The crack tip speeds were 300, 400, and 

600 m/s, and the experimental signals are from the peak detector. ...144
xviii

Figure 110. Temperature increase predicted by the dissipative 

cohesive zone model compared to experimental data for 400 m/s 

crack growth in PMMA..............................................................................145 

Figure 111. Temperature increase predicted by the dissipative 

cohesive zone model compared to experimental data for 600 m/s 

crack growth in PMMA..............................................................................145

Figure 112. Comparison of the internal heat rate predicted by using 

the quad-high stress distribution in the dissipative cohesive zone 

model with experimental data for low, mid, and high crack tip 

speed fracture of PMMA............................................................................146 

Figure 113. Distribution of temperature among the vertically oriented 

infrared detector elements at the time of peak temperature. ...............148 

Figure 114. Contours of temperature increase in PMMA for a 

propagating crack tip and trailing thermal wake. Crack tip 

propagation speed is ~300 m/s. The crack opening displacement is 

evident by the shape of the trailing wake region. ..................................149 

Figure 115. Depiction of craze thickening, splitting, and crack opening 

relative to the stationary infrared detector elements for a craze 

passing through the region monitored by an element. Bottom of 

figure is representative data for the configuration illustrated. ...........150 

Figure 116. Depiction of craze thickening, splitting, and crack opening 

relative to the stationary infrared detector elements for a craze 

passing between two detector elements. Bottom of figure is 

representative data for the configuration illustrated.............................152 

Figure 117. Depiction of crack opening relative to the stationary 

infrared detector elements for a craze passing below the detector 

element array. Bottom of figure is representative data for the 

configuration illustrated. ...........................................................................153 

Figure 118. Crack face opening speed for fracture of PMMA with a 

crack tip speed of 300 m/s. Experimental data derived from 

temperature measurements are compared to analytical expression 

from Freund (1990). ....................................................................................154 

Figure 119. Variation of density with absolute material temperature 

for PMMA (Domininghaus 1988). ............................................................157 

Figure 120. Variation of specific heat with material absolute 

temperature for PMMA (Domininghaus 1988). .....................................157 
xix

List of Tables 

Table 1. Rate effects on elastic moduli of PMMA and PC............................17 

Table 2. Polymer properties..............................................................................17 

Table 3. Uniaxial high strain-rate yield stresses and fraction of plastic 

work converted to heat (β) from Li and Lambros (2001). .....................107 

Table 4. Parameter values used for the parametric study..........................131 
xx 

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1

1. Introduction 

1.1 Background and Motivation 

Polymeric materials are finding increased use in applications that subject the 

materials to intense, transient loadings. Examples of this include combat aircraft 

canopies that must withstand bird impacts and transparent face shields used by 

explosive ordnance disposal teams that cannot be perforated by small fragments. 

The design and evaluation of these devices is typically done through 

experimentation and numerical modeling because of the complexities of the 

transient loading and failure processes. The cost of performing extensive, 

detailed experiments on complex systems is becoming prohibitive, leading to an 

increased reliance on numerical simulations. Because of this, the modeling of 

localized failure in materials subject to extreme loading conditions has received 

increased attention during the past few years. In particular, emphasis has been 

placed on developing suitable crack- and shear-band initiation, growth, and 

arrest models for finite-element implementation. Key to achieving this is a 

thorough understanding of the failure event itself. This task is further 

complicated for failure events that are highly transient because of the small time 

scales involved. 

As an elastic-plastic material undergoes plastic deformation, a portion of the 

work done on this material is stored internally, and the remainder is dissipated, 

usually in the form of heat. If the deformation event is rapid compared to the 

time required for conduction to take place, significant temperature increases may 

result, thus affecting material response. As such, the coupling between the 

deformation mechanics and thermal heating for high-loading rate events must be 

accounted for in any modeling. This is especially true for materials of 

low-thermal diffusivity such as polymers because the heat generated during 

localized deformation does not have time to conduct to surrounding regions. In 

addition, polymers are more susceptible to thermal effects than metals or 

ceramics in the sense that they exhibit larger stiffness and strength variations 

over smaller temperature changes. Thus, not only must a constitutive and failure 

model account for rate effects, but it must also account for the effects of 

temperature changes on material properties from thermomechanical heating. 

During fracture, a process zone exists ahead of the propagating crack tip, in 

which the physical mechanisms associated with failure occur. If the material is 

brittle, the formation of new surfaces occurs within the process zone. The energy 

required for surface creation is twice (because two surfaces are created during 

fracture) the material’s surface energy, γ, which is a material property. 

Surprisingly, studies have shown that crack tip heating does occur in what are 
2 

often thought of as brittle polymers during dynamic fracture (Döll 1976; Fuller 

et al. 1975), implying that additional energy beyond that needed for the creation 

of new surfaces is required to fracture such brittle materials. Sun and Hsu (1996) 

noted that the energy for opening mode fracture of a brittle material can in some 

cases be several orders of magnitude larger than 2γ,. The excess energy is used in 

the process zone for microstructure breakdown and molecular bond scission and 

is eventually dissipated as heat. Because crack tip speeds during dynamic 

fracture are a significant fraction of the material dilatational wave speed, the 

generated heat does not have time to conduct to surrounding regions. This 

condition results in a temperature increase inside the process zone of the 

material. For a brittle polymeric material (i.e., one of low thermal diffusivity), a 

high crack tip temperature can lead to thermal softening that blunts the 

advancing crack tip and effectively “toughens” the material. For a ductile 

material, plastic deformation surrounds the crack tip process zone, contributing 

to the overall temperature increase. Thus, the temperature field surrounding a 

propagating crack is related to the dissipation of mechanical work that went into 

deformation and failure of the material. Measurement of this field at sufficient 

resolution can provide useful insight into the physical mechanisms within the 

process zone. Additionally, a detailed description of mechanical work 

dissipation will enable a crack tip failure law, or cohesive law, to be developed. 

Additional motivation for measuring the temperature field around a propagating 

crack is found in the shear-dominated fracture experiments of Ravi-Chandar 

et al. (2000) and Rosakis et al. (1999). These experiments, discussed in detail in 

the next section and in section 4, used high-speed photography to examine the 

stress field surrounding a propagating shear crack in brittle polymers. The 

resulting fracture surfaces from the Ravi-Chandar et al. (2000) experiments 

showed distinct evidence of polymer melting: thin strands of material were 

drawn along the crack surface. The stress fields recorded by Rosakis et al. (1999) 

showed evidence of frictional crack surface contact aft of the advancing crack tip. 

It is not clear from these studies if the melted strands on the fracture surface were 

from frictional sliding or from the fracture process itself, much less what the 

dissipation mechanisms are for this mode of fracture. Temperature 

measurements during shear-dominated dynamic fracture will provide additional 

details and insight into these processes. 

In the present work, the temperature field surrounding high-speed cracks in two 

amorphous polymers were studied. Polymethyl methacrylate (PMMA), a 

nominally brittle material commonly known as acrylic, and polydian carbonate, 

a nominally ductile material known as polycarbonate (PC), were used in the 

study. Two different modes of dynamic fracture were examined, opening and 

shear-dominated. Measurements of crack tip speed, dynamic fracture toughness, 

and temperature surrounding the propagating crack were made to establish the 

dissipative nature of the process zones for each material and mode of fracture. 
3

Microscopy analyses of the resulting fracture surfaces provided additional 

physical insight into the dissipation mechanics involved. The contribution of 

heating from plastic deformation to the overall temperature field for the ductile 

polymer was quantified in a combined experimental-numerical effort. The body 

of experimental data (insights gained during the study) and measurements of the 

dynamic energy release rate during fracture led to the development of a 

dissipative cohesive material model applicable to opening mode fracture of 

PMMA. In the next section, a literature review is presented to introduce many of 

the concepts and to survey past studies of crack tip heating. 

1.2 Literature Review 

1.2.1 Mechanics of Propagating Cracks 

Asymptotic Fields 

The two-dimensional (2-D) asymptotic stress field surrounding a crack tip 

propagating dynamically with constant speed is given as (Freund 1990) 

( , ) , 2 ( , ) 2 ( , ) 2 ~ a

r

K

a

r

K

a

r

K III

ij

d

II III

ij

d

I II

ij

d

I

ij & & θ & π

θ

π

θ

π

σ Σ + Σ + Σ (1) 

for r → 0, where (r,θ) are polar coordinates in the plane of deformation centered 

at the crack tip, d KI , d KII , and d KIII are scalar multipliers called the dynamic 

stress intensity factors corresponding to mode I, mode II, and mode III loading 

respectively, a& is the crack tip speed, and ( , a) ij Σ θ & are known angular functions 

for each mode of fracture. The three modes of loading that can be applied to a 

2-D crack are shown in Figure 1. In this study, mode I and predominantly mode 

II fractures are examined. A significant aspect of equation (1) is the stress 

singularity that develops near the crack tip. In real material systems, stress 

singularities do not exist, the material near the crack tip deforms and fails, 

eliminating the stress singularity. 

It is often more convenient to work with energy quantities instead of stress or 

deformation. The dynamic energy release rate, G, is defined as the rate of 

mechanical energy that flows into the advancing crack tip per unit crack 

advance. Freund (1990) derived an expression for plane stress relating the 

dynamic energy release rate to the dynamic stress intensity factors for linear 

elastodynamic crack growth as 

( ) ( ) ( ) , 2

( ) 2 ( ) 2 ( ) 2 d

III

d III

II

d II

I

I K A a K

E

A a K

E

A a G

μ

& & & = + + (2) 

, 

,
4 

Figure 1. Three modes of crack loading. 

where μ is the shear modulus, E is Young’s modulus, and AI, AII, and AIII are 

universal functions that do not depend on loading or body geometry. 

Expressions for the universal functions can be found in Freund (1990). The 

energy release rate is essentially the amount of energy that must be supplied to 

the crack tip for it to propagate at a given speed and includes the energy needed 

to create free surfaces as well as that needed to alter the material at the macroscale (e.g., plasticity) and microscale levels (e.g., molecular chain realignment). 

Döll (1976) and Sharon et al. (1996) found G to increase nonlinearly with 

crack tip speed for mode I experiments using PMMA. These studies showed G

to asymptote toward infinity when the crack tip speed approached 60% of the 

material’s Rayleigh wave speed, cR. Zehnder and Rosakis (1990) have shown the 

limiting crack tip speed in 4340 steel to also be approximately 0.6 cR. When an 

attempt is made to drive the crack faster, the single crack will branch, or 

bifurcate symmetrically, into two cracks because of a change in the maximum 

elastodynamic hoop stress direction (Yoffe 1951), thereby splitting the available 

energy between two crack tips. Sharon et al. (1996) observed short microcracks, 

which they termed failed branching, that increased in occurrence as G was 

increased. It was reasoned that this accounted for the nonlinear relationship 

between G and crack tip speed. 

1.2.1.1 Cohesive Failure Model 

The notion of a cohesive zone at the tip of a crack was first introduced by 

Barenblatt (1959) and Dugdale (1960) to analytically eliminate the stress 

singularity at the tip of a crack under mode I loading. Freund (1990) extended 

the cohesive zone idea to the dynamic crack growth case. A cohesive zone is 

introduced as a line of length L that extends from the crack tip to model inelastic 

Mode I Mode II Mode III
5

material behavior ahead of the crack tip (as shown in Figure 2). Cohesive 

stresses act along the cohesive line to resist crack opening. The stress field 

outside this region has a stress intensity factor KI appl. Within this region, the 

opening of the crack is resisted by a cohesive stress denoted σ(δ), where δ is the 

opening displacement within the cohesive zone. The cohesive traction leads to a 

negative stress intensity factor KI coh at the crack tip. The total stress intensity 

factor is the sum of the applied and cohesive stress intensity factors. The stress 

singularity is eliminated by selecting a cohesive zone length, L, such that the 

combined stress intensity factor is zero at the crack tip. The expression relating 

the cohesive stress to the cohesive zone length for plane stress conditions is given 

by Freund (1990) as 

[ ] , 2 ( ) ( ) ( )

2

0 0

∫ ∫ 



 



 =

c L

I dl

l

l

E

A a d

δ

σ δ

π

σ δ δ & (3) 

where δc is the maximum opening displacement in the cohesive zone (see 

Figure 2). Furthermore, the cohesive stress is related to the dynamic energy 

release rate as 

( ) .

0

∫ =

c

G d

δ

σ δ δ (4) 

Figure 2. Schematic illustration of a cohesive zone idealization at the tip of a crack to 

model the inelastic material response. The applied cohesive stress, σ(δ), 

cancels the stress singularity at the crack tip. 

The cohesive zone notion is ideal for the present study because the cohesive zone 

is by design the only region where mechanically dissipative processes occur. 

Thus, if the temperature field can be measured along with G, the necessary 

ingredients are in hand to assemble a meaningful cohesive model based on the 

actual dissipation from a dynamic fracture event. This is in contrast to earlier 

cohesive models (Tvergaard and Hutchinson 1992; Needleman 1992; Xu and 

,

.

σc

L

δc

Crack Faces Crack

Tip

Cohesive

Zone

Tip

σ(δ)
6 

Needleman 1994; Tvergaard and Hutchinson 1996; Camacho and Ortiz 1996) 

where the traction-separation relation σ(δ) was based on theoretical arguments 

or postmortem fracture surface observations and not on experimental 

measurements taken during the decohesion process. 

The cohesive zone concept is also attractive for this study because the 

phenomenon of crazing that causes brittle fracture in amorphous polymers 

conceptually resembles a cohesive zone. Amorphous polymers consist of long 

molecular chains randomly oriented and entangled. Their mechanical response 

is governed by two types of bonds at the microscopic level: covalent bonds 

between the atoms themselves and van der Waals forces between molecule 

segments. The viscoelastic behavior of polymers is a direct result of the forces 

between the entangled molecular chains. As an amorphous polymer cools from 

the liquid state, it remains viscous until it reaches the glass transition 

temperature, Tg, at which time the relative motion of the molecules becomes 

restricted. Amorphous polymers do not have a defined melt temperature. 

Unlike crystalline materials, there is not a threshold temperature that separates 

fluid and solids states. When heated beyond Tg, the material becomes less 

viscous until it eventually reaches a fluidlike state. 

Polymer crazing is the disentanglement and alignment of molecular chains into 

large fibrils, or groups of chains, as a result of high-tensile loading. Voids form 

between the fibrils due to the relatively weaker van der Waals forces. Kambour 

(1964, 1965, 1966a, 1966b) observed the formation of crazes at the tip of statically 

grown cracks in glassy polymers using optical methods. Subsequent studies 

have concluded that brittle fracture in polymers is the result of craze formation 

followed by fibril breakdown (molecular chain scission) and fibril stub recoil 

(Beahan et al. 1973; Murray and Hull 1970; Kramer 1983; Kramer 1984; Yang 

et al. 1986; Henkee and Kramer 1984; Donald and Kramer 1982; Berger 1990). 

Figure 3 depicts the craze formation process ahead of a propagating crack tip. 

When a polymer well below the glass transition temperature is subjected to an 

imposed strain, a competition between classic shear yielding and craze formation 

occurs. A large hydrostatic tensile component in the stress tensor is conducive to 

crazing, while shear yielding is favored by a large deviatoric stress component 

(Anderson 1995). Ravi-Chandar (1995) found the competition between plasticity 

and craze formation to be load-rate sensitive, with craze formation favored at 

high load rates. 

1.2.1.2 Shear-Dominated Dynamic Crack Growth 

There recently have been numerous studies on the mechanics of 

shear-dominated dynamic fracture of materials. Typically, mode II loading will 

cause a crack to immediately kink at an angle of approximately 70° relative to the 
7

Figure 3. Crazing process at the tip of an advancing crack tip. Stresses ahead of the crack 

tip cause molecular chain alignment to occur. Further stretching of the 

material causes the aligned molecules to form large bundles called fibrils and 

the creation of small voids between fibrils. The location of molecular chain 

scission is the crack tip. Fibril recoil occurs as the crack faces separate. 

original crack direction. In doing so, the crack propagates under locally mode I 

conditions. This is due to the fracture toughness for mode I being lower than

mode II. The fracture mechanisms associated with mode I dynamic crack growth 

have lower energy requirements than do those associated with mode II. 

Broberg (1987) found the mode I toughness of PMMA to be 2.5 times lower than 

the mode II toughness. The kink angle of approximately 70° was found by 

Erdogan and Sih (1963) to be the result of the maximum hoop stress direction for 

shear-loading conditions. 

There are generally three ways to suppress crack kinking during sheardominated crack loading. The first method is to apply sufficient in-plane 

compressive tractions to eliminate the possibility of opening mode cracking. 

This was demonstrated by Broberg (1987) who successfully grew mode II cracks 

in PMMA. The second method is to exploit a failure mode transition that occurs 

in ductile materials subjected to dynamic shear loading. This was demonstrated 

by Kalthoff and Winkler (1987) for steel and Ravi-Chandar (1995) for 

Voids

Aligned

Molecules

Crack

Face

Recoiling

Fibril

Stubs

Fibrils

Aligned

Molecules

Crack Propagation

Direction

Amorphous

Region

Voids

Open

Space
8 

PC. These studies used the asymmetric impact-loading geometry introduced by 

Kalthoff and Winkler (1987) and shown in Figure 4, to generate a dynamic shear 

crack. For ductile materials, a failure mode transition from mode I crack kinking 

to highly localized adiabatic shearing, i.e., adiabatic shear band formation (as 

defined in Duffy [1984]), was found to occur when the impact speed of the 

projectile exceeded a threshold value. This transition did not occur in brittle 

materials, crack kinking occurred regardless of projectile impact speed. 

Figure 4. Asymmetric loading geometries used to generate an adiabatic shear band and a 

shear-dominated crack tip. The left is the method introduced by Kalthoff and 

Winkler (1987). The right is the configuration used by Rosakis et al. (1999) and 

Ravi-Chandar et al. (2000). 

The third method for suppressing crack kinking is to introduce a path of 

weakened material, effectively reducing the mode II fracture toughness. This 

approach has been used successfully by Rosakis et al. (1999) and Ravi-Chandar 

et al. (2000), and is the method used in this study. A geometric weakening was 

implemented by Ravi-Chandar et al. (2000). Grooves were machined along the 

sides of edge-cracked panels of PMMA, providing a preferred crack propagation 

path. Rosakis et al. (1999) on the other hand were able to generate sheardominated fracture of weakly bonded Homalite 100 panels. Transient sheardominated loading was achieved using a slight modification of the Kalthoff and 

Winkler (1987) asymmetric loading configuration, also shown in Figure 4. 

High-speed photoelasticity was used by Ravi-Chandar et al. (2000) to measure 

the shear stresses in panels of PMMA. Rosakis et al. (1999) used the method of 

coherent gradient sensing (Rosakis 1993) to record the maximum in-plane 

principal stresses surrounding the propagating shear crack in the Homalite 100 

panels. In both cases, intersonic crack tip speeds, i.e., lying between the shear 

and dilatational wave speeds, were observed. This aspect of shear crack 

Projectile 

Projectile

Loading Geometry of 

Kalthoff and Winkler (1987)

Loading Geometry of 

Rosakis et al . (1999) and 

Ravi-Chandar et al . (2000) 
9

propagation will be discussed further in section 4. The fracture surfaces from the 

PMMA experiments of Ravi-Chandar et al. (2000) showed signs of polymer 

melting. Thin strands of material were deposited on the fracture surface oriented 

along the crack propagation direction. The stress field images of Rosakis et al. 

(1999) showed evidence of crack face contact slightly aft of the crack tip. It was 

speculated that this was frictional sliding of the crack faces, but it occurred over a 

small portion of the crack face. As will be seen in section 4, the crack face contact 

results in significant heating and is likely a result of frictional sliding.

1.2.2 Thermomechanical Coupling 

During dynamic fracture, energy flows into the region surrounding the 

propagating crack tip. This energy originates from the applied external loading 

and is used for two things, plastic deformation and fracture. Part of the energy 

that goes into plastic deformation is stored internally and the remainder is 

dissipated as heat, referred to as thermoplastic heating. The portion that is 

stored internally remains in the material microstructure and represents a change 

in internal energy. From a modeling perspective, it will be convenient to 

consider energy and work rates. The fraction of plastic work rate that goes to 

dissipative heating is denoted as β. The value of β has been the source of many 

investigations that are reviewed later in this section. Generally, β is about 0.9 for 

metals and 0.5 for polymers. 

Of the energy that is used for fracture, some is used to create new surfaces. This 

energy requirement is due to the increased stored energy of a free surface and is 

a material property, denoted as γ. Because two surfaces are created during 

fracture, 2γ is needed for fracture. The remaining energy used for fracture 

manifests itself in the form of internal heating. This quantity, denoted as q& , is 

termed the thermofracture heating. If the deformation is sufficiently transient 

such that thermal convection and radiation to the surrounding environment and 

thermal conduction within the body does not have time to occur, a local increase 

in temperature will result. The focus of this study is to examine the 

thermoplastic and thermofracture events, with a particular emphasis on the 

thermofracture event. 

For a material exhibiting plasticity, the relationship between heat generation and 

mechanical fields during time-dependent deformation and fracture can be 

incorporated into the heat energy equation as 

, (3 2 ) , kT T0 q c pT p

ij ij

e

ii kk & −α λ + μ ε& + βσ ε& + & = ρ (5) 

where k is the thermal conductivity, T is the absolute temperature, T0 is the initial 

temperature, α is the coefficient of thermal expansion, cp is the specific heat, λ

and μ are the Lamé elastic constants, ρ is the density, σij are the stress 

components, e

ij ε and pij ε are the elastic and plastic strain components, 

, 
10 

respectively, β is the fraction of plastic work rate that is converted to heat, and q&

is the rate of internal heat generation. The dot denotes differentiation with time. 

The first term on the left hand side is due to parabolic heat conduction, the 

second term is the reversible thermoelastic effect, and the third term represents 

the rate of plastic work converted to heat. The rate of internal heat generation 

term, q& , encompasses all dissipative processes present during fracture that 

generate heat, excluding gross plastic deformation. Sun and Hsu (1996) provide 

an analytic expression, of the form ( 2 ) ( ) ( ) 0 0 G − γ a&δ x − x δ y − y , for q& , under 

the conditions of stable crack growth, where G is the Irwin strain energy release 

rate, and γ is the work to create a single fracture surface, δ is the Dirac-delta 

function, x and y are coordinates, and the zero subscript denotes the current 

crack tip coordinates. Physically, the term (G – 2γ) represents energy that has 

flowed through the crack tip region (G) but is in excess of that needed for crack 

surface formation (2γ). This excess energy would be dissipated as heat. The 

analytic expression for q& , given above by Sun and Hsu (1996), conceptually 

makes sense; however, it is incorrect. The expression has units of J/m s, whereas 

the governing equation for which it was derived requires it to have the units of 

J/m3 s. 

The heat energy equation given by equation (5) was derived using a linear 

thermoelastic stress-strain relation given as 

2 (3 2 ) ( ) , ij = ij kk + μ ij − λ + μ δ ijα T −T0 σ λδ ε ε (6) 

where T0 is the initial temperature. This elastic constitutive relation is fully 

coupled in the sense that an imposed stress or strain will alter the temperature, 

and an imposed temperature will produce a stress and/or strain. The heat 

energy equation given by equation (5) is not fully coupled, however, because the 

dissipative term associated with plastic deformation is not fully coupled. To 

make equation (5) fully coupled would require a coupled thermoplastic 

constitutive relation. Additionally, the material properties would need their 

temperature dependence, if any, also included. 

For a viscoelastic material that does not exhibit plastic deformation, equation (5) 

can be rewritten as (Boley and Weiner 1997): 

, (3 2 ) , kT T0 s d q c pT l

ij

l

ij

e

ii kk & −α λ + μ ε& + + & = ρ (7) 

where sij is the deviatoric stress tensor, dij is the viscous strain tensor, and 

superscript l is a summation index. However, in this study polymers well below 

the glass transition temperature are investigated at very high load rates. Thus, 

equation (5) is sufficient for the investigation of thermomechanical effects, 

because the viscous dissipation strain, dij, would be small. 

If the duration of the fracture event is sufficiently short, the thermal conduction 

term in equation (5) becomes negligible (i.e., nearly adiabatic conditions prevail). 

, 

, 
11

Furthermore, the thermoelastic term is usually small compared to the 

thermoplastic term (shown in section 3 for our case) and can also be neglected. 

This reduces equation (5) to 

q c T . p

p

ij ij & βσ ε& + & = ρ (8) 

Equation (8) provides the coupling between temperature rise in a material and 

dissipative processes that generate heat during highly transient deformation. 

The thermal dissipation terms in equation (8) can be quantified by measuring the 

temperature rise of material in the vicinity of a propagating crack tip. Thus, if a 

material is known to exhibit negligible plastic deformation at room temperature 

and quasi-static loading rates (~10–3!10–2 s–1), referred to herein as a nominally 

brittle material, temperature measurements of the propagating crack tip region 

will lead to quantifying q& . Conversely, if a material is known to exhibit 

significant plastic deformation at room temperature (referred to as a nominally 

ductile material), temperature measurements will lead to the total thermal 

dissipation, which includes the plastic work rate and the additional dissipation 

mechanisms. Quantifying and understanding these two terms during dynamic 

fracture of amorphous polymers is the primary focus of this study. 

The question of what fraction of plastic work is converted into heat (i.e., β in 

equations [8]) has been under investigation for many years. A large volume of 

research on this topic has concentrated on metals. Most studies conclude that the 

fraction of plastic work rate converted to heat for metals is about 0.9. The 

dependence of β on strain rate remains controversial. Farren and Taylor (1925) 

and Taylor and Quinney (1934) found β to be ~0.85 for steel and copper using 

thermocouples and calorimetry techniques. Mason et al. (1994) and Hodowany 

et al. (2000) found β to vary from 0.6 to 0.9 for steel, aluminum, and titanium 

alloys and to be dependent on strain and strain rate. Kapoor and Nemat-Nasser 

(1998) reported all of the mechanical work to be converted to heat (i.e., β = 1.0) 

for steel, aluminum, titanium, and tantalum alloys. 

In contrast, limited work has been done on the determination of β for polymers. 

During the investigation of cold drawing processes of PC. Adams and Farris 

(1998) performed a series of tests at low strain rates (from 0.18 to 1.8 min-1) and 

studied the energy conversion between mechanical work, heat, and internal 

energy using deformation calorimetry. They found 50%–80% of the mechanical 

work was dissipated into the environment as heat while the rest was stored as 

internal energy in the material. Rittel (1999) conducted a study of heat 

generation during high-rate deformation of PC. He found β to vary from 0.4 to 

1.0, depending on strain rate. More recently, Li and Lambros (2001) found β for 

PC to range from 0.5 to 0.6 during high-rate compression. For PMMA, β is not 

defined because it is nominally brittle at room temperature and high loading 

rates. 

. 
12 

1.2.3 Crack Tip Temperature Measurements 

Early estimates of opening mode crack tip temperatures in the nominally brittle 

amorphous polymer PMMA were deduced by Fuller et al. (1975) using a single 

unfocused infrared detector along with a thermocouple and liquid crystal films. 

The actual temperature measurements made by Fuller et al. (1975) were 

peripheral to the crack tip and did not constitute a temperature field 

measurement. A transient heat conduction analysis was used to deduce the 

crack tip temperature. This analysis required an estimate of the dissipation 

region responsible for heat generation, which Fuller et al. (1975) assumed to 

extend below the fracture surface by no more than 3 μm. With this assumption, 

Fuller et al. (1975) reported a temperature rise above room temperature of ~500 K 

for cracks propagating within a velocity range of 200–650 m/s. However, as will 

be seen later, the region responsible for heat generation actually extends ~100 μm 

below the fracture surface for PMMA used in the present study. The larger 

dissipation region will significantly lower the estimated crack tip temperature 

using the technique of Fuller et al. (1975). In sharp contrast, Döll (1976) 

measured a temperature increase of ~0.85 K at a single location using an 

embedded thermocouple a short distance away from the opening mode crack tip 

path in PMMA. It should be noted that the peak temperature recorded by Döll 

was 1.7 s after crack initiation, indicating that thermal conduction had occurred. 

Rittel (1998) also used a single embedded thermocouple in PMMA and recorded 

a peak temperature increase of ~50 K during dynamic opening mode fracture. 

Thus, the limited thermal data for opening mode fracture of polymers suggests 

that a temperature increase does occur, but that its magnitude is considerably 

lower than the temperature increase observed in metals. In addition, it should be 

pointed out that the studies of Fuller et al. (1975), Döll (1976), and Rittel (1998) 

provide measurements at only one point on the sample. 

To date, there have not been any published results of the temperature fields 

during either opening or shear-dominated crack growth in polymers. However, 

studies have been made to measure the temperature of adiabatic shear bands. 

Using the loading geometry introduced by Kalthoff and Winkler (1987), Zhou 

et al. (1996a) and Mason et al. (1994) studied the temperature rise surrounding 

adiabatic shear bands in C-300 maraging steel and 4340 steel. The adiabatic 

shear band that developed in the metal during their experiments was followed 

by a dynamically propagating opening crack. A similar experiment was 

performed by Rittel (2000) to measure the temperature surrounding a 

dynamically propagating adiabatic shear band in PC. Using a single embedded 

thermocouple, he reported a temperature increase of 75 K when the adiabatic 

shear band propagated directly through the thermocouple. However, 
13

this study focused on the shear band and did not examine the fracture process 

itself. Thus, little is known regarding the dissipative processes present during 

the shear fracture of ductile and brittle polymers. 

1.3 Research Objectives 

The present study improves upon the previous temperature measurements by 

directly measuring the temperature field surrounding the propagating crack tip 

in amorphous polymers. This has been done for ductile metals using infrared 

detector arrays (Zehnder and Rosakis 1991; Kallivayalil and Zehnder 1994; 

Mason and Rosakis 1992; Zehnder et al. 2000). This study will use a onedimensional (1-D) infrared detector array to measure the temperature field for 

not only ductile, but also brittle polymeric materials. 

The overall goal of this research is to provide a better understanding of the 

dissipative processes that occur during high-speed opening and sheardominated fracture of amorphous polymers. The specific objectives of the 

current study are as follows: 

• Quantify in high resolution the temperature field surrounding a 

dynamically propagating crack tip in a nominally brittle and nominally 

ductile amorphous polymer. 

• Identify the portion of heating due to plastic deformation in the overall 

crack tip temperature field for the ductile polymer. 

• Provide insight into the process zone for shear-dominated fracture of 

amorphous polymers. 

• Develop a dissipative, cohesive-based failure model for predicting heat 

generation in a brittle polymer. 

These objectives will be achieved through comprehensive experimentation, 

finite-element analysis, fracture surface microscopy, and analytical modeling. 

Although aspects of materials science and micromechanics are used to provide a 

deeper understanding of the process zone physics, an attempt is made here to 

cast the modeling and analyses in a continuum mechanics framework. 

This study presents significant contributions to the area of dynamic fracture. 

Crack tip temperature fields are measured in polymeric materials for the first 

time. More significantly, the temperature field of a shear crack was obtained for 

the first time in any material. For mode I cracks, the crack tip speed, temperature 

field, and fracture energy are measured simultaneously for the very first time, 

offering the ability to directly correlate the crack tip temperature field with 

crack tip energy and kinematics. Finally, an appropriate form for a dissipative 

crack tip cohesive law is obtained and compares well with the thermal 

measurements made for low-speed opening mode fracture in a brittle polymer. 
14 

2. Experimental Techniques 

2.1 Material Selection 

The polymer materials used for this study were PMMA and extruded PC. 

PMMA exhibits negligible plastic deformation and is prone to craze formation 

when loaded beyond its elastic limit at room temperature (Kambour 1964). 

Polycarbonate is a relatively ductile material that exhibits significant plastic 

deformation under large loads (Adams and Farris 1998). The PMMA used for 

most of the study was obtained from the Polycast Technology Corporation under 

the trade name PolyCast. A second smaller supply of PMMA was obtained from 

CYRO Industries under the trade name Acrylite FF. The two sources of PMMA 

provided material with significantly different molecular weights, 1,400,000 and 

140,000, respectively. In the subsequent text, the 1,400,000 molecular weight 

PMMA will be referred to as high molecular weight, and the 140,000 PMMA will 

be referred to as low molecular weight. The PC material was Hyzod from 

Sheffield Plastics. All materials were purchased as flat panels with a nominal 

thickness of 8.5 mm. 

Both materials are transparent to radiation in the visible and near infrared 

spectrum, but are opaque to radiation in the mid- and far-infrared spectrum 

(Incropera and DeWitt 1980). The transmissivity of each material is about 90% 

for radiation in the 0.4–3 μm wavelength range and quickly drops to 0% outside 

this wavelength band. As will be seen later in this section, heating of the 

material will cause radiation emission in the 6–12 μm range. This is well outside 

the radiation transmissivity wavelength band for these materials. This means 

that the radiation that is measured originates from the surface of the specimen 

and not the interior. An experiment is described in section 2.6.4 that verifies this. 

The technique used to make the PolyCast PMMA is a casting technique that 

renders the molecular chains completely unoriented. Acrylite FF PMMA is made 

in such a way that a slight amount of chain orientation does occur and the 

molecular weight is significantly reduced. The extrusion process used in 

forming PC results in considerable orientation of molecular chains. The fracture 

experiments were performed such that the crack propagated along the molecular 

chain orientation direction for Acrylite FF PMMA and perpendicular to the 

extrusion-oriented molecular chains for PC. The temperature increase above 

room temperature necessary to reach the glass transition temperature of PC is 

125 K (Domininghaus 1988). The corresponding temperature increase for PMMA 

is 90 K (Domininghaus 1988). 
15

The analyses used throughout this study require known values for the 

mechanical and thermal properties of PMMA and PC. The needed properties 

were obtained either from the literature or were determined as part of this study. 

Young’s modulus, E, was directly measured from a series of standard tension 

tests performed in accordance with the American Society for Testing and 

Materials (ASTM) D638 standard. Tensile test specimens were prepared from 

the PC and PolyCast PMMA material. The PC specimen was made such that the 

molecular chain orientation relative to the tensile traction direction would be the 

same as for the dynamic fracture experiments, i.e., perpendicular to applied load. 

Quasi-static and low-rate tests were performed on a conventional load frame 

instrumented with a load cell to measure force and a clip-on displacement gage 

to determine specimen strain. The strain rates were either 4.5 × 10-4 s-1 or 0.1 s-1. 

The resulting stress-strain curves from these tests are shown in Figures 5 and 6 

for PMMA and PC, respectively. Young’s modulus, determined in the manner 

specified in ASTM D638, is listed in Table 1. The modulus for PMMA was found 

to be rate dependent, with the material behaving stiffer at higher loading rates. 

The PC modulus was independent of loading rate for the rates tested. These 

results are consistent with those of Theocaris and Andrianopoulos (1982) and Li 

and Lambros (2001), who examined the elastic constants of PMMA and PC at 

intermediate and high loading rates. Theocaris and Andrianopoulos (1982) 

observed the high-rate elastic modulus for PMMA to increase 90% for a strainrate increase from quasi-static to 60 s-1. The modulus for PC was found to 

increase 6% for a similar increase in rate. Li and Lambros (2001) reported a 

similar increase in the modulus for PMMA as the strain rate increased from 

quasi-static to 5000 s-1. Given that high strain rates accompany dynamic crack 

growth, the high-rate Young’s moduli for PMMA and PC are also included in 

Table 1 and were the moduli used throughout the study. Additional mechanical 

and thermal properties obtained from the literature, including the dilatational, 

shear, and Rayleigh wave speeds (cd , cs , and cR , respectively) are listed in Table 

2. It should be noted that Theocaris and Andrianopoulos (1982) found Poisson’s 

ratio, ν, for both materials to be independent of load rate. 

2.2 Experimental Configuration and Impact Event 

Opening and shear-dominated dynamic fracture experiments were performed at 

room temperature on PMMA and PC. Single edge notched panels with a 

nominal height of 292 mm, length of 127 mm, and thickness of 8.5 mm were 

impacted by a steel projectile having the shape of a right circular cylinder with a 

length of 75 mm and a diameter of 38 mm. A starter notch (25 mm long and 

1–mm wide) was cut into one edge of each panel with a band saw. A razor blade 

was used to create a sharp end to the otherwise blunt notch. The polymer 

specimen was placed on two cantilevered glass microscope slides to provide a 
16 

Figure 5. Low-rate tensile stress-strain curves for high molecular weight PMMA 

showing rate sensitivity of Young’s modulus and ultimate strength. Two 

tests were performed for each strain rate. Brittle failure occurred during each 

test. 

Figure 6. Low-rate tensile stress-strain curves for PC showing strain-rate sensitivity of 

yield strength. Young’s modulus was not rate sensitive. Two tests were 

performed for each strain rate. 

Engineering Strain

0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225

Engineering Stress (MPa)

0

10

20

30

40

50

60

70

80

5 x 10-4 s-1

0.12 s-1

Engineering Strain

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Engineering Stress (MPa)

0

20

40

60

80

100

120

1 x 10-4 s-1

4.5 x 10-4 s-1

0.1 s-1
17

Table 1. Rate effects on elastic moduli of PMMA and PC. 

E (GPa) 

4 1 4.5 10 s − − ε& = × 1 0.1s − ε& = 1 60 s − ε& =

PMMA 2.9 4.0 5.5 

PC 2.3 2.3 2.5 

Table 2. Polymer properties. 

ρ

(kg/m3

) 

ν k

(W/m⋅K) 

cp

(J/kg⋅K) 

cd

a

(m/s) 

cs

a

(m/s) 

cR

a

(m/s) 

PMMA 1190b 0.36b 0.19b 1470b 2330c 1330c 1243c

PC 1200b 0.31b 0.21b 1170b 1750d 1075d 1000d

a Dynamic plane stress values. 

b Domininghaus (1988). 

c Singh et al. (1997). 

d Ravi-Chandar et al. (2000). 

break-away support that approximates traction-free boundary conditions. This 

was done for two important reasons: First, it greatly simplifies the boundary 

conditions for finite-element modeling performed later. Second, it was found 

that placing the specimen directly on a steel stand provided a path for energy to 

flow out of the specimen, reducing the energy available to the propagating 

crack tip. High crack speeds and crack branching, discussed in section 3, 

occurred only when the specimen was isolated from the steel stand with the 

break-away supports. The impact event occurred inside an impact chamber to 

contain debris. Opening mode dynamic fracture was achieved by impacting the 

panel in a symmetric manner as shown in Figure 7. The edge opposite the starter 

notch was impacted. 

Shear-dominated fracture was achieved by asymmetric impact with the panel 

edge containing the starter notch, as shown in Figure 8. The experimental 

configuration was adopted from Ravi-Chandar et al. (2000). Grooves were 

milled along each side of the specimen to provide a preferential crack 

propagation path and to suppress crack kinking and branching. The grooves 

each had a width of 3.2 mm, a depth of 2.5 mm, and the bottom surface of each 

groove was polished smooth with 2000 grit sand paper. Approximately 60% of 

the specimen thickness was removed along the groves, consistent with the 

amount Ravi-Chandar et al. (2000) found necessary to ensure that a 

shear-dominated crack would propagate. It is worth noting that Ravi-Chandar et 

al. also observed that an opening crack at an angle to the initial starter notch 

propagated simultaneously into the upper half of the specimen. Details of the 

Ravi-Chandar et al. (2000) experiments are provided in section 4. 
18 

Figure 7. Opening mode dynamic fracture experiment configuration also showing 

infrared detector coverage and crack propagation gage. Crack propagates 

from right to left. 

Figure 8. Shear-dominated dynamic fracture experiment configuration also showing 

infrared detector coverage and painted wires for coarse crack speed 

measurement. Crack propagates from left to right. 

4

6

8

14

10

12

Steel Projectile

292 mm

20 mm

Painted

Break

Wires IR Detector

Focus

127 mm

8.5 mm

0.08 mm

0.08 mm

0.02 mm

1.30 mm a@

Starter

Notch

3

5

7

13

9

11

0.10 mm

2

15

Steel Projectile

292 mm

20 mm

3 Break

Wires

IR Detector

Focus

127 mm

8.5 mm

a@

3.2 mm

Groove Depth: 2.5 mm Both Sides

3.2 mm

Groove

Starter

Notch

4

6

8

14

10

12

0.08 mm

0.08 mm

0.02 mm

1.30 mm

3

5

7

13

9

11

0.10 mm

2

15
19

The projectile speed was low enough to prevent deformation of the projectile 

during impact (i.e., it served as a rigid body to generate a stress pulse at the 

projectile-panel interface). An alignment procedure was followed prior to each 

experiment to ensure that a planar impact occurred between the flat front face of 

the projectile and the edge of the polymer panel. The asymmetric impact that 

occurs during the shear-dominated experiments causes a compressive stress 

pulse to propagate from the projectile-panel contact region. Because the loading 

is asymmetric relative to the starter notch, a shear-dominated stress field 

develops at the notch tip after the passage of the stress wave that leads to crack 

initiation and propagation. During the opening mode experiments, the stress 

pulse resulting from symmetric impact travels to the opposite end of the panel 

and reflects as a tensile pulse. As the reflected wave passes the starter notch, the 

presence of traction free surfaces belonging to the notch induce an opening mode 

(mode I) stress field at the notch tip, where eventually an opening, or mode I, 

dynamic fracture event commences. Explicit finite-element simulations of the 

initial impact and subsequent propagation of waves into a notched panel of 

PMMA to confirm the explanation of events leading to crack initiation were 

performed using the ABAQUS finite-element code (Hibbitt et al. 1998). Details of 

the finite-element simulations are provided in section 2.6.2. These calculations, 

combined with crack initiation times obtained from initial experiments, indicated 

that significant material motion in the horizontal direction would occur prior to 

the crack tip reaching the desired region of interest. This issue is of considerable 

consequence for infrared detector alignment and is discussed further in section 

2.6.2. 

2.3 Gas Gun 

The projectile was accelerated with a compressed gas gun to a speed ranging 

between 8 and 40 m/s. The gun, shown in Figure 9, has a 50-mm inside diameter 

and a 2.1-m-long launch tube. The breech of the gun is a high-pressure reservoir 

with a piston assembly inside. The interior diameter of the breech is 180 mm. 

The piston consists of two disks connected with a threaded rod. The rear disk 

has a diameter of 178 mm, which allows it to freely move and air to pass by. The 

front disk has a diameter of 60 mm. Prior to each experiment, the piston was 

moved to the forward position to seal the breech from the launch tube. Highpressure air was then pumped into the breech chamber. A valve on the rear of 

the breech was opened, allowing the high-pressure air to vent to ambient. The 

resulting pressure differential inside the breech caused the piston assembly to 

move towards the rear of the breech, breaking the seal to the launch tube. The 

remaining high-pressure air then moved into the launch tube, accelerating the 

projectile to the desired speed. A typical breech pressure needed to obtain a 

projectile speed of 40 m/s was 350 MPa. The 75-mm-long, 38-mm-diameter 

projectile was made from maraging C350 steel. A PC jacket was 
20 

Figure 9. Photograph of gas gun, impact chamber, and infrared detector system. 

placed around the steel projectile to match the bullet diameter with the launch 

tube diameter. The jacket had a length of 50 mm and was placed such that the 

rear of the jacket was flush with the rear of the steel projectile. This ensured the 

jacket did not come in contact with the PMMA or PC test specimen. 

The energy available for high-speed crack growth comes from the kinetic energy 

of the impacting steel projectile. In order to quantify the conditions that led to 

crack growth and to model the impact and crack growth event numerically, 

projectile speed must be measured. Projectile speed at the gun muzzle was 

measured with the interruption of two infrared beams in the travel path of the 

projectile, illustrated in Figure 10. Two sets of holes were drilled though the 

launch tube wall at the muzzle end to accept two commercially available IR 

emitter and detector pairs. The two infrared emitters, separated 50 mm along the 

projectile travel direction, were constantly emitting infrared energy with a 

wavelength of 940 nm. The detectors were located on the opposite side of the 

launch tube wall and registered the infrared energy. As the projectile traveled 

past this region of the gun tube, it interrupted the first infrared beam, which 

began an external time interval counter. Interruption of the second beam 

stopped the counter. The projectile speed was determined by dividing the 

distance between break beams by the time required for the projectile to traverse 

this distance. 

2.4 Projectile Contact, Crack Initiation, and Crack-Speed 

Measurements 

Projectile contact with the test specimen was determined with a voltage 

completion circuit. Two bare-ended wires were attached to the test specimen at 

Impact Chamber

Gas

Gun

Infrared

Detector
21

Figure 10. Projectile velocity measurement system using two infrared break-beams at the 

muzzle of the launch tube. Projectile interrupts infrared beam, causing 

infrared detectors to change response. 

the location of projectile contact. The bare ends were separated by several 

millimeters, preventing electrical continuity. The opposite ends of the wires 

were connected to a 10 kΩ resistor in series with a 9 V battery. Upon impacting 

the test specimen, the steel face of the projectile made contact with the two bare 

wire ends and completed the electrical circuit. A 500 MHz digital storage 

oscilloscope (Tektronix TDS 540B) recorded the time of circuit completion by 

monitoring the voltage across the resistor. This technique served as a very 

reliable method for determining the time of initial projectile contact with the test 

specimen and served as the trigger source for all of the transient recorders used 

for each experiment. 

Crack initiation time was measured with a break wire painted on the specimen to 

complete a voltage interruption circuit. A fine line of conducting silver paint 

(SPI Corp. product no. 5001) was hand painted on the test specimen at the initial 

crack tip location such that it would be severed when the crack initiated (i.e., the 

paint line was perpendicular to the crack propagation direction). Wires were 

attached to the paint line and connected to a 10 kΩ resistor in series with a 9 V 

battery. Initiation of crack propagation severed the painted line, opening the 

electrical circuit. The time of crack initiation was determined by monitoring the 

voltage across the resistor with a digital storage oscilloscope. 

Steel Projectile

Infrared

Emitters

Infrared

Detectors

PC

Sleeve

Launch Tube
22 

There are two techniques that can be used to directly measure the speed of a 

propagating crack tip, high-speed photography, and electrical resistance grids. 

High-speed photographic techniques require an intense light source to illuminate 

the crack tip region in either a front-lit or back-lit configuration, and a multiframe camera with an extremely fast shutter and framing times. Because 

crack tip speeds can be on the order of 103 m/s, framing times of less than one 

microsecond and shutter speeds a fraction of this are necessary to obtain precise 

crack-speed measurements. A variation of this technique is to use an open 

shutter camera with a pulsing light source (usually a laser) that serves as the 

shutter. Crack-speed measurements can be made simultaneously with 

photoelasticity or other full-field imaging techniques provided the crack tip is 

clearly discernable on the sequence of full-field images. The experiments of 

Ravi-Chandar et al. (2000) used high-speed photoelasticity to measure crack tip 

speed during shear-dominated dynamic crack growth. Rosakis (1993) and 

Rosakis et al. (1999) used the high-speed photographic technique of coherent 

gradient sensing to simultaneously measure crack tip speed and full-field inplane principal stresses. 

The electrical resistance grid technique uses a number of electrical wires laid 

across the path of the crack. As the crack propagates, it severs the wires 

sequentially and provides an electrical signal which can then be used to 

determine the crack position and speed with time. This technique was used by 

Dulaney and Brace (1960), Cotterell (1965), Anthony et al. (1970), and Paxson and 

Lucas (1973) with great success. A drawback to this technique is that the 

resolution of the grid dictates the resolution of the speed measurement, and it is 

difficult to create very fine grids. This can be improved by using an electrically 

conductive film instead of discrete wires (Stalder et al. 1983; Fineberg et al. 1991). 

This approach offers the continuous measurement of crack tip position. 

However, it does require depositing a conductive film over a portion of the test 

specimen, which can be quite challenging. 

Crack propagation speed was measured in this study with the electrical 

resistance grid technique. The crack propagation speed measurements made by 

Ravi-Chandar et al. (2000) in PMMA and PC using high-speed photography 

were used to design a conductive grid of suitable spatial coverage and 

resolution. Details of this gage, including the circuitry, gage size, and 

fabrication, are provided below. 

A series of break wires painted on the side of the test specimen, as shown in 

Figure 11, was used to complete a voltage interruption circuit. Two designs were 

used, one having 11 painted lines, and the other having 21 lines. Each line, or 

wire, was 0.3 mm thick, and the spacing between the middle of adjacent lines 

was either 1.5 or 3 mm. The 21-line gage is shown in Figure 11. The painted 
23

Figure 11. Crack propagation gage constructed from thin lines of conducting silver paint. 

Top illustration shows gage configuration with the crack propagating from 

left to right, severing the painted lines sequentially. Bottom photograph is of 

a completed gage on a PC fracture specimen. Crack initiation gage is also 

shown. 

lines were connected as a single parallel resistor circuit and then included in the 

circuit shown in Figure 12. The measured voltage output, E0, for this circuit is 

given by the following relation: 

i

s

p

l l

p

l l E

R

R

R E





































+













 =

∑

∑

=

=

1

1

0

1

1

1

, (9) 

Lead Wires

Intended

Crack Path

Painted

Lines

Crack Initiation

Circuit

1.5 mm

30 mm

Notch

Tip

21 Painted Lines

0.3 mm thick

To Matching Resistors

Notch

Tip
24 

Figure 12. Schematic of the crack propagation speed measurement circuit using painted 

silver lines on the polymer test specimen. The propagating crack sequentially 

severs the painted lines, changing the overall resistance of the circuit. 

where Ei is the voltage of the power supply (9 V), Rl are the resistances of the 

painted line paths remaining intact, p is the number of intact lines, and Rs is a 

series resistor. As the crack propagates, it severs each of the painted lines one at 

a time, changing the overall resistance of the parallel resistor portion of the 

circuit. This, in turn, causes an abrupt change in the monitored voltage, E0. The 

time between abrupt changes in E0, along with the known spacing between 

painted lines, enables the crack propagation speed to be determined. 

An accurate crack speed requires the lines to be placed on the test specimen with 

precision. This was accomplished with a custom made template. A 0.2-mmthick piece of spring steel was made into a template for painting the silver lines 

by cutting the desired line pattern in the steel with a numerically controlled 

electro-discharge machining technique. Eleven lines were cut, each line having a 

width of 0.3 mm and a centerline-to-centerline spacing between lines of 3.0 mm. 

A second template with 21 lines having a line-to-line spacing of 1.5 mm was used 

to double the resolution of the crack-speed measurements. The first line was 

placed 3 mm from the initial notch tip for both templates. To keep the template 

from shifting and the paint from wicking between the template and test 

specimen during paint application, a spray adhesive was applied to the template 

prior to placing it on the specimen. Ends of the painted lines were either 

grouped together to form a single conductor or connected to a resistor, as shown 

in Figure 12. The resistance for each line was selected such that the same abrupt 

change in E0 occurred as each painted line was sequentially severed. A 

photograph of a completed 21-line gage on a PC test specimen is shown in 

Figure 11. 

Rs

To

Oscilloscope

R Rn R1 R2 R3 R4 Rn-2 n-1

Painted Lines

on Test

Specimen

....

. . . . . . .

. . . . . .
25

The groove that was present on the test specimen used for the shear-dominated 

fracture experiments prevented a template from being used to paint the 

conductive lines. For these experiments, only three wires were hand painted and 

used in circuits identical to the one used for determining crack initiation time 

(see Figure 8). This enabled a coarse estimate of the crack tip speed to be made. 

The Ravi-Chandar et al. (2000) data, obtained under crack growth conditions 

similar to this study, were used to verify the coarse speed measurements for the 

shear crack. 

2.5 Dynamic Energy Release Rate Measurements 

The total energy that flows into the crack tip region during dynamic fracture is 

the dynamic energy release rate, G, discussed in section 1. This energy 

represents to total energy available for fracture and is partitioned between plastic 

work and fracture energy. An accurate determination of G is critical for the 

cohesive zone modeling that is performed in section 6. The dynamic energy 

release rate has been measured for various crack tip speeds in PMMA by Döll 

(1976) and Sharon et al. (1996). These studies used a single-edge-notched plate 

specimen under quasi-static loading. If the initial notch tip is blunt, significant 

energy is stored in the material prior to crack initiation. This stored energy then 

flows into the crack tip region, driving it to high-speed. Analytic expressions for 

G are available for the configuration used by Döll (1976) and Sharon et al. (1996), 

and require accurate measurement of the crack tip speed. Unfortunately, the 

results reported by Döll (1976) and Sharon et al. (1996) are not consistent. The 

energy values reported by Sharon et al. (1996) are higher than those of Döll 

(1976) for the same crack tip speed, often by a factor of 2 or more. For this 

reason, independent measurements of G were made during this study. 

Furthermore, these measurements were made simultaneously with the 

temperature field measurements, something that had never been done before. 

The simultaneous measurement left no doubt of the energy responsible for a 

given temperature field. 

The dynamic energy release rate, G, was determined for a selected number of 

mode I experiments using a strain gage technique. The approach used in this 

study is identical to that published by Khanna and Shukla (1995), which was 

built upon the pioneering work of Sanford and Dally (1987) and Sanford et al. 

(1990) and is summarized next for completeness. 

A series solution for the stresses near the tip of a crack propagating at constant 

speed was derived by Irwin (1980). Chona (1987) added to this expression the 

higher order terms needed to account for the influence of finite boundaries. For 

a linear elastic, isotropic material, the strain, ε x′x′ , sensed by a strain gage 

oriented at an angle α as shown in Figure 13 is (using the notation of Khanna and 

Shukla [1995]), 
26 

Figure 13. Strain gage coordinate system showing the gage orientation angle, α. Strain 

gage is fixed in space and the coordinate system origin is attached to the 

propagating crack tip that moves from left to right. 

2με κ (λ λ ){} } Re ( ) Re ( ) (1 λ ){Re 1( 1 ) Re 1 ( 1 ) cos 2α 2

1 1 1 1 1

2

2

2

1 z z z z x x  + + Γ + Ψ



 ′ ′ = Γ + Ψ

α λ α

λ

λ λ Re ( ) cos 2 (1 ) Re ( ) cos 2

(1 )

4

2 2

2

2 2 2 2

2

1 2 Γ z − + Ψ z

+ −

+ 2λ1{ } − Im Γ1 (z1 ) + Im Γ2 (z2 ) − Im Ψ1 (z1 ) sin 2α

Im ( )sin 2 , 4

(1 )

2 2

1 2

2 2

2





 Ψ +

+ α

λ λ

λ

z (10) 

where 

, 1

1 , 4 (1 )

1

2 2

1 2 2

2

2

1

ν

ν

κ

λ λ λ

λ β

+

− = − +

+ =

and μ and ν are the shear modulus and Poisson’s ratio, respectively. The 

velocity-dependent functions, λ1 and λ2, are defined as 

1 ,

2

2

1 













 = −

d c

a& λ (11a) 

1 ,

2

2

2 













 = −

s c

a& λ (11b) 

, 

, 

, 

, 

x

y

Strain

Gage

r

θ

x’

y’

α

yg
27

where a& is the crack propagation speed and cd and cs are the longitudinal and 

shear wave speeds respectively. The complex valued functions Γ1, Γ2, Ψ1, and Ψ2, 

are defined in a series form as 

C z k , n 0

n 1/2

n k (z ) ; 1,2 (nosum) Γk k = ∑ =

∞

=

− (12a) 

D z k , m 0

m

n k (z ) ; 1,2 (nosum) Ψk k = ∑ =

∞

=

 (12b) 

where the velocity transformed variables zk are defined in Figure 14 as 

z x iλ y r e k , iθθ

k = + k = k ; = 1,2 (13) 

and Cn and Dn are constant coefficients depending on applied loading and 

geometry. The stress intensity factor is given in terms of the first coefficient of 

the Γk series (equation 12a) by 

2 . K C0 π d

I = (14) 

A three-parameter representation of the strain field is obtained by setting n = 1 

and m = 0 in equation (12a) and (12b), yielding 



















 ′ ′ = − + + +

2

2 sin 2 sin

2 (1 ) cos 2 cos2 ( ) cos

1 2 1

1

2 1

1

2 1

2

2

1

1

0 1

φ λ α φ λ α φ κ λ λ

ρ

με x x C β

















 +

+ − 2

2 sin 2 sin

2

cos 2 cos

(1 )

1 4 2

1

2

2

2

1 2

2

φ λ α φ α

λ

λ λ

ρ



















 − − + + − 2

2 sin 2 sin

2 (1 ) cos 2 cos2 ( ) cos

1 1

1

2 1

1

2 1

2

2

1

1

1 1

φ λ α φ λ α φ κ λ λ

ρ

C β

















 − + − 2

2 sin 2 sin

2

cos 2 cos

(1 )

1 4 2

1

2

2

2

1 2

2

φ λ α φ α

λ

λ λ

ρ

 D [( )( cos 2 )]. 2

2

2 + 0β1 λ1 − λ κ + α (15) 

The coefficient of the D0 term in equation (15) can be set to zero if 

cos 2α = −κ . (16) 

Note that the solution of equation (16) for the gage orientation angle α yields two 

values, an acute and an obtuse angle. When the obtuse angle is used, the peak in 

the strain profile occurs when the crack is directly below the strain gage. At this 

instant, / 2 θ = φ1 = φ2 = π (see Figure 14). 

, 

, 

, 
28 

Figure 14. Coordinate system used in the constant crack tip speed analysis for 

determining the dynamic energy release rate from a strain gage. 

Coordinate system origin is at the crack tip and propagates with the crack. 

The first term of equation (15) gives an accurate representation of the strain field 

around the crack tip provided the strain gage is located within the singularitydominated region. This eliminates the influence of the second term. By placing 

the strain gage sufficiently close to the crack tip at an obtuse angle α and using 

equation (14), equation (15) gives an expression for the dynamic stress intensity 

factor, d KI : 

 { } 

 = ′ ′ κ λ − λ + + λ α + λ α

λ μ ε πβ ( ) (1 ) cos 2 2 sin 2 1 4 ( ) 1

1

2

1

2

2

2

1

1 1

x x p g

d

I K y

,

1

2 1

2

1 2

2

cos 2 2 sin 2

1

1 4 −





















α + λ α

+ λ

λ λ

λ − (17) 

where x x p ( ) ε ′ ′ is the peak strain obtained during the experiment and yg is the 

perpendicular distance from the crack path to the strain gage centroid. 

As mentioned earlier, there are two angles that satisfy equation (16), an acute 

and an obtuse angle. The advantage of using the obtuse angle is that the crack is 

directly under the strain gage at peak strain, which aids in coordinating the 

crack tip position with the crack propagation gage and the thermal 

x

y

z

z1

z2

.

.

. Crack Face

r

ρ1

ρ2

θ φ1 φ2

, 
29

measurements. The location of the peak strain for the two angles is shown in 

Figure 15, where the first term of equation (15) is plotted as a function of distance 

along the crack plane (x = 0 corresponds to the crack tip position). For 

illustrative purposes, a crack propagating at 400 m/s in PMMA was used, along 

with an assumed d KI value of 2.2 MPa·m1/2. The corresponding obtuse and 

acute angles are 121° and 59°, respectively. The peak strain for the acute angle 

occurs about 5 mm ahead of the crack tip and is 30% lower in magnitude. The 

sensitivity of the obtuse angle solution of equation (15) to the crack tip speed is 

shown in Figure 16. The strain variation with position is shown for crack tip 

speeds ranging from 200 to 800 m/s. PMMA material properties were used. 

Sensitivity is seen to improve for the higher crack speeds, although there is some 

difference in peak strains for lower crack speeds. Figure 16 indicates that the 

strain gage technique does provide sufficient sensitivity to crack tip speed. 

Figure 15. Effect of the strain gage orientation on the location of peak strain as the crack 

propagates past the gage at 400 m/s in PMMA. Crack tip is located at x = 0. 

Acute angle results in the peak strain occurring 5 mm prior to crack passage, 

whereas the obtuse angle gives a peak strain when the crack is directly 

beneath the gage. 

x (mm)

-30 -20 -10 0 10 20 30

Strain (microstrain)

-500

0

500

1000

1500

2000

2500

Obtuse Angle

Acute Angle
30 

Figure 16. Sensitivity of strain gage response to crack tip speed for PMMA. Obtuse 

angle orientation is used. 

The conversion of d KI to the dynamic energy release rate G for plane stress is 

given by Freund (1990) as 

( ) , ( ) d 2

I

I K

E

A a G & = (18) 

where E is Young’s modulus (dynamic value for polymers) and AI is the mode I 

universal function 

, (1 ) 2

1

2

c D

a A

s

I

ν

λ

− = & (19) 

where 2 2

1 2 2 D = 4λ λ − (1+ λ ) and λ1 and λ2 are defined by equations (11a) and 

(11b), and cs is the material shear wave speed. Implicit in the above derivation is 

that the material is isotropic and linear elastic, that the strain gage is located 

within the singularity-dominated zone, and that steady-state plane stress 

conditions exist. 

Strain measurements were made with a conventional foil grid strain gage 

(Measurements Group Inc. No. CEA-13-125UN-350) configured as a quarterWheatstone bridge. The strain gage had an active grid length and width of 

3.18 mm and 2.54 mm, respectively, and a gage factor, F, of 2.11. The bridge 

completion circuit and associated signal conditioning amplifiers were Pacific 

Instruments Model 3210 and had a bandwidth of 150 kHz. The signal from the 

x (mm)

-8 -6 -4 -2 0 2 4 6 8

Strain (microstrain)

500

1000

1500

2000

2500

3000

3500

800 m/s

600 m/s

400 m/s

200 m/s

, 

, 
31

amplifier was recorded with a Tektronix TDS 540B 500 MHz digital storage 

oscilloscope. The bridge excitation voltage was 4 V. The voltage signal from the 

signal conditioning amplifier was converted to strain using the standard quarterbridge equation (Measurements Group Inc. 1982) 

10 ,

(1000 2 )

4

ˆ 6

0

0

×

−

=

E

E

F

E

E

ε (20) 

where E0 is the output voltage in mV after compensating for the amplifier gain, E

is the bridge excitation voltage in mV, F is the gage factor, and εˆ is the indicated 

strain, given in microstrain. The output from a quarter Wheatstone bridge circuit 

becomes nonlinear for large changes in the strain gage resistance, i.e., large 

strains. The indicated strain given by equation (20) can be corrected for this 

nonlinearity by the following relation (Measurements Group Inc. 1982) to give 

true strain, ε: 

10 . 2 ˆ 10

2 ˆ 10 6

6

6

× − ×

× = −

−

ε

ε

ε

F (21) 

Two strain gages were attached to the polymer test specimens as shown in 

Figure 17. The obtuse angle α used for PMMA and PC was 121° and 119°, 

respectively. The gages were placed with their centroid at 7 mm above the crack 

propagation path. Because the actual crack path varied somewhat, this distance 

was measured for each gage after the test, and this measured distance is what 

was used for yg in equation (17). The gages were located 7 mm and 15 mm aft of 

the initial notch tip, as shown in Figure 17. The adhesive used to bond the strain 

gages to the polymer test specimen was Measurements Group Inc. M-Bond 200. 

The gages were placed on the side opposite the crack propagation gage. 

After each test, the recorded voltage signal was converted to strain using 

equations (20) and (21). The maximum value for each gage was then used in 

equations (17) and (18) to give the dynamic energy release rate, G. Typical 

measurements for G and corresponding crack tip speed are presented in 

section 3. 

2.6 Thermal Measurements 

In addition to the measurements of crack tip speed and dynamic energy release 

rate, simultaneous temperature field measurements were also made. There are 

numerous experimental techniques, and transducers are available for measuring 

the temperature of a body. These techniques fall into four general categories: (1) 

thermistors, or technology based on materials whose electrical impedance 

changes with temperature, (2) thermocouples, (3) methods that rely on direct 

, 
32 

Figure 17. Position and orientation of strain gages used to measure the dynamic energy 

release rate during opening mode dynamic fracture. Angle shown is for 

PMMA (119° was used for PC). Crack propagates from left to right. 

heat transfer, such as thermally sensitive films, and (4) photon detectors, which 

are semi-conductor devices that produce voltage or current flow when excited by 

photons having energy at particular wavelength bands. Each method has a 

characteristic response time and spatial resolution. Thermistors are relatively 

slow devices, relying on thermal conduction to heat a resistor. They are useful 

for discrete point measurements of temperature within an existing temperature 

field and must be affixed to the material of interest. Thermocouples are devices 

that consist of two dissimilar metallic conductors joined at their ends. When the 

metal conductors are exposed to a heat flux, they heat to different temperatures 

(because of differing thermal properties) and an electromotive force proportional 

to the temperature difference is induced. If the junction of the two conductors is 

very small, response times on the order of microseconds can be achieved. Like 

thermistors, they measure the temperature of a discrete point and must be 

attached to the test specimen. Rittel (1998, 1999, 2000) and Döll (1976) used 

miniature thermocouples to make discrete point measurements of the 

temperature increase in polymers during dynamic deformation. Thermally, 

sensitive films are applied directly to the test specimen and change colors in 

response to the temperature of the underlying surface. Calibration and 

interpretation of the colors is subjective. They were used by Fuller et al. (1975) to 

measure the temperature increase of a propagating crack in PMMA. The newest 

generation of commercially available photon detectors are highly sensitive to 

radiation in a number of wavelength bands, measure temperatures remotely, and 

have response times less than 0.5 ms. Additionally, the detectors can be 

configured in 1-D and 2-D arrays for measuring temperature 

7 mm 8 mm 7 mm

121 deg. Strain Gages

Notch Tip Crack Path
33

fields. The disadvantage of this technique is their high costs. Proton detectors 

have been used to measure the temperature field of cracks propagating in ductile 

metals (Zehnder and Rosakis 1991; Mason and Rosakis 1992). 

For the measurement of the temperature field surrounding a rapidly propagating 

crack tip, a high resolution, fast response method is needed. Additionally, a 

method that does not influence the dynamic fracture event, such as drilling holes 

or attaching mass to the surface, is desired. Detectors that respond to photons 

are ideally suited for crack tip temperature measurements because they can be 

made very small (less than 100 μm) and have a response time of less than 0.5 ms. 

The photon detector method was used in this study for temperature field 

measurements. 

Electromagnetic radiation is emitted from any body that has a temperature 

greater than absolute zero. The particular wavelength of energy emitted by a 

body is related to its absolute temperature. For a black body emitter, the relation 

between spectral emissive power, Eλ, wavelength, λ, and absolute temperature, 

T, is given by the Planck distribution (Planck 1959), 















 −

=

1

2

5

1

T

C

e

C E

λ

λ

λ

, (22) 

where the first and second radiation constants, C1 and C2, respectively, are 

3.742 × 108 W⋅μm4/m2 and 1.439 × 104μm⋅K. Equation (22) is plotted in Figure 18 

for several selected temperatures. Figure 18 shows that the spectral region in 

which the radiation is concentrated depends on temperature. Hotter objects emit 

energy of a shorter wavelength. Additionally, the peak in spectral emissive 

power is higher for hotter objects. In the context of crack tip temperatures, 

Figure 18 indicates that lower crack tip temperatures result in the emission of 

longer wavelength radiation at lower power levels. Zehnder and Rosakis (1993) 

have compiled the spectral responsivity of several common photon detectors. 

This comparison, illustrated in Figure 19, shows that a detector made from one 

material, such as InAs, can be very sensitive at a particular wavelength band, but 

may be completely unresponsive to radiation at a different wavelength band. 

The maximum temperature in the crack tip field anticipated for this study is on 

the order of 300 K. Using Figure 18, this temperature gives peak spectral 

emissive power at ~10-μm wavelength. The ideal detector for this wavelength 

indicated by Figure 19 is HgCdTe. For this reason, a custom-built HgCdTe 

detector was procured for the study of polymer heating during high-rate 

deformation and fracture. 
34 

Figure 18. Planck distribution for a black body emitter at several temperatures showing 

the shift in emitted radiation wavelength as the body is heated. 

Figure 19. Spectral responsivity of several common photon detectors (Zehnder and 

Rosakis 1993). 

Wavelength λ (μm)

0.1 0.2 0.4 0.6 1 2 4 6 10 20 40 60 100

Spectral Emissive Power, 

Eλ (W/m2 μm)

10-3

10-2

10-1

100

101

102

103

104

105

106

107

300 K

800 K

1300 K

1800 K
35

2.6.1 Infrared Detector System and Operation 

The temperature field surrounding the propagating crack tip and the trailing 

thermal wake was measured in real time with a high-speed infrared detector and 

focusing optics. The detector was manufactured by the Fermionics Corporation 

and consists of 16-square HgCdTe (mercury cadmium telluride) photovoltaic 

elements of size 80 μm, arranged in a linear array with a spacing between 

elements of 20 μm. The detector elements are sensitive to radiation in the 

mid-infrared spectrum, 6-12 μm wavelength, corresponding to black-body 

temperatures ranging from 300 to 400 K (Incropera and deWitt 1981). The linear 

array is cooled to 77 K with liquid nitrogen to increase the signal-to-noise ratio of 

the detector system. The detector was connected to a 16 channel, 10 MHz 

preamplifier that was also custom made by Fermionics Corporation, and the 

system output was recorded with digital storage oscilloscopes having a 

bandwidth of 500 MHz (Tektronix TDS 540B). The response time of the detector 

and preamplifier was less than 0.25 μs, which provided sufficient temporal 

resolution to capture the expected transient thermal event. 

The radiation emitted from the specimen surface was focused on the detector 

elements by what is essentially a short focal length Newtonian telescope similar 

to the optical configuration used by Hodowany et al. (2000) and is depicted in 

Figure 20. A spherical mirror served as the focusing element and a square flat 

mirror was used to fold the optical axis. The spherical mirror has a diameter and 

focal length of 150 mm. The height and width of the flat mirror was 30 mm. The 

surfaces of the spherical and flat mirrors were gold plated to minimize reflective 

losses. The optical configuration allows the magnification to be varied by 

adjusting the distances of the detector and polymer test specimen to the spherical 

mirror according to the mirror equation (O’Shea 1985): 

, 1 1 1

1 0 t t f − = (23) 

where -t0 is the distance from the detector element to the spherical mirror, t1 is 

the distance from the polymer test specimen to the spherical mirror, and f is the 

focal length of the mirror. The magnification, M, is given as (O’Shea 1985) 

.

0

1

t

t M = (24) 

Note that the magnification is negative because t0 is always negative (an artifact 

of the sign convention used in the optics community), indicating that the image 

is inverted by the optical system. Hence, a magnification of !1, which was the 

primary configuration used for most of this study, is achieved when the absolute 

distances t0 and t1 are both 300 mm. This magnification gave a detector spatial 

resolution of 80 μm. 

, 
36 

Figure 20. Optical configuration used to measure the temperature field surrounding a 

propagating crack tip. Infrared detector array coverage is also shown. 

The spherical mirror was rotated about 10° off of the optical axis it forms with 

the polymer specimen, as shown in Figure 20, which introduced a small amount 

of optical distortion. This was necessary because of the need to place the flat 

mirror in front of the spherical mirror in order to fold the optic axis of the 

reflected infrared radiation. If the spherical mirror was not rotated, the flat 

mirror would have to be placed directly in front of it, blocking all of the incoming 

energy (for the case of M = !1). By rotating the spherical mirror a small amount, 

the reflected beam is cast slightly off to the side, which eliminates the need to 

place the flat mirror directly in the incident energy path. The amount of 

distortion was qualitatively identified by placing a clear piece of plastic with a 

grid printed on it at the polymer specimen location and examining the projected 

image at the IR detector with a white screen. An incandescent light bulb was 

used as the illuminating light source behind the clear plastic. In this way, the 

rotation of the spherical mirror and the placement of the flat mirror could be 

adjusted to give good optical throughput while minimizing the optical 

distortion. Although this was done using white light rather than radiation in the 

infrared wavelength, the mirror position minimizing distortion is identical. This 

is due to the fact mirrors were used to focus the system, which reflect radiation in 

the same direction regardless of wavelength. 

The linear array was placed so that it was perpendicular to the crack propagation 

direction, as shown in Figures 6 and 7. Up to 14 of the 16 detector elements were 

used in the array during any given test. The resulting area observed by the 

detector array for the opening mode and shear-dominated experiments is shown 

Planar

Mirror

Liquid Nitrogen

Dewar

Detector

Array Spherical

Mirror

Specimen

10 deg.

Projectile

Alignment

Laser

Laser Beam

2 3 14 15 ....

1.3 mm

0.08 mm

0.02 mm

Detector Coverage
37

in the insets of Figures 6 and 7, respectively. For most experiments, a location 

~15–20 mm from the initial notch tip was monitored by the detector. If strain 

gages were present, this location was adjusted to coincide with one of the gage 

locations. 

The region monitored by the detector elements for the shear-dominated 

experiments was on the grooved surface and was always contained within the 

width of the groove because surface emissivity greatly depends on surface 

roughness. Variations of surface emissivity were minimized by polishing the 

grooves with 2000-grit sandpaper to achieve a smooth surface. This was also 

done for the calibration specimens discussed in section 2.6.3. 

Some of the experiments discussed in sections 3 and 4 resulted in temperature 

increases that were less than 20 K above ambient. The infrared detector output 

for these low temperature events had a considerable level of noise. A digital 

filtering scheme was used to improve the signal to noise ratio of the thermal 

signals. The infrared detector output that was digitized and stored with a digital 

storage oscilloscope was transferred to a computer. The signals were then 

processed with the software package, TableCurve 2D, where they were 

smoothed in the time domain using a 4-point polynomial interpolation routine 

resident in the software (Jandel Scientific 1994). The smoothing algorithm 

performs five passes through the data. One pass of the algorithm consists of 

moving across the data and, at each time point, performing a polynomial 

interpolation using the two points lower in time and two points greater in time. 

The y-value (detector voltage output) from the interpolation replaces the original 

y. The algorithm was designed for low levels of smoothing. It is nearly 

impossible to oversmooth the data using this approach. To ensure this, all 

smoothing was performed interactively to verify features of the original thermal 

signals were not lost. 

2.6.2 Alignment and Focusing Procedure 

Precise alignment and focus of the infrared detector system are critical to 

achieving high-resolution measurements of the crack tip temperature field. 

During the initial equipment set up process, the axis of the gun launch tube was 

placed level and perpendicular to a leveled sighting laser used subsequently to 

aid in alignment and focusing tasks. The intersection of the 1-mm-diameter laser 

beam with the launch tube centerline is the location where the infrared detector 

system was focused. The initial coarse alignment was achieved by placing a 

clear piece of plastic with markings on it at the intended focus point and shining 

an incandescent light through it towards the infrared detector system. The 

projected image at the infrared detector was visually monitored while the 

mirrors were adjusted on an optical breadboard to obtain the correct 

magnification, maximize brightness, and minimize optical distortion due to 

spherical mirror rotation. Once the mirror locations were fixed, the infrared 
38 

detector unit was placed on a translation stage that permitted precise movement 

in three orthogonal directions, and this assembly was then placed on the 

breadboard at the appropriate location. With the optical components in their 

proper place, a final alignment was performed using a hot-wire technique that is 

discussed next. 

A hot-wire jig was made by taking a polymer test specimen and drilling two 

holes in it, as shown in Figure 21. A 76-μm-diameter Chromel-A wire was 

placed across each of the holes and held tightly by wrapping it around screws 

located at the periphery of the holes. The wire was oriented horizontally across 

one hole and vertically across the other. To hold the hot-wire jig in place, a 

fixture was attached to the inside of the impact chamber and served as a guide 

for the jig (and polymer test specimens) to lean against. The fixture made contact 

with the jig side at three points, one near the top and two at the bottom, and was 

made from PC. This fixture ensured that when the hot-wire jig was removed 

after the focusing process was completed, the test specimens were placed at 

exactly the same position and remained in focus. A scissors jack with 

cantilevered glass microscope slides was placed underneath the jig to allow 

vertical positioning. With the jig in place inside the impact chamber, the 

horizontal wire was placed so that it intersected the center of the alignment laser 

beam. The wire was then heated by passing ~0.4 A of current through it, which 

caused emission of infrared radiation from the wire. 

Figure 21. Hot-wire jig used for alignment and focusing the infrared detector system. 

The 76-μm-diameter chromel-A wire serves as an infrared radiation source by 

passing current through it. 

Chromel-A

Wire

Hold-Down

Screw Hole

PMMA

Specimen
39

The infrared detector preamplifier does not respond to a constant or slowly 

varying signal, so a rotating chopper wheel was placed in the optical path 

between the hot wire and the detector to modulate the infrared signal at ~50 Hz. 

Modulation of the infrared signal in this manner is standard practice with 

infrared detection instrumentation. The detector output was then monitored 

with digital storage oscilloscopes automatically measuring the amplitude of the 

resulting square wave signal. Because the detector array is vertical and the hot 

wire is horizontal and smaller than a single detector element (wire diameter is 

76 μm and the detector element is 80 μm), proper focus is achieved when only 

one detector element responds with a square wave signal. The translation stage 

underneath the detector is adjusted to obtain this condition. Once focused, the 

detector array was adjusted vertically so the hot wire was imaged onto the 

element in the middle of the array. The vertical hot wire is then placed so that it 

intersects the alignment laser beam, and the process is repeated. However, with 

the detector array and hot wire both being vertical, alignment is achieved when 

all detector channels register a square wave signal. The hot-wire alignment and 

focusing procedure was performed prior to each experiment. 

Once the infrared detector system was focused and aligned, the instrumented 

polymer test specimen was placed in the impact chamber against the alignment 

fixture. The detector aim point on the specimen was marked prior to this with an 

ink marker on the side opposite that monitored by the infrared detector system. 

The specimen was moved vertically by the scissors jack and slid horizontally 

across the glass microscope slides until the marked observation point was in the 

center of the laser beam. This process, although laborious, ensured proper 

alignment and focus for each experiment. 

2.6.3 Analysis of Specimen Translation 

The detector aim point marked on the specimen was not the desired observation 

point because of the significant particle motion that occurs prior to the crack 

reaching this region. Impact-induced stress waves create a particle motion field 

that results in significant displacements of regions ahead of the notch tip. 

Finite-element simulations of the mode I specimen configurations discussed in 

section 2.2 were used to quantify this motion. 

Explicit linear elastic plane stress simulations of the impact event were 

performed with the finite-element code ABAQUS. Detailed descriptions of the 

finite-element code and modeling procedure are presented in section 5. Both the 

polymer specimen and the steel projectile were included in the simulation. The 

finite-element mesh used to discretize the polymer specimen and steel projectile 

consisted of 2-D plane stress quadrilateral elements with a size of 

5 mm each. The notch tip in the fracture specimen was modeled as a wedge with 
40 

straight faces, having an opening of 2 mm at the specimen edge and coming to a 

point at the notch tip 25 mm from the specimen edge. Figure 22 shows the 

finite-element mesh used. Because the configuration is symmetric, only the top 

half of along the plane of symmetry were imposed to prevent node motion 

perpendicular to the axis of symmetry. The polymer material was modeled as 

linear elastic, using the material properties listed in Tables 1 and 2. The steel 

projectile was also modeled as linear elastic with a modulus of 200 GPa, density 

of 7850 kg/m3, and Poisson’s ratio of 0.3. The interface between the projectile 

and polymer specimen was a frictionless contact zone that permitted the 

exchange of energy and momentum. The remaining boundaries of the projectile 

and specimen were traction free. Because the simulation was explicit, additional 

boundary conditions were not required (i.e., the specimen was held in place by 

its the polymer specimen and steel projectile were modeled. Boundary 

conditions own inertia). The projectile speed was varied from 8 to 40 m/s. The 

goal of the simulation was to identify the horizontal translation of material 

located 10 mm in front of the initial notch tip. The translation occurs from 

momentum transfer from the projectile through wave propagation. In this spirit, 

the crack was not permitted to propagate and the stresses at the notch tip were 

permitted to grow unbounded. A convergence study was not performed 

because the mesh used was fine enough for the wave speeds obtained in the 

simulation to be identical to those of the material listed in Table 2. 

The results are shown in Figure 23, which plots the motion of material 10 mm 

ahead of the initial notch tip as a function of time for impact velocities ranging 

from 8 to 40 m/s. Given that typical crack initiation times for PMMA are on the 

order of 100 μs, Figure 23 indicates that motion ranging from 1 to 4 mm can 

occur. The aim point marked on the test specimen was compensated to account 

for this motion. This adjustment increased the likelihood that the temperature 

measurement location would coincide with the strain gage location discussed in 

section 2.5, and helped reconcile timing data from the crack propagation gage 

and the infrared detector signals to identify the true crack tip location. 

2.6.4 Calibration 

The output of the infrared detector system is in the form of voltage. As such, a 

calibration between output voltage and specimen temperature is needed. 

Zehnder and Rosakis (1993) discuss a calibration method based on the detector 

spectral response and the test specimen surface emissivity. Unfortunately, these 

quantities are difficult to quantify. Therefore, an empirical calibration was 

performed to provide the necessary relation between detector output and 

specimen temperature. 

Specimens with a thickness identical to those used in the fracture experiments 

were used to calibrate the infrared detector system. This included using 
41

Figure 22. Finite-element mesh and boundary conditions used for numerical simulation 

of the projectile impacting the polymer specimen. Steel projectile moves from 

left to right. 

Figure 23. Finite-element results showing the horizontal displacement of a point 10 mm 

ahead of the initial notch tip due to projectile impact at various speeds. 

Initial

Notch

Steel

Projectile

Frictionless

Impact

Interface

Plane of Symmetry

Boundary Conditions

PMMA

Specimen

Time (μs)

0 25 50 75 100 125 150 175 200

Horizontal Displacement (mm)

0

1

2

3

4

5

6

40 m/s

20 m/s

10 m/s

8 m/s
42 

specimens with a polished groove to provide a suitable calibration for the 

shear-dominated experiments. The grooves on these specimens were polished 

with 2000-grit sandpaper to achieve a nearly mirror-smooth appearance. This 

was necessary to minimize the variation in surface emissivity, which the infrared 

detector is sensitive to. Calibration consisted of placing either an oven-heated or 

ice bath-cooled polymer specimen in the impact chamber at the object plane of 

the detector optics and recording the infrared detector system output for each 

detector channel as the specimen returned to room temperature. The calibration 

specimen was held in place with a PMMA fixture attached to the muzzle end of 

the gun. The detector was aligned and focused on this fixture using the hot wire 

method discussed in section 2.6.2. Because the detector preamplifier does not 

respond to a constant or slowly varying signal, a rotating chopper wheel was 

placed in the optical path between the calibration specimen and the detector to 

modulate the infrared signal. The actual temperature of the specimen during the 

calibration process was measured with a K-type (Chromel-Alumel junction) 

thermocouple embedded in the specimen. Prior to calibration, the thermocouple 

was placed in a small hole drilled in the calibration specimen and then potted 

with a material of similar thermal diffusivity (PMMA monomer with the 

appropriate catalyst/hardening agent, Weld-On 10 adhesive from San Diego 

Plastics). The thermocouple was within 1 mm from the surface that the detector 

system monitored. Multiple calibrations were performed for each specimen and 

optical configuration was performed to ensure repeatability. 

From the calibration data, a correlation between detector output and specimen 

surface temperature was obtained for each detector channel, each optical 

configuration, and each material used throughout the study. A hyperbolic curve 

of the form 

cx

b x

ax

y x +

+

( ) = , (25) 

where y(x) is the temperature increase, x is the detector output, and a, b, and c are 

fitting parameters, was fitted to the data for each channel using the curve fitting 

routines in the SigmaPlot software package (SPSS Inc. 2000). This provided an 

analytic expression for converting the detector output to temperature increase. 

Typical calibration curves for both PC and PMMA, for one channel of the 

detector system are shown in Figures 24 and 25, with and without grooves, 

respectively. Note that the temperature increase denoted in Figures 24 and 25 

(and all subsequent figures depicting temperature) corresponds to a temperature 

increase above ambient room temperature, not absolute temperature. The 

calibration for the grooved specimens was found to be slightly different from the 

non-grooved specimens (attributable to the different surface emissivity), but was 

highly repeatable. 
43

Figure 24. Typical calibration for one channel of the infrared detector system showing 

results for PMMA and PC. 

Figure 25. Typical calibration for one channel of the infrared detector system showing 

results for PMMA and PC using a grooved specimen. Groove was polished 

with 2000-grit sandpaper to achieve smooth finish. 

Detector Channel Output (mV)

-50 0 50 100 150 200 250 300 350 400

∆T (K)

-20

0

20

40

60

80

100

120

140

PC - No Groove

PMMA - No Groove

Detector Channel Output (mV)

-50 0 50 100 150 200 250 300 350 400

∆T (K)

-20

0

20

40

60

80

100

120

140

PMMA with Groove

PC with Groove
44 

2.6.5 Sources of Potential Error 

There are a number of potential error sources associated with the use of a 

focused infrared detector system for measuring the temperature field 

surrounding a propagating crack tip. These sources include a phenomenon 

known as detector cross talk, limitations of the optical depth of field, detector 

resolution, radiation transmissivity, and calibration issues. Each of these sources 

is explored in this section. Additionally, the notion of adiabatic heating near the 

crack tip is confirmed with an analysis presented at the end of this section, thus 

providing the necessary foundation for proper interpretation of the thermal 

measurements in the sections that follow. 

2.6.5.1 Detector Cross Talk 

The infrared detector used in this study has individual semiconductor elements 

spaced only 20 μm apart. If radiation falling upon one detector element results in 

neighboring elements responding because of their close proximity, a 

phenomenon known as cross talk has occurred. Reasons for this include 

electromagnetic field effects, focusing errors, and optical aberrations. Significant 

detector cross talk effectively reduces the system spatial resolution by smearing a 

well-defined signal over numerous detectors. The detector-element cross talk 

was determined by placing a heated wire with a diameter less than the 

detector-element size at the test specimen location and recording the voltage 

output for each channel of the detector system after it was carefully focused. 

Figure 26 shows the output for the detector array focused on a 76-μm wire. 

Detector cross talk was found to be ~75%–10%. 

2.6.5.2 Depth of Field 

The depth of focus of the optical system was also determined using a small 

diameter-heated wire. With the hot wire at the test specimen location, the 

detector was moved along the optic axis using a translation stage to bring it in 

and out of focus. The output of the detector array was recorded as the 

translation occurred. Figure 27 shows how the image of the wire began to spread 

to other detector elements as the system became unfocused. The results 

indicated that the optical configuration had approximately a 1-mm depth of 

field. A consequence of this small depth of field is that the specimen positioning 

fixture discussed in section 2.6.2 was required for accurate placement of each test 

specimen. 

2.6.5.3 Spherical Aberrations 

Spherical aberrations occur when a large diameter-focusing element, such as the 

spherical mirror used in this study, is used in a short focal-length optical system. 

The aberration is due to light from the outer and inner regions of the mirror 
45

Figure 26. Infrared detector element cross talk observed using a 76-m heated wire. 

Cross talk of 7%–10% was recorded. 

Figure 27. Detector output measured as the detector array was translated in and out of 

focus to obtain an estimate of the infrared measurement system depth of 

focus. 

Detector Element Number

5 6 7 8 9 10 11

Detector Output (mV)

0

10

20

30

40

50

Distance Along Optic Axis (mm)

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Detector Output (mV)

0

10

20

30

40

50

Element 8

Element 7

Element 9

Depth of Focus
46 

being focused at different locations. A qualitative estimate of the degree of 

spherical aberration present in the optical configuration was made by covering 

the outer regions of the spherical mirror during the detector cross talk 

assessment. No difference in the level of detector cross talk was noted when 

various apertures were placed on front of the mirror. Hence, the optical 

configuration was free of significant spherical aberration. 

2.6.5.4 Surface vs. Internal Heating 

It was asserted in section 2.1 that both materials are not transparent to radiation 

in the mid- to far infrared spectrum. This was verified by placing a 3-mm-thick 

piece of each material in the optical of a hot wire (and chopping wheel) and the 

focused detector. The detector response indicated that thin sheets of PC and 

PMMA completely blocked the transmission of infrared energy from the hot wire 

to the detector array, thus confirming the assertion. As a result of this, only 

surface temperature measurements can be made with the infrared detector 

system. Near the surface, a state of plane stress exists. In the interior of the 

polymer specimen near the crack front, the stress state changes to approximately 

plane strain (Krishmaswamy et al. 1991). An implication of this is that the 

temperature at the crack tip will likely not be uniform through the specimen 

thickness. As such, the temperature measurements made in this study are likely 

to be different from those made with an embedded thermocouple, such as the 

kind used in Rittel (1998). In addition, local temperatures will be underestimated 

using this measurement technique if the heat source of interest is less than the 

detector element size, because each detector records the average temperature of 

an 80-μm × 80-μm area. If the crack tip process zone has a lateral span less than 

80 μm, the true temperature can be determined only if the size of the zone is 

known. This is discussed further in section 6. 

2.6.5.5 Temporal Resolution 

Errors in the temporal resolution of the detector array will occur if the time for 

the thermal source to propagate a distance equal to a detector-element width is 

less than the detector response time. Such a situation will lead to the detector 

increasing the time duration of the actual thermal signal. The time for an abrupt 

thermal pulse propagating at 600 m/s (the approximate limiting crack speed in 

PMMA and PC) to travel an element distance of 80 μm is 0.13 μs, which is 

slightly less than the system response time of 0.25 μs. Thus, if the thermal signals 

encountered during the experiments were abrupt (i.e., a step-shaped pulse), a 

correction would be required to resolve the true rise time. However, as will be 

seen later, the thermal signals from the experiments had rise times an order of 

magnitude greater than the infrared detector system response time. This 

suggests the temporal resolution is limited only by the detector amplifier 

response time. 
47

2.6.5.6 Infrared Emission from Heated Surroundings 

Kapoor and Nemat-Nasser (1998) investigated the fraction of plastic work 

converted to heat for several metals. Their experimental approach used a 

focused infrared detector to directly measure the surface temperature of the 

deforming specimen, and a Hopkinson bar technique to indirectly determine 

specimen temperature. The two methods gave differing temperatures. The 

infrared detector calibration procedure, similar to the one used in this study, was 

offered as a possible error source that led to the temperature discrepancy. It was 

suggested that the hot calibration specimen might produce excessive heating of 

the surrounding air during the relatively slow calibration procedure, leading to 

additional infrared energy being emitted. To investigate this possible error 

source for the infrared detector configuration used in the present study, air 

heated to 175 K above room temperature was suddenly permitted to flow past 

the focus point of the optical configuration. Virtually no change in signal from 

ambient was recorded by the detector system (compared to actual temperatures 

recorded in the fracture experiments). Hence, the infrared contribution from 

heated air during the calibration is negligible, at least for the optical 

configuration used in the present study. 

2.6.5.7 Effect of Material Damage on Surface Emissivity 

The surface emissivity of the craze material in the crack tip process zone may be 

considerably different from that of the calibration sample, leading to errors in 

converting the detector output to temperature. This was investigated by 

focusing the detector system on a heated piece of PMMA containing an arrested 

crack. The detector was translated vertically while maintaining focus to 

determine if detector output changed when the elements moved from the 

original PMMA surface to the process zone. Although no difference in detector 

output was observed, this test was not conclusive because of potential 

differences in the process zone of a stationary, arrested crack, and an active, 

growing crack. This remains a potential error source. 

2.6.5.8 Adiabatic Conditions 

Interpretation of the recorded infrared detector signals is based on an 

assumption that the failure event is nearly adiabatic. The condition of adiabatic 

heating presumes that the time intervals of interest in the experiments are 

sufficiently small to neglect thermal conduction and the loss of heat to 

convection and radiation. An upper bound of the heat loss due to convection 

and radiation effects can be made from the rate of calibration specimen cooling. 

For a calibration specimen with an initial temperature 80 K above room 

temperature (typical temperature increases measured during the fracture 

experiments) the maximum cooling rate was approximately 1 K/s. Given that 

the time intervals of interest in this study were on the order of 100 μs, cooling 
48 

due to radiation and convection would result in a temperature decrease of 

0.001 K. Thus, although radiation losses do occur (in fact that is what is detected 

by the highly sensitive infrared detector system), the losses are negligible from 

an engineering perspective. 

As will be seen in the following sections, the measured temperature field 

surrounding a propagating crack tip is highly localized. This leads to the 

potential for heat conduction to occur during the time interval of interest, 

invalidating the adiabatic heating assumption. The heat conduction solution for 

a semi-infinite solid initially at ambient temperature whose entire surface is 

suddenly heated to a constant temperature was used. The solution for this 

situation is given by Poulikakos (1994) as 

0 0

4 t

x T(x,t) (Ti T )erf + T 













 = − η

, (26) 

where T(x,t) is the dependent temperature sought, T0 is the constant temperature 

suddenly applied to the entire free surface at time t = 0, Ti is the initial ambient 

temperature, x is the distance in the solid from the free surface, η is the thermal 

diffusivity defined as k/ρcp, and erf is the error function. This relation is plotted 

in Figure 28 for several different locations below a PMMA crack surface 

suddenly heated to a temperature 100 K above ambient room temperature. As a 

specific example, consider a location 0.1 mm below the fracture surface in 

PMMA. For a surface with a sudden temperature increase 100 K above ambient, 

the time required to raise the temperature 1 K at the location 0.1 mm below the 

surface is 6.72 ms for PMMA and 4.88 ms for PC. These times are far in excess of 

the time intervals of interest in these experiments. This analysis indicates that 

adiabatic heating is a reasonable assumption. 

3. Opening Mode Experiment 

3.1 Introduction 

As mentioned in section 1, a competition between classic shear yielding and 

craze formation occurs when a polymer well below the glass transition 

temperature is subjected to an imposed strain. During opening mode fracture of 

brittle materials, the failure mechanism was noted by Kambour (1964) to be from 

craze formation and breakdown. As the fracture event becomes increasingly 

dynamic, Kambour et al. (1978) found that craze formation becomes the failure 

mechanism for both brittle and ductile materials, although plasticity is still 
49

Figure 28. Temperature increase of several x locations below the surface of a semiinfinite PMMA solid with a suddenly applied surface temperature. Initial 

ambient temperature of solid was 293 K and suddenly applied constant 

surface temperature was 393 K. 

present at the crack tip region. Increasing the energy available for crack growth 

(the energy release rate, G) causes the crack tip speed to increase until a limiting 

speed of about 60% cR is reached. Attempts to drive the crack faster cause it to 

split into two advancing cracks (Yoffe 1951), effectively decreasing the energy 

flowing into each crack. Döll (1976) and Sharon et al. (1996) found G to increase 

nonlinearly with crack tip speed for mode I experiments using PMMA. These 

studies showed G to asymptote towards infinity when the crack tip speed 

approached 60% cR. 

The fracture surface of brittle polymers was observed by Döll (1989) to roughen 

as the crack tip speed increases. The roughness has been attributed to an 

evolution of the crack tip structure from a single craze front to multiple, layered 

craze fronts propagating simultaneously. Sharon et al. (1996) measured the 

increase in surface area for PMMA resulting from high-speed crack growth. 

They found the surface area to increase sixfold and concluded the increased 

energy required for fast crack growth is used solely for the creation of new 

surfaces. Sun and Hsu (1996) noted the energy needed for fracture in brittle 

materials is several orders of magnitude greater than the surface energy, γ, 

suggesting that the increased surface energy requirements alone do not account 

for the energy increases needed for high-speed crack growth. 

Time (ms)

0 5 10 15 20 25

∆T (K)

0

1

2

3

4

5

x = 0.1 mm

x = 0.15 mm

x = 0.2 mm
50 

Studies have shown that crack tip heating does occur in brittle materials during 

dynamic fracture (Döll 1976; Fuller et al. 1975), implying that additional energy 

beyond that needed for the creation of new surfaces is required to fracture a 

brittle material, as noted by Sun and Hsu (1996). The excess energy is used in the 

process zone for microstructure breakdown and molecular bond scission, leading 

to the generation of heat. Because crack tip speeds during dynamic fracture are a 

significant fraction of the material dilatational wave speed, the generated heat 

does not have time to conduct to surrounding regions. This condition results in a 

temperature increase inside the process zone of the polymer. For a ductile 

material, plastic deformation surrounds the crack tip process zone, contributing 

to the overall temperature increase. Thus, the temperature field surrounding a 

propagating crack is related to the dissipation of mechanical work that went into 

deformation and failure of the material. Measurement of the crack-driving 

energy and crack tip kinematics, along with the resulting temperature field at 

sufficient resolution can provide useful insight into the physical mechanisms 

within the process zone. 

The study of opening mode fracture in PC and PMMA involved examining the 

crack tip propagation behavior, the flow of energy into the propagating 

crack tip, measurement of the thermal field surrounding the crack tip, and 

subsequent examination of the fracture surfaces. This combined approach led to 

an understanding of the crack tip process zone that would be useful for 

subsequent modeling efforts. 

The experiments are conceptually simple; however, in practice, there were 

numerous challenges. Measurement of the temperature field surrounding a 

propagating crack tip with a focused infrared detector array can be a difficult 

task because of the limited array size. For the configuration used in this study, 

the detector will miss the passage of the crack if the propagation path differs 

from the intended path by as little as 1 mm. To minimize path deviation, 

shallow grooves could be machined along the side of the test specimen to 

provide a preferential path. Unfortunately, this would alter the stress field and 

possibly the process zone. The approach used in this study was to perform a 

large number of experiments using polymer specimens free of any geometric or 

material alterations along the intended crack path. With a large enough group of 

experiments, the crack tip temperature field would be captured by the detector 

array for a sufficient number of experiments to draw scientifically sound 

conclusions. This resulted in well over 100 opening mode experiments being 

performed during the course of the study. 

This section discusses the results of these experiments by way of presenting data 

from a few selected tests that are representative of the entire collection of 

successful experiments. The section is organized by the type of measurement or 

phenomenon being observed. The results for both PMMA and PC are included 

in each section to better aid in comparing the relative behavior of the brittle and 
51

ductile materials. Unless otherwise noted, the results presented for PMMA are 

for high molecular weight PMMA. The results presented for low molecular 

weight PMMA will be clearly indicated as such. 

3.2 Crack Tip Speed Measurements 

The energy that drives a crack to high-speed comes from the kinetic energy of the 

impacting projectile, as discussed in section 2. However, a one-to-one 

correspondence of crack tip speed with impact velocity does not exist. Because it 

is desired to quantify the conditions responsible for the crack tip temperature 

fields presented later in this section, measurements of the crack tip propagation 

history were made. The crack propagation gage discussed in section 2 was used 

to determine the crack tip location during propagation. Recall the principle of 

operation is the severing of consecutive electrically conductive wires painted on 

the test specimen as the crack propagates. The electrical circuit for the gage was 

discussed in section 2.4. The circuit output was recorded on a digital storage 

oscilloscope. A typical output record from the gage is shown in Figure 29. The 

severing of a wire corresponds to an abrupt increase in circuit voltage. The 

sequence of wire break times was used to locate the crack tip position within the 

region monitored by the propagation gage. This was possible because the 

location of the wires on the test specimen was known a priori. The position 

history of a crack propagating in PMMA is shown in Figure 30. Two examples 

are presented in Figure 30, one corresponding to a projectile impact speed of 40 

m/s, and the other corresponding to an impact speed of 8 m/s. As mentioned 

earlier, a higher impact speed generally results in a higher crack tip speed. This 

is evident in Figure 30 by the different slopes of the two experiments shown. The 

crack tip speed was determined by numerically differentiating the crack tip 

position vs. time data shown in Figure 30. A central differencing scheme was 

used. The crack tip speed determined in this manner from the data in Figure 30 

is shown in Figure 31. Both cracks accelerate to a nearly steady-state 

propagation speed, dependent upon the projectile-impact speed. Although 

numerical differentiation of data signals can amplify the data scatter, the 

resolution of the propagation grids used in this study kept the data scatter to an 

acceptable level. The error bar shown in Figure 31 has a magnitude typical of 

crack tip speed measurement uncertainty in dynamic fracture mechanics 

(Lambros and Rosakis 1995). 

Typical crack tip speed history obtained from break-wire signals for numerous 

opening mode fracture experiments on both polymers are shown in Figure 32. 

The data set in Figure 32 corresponds to experiments with projectile impact 

speeds of ~35–40 m/s for PMMA and 25–40 m/s for PC. The uncertainty in 

crack tip speed is approximately ± 50 m/s (from Figure 31). In PMMA the crack 

tip accelerates to a terminal speed of ~680 m/s, or 0.55 cR. Terminal speed is 

achieved after the crack tip has traveled 10–15 mm. Post-mortem 
52 

Figure 29. Typical crack propagation gage circuit output. The abrupt increases in 

voltage are generated by the painted wires being severed by the propagating 

crack. 

Figure 30. Crack tip position history obtained from the crack propagation gage timing 

data for PMMA. Two different experiments are shown, corresponding to 

different projectile-impact speeds. 

Time (μs)

160 170 180 190 200 210 220 230 240

Voltage (V)

0

1

2

3

4

5

6

Time after Crack Initiation (μs)

0 20 40 60 80 100 120

Crack length (mm)

0

5

10

15

20

25

30

35

8 m/s Impact

Speed

40 m/s Impact

Speed
53

Figure 31. Crack tip propagation speed in PMMA for two experiments with different 

projectile-impact speeds. Propagation speed was determined from crack 

propagation gage timing data and the distance between the painted wires. 

Error bar shows typical measurement uncertainty for crack tip speed 

(estimated from Lambros and Rosakis [1995]). 

examination of the PMMA specimens indicated that crack branching occurred 

20–25 mm from the starter notch tip. This correlates to the region where the 

terminal crack tip speed was sustained and is consistent with theoretical crack tip 

speeds required to produce branching (Broberg 1960). For the case of PC, the 

terminal crack tip speed is ~550 m/s (0.55 cR ). Crack branching also occurred in 

PC after terminal speed was achieved. 

Impact speeds less than 35 m/s for PMMA and 25 m/s for PC, but above 10 m/s, 

resulted in terminal speeds identical to the high impact speed experiments. 

However, the distance the crack propagates prior to reaching terminal speed 

increases. As the impact speed is lowered to values of ~10 m/s in PMMA, the 

crack no longer accelerates to a speed of 0.55 cR. Instead, a lower terminal speed 

of ~300 m/s (0.24 cR) is achieved (as shown in Figure 31) and the crack does not 

branch. 

Measurements of the crack tip speed in low molecular weight PMMA were not 

made because the test specimens used during the low molecular weight PMMA 

experiments were not instrumented. However, differences in crack branching 

were observed. The crack in low molecular weight PMMA was observed to 

branch much earlier than the high molecular weight PMMA, whereas branching 

occurred 20–25 mm from the notch tip for the high molecular weight 

Time after Crack Initiation (μs)

0 20 40 60 80 100 120

Crack Tip Speed (m/s)

0

200

400

600

800

1000

8 m/s Impact

Speed

40 m/s Impact

Speed

Estimated

Magnitude of

Uncertainty
54 

Figure 32. Opening mode crack tip speeds for PC and PMMA as measured by the crack 

propagation gage for experiments with a projectile-impact speed of 40 m/s. 

experiments, it occurred at ~10 mm from the notch in the low molecular weight 

tests. It is not known if branching in the low molecular weight material occurs at 

a lower speed or if the crack tip accelerated to the same terminal speed as the 

high molecular weight material, but over a shorter distance. 

3.3 Dynamic Energy Release Rate Measurements 

The dynamic energy release rate was measured using the strain gage technique 

discussed in section 2. The output voltage from the strain-gage signal 

conditioning amplifier was converted to strain using equations (20) and (21). 

Typical signals recorded from the strain gage system (after converting voltage to 

strain) for PC and PMMA with crack speeds near the terminal speed are shown 

in Figures 33 and 34, respectively. The two strain gages were located 7 mm and 

15 mm from the initial notch tip (see Figure 17). The peak strains for the two 

gages were not identical for either material, indicating the crack speed was not 

the same at the two strain-gage locations. The timing data from the crack 

propagation gage indicated the crack tip was still accelerating at the first gage 

location, but had achieved steady-state speed before reaching the second gage 

location. The condition of steady-state crack propagation (i.e., no crack 

branching) in PMMA was verified for the low impact speed case using three 

strain gages located 7, 15, and 25 mm from the notch tip. The strain signals for 

this case (projectile impact speed of 8 m/s) are shown in Figure 35. 

Distance from Notch Tip (mm)

0 5 10 15 20 25 30

Crack Tip Speed (m/s)

0

100

200

300

400

500

600

700

800

PC

PMMA

PC

PMMA
55

Figure 33. Measured strain from the strain gage circuits for a crack propagating at 0.55 cR

in PC. Distance between strain gages was 8 mm. 

Figure 34. Measured strain from the strain gage circuits for a crack propagating at 0.55 cR

in PMMA. Separation between strain gages was 8 mm. 

Time (μs)

50 75 100 125 150 175

Strain (με) -2000

-1000

0

1000

2000

3000

4000

5000

6000

Gage 1

Gage 2

Time (μs)

50 100 150 200 250 300 350

Strain (με)

-500

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Gage 1

Gage 2 Gage 3
56 

Figure 35. Measured strain from the strain gage circuits for a crack propagating in 

PMMA. The crack accelerated to a terminal speed of ~450 m/s. 

The projectile impact speed was 8 m/s. The gages located 15 and 25 mm from 

the notch tip have the same peak strain, indicating steady-state crack 

propagation. Timing data from the crack propagation gage was consistent with 

the time of peak strain for the mode I experiments. 

Calculation of the dynamic energy release rate, G, requires not only peak strain 

values, but also crack tip speed, a& , and the perpendicular distance between the 

strain-gage centroid and crack propagation path, yg. The crack tip speed was 

obtained from the crack propagation gage as seen in Figures 31 and 32. The 

distance between the strain gage and crack path, yg, was measured after each 

experiment for each gage. These values were then used in equations (17) and 

(18) to give G. The calculated values of G are shown as a function of crack tip 

speed for PC and PMMA in Figure 36. Note that measurements of G for PC were 

limited to the terminal crack speed. The values reported here are consistent with 

those reported for PMMA by Döll (1976). 

3.4 Temperature Field Measurements 

Previous measurements of opening mode crack tip temperature in PMMA were 

made by Fuller et al. (1975), Döll (1976), and Rittel (1998). As discussed in 

section 1, Fuller et al. (1975) used a thermocouple and heat sensitive film to 

determine the total heat evolved at points along the crack path in one series of 

experiments and an unfocused infrared detector to measure the radiation 

Time (μs)

50 100 150 200 250 300

Strain (με) -2000

0

2000

4000

6000

8000

Gage 1

Gage 2
57

Figure 36. Dynamic energy release rate for various crack tip speeds in PC and PMMA. 

emitted from the entire heat affected zone at the crack tip during a different set of 

experiments. The results from both sets of experiments were combined to 

deduce a maximum crack tip temperature increase of 500 K. Using 

thermocouples embedded a short distance from the crack path in PMMA, Döll 

(1976) measured a temperature increase of ~0.85 K at a single location. However 

significant heat conduction had occurred by the time the peak temperature was 

recorded. Rittel (1998), also using an embedded thermocouple in PMMA, 

recorded a peak temperature increase of ~50 K during dynamic opening mode 

fracture. 

The previous approaches to measuring the temperature of a crack tip 

propagating in a polymeric material were limited to discrete point measurements 

or collection of the entire heat flux from a large region. Furthermore, the 

techniques were invasive in the sense that a thermocouple was embedded in the 

material that the crack was to propagate through, possibly affecting the 

propagation mechanics. This study improves upon the previous temperature 

measurements by directly measuring the temperature field surrounding the 

propagating crack tip in amorphous polymers using a noninvasive approach. 

This has been done for ductile metals using infrared detector arrays (Zehnder 

and Rosakis 1991; Kallivayalil and Zehnder 1994; Mason and Rosakis 1992; 

Zehnder et al. 2000). This study will use a 1-D infrared detector array to measure 

the temperature field for not only ductile, but also brittle polymeric materials. 

Crack Tip Speed (m/s)

200 250 300 350 400 450 500 550 600

G (J/m2

) 1000

2000

3000

4000

5000

6000

7000

PC

PMMA
58 

The infrared detector used for this study was discussed in section 2. All of the 

temperature field measurements were made with the detector optics configured 

for a magnification of –1 (negative implying inverted image). The infrared 

detector system monitored a location on the polymer panel within the region 

covered by the crack propagation gage (when used), but prior to the location 

where branching occurred. Recall from section 3.2 that the crack tip accelerated 

to a terminal speed over a short distance. This distance was influenced by the 

impact velocity. As such, temperature measurements made at crack tip speeds 

less than terminal could be made by reducing the impact speed and moving the 

detector focus point closer to the initial notch tip. This approach, although not 

precise and requiring significant trial and error, did permit the temperature 

fields of slower crack tips to be measured. The temperature measurements of 

cracks propagating in PC are presented first in this section, followed by the 

measurements made using PMMA specimens. This section concludes with a 

discussion of the general features found in the temperature signals and 

comparison of the measurements made in this study with those of others in the 

literature. 

3.4.1 PC Experiments 

Figure 37 shows the infrared detector signals (after calibration and data 

smoothing, described in section 2) for a crack propagating in PC at 

approximately its terminal speed. The quantity ∆T shown in Figure 37 on the 

vertical axis represents temperature increase over ambient. The projectile-impact 

speed was 25 m/s. No increase in temperature is observed before the time 

window shown in Figure 37, indicating that stress wave propagation in the 

polymer specimen does not cause dissipative heating. Detector 11 recorded the 

highest temperature increase for this particular experiment. The maximum 

temperature increase over ambient was 105 K. The subsequent detector elements 

flanking detector 11 registered decreasing peak temperatures. Each of the 

temperature curves in Figure 37 show an increase in temperature followed by a 

slower, monotonic decrease. There is a small amount of thermoelastic cooling 

that precedes crack tip heating. This was detected by the infrared measurement 

system. Figure 38 presents the infrared detector output for only elements 10 and 

11 from Figure 37. With the traces from the additional detector elements 

removed, the thermoelastic cooling ahead of the crack tip is clearly visible. The 

amount of cooling was ~5 K below ambient. Thermoelastic cooling was 

observed on nearly all mode I experiments for both PC and PMMA. A 

theoretical estimate of thermoelastic cooling is given in section 3.4.3. 

The crack propagation gage and strain gages provide data relative to a frame of 

reference attached to the polymer specimen. The infrared detector is fixed in 

space and monitors the passage of the crack tip as the crack and polymer 
59

Figure 37. Thermal detector output (after calibration) for opening mode fracture of PC 

with a crack tip speed of 0.55 cR. 

Figure 38. Thermal detector output from two detector elements for an opening mode 

crack propagating in PC at a terminal speed of 0.55 cR. Thermoelastic cooling 

is observed prior to dissipative crack tip heating. 

Time (μs)

200 205 210 215 220 225 230 235 240 245 250

∆Τ (Κ)

0

20

40

60

80

100

120

3

4

11

5

6

7

8

14

13

9 12

10

Time (μs)

195 200 205 210 215 220 225 230 235 240 245

∆T (K)

-10

0

10

20

30

40

50

60

70

80

90

100

110

11

10

Thermoelastic

Cooling
60 

specimen translates before it. The impact-induced specimen translation was 

discussed in section 2, along with finite-element simulations to quantify this 

specimen translate before it. The impact-induced specimen translation was 

motion. The translation data in Figure 23 was used to reconcile the timing of 

events measured with the crack propagation gage, strain gages, and the infrared 

detector. The disturbance that severed the painted lines of the propagation gage 

is located at ~213 μs on the temperature plots shown in Figures 37 and 38. It 

does not correspond to the peak temperature. Although the location of the strain 

gages seldom coincided with the location that the infrared detector monitored at 

the time of material heating (due to variations in crack initiation time), the times 

of peak strain did correlate with the crack propagation gage. It must be kept in 

mind that some of the temperature signal rise time may be artificial. This 

possibility is discussed in detail in section 6. 

The temperature profile for a crack propagating in PC at 450 m/s (0.45 cR) was 

obtained by decreasing the projectile-impact speed to 25 m/s and moving the 

detector so that it monitored a region 5 mm from the initial notch tip. The results 

from this experiment are shown in Figure 39. For this lower speed case, the 

maximum crack tip temperature increase was 57 K and was measured by 

detectors 7 and 8. In this case, the crack propagated virtually in the middle of the 

gap between detectors 7 and 8. Flanking elements 6 and 9 also have similar 

signals, as do elements 5 and 10. The crack propagation gage registered the 

presence of a disturbance at 196 μs in Figure 39, well ahead of the peak 

temperature. Attempts to capture the temperature field for a crack propagating 

slower than 450 m/s were unsuccessful because impact speeds less than 25 m/s 

were unable to initiate a crack in the ductile PC. 

The approximate temperature beyond which the molecular chains of an 

amorphous polymer begin to move (albeit in a highly viscous manner) is the 

glass transition temperature, Tg. For PC, the temperature increase above room 

temperature necessary to reach Tg is 125 K. The temperature increase measured 

at the tip of a mode I crack propagating at a speed of 0.55 cR was 105 K, just 

below Tg. 

There is a noticeable difference in the rise time of the temperatures presented in 

Figures 37 and 39. Because the crack tips were propagating at different speeds 

for these two figures, a direct comparison of rise time is inappropriate. A valid 

comparison can be made after the time scales in Figure 37 and 39 have been 

converted to length, which is discussed next. 

The physical size of the regions where crack tip plasticity and internal heat 

generation occur is crucial to developing an understanding of the process zone 

for high-speed crack propagation. The size of these regions can be determined 

through the temperature measurements given their dissipative nature. 

Determination of physical size entails a measurement of the length along the 
61

Figure 39. Thermal detector output (after calibration) for opening mode fracture of PC 

with a crack tip speed of 0.45 cR. 

propagation direction as well as a dimension transverse to crack tip motion. The 

infrared detector monitors the temperature field surrounding a propagating 

crack tip with a series of closely spaced detectors oriented transversely to the 

crack propagation direction. The spacing of the detector elements in the array 

provides the transverse dimension of the thermal field. The lengthwise 

dimension can be obtained by a time-length transformation exploiting a 

steady-state crack growth assumption. For a constant crack tip speed, a& , the 

transformation of time to length is given by 

ξ = a&t , (27) 

where ξ is length and t is time. This transformation was applied to the PC 

temperature data shown in Figures 37 and 39 to convert the time scales to length. 

Contours of the temperature field were then constructed knowing the vertical 

spacing of the detector elements in the array. The temperature contours for the 

high- and low-speed PC experiments are shown Figures 40 and 41, respectively. 

A violation of the steady crack growth assumption used in equation (27) would 

affect only the length of the contours in the propagation direction; the vertical 

dimensions would not be affected. The peak temperature is indicated by the left 

side of a horizontal dashed line included in the figures. It should be kept in 

mind when looking at the contour figures that the vertical distance between 

adjacent detector element centers is 0.10 mm, and only the region near the 

crack tip is shown. The ductile nature of PC is evident by the large region of 

temperature increase surrounding the crack tip in Figures 40 and 41, suggesting 

that plastic deformation is a significant thermal dissipation mechanism. The 

Time (μs)

185 190 195 200 205 210 215 220 225

∆T (K)

-10

0

10

20

30

40

50

60

70

3

5 4

6

7

8

9

10

11

12

13

14

, 
62 

Figure 40. Temperature contours for opening mode fracture of PC with a crack tip speed 

of 0.55 cR. The large lateral span on the heat-affected region suggests plastic 

deformation is a significant heating mechanism near the propagating crack 

tip. 

Figure 41. Temperature contours for opening mode fracture of PC with a crack tip speed 

of 0.45 cR. The lateral span on the heat-affected region is smaller than the 

higher speed crack tip experiment. 

Length along Propagation Direction (mm)

109 110 111 112 113 114 115 116

Vertical Dimension (mm)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

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10

20

50

40 30

80

70 60

90

100

100

90

80

70 60 50 40 30 20 10

40 20 30

10

Propagation

Direction

Length along Propagation Direction (mm)

87 88 89 90 91 92

Vertical Dimension (mm)

0.0

0.1

0.2

0.3

0.4

0.5

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0.7

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0.9

1.0

1.1

20

10

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50

40

50

40 30

20

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50 10 20 3040

Propagation

Direction
63

lateral span of the heat-affected zone in the high-speed case is ~1 mm, whereas 

the width for the lower speed case is about 0.65 mm. The length of the zone is 

shorted for the lower speed case as well. These two experiments indicate that the 

thermally dissipative zone at the tip of a crack propagating in PC increases in 

size as the crack speed increases. This will be discussed further in section 3.5. 

An estimate of the plastic deformation zone can be made using the asymptotic 

nature of the crack tip stress and the measured energy release rate. Anderson 

(1995) presents an analytic expression for the size of the crack tip plastic region 

for an elastic-plastic material under plane stress quasi-static mode I loading as 

, 1

2















 =

y

I

p

K

r

π σ

 (28) 

where rp is the radius of the plastic zone, KI is the quasi-static mode I stress 

intensity factor, and σy is the yield strength of the elastic-plastic material. An 

estimate of the plastic zone size will be made by replacing the quasi-static stress 

intensity factor with a dynamic value. The relationship between the dynamic 

energy release rate and the dynamic stress intensity factor was given by 

equations (18) and (19). The dynamic energy release rate measured for a crack 

propagating in PC with a speed of 550 m/s was 6600 J/m2. Using the material 

properties for PC in Tables 2.1 and 2.2 and a yield stress of 70 MPa (see Figure 6), 

the radius of the plastic zone is estimated by equation (28) as 0.7 mm, giving a 

total lateral span of 1.4 mm. The predicted lateral span compares to a heataffected zone shown in Figure 40 of about 1 mm. Given that equation (28) is an 

engineering approximation, the two measures of plastic-zone size are in 

reasonable agreement. 

3.4.2 PMMA Experiments 

Figure 42 shows the temperature variation obtained from a crack propagating at 

~0.55 cR in high MW PMMA. As was the case for PC, dissipative heating due to 

stress wave propagation was not observed. The maximum temperature increase 

recorded here was 66 K. Despite the fact that PMMA is brittle at room 

temperature, the temperature curves have the same general characteristics as 

those observed for PC. In this case the crack tip passed near detector element 11. 

Note, however, that the detector elements immediately flanking element 11 (i.e. 

elements 10 and 12) respond slower and reach a lower peak temperature than the 

corresponding ones in PC (i.e., elements 10 and 12 in Figure 37). This suggests a 

more localized thermal signal in the case of PMMA. Spatial resolution of the 

detector array becomes an issue for localized thermal sources, such as that found 

, 
64 

Figure 42. Thermal detector output (after calibration) for opening mode fracture of 

PMMA with a crack tip speed of 0.55 cR. 

for PMMA. If the source is small enough, it could pass between two adjacent 

detector elements and only its thermal wake would be sensed. This is illustrated 

in Figure 43, which shows the temperatures recorded for an experiment similar 

to the one shown in Figure 43, except this time, the thermal source propagated 

between two detector elements instead of directly across an element. 

As opposed to PC, it was possible to dynamically propagate cracks in PMMA at 

a relatively slow speed. A projectile impact speed of 8 m/s was sufficient to 

cause a constant crack growth speed of 250–300 m/s (0.20–0.24 cR). The 

temperature curves for a PMMA experiment with a crack propagating 250 m/s 

are shown in Figure 44. A maximum temperature increase of 23 K was 

registered by detector element 9. Flanking detector elements did not begin to 

register any temperature increase until after the peak temperature on element 9 

was obtained. This indicates that the thermal source responsible for the 

temperature increase had a lateral span (dimension along the detector array 

direction) approximately the same as a single detector element or less. This will 

be shown more precisely later when looking at the temperature contours 

surrounding the crack tip. The flanking elements detected the passage of the 

crack faces as the crack opened. Using the present loading system, it was not 

possible to dynamically propagate cracks in PMMA slower than 250 m/s. 

Time (μs)

105 110 115 120 125 130 135

∆T (K)

0

10

20

30

40

50

60

70

12

11

13

10

14

15

9

8
65

Figure 43. Thermal detector output (after calibration) of opening mode fracture of 

PMMA with the crack tip propagating between two detector elements. 

Figure 44. Thermal detector output (after calibration) for opening mode fracture of 

PMMA with a crack tip speed of 0.24 cR. 

Time (μs)

120 125 130 135 140 145 150

∆T (K)

0

10

20

30

40

50

60

70

6

7

5 8

4 9

10

3

Time (μs)

260 270 280 290 300 310 320 330 340 350 360

∆T (K)

-5

0

5

10

15

20

25

9

8

10
66 

The thermal signal for a crack propagating at low speed in low molecular weight 

PMMA is shown in Figure 45. Although the speed was not measured directly, it 

was estimated to be ~300 m/s by observation of the resulting fracture surface 

(discussed in section 3.6). The maximum temperature increase measured for this 

material was only 8 K, and only one detector element sensed the passage of the 

crack tip. With such a low-temperature increase, the signal-to-noise ratio of the 

detector output is very low. For this reason, only the detector that responded to 

the crack passage is shown in Figure 45. Note that the signal shown is also quite 

noisy. The recorded temperature for the low molecular weight PMMA was only 

35% that of the high molecular weight PMMA under similar crack growth 

conditions. This limited data set suggests that the magnitude of the temperature 

field surrounding a propagating crack is dependent on molecular weight and is 

consistent with the trend Döll (1976) found to exist between energy release rate, 

G, and molecular weight. 

Figure 45. Thermal detector output (after calibration) for opening mode fracture of low 

molecular weight PMMA with a crack tip speed of 0.24 cR. 

A final series of experiments using PMMA was performed with the detector 

array rotated to a horizontal position. This was done to monitor the thermal 

source at various locations along the propagation direction instead of 

perpendicular to it, and as such, it provided a measure of crack tip speed. These 

experiments required many repetitions because the crack path deviation had to 

be within 80 m for the array to register its passage. The results from an 

Time (ms)

220 225 230 235 240 245 250 255 260 265 270

∆T (K)

-2

0

2

4

6

8

10

12
67

experiment for which the projectile-impact speed was 25 m/s are presented in 

Figure 46. The temperature from detector elements 2 and 15 are shown. Both 

elements detected a thermal source with a temperature increase of ~10 K, 

followed by a plateau and late time rise. Although the latter features of the 

signals are not well understood, the initial portion corresponds to crack tip 

passage. It is not known if the crack tip passed directly over the detector array or 

if it traversed slightly above or below. However, the measured temperature is 

lower than what had been recorded for experiments of similar impact speed, but 

with the detector array in the vertical position, suggesting that the crack tip was 

not directly over the array. Timing of the initial portion of temperature increase 

for each element indicated a crack tip speed of 400 m/s. The important aspect of 

Figure 46 is the qualitative agreement between the two signals, confirming the 

thermal source in PMMA does not fluctuate significantly. 

Figure 46. Thermal detector output (after calibration) for opening mode fracture of 

PMMA with the detector array rotated to the horizontal position. Crack tip 

may not have passed directly through the region monitored by the infrared 

detector system. Crack tip propagation speed is 0.57 cR. 

A temperature contour plot was generated for the PMMA high crack tip speed 

experiment shown in Figure 42 using the same procedure and assumptions used 

for generating the PC contour plots. The temperature contour for PMMA, shown 

in Figure 47, has a lateral span at peak temperature of ~0.3 mm (compared to 

0.65 mm and 1 mm for PC). The region of temperature increase is more localized 

and will be shown in section 3.6 to correlate with crazing along 

Time (μs)

110 115 120 125 130 135 140 145 150 155 160

∆T (K)

-5

0

5

10

15

20

25

15

System

Noise

2
68 

Figure 47. Temperature contours for opening mode fracture of PMMA with a crack tip 

speed of 0.55 cR. The contours are highly localized compared to those of PC, 

suggesting that plastic deformation does not contribute to the crack tip 

temperature increase. 

multiple layers. A temperature contour plot could not be generated for the 

low-speed PMMA experiment because only one detector responded at the time 

of crack tip passage. Given this, the detected heat source may be smaller than 

the detector height (i.e., 0.08 mm), which will cause the temperature 

measurement system to underestimate the true temperature. This is because the 

detector element, being of finite size, averages the radiation incident over it’s 

entire area. This will be addressed in detail in section 6. 

3.4.3 Discussion of Temperature Field Features 

The temperature curves for both materials show a rise in temperature followed 

by a slower decrease. As suggested by Mason and Rosakis (1992), a likely reason 

for the decrease in temperature is crack opening. If the crack is assumed not to 

open and adiabatic conditions exist (due to the short times involved), the 

temperature curves would be expected to increase to a peak value and remain 

constant in the trailing thermal wake (as will be seen later in the sheardominated fracture experiments where there is no crack face opening this is 

precisely what happens [see Figures 62a and 62b]). This would continue until 

sufficient time elapses for thermal conduction to occur. If, on the other hand, 

crack opening occurs, the temperature recorded would decrease as the detector 

Length along Propagation Direction (mm)

67.0 67.5 68.0 68.5 69.0 69.5 70.0

Vertical Dimension (mm)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

10

50 40 30 20

60

60 50 40 30 20

10

10 20 50 60

Propagation

Direction
69

began to view the air gap between separated fracture surfaces. The detector 

element sequence for temperature decrease is consistent with this possible cause. 

The effect of crack face motion on the temperature measurements will be 

pursued in section 6. 

There is an additional early time feature present on nearly all of the temperature 

curves. Immediately prior to the initial increase in temperature, a small amount 

of cooling was recorded. This was shown in Figure 38 for PC. The amount of 

cooling in PC was about 5 K. For PMMA, the decrease in temperature was less 

than 2 K and was barely discernable from signal noise. This is the thermoelastic 

cooling effect discussed in detail by Rittel (1998). It results from the large tensile 

triaxial stress state developed near the crack tip prior to initiation. The 

magnitude of the cooling observed in this study, however, is much smaller than 

the 60-K cooling discussed by Rittel (1998). 

An estimate of the thermoelastic cooling can be made using the heat energy 

equation given by equation (5) in section 1. Considering only the elastic 

components and assuming adiabatic conditions, the heat energy equation 

simplifies to 

(3 2 ) , 0

e

pT T kk ρc α λ μ ε& & = − + (29) 

where the parameters were defined in section 1. The Lamé elastic constants λ

and μ are related to material properties through the relations 

( )( ) , 1 ν 1 2ν

ν λ

+ − = E (30) 

( ) . 2 1 ν

μ

+ = E (31) 

Integrating equation (29) and solving for the temperature increase, ∆T, yields 

, (3 2 ) 0

p

e

kk

c

T T

ρ

−α λ + μ ε ∆ = (32) 

where the initial state of the fracture specimen was assumed strain free. Material 

strain is related to stress through the isotropic form of Hooke’s law: 

, 1

ij ij ij kk E E

δ σ

ν

σ

ν

ε − + = (33) 

where δ ij is the Kronecker ∆ function. The maximum amount of thermoelastic 

cooling will occur near the boundary between elastic and inelastic material 

behavior ahead of the propagating crack tip. Assuming plane stress conditions, 

the stress state at the location of maximum elastic stress will be σ 11 = σ 22 = σ c , 

where σ c is a critical stress, and the other stress components equal to zero. The 

, 

, 

, 

, 
70 

critical stress for PC is the yield stress, observed in the uniaxial stress-strain 

curve of Figure 6 to be ~70 MPa. For PMMA, the critical stress is the stress 

beyond which inelastic behavior occurs. This is discussed in detail in section 6, 

where a value of 120 MPa is used. These critical stresses were used in equations 

(32) and (33) to give an estimate of the amount of thermoelastic cooling for each 

material. Using a coefficient of thermal expansion of 65 × 10-6 K-1 for PC and 

70 × 10-6 K-1 for PMMA (Domininghaus 1988) along with the material properties 

listed in Tables 1 and 2, the amount of thermoelastic cooling given by 

equation (33) was 1.8 K for PC and 2.6 K for PMMA. These estimates are near 

the values observed on the infrared detector signals and are considerably less in 

magnitude than the 60 K cooling reported by Rittel (1998). 

The measured maximum temperature increase surrounding the opening mode 

crack tip in PMMA differs from that reported by Fuller et al. (1975), who cite a 

temperature increase at the crack tip of 500 K. The crack tip temperature was 

deduced by Fuller et al. (1975) using a single unfocused infrared detector along 

with a thermocouple and liquid crystal films. The actual temperature 

measurements were peripheral to the crack tip and did not constitute a 

temperature field measurement. A transient heat conduction analysis was used 

to deduce the crack tip temperature. This analysis required an estimate of the 

dissipation region responsible for heat generation, which Fuller et al. (1975) 

assumed to extend below the fracture surface by no more than 3 μm. With this 

assumption, Fuller et al. (1975) reported a temperature rise above room 

temperature of ~500 K for cracks propagating within a velocity range of 

200–650 m/s. Results from the fracture surface micrography examination 

presented in section 3.6 indicate that the region responsible for heat generation 

during high-speed fracture in PMMA consists of multi-layered crazes that form a 

“craze band.” These features extend below the surface 100-200 μm for cracks 

that were near the terminal speed. This larger heat affected zone would 

significantly lower the estimated crack tip temperature if using the method of 

Fuller et al. (1975). The dependence of the technique used by Fuller et al. (1975) 

on the estimated craze width accounts for the discrepancy of their results with 

the more direct measurements performed in this study. 

3.5 Thermal Dissipation of Mechanical Work 

The heat energy equation given by equation (5) for an adiabatic event with 

negligible thermoelastic cooling reduces to the form 

q c pT p

ij ij & βσ ε& + & = ρ , (34) 

with the parameters having been defined in section 1.2.2. The two quantities on 

the left hand side of equation (34) represent the heat generation due to plastic 

work ( p

ij ij βσ ε& ) and the internal heat generation from a dissipative crack tip 

process zone ( q& ). As presented in Zehnder and Rosakis (1993), these quantities 
71

can be extracted from the recorded infrared detector signals by numerically 

differentiating the temperature data and multiplying by the polymer specific 

heat and density (given in Table 2). This was done for four of the experiments 

presented in section 3.4, specifically the two PC experiments shown in Figures 37 

and 39 and the high and low crack speed PMMA experiments shown in 

Figures 42 and 44. Numerical differentiation was performed on the temperature 

data using the following algorithm: 

i i

i i

t t

T t T t

T t −

− ≈

+

+

1

1 ( ) ( ) ( ) & , (35) 

where T(ti+1) and T(ti) are the temperatures corresponding to the times t i+1 and ti, 

and i signifies the sequential ordering of temperature with time. Differentiation 

using equation (35) was a trivial matter because the data was collected with a 

digital storage oscilloscope and stored as a sequence of time-temperature pairs. 

The large lateral span of the temperature contours for PC, shown in Figures 3.12 

and 41, is indicative of significant plastic deformation. As such, the plastic work 

rate was assumed to be the only dissipation term for PC. A value of 0.5 was 

assumed for β (Li and Lambros 2001). Contours of plastic work rate for the high 

and low crack speed PC experiments are presented in Figures 48 and 49. 

Increasing the crack propagation speed from 450 to 550 m/s results in the plastic 

work rate zone increasing in size. The peak work rate was 94 × 1012 J/m3s for the 

low-speed case and 106 × 1012 J/m3s for the high-speed experiment. The lateral 

span increased from 0.3 mm to ~0.42 mm as the crack speed increased from 450 

to 550 m/s. The length of the plastic work rate region increases as well. 

The highly localized temperature signals recorded for PMMA suggest that 

plastic deformation is not a dominant dissipative mechanism associated with 

fracture of PMMA. For a purely brittle material, all of the fracture energy is used 

to create new surfaces. Under these conditions, G = 2γ, there is no excess energy 

available to dissipate as heat. Yet, opening mode dynamic fracture of brittle 

PMMA did result in crack tip heating. This suggests that the internal heating 

term q& , is of appreciable magnitude in equation (34). Given this, internal 

heating was assumed to be the sole contributor to thermal heating for PMMA 

when calculating the dissipation terms in equation (34). A contour plot of the 

rate of internal heating for the PMMA high crack speed experiment is shown in 

Figure 50. The maximum rate of heat generation is 75 × 1012 J/m3s, and the 

width of the zone is ~100 μm. The low crack speed PMMA experiment resulted 

in only one detector element sensing the passage of the crack tip. Although the 

rate of internal heat generation could be determined, a contour plot could not be 

constructed. Moreover, the fact that only one detector responded implies that 

the size of the thermally active region is at most 80 μm, (i.e., one detector element 
72 

Figure 48. Rate of plastic work contours for opening mode fracture of PC with a crack tip 

speed of 0.55 cR. The heat source was assumed to be only ductile plastic 

deformation with a β of 0.5. 

Figure 49. Rate of plastic work contours for opening mode fracture of PC with a crack tip 

speed of 0.45 cR. The heat source was assumed to be only ductile plastic 

deformation with a β of 0.5. The span of the contours are smaller than for the 

higher crack tip speed experiment, illustrating the influence of crack 

propagation speed. 

Length Along Propagation Direction (mm)

109.00 109.25 109.50 109.75 110.00 110.25 110.50 110.75 111.00 111.25

Vertical Dimension (mm)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

30

30

30

40

50 40

40

60

70

80 100 10090 90 30 40 50 60

Propagation

Direction x 1012 J/m3

s

Length Along Propagation Direction (mm)

87.50 87.75 88.00 88.25 88.50 88.75 89.00 89.25 89.50 89.75

Vertical Dimension (mm)

0.0

0.1

0.2

0.3

0.4

0.5

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0.8

0.9

1.0

1.1

30

30

30

40

40

60

50

60

7060 80 90 90 60 70 80

50

x 1012 J/m3s

Propagation

Direction
73

Figure 50. Rate of internal heat generation contours for opening mode fracture of PMMA 

with a crack tip speed of 0.55 cR. Internal heating was assumed to be the only 

source of crack tip heating. 

height). The maximum rate of heat generation for this case was 11 × 1012 J/m3s. 

A 60% reduction in crack tip speed reduced the magnitude of the internal heat 

source by ~85%. However, a word of caution is due here. The high-speed 

PMMA case resulted in a heat-affected region larger than one detector element 

(as shown in the temperature contour plot in Figure 47). The heat source 

responsible for the temperature field will be shown in the next section to be from 

multilayered crazing. The low-speed PMMA experiment showed a temperature 

field that was more localized. Only one detector element responded. This 

situation must be treated with caution. The heat source may be smaller than the 

detector area, which will cause the temperature measurement system to 

underestimate the true temperature. This is because the detector element, being 

of finite size, averages the radiation incident over the entire detector area. This 

will be addressed in detail in section 6. The calculated rate of internal heating 

(11 × 1012 J/m3s) may also be incorrect because the temperature may be 

underestimated. If the thermal source is smaller than the detector element size, 

the internal heating rate will be higher than calculated here. 

Comparing the general features of the heat source contours of PC and PMMA 

reveals a fundamental difference in their structure. The PC contours have 

significant lateral span, whereas the PMMA contours are much more localized. 

Length along Propagation Direction (mm)

61.00 61.25 61.50 61.75 62.00 62.25 62.50 62.75 63.00 63.25

Vertical Dimension (mm)

0.0

0.2

0.4

0.6

0.8

1.0

30

40 50

40 70 60

x 1012 J/m3s

Propagation

Direction
74 

This is further evidence that the crack tip process zone for PMMA does not have 

significant plastic deformation associated with it, whereas the opposite is true for 

PC. The next section will present the findings from a detailed examination of the 

resulting fracture surfaces for both materials and will add yet additional 

evidence of the mechanisms associated with the crack tip process zones. 

3.6 Fracture Surface Micrography 

The fracture surfaces resulting from high-speed crack growth in both polymers 

were glassy smooth in appearance near the starter notch tip and became 

increasingly rough farther from the notch as crack tip speed increased, consistent 

with the observations of Döll (1989) for accelerating cracks. A photograph of 

typical fracture surfaces for high-speed crack growth in PC is shown in 

Figure 51. The PC fracture surface clearly shows the rough region to be confined 

to the middle of the specimen thickness where a state of plane strain existed 

during fracture. The region near the surface monitored by the infrared detector 

remained smooth. Kambour (1964) associated the glassy smooth region with the 

formation of a single polymer craze. Scanning electron microscope images 

(SEM) of the smooth and rough regions of the PC fracture surface are shown in 

Figure 52. The smooth surface shown in the bottom image of Figure 52 is 

essentially flat and featureless, confirming that it was formed by a single craze. 

Because this is the region viewed by the infrared detector system, the large heataffected region for PC is likely from high-rate plastic deformation only. An 

image of the rough region is shown on the top of Figure 52. This portion of the 

fracture surface shows multiple layers of material to have partially separated by 

layered crazing and then deformed by what appears to be gross-plastic 

deformation. Given the limited depth of focus of the infrared detector optics, it is 

unlikely that the detector observed this event. The images do suggest that the 

detector sensed the plastic deformation occurring ahead of the crack tip and the 

crazing process along a single plane. 

The evolution of fracture surface roughness for the high-crack growth speed 

PMMA specimen, shown in the photograph of Figure 53 was quite different than 

was observed for PC. The transition from smooth to rough occurred somewhat 

uniformly through the specimen thickness i.e., the rough region was not 

confined to the middle of the thickness as it was for PC). SEM images of the 

smooth and rough fracture surface regions for PMMA are shown in Figure 54. 

As was the case for the smooth PC fracture surface, the smooth region for PMMA 

is featureless and is consistent with the formation of a single craze. When 

capturing the thermal signals for the high-crack speed case, the detector 

monitored a location on the specimen where the fracture surface was rough. 

Using an optical microscope, the PMMA fracture surface roughness near the 

region of infrared observation was measured to be ~0.10 mm. This roughness 
75

Figure 51. Photograph of PC fracture surface showing unique regions of smooth and 

rough surface texture. 

was due to the development of multiple craze fronts below the exposed surface. 

Sharon et al. (1996) reported an increasing degree of microcracking in their high 

crack speed PMMA experiments, which is consistent with this observation. The 

SEM picture of this region, shown on the top of Figure 54, was taken at 45° to the 

fracture surface and shows these multiple craze fronts and the surface roughness 

that their interaction has produced. The inset for the rough region image shows 

the surface of one of the resulting multiple craze layers. The partially removed 

layers do not show signs of plastic deformation as was observed for PC. The pair 

of mating PMMA fracture surfaces thus indicate that the dynamic fracture 

process zone responsible for multiple crazing has a width of about 100–200 μm. 

Because PMMA exhibits little ductility at room temperature, it would be 

expected that any thermomechanical heating in this case would be limited only 

to this 100–200 μm “craze band.” The temperature and internal heating contours 

shown in Figures 47 and 50 are consistent with this explanation. 

In addition, the experiments that produced low-crack tip speeds (250 m/s) 

yielded fracture surfaces that were completely smooth, the crack tip never 

entered the speed regime that causes surface roughness. Figure 55 shows a 

photograph of this glassy smooth surface that corresponds to the passage of one 

single craze. Recall that in these experiments only one detector responded. 

Given this, it is possible that in these experiments, the craze lateral width was 

Initial Notch

Tip

Smooth

Surface Rough

Surface

Crack Propagation

Direction
76 

Figure 52. SEM images of the smooth and rough regions of a PC fracture surface. The 

crack propagated from the bottom of the figure to the top. 

less than 80 μm, the size of a single detector element. If this was in fact the case, 

the temperature reported for the slow-speed PMMA experiment is an 

underestimate because the detector element output is proportional to the 

radiation falling upon it’s total area. The true temperature could be calculated 

only if the lateral width of the craze was known. This is discussed further in 

section 6. 

100 μm

40 μm

Crack

Growth

Direction

Smooth

Rough

Notch

Tip

Top View

of Fracture

Surface
77

Figure 53. Photograph of a PMMA fracture surface showing the transition of surface 

roughness and eventual branching. 

The fracture surfaces for the low molecular weight PMMA were qualitatively the 

same for all crack speeds as discussed above, except the onset of microcracking 

and crack-branching occurred earlier. The observations described above enabled 

an estimate of crack tip speed to be made based on the appearance of the fracture 

surface for polymer specimens that were not instrumented with a crack 

propagation gage. 

3.7 Additional Comments on Opening Mode Crack Tip Behavior 

To complete the discussion of opening mode fracture behavior of PC and 

PMMA, the dependence of crack tip temperature on the dynamic energy release 

rate is obtained, followed by a brief summary of the crack tip process zone 

mechanisms. The dependence of the maximum recorded temperature change on 

crack tip propagation speed is shown in Figure 56 for both PC and PMMA. The 

maximum crack tip temperature increase that was recorded is plotted as a 

function of the crack tip speed, a& , normalized by cR for each material. The data 

indicate a trend of increasing crack tip temperature with increasing speed. A 

similar result was observed in metals by Mason and Rosakis (1992) and is 

consistent with the trend of increasing energy flux into the crack tip process zone 

with increasing crack speed in PMMA reported by Sharon et al. (1996). 

However, Sharon et al. (1996) noted that near the terminal crack tip speed, 

increasing the energy flux into the crack tip did not result in a commensurate 

increase in crack tip speed. Instead, they found that energy was dissipated in the 

Initial Notch

Tip

Smooth

Surface

Rough

Surface

Crack Propagation

Direction

Branch

Location
78 

Figure 54. SEM image of the smooth and rough regions of a high-speed crack growth 

PMMA fracture surface. The crack propagated from the bottom of the figure 

to the top. The rough region has a surface roughness of about 100–200 μm. 

form of new surface creation (microcracking in the case of PMMA), thereby 

limiting the maximum achievable crack tip speed. As discussed in section 3.1, 

Sharon et al. (1996) measured a sixfold increase in surface area from the 

microcracking. For a perfectly brittle material, all of the fracture energy is used 

to create new surfaces. However, Sun and Hsu (1996) noted that the energy 

needed for fracture in some nominally brittle materials is several orders of 

magnitude greater than the surface energy. Despite the increase in surface 

40 μm

100 m Crack

Growth

Direction

Smooth

Rough

Notch

Tip

Top View

of Fracture

Surface

10 μm
79

Figure 55. Photograph of a low-speed crack growth PMMA fracture surface showing a 

glassy smooth surface corresponding to the passage of a single craze. 

Figure 56. Opening mode maximum crack tip temperature as dependent on crack tip 

speed. 

Initial Notch

Tip

Smooth

Surface Crack Propagation

Direction

a / cR

0.0 0.2 0.4 0.6 0.8

Maximum 

∆T (K)

0

20

40

60

80

100

120

PC

PMMA

.
80 

energy requirements due to microcracking, there is still a significant excess of 

fracture energy. The opening mode temperature data indicate that the increase 

in energy flux to the crack tip goes into more than just creating new surfaces, as 

stipulated by Sharon et al. (1996), but is also dissipated as heat. This is illustrated 

in Figure 57 which shows the relation between the maximum temperature 

increase with a dynamic energy release rate for PMMA. This curve was 

generated by combining polynomial curves that were fitted to the data presented 

in Figures 36 and 56. No line is shown for PC because of the limited dynamic 

energy release rate data available (see Figure 36). The rate of temperature 

increase for PMMA becomes less as G is increased. It is speculated that this 

occurs because some of the available energy is diverted into creating an 

increasing number of microbranches and also because the temperature is 

approaching the material glass transition temperature. The trend observed in 

Figure 57 establishes a link between the dissipation of heat in a craze-dominant 

crack tip process zone and the amount of energy flowing into the process zone. 

The process zone at the tip of a propagating crack in PC appears to contain a 

single craze preceded by gross-plastic deformation. The middle of the specimen 

thickness shows evidence of multiple craze layers with plastic deformation 

occurring within the craze region. As the crack tip speed increases towards a 

limiting speed of 0.55 cR, the rate of energy dissipated as heat increases slightly 

but the process zone itself increases in size, yielding a net increase in crack tip 

temperature. For low-speed crack growth in PMMA, the crack tip process zone 

consists of a single craze. As the crack tip speed increases and more energy 

flows into the process zone, the single craze crack tip changes to one consisting 

of layered, multiple crazes propagating simultaneously. 

The question of how much heating in PC can be attributed to plastic work will be 

addressed in section 5, where results from the experiments presented in this 

section will be combined with explicit finite-element simulations of a dynamic 

crack growth event. The opening mode PMMA results will be used in section 6 

to build a crack tip thermomechanical constitutive model, or cohesive zone, for 

high-speed crack propagation in PMMA. The data and insights presented in this 

section are critical to these endeavors. 

4. Shear-Dominated Fracture Experiments 

4.1 Introduction 

Shear-dominated dynamic crack propagation has been the focus of numerous 

studies in recent years, on the basis of the realization that shear-dominated crack 
81

Figure 57. Empirical relationship between maximum crack tip temperature and dynamic 

energy release rate for opening mode fracture of PMMA. 

growth is possible materials possessing inhomogeneous toughness variation 

(e.g., interfaces) and/or under large confining pressure (e.g., earthquake events). 

In a material of homogeneous toughness, mode II loading has been observed to 

cause a crack to immediately kink and propagate at an angle of ~70° relative to 

the initial crack line. Such crack kinking under dynamic shear conditions can, 

however, be suppressed. 

As discussed in section 1, several approaches have been successfully used to 

suppress crack kinking, allowing the initiation and growth of dynamic shear 

cracks in laboratory situations. The method for suppressing crack kinking used 

in the present study is to introduce a line of weakened material, effectively 

reducing the shear fracture toughness along that line. This approach has been 

used successfully by Rosakis et al. (1999) and Ravi-Chandar et al. (2000). Rosakis 

et al. (1999) were able to generate shear-dominated dynamic fracture of weakly 

bonded Homalite 100 panels. Bonding was achieved by allowing partial healing 

of the interface between the two panels. Techniques for interface healing are 

described by Wool (1995). In contrast, geometric weakening was implemented 

by Ravi-Chandar et al. (2000), who machined grooves along the sides of edge 

cracked panels of PMMA, thereby providing a preferred crack propagation path. 

Asymmetric loading for the Rosakis et al. (1999) and Ravi-Chandar et al. (2000) 

experiments was achieved using a configuration similar to the Kalthoff and 

Winkler (1987) configuration discussed in section 1. Crack tip speeds between 

G (J/m2)

2000 3000 4000 5000 6000 7000 8000

Maximum 

∆T (K)

0

20

40

60

80
82 

the shear and dilatational wave speeds were observed for the experiments of 

both Rosakis et al. (1999) and Ravi-Chandar et al. (2000). In addition, the fracture 

surfaces from the PMMA experiments of Ravi-Chandar et al. (2000) showed 

signs of polymer melting. The shear-dominated fracture experiments of Rosakis 

et al. (1999) also recorded a peculiar phenomenon occurring at the crack tip: the 

existence of a finite-contact region immediately behind the advancing crack tip, 

speculated to be crack face frictional sliding. The limiting speed for mode II 

crack growth was studied by Andrews (1976) and Burridge at al. (1979) using 

numerical techniques. Both of these studies modeled the material along the 

intended crack path with a cohesive traction law that in effect introduced a 

weakened interface. Andrews (1976) found the crack tip speed to be limited to 

s 2c . Burridge et al. (1979) found conditions for stable crack growth to exist for 

s d 2c < a& < c , unstable conditions for R a& < c and s s c < a& < 2c , and a 

forbidden crack speed regime of R s c < a& < c . The experiments of Rosakis et al. 

(1999) using panels Homalite and Ravi-Chandar et al. (2000) using PMMA 

resulted in a stable crack speed near s 2c . 

Motivated by the observations of surface melting observed by Ravi-Chandar 

et al. (2000), the work presented in this study provides the first-ever 

measurement of the thermal dissipation associated with shear-dominated 

fracture in polymers. It was mentioned in section 1 that thermocouple probes 

have been used to measure the temperature at a single location during the 

propagation of an adiabatic shear band in PC (Rittel 2000). The work 

documented in this section goes well beyond previous studies by providing high 

resolution real-time multi-point measurements of temperature surrounding the 

propagating crack tip. These measurements render significant insight of the 

crack tip process region for shear-dominated fracture. 

4.2 Crack Tip Speed Measurements 

The experiments of Ravi-Chandar et al. (2000) and Rosakis et al. (1999), along 

with the computations of Andrews (1976) and Burridge et al. (1979), indicate that 

intersonic crack tip speeds are possible for a dynamically propagating shear 

crack. Given this possibility and the desire to quantify the crack tip kinematics 

associated with crack tip heating, the crack tip speed was determined for the 

shear-dominated fracture experiments. Crack tip speed was measured in a way 

similar to the opening mode experiments, except three hand-painted lines were 

used instead of the 11 or 21 lines applied with a precision template (see Figure 8). 

This was necessary because of the groove machined into the polymer specimen. 

One line was painted at the initial notch tip, and the other two were painted 

15 and 25 mm from the notch. The region monitored by the infrared detector 

was half way between these two wires. The impact speed for the sheardominated experiments was ~40 m/s. The signal from the crack propagation 
83

circuit for a typical PMMA experiment is shown in Figure 58. The passage of the 

crack tip is evident by the abrupt decrease in circuit voltage for each of the three 

wires. The crack propagation gage data indicated that the disturbance that 

severed the painted lines was propagating at ~0.6 cs for PC (650 m/s) and varied 

from 1.1 cs to s 2c for PMMA (1440 to 1880 m/s). As will be seen in the next 

section, an adiabatic shear band formed ahead of the shear crack in PC. As such, 

the speed measured in PC is most likely that of the shear band and not the actual 

crack tip. The crack tip speed measured for PMMA was at or below the speeds 

reported by Ravi-Chandar et al. (2000) who used high-speed photography to 

measure a crack tip speed of s 2c in nearly the same experimental 

configuration. The measured speed reported here is reasonably close to that of 

Ravi-Chandar et al. (2000) given the coarse resolution of the propagation gage for 

these experiments. 

Figure 58. Typical output from the break-wire circuit used to determine the crack tip 

speed for shear-dominated dynamic fracture of PMMA. Output from the 

crack initiation and two crack-speed measurement circuits is shown. 

4.3 Temperature Field Measurements 

Although limited studies exist in the temperature measurement of opening mode 

dynamic fracture experiments (Fuller et al. 1975; Döll 1976; Rittel 1998; Mason 

and Rosakis 1992; Zehnder and Rosakis 1991), there have not been any studies of 

the crack tip heating associated with shear-dominated dynamic fracture for any 

material. In the present study, the temperature was measured in a manner 

similar to that used for the opening mode fracture experiments discussed in 

Time (μs)

40 45 50 55 60 65 70 75 80

Circuit Voltage (V)

0

2

4

6

8

Wire #1

Wire #2

Crack

Initiation
84 

section 3. The infrared detector system monitored a location on the polymer 

panel expected to be within the constant crack tip speed regime based upon 

crack tip speeds reported by Ravi-Chandar et al. (2000). Figure 8 shows the 

experimental setup and the location of temperature observation. It should be 

noted that the initial observation point was not the material point where the 

measurement was actually made during the experiment because of global 

specimen motion in front of the stationary detectors. As in the opening mode 

case, timing from each experiment correlating the time of crack passage, 

specimen dimensions, and wave speeds of the materials allow for the estimation 

of the propagating crack tip location with respect to the stationary infrared 

detector array. 

4.3.1 PC Experiment 

The infrared detector signals (after calibration and data smoothing, discussed in 

section 2) for shear-dominated fracture of PC are shown in Figure 59. Each of the 

12 curves in Figure 59 is identified by the detector element it corresponds to. 

Once again, it should be noted that the vertical axis indicates a change of 

temperature ∆T from ambient. For the representative experiment shown, 

detector 12 recorded the highest temperature increase, followed by flanking 

elements with lower temperatures and delayed temperature increases. This 

suggests that the crack tip passed through the area that detector 12 was 

monitoring or very close to it. The maximum temperature increase recorded by 

detector element 12 was 142 K, which is 17 K higher than the glass transition 

temperature for this material. The nominally ductile (at room temperature) PC 

polymer is known to produce adiabatic shear bands under shear loading (RaviChandar 1995), thereby introducing an energy dissipation mechanism that 

decreases the flux of energy to the crack tip, preventing the crack tip from 

propagating faster than cs. This dissipation mechanism is evident in the 

temperature measurements for PC, shown in Figure 59. A slow, gradual 45-K 

increase in temperature occurs over 180 μs and is followed by a distinct, large 

increase in temperature. This large temperature increase, corresponding to the 

thermal dissipation of a propagating crack, has a magnitude of about 100 K and 

is of the same magnitude found for the opening mode fracture of PC. It is 

believed that the second peak in the temperature signal is from frictional heating 

generated by global relative motion of the two, now fractured, halves. Timing 

data from the crack propagation gage indicated that the adiabatic shear band 

was responsible for severing the painted wires (in Figure 59 the second wire was 

severed at 113 μs). Thus, the crack propagation gage for the case of PC recorded 

the propagation speed of the adiabatic shear band. 

The initial portion of the temperature curves shown in Figure 59 are qualitatively 

similar to the temperature curves of propagating adiabatic shear bands in C-300 

maraging steel obtained by Zhou et al. (1996a). In light of this, and the fact that 
85

Figure 59. Typical thermal detector output (after calibration) for shear-dominated 

fracture of PC with side grooves to suppress crack kinking. ∆T denotes 

temperature change from ambient, and t = 0 corresponds to time of impact. 

PC can produce adiabatic shear bands (Ravi-Chandar 1995), an experiment to 

obtain the thermal signature of an adiabatic shear band and arrested crack in PC 

was performed by impacting a panel of PC with no side grooves at a speed of 

45 m/s. This was done to verify that the feature preceding the crack tip in 

Figure 59 is in fact due to an adiabatic shear band. A slightly different optical 

configuration with only eight detector elements was used. The detector array 

monitored a region 5 mm from the starter notch tip (as opposed to ~20 mm for 

the previously discussed experiment). Upon impact, an adiabatic shear band 

formed ahead of the starter notch, followed by an arrested crack. The 

temperature signals from such an experiment are shown in Figure 60. The 

results are similar to the PC adiabatic shear band temperatures reported by Rittel 

(2000) using a single thermocouple measurement technique. The same time and 

temperature scales that were used in Figure 59 are used in Figure 60 to aid in 

their comparison. The two figures have similar features, especially during the 

early time portion of the temperature record (first several hundred 

microseconds). The primary difference is the lower peak temperature in 

Figure 60 and a shorter time variation. The adiabatic shear band experiment 

shown produced a peak temperature increase of 118 K, whereas the peak 

temperature increase for the shear crack experiment was 142 K. An exact 

comparison should not be made because the adiabatic shear band test specimen 

did not have side grooves. The similarities do suggest that the PC did generate 

an adiabatic shear band that was followed by a shear-dominated crack. In the 

adiabatic shear band experiment shown in Figure 60, however, the crack did not 

propagate beyond a few millimeters and the specimen was not separated in half. 

Time (μs)

50 75 100 125 150 175 200 225 250 275 300 325 350

∆T (K)

0

20

40

60

80

100

120

140

160

5

9

4

7

6

12

8

14

10

13

11

3
86 

Figure 60. Temperature history of an adiabatic shear band and arrested crack in a PC 

specimen without side grooves. ∆T denotes temperature change from 

ambient, and t = 0 corresponds to time of impact. 

Thus frictional heat generation, either during crack growth or because of 

subsequent global relative motion, would be nonexistent. This further suggests 

that the second peak in the experiment of Figure 59 is due to crack face contact 

and friction. 

Assuming steady-state conditions, the temporal evolution in temperature can be 

converted to a spatial one in the manner described in section 3.4.1. The resulting 

temperature contour plot for the PC experiment is shown in Figure 61. The peak 

temperature is indicated by the left side of a horizontal dashed line included in 

the figures. It should be kept in mind when looking at the contour figure that the 

vertical distance between adjacent detector element centers is 0.10 mm. Even 

though in this work only an average value of crack tip speed is measured, the 

real-time high-speed photography results of Ravi-Chandar et al. (2000) indicate 

that crack tip speed is approximately constant, at least during part of the crack 

growth event. A violation of the steady-state crack growth condition assumed in 

generating Figure 61 would affect only the length of the contours in the 

propagation direction; the vertical dimensions would not be affected. Only the 

region near the crack tip is shown in Figure 61, the portion attributable to the 

adiabatic shear band which would be spatially larger is not shown. The vertical 

spatial extent of the PC temperature contours is significant, reaching ~1 mm near 

the peak temperature location. This event is similar to what was found for 

opening mode fracture of PC in section 3. The large lateral dimension of the PC 

Time (μs)

50 75 100 125 150 175 200 225 250 275 300 325 350

∆T (K)

0

20

40

60

80

100

120

140

160

16

8

10

6

12
87

Figure 61. Temperature contours around crack tip region for shear-dominated fracture of 

PC. Temperature increase from room temperature (K). Steady-state crack 

growth has been assumed. 

temperature field implies that plastic deformation is a significant mechanism for 

the dissipation of mechanical work during shear-dominated dynamic fracture 

(aside from the adiabatic shear band that precedes the crack tip). 

4.3.2 PMMA Experiments 

In the shear-dominated experiments, PMMA exhibited a very different thermal 

signature than PC. A typical temperature record for PMMA is shown in 

Figures 62a and 62b. Because PMMA responds in a brittle fashion at room 

temperature, no early adiabatic shear band dissipation is seen (as in Figure 59 for 

PC). Instead, the temperature is observed to abruptly increase by ~85 K 

(approximate increase necessary to reach the glass transition temperature) in 

7 μs. Detector element 8 recorded the highest initial temperature increase. The 

flanking elements detected a significantly smaller temperature increase denoting 

a highly localized event. After the initial temperature increase, no additional 

thermal dissipation is evident for approximately 30 μs. The constant 

temperature regime is then followed by a slow increase in temperature that goes 

well above the glass transition temperature of PMMA (see Figure 62a). This 

temperature profile suggests that the first increase in temperature is due to the 

thermal dissipation associated with the passage of a shear-dominated crack, and 

the later high temperature event is frictional heating. Ravi-Chandar et al. (2000) 

observed in their shear-dominated fracture experiments that the PMMA 

specimens were cut into two halves very quickly because of the high crack tip 

speeds. With this in mind, the late time frictional heating observed in this study 

is likely associated with the global motion of the two specimen halves. 

Length Along Propagation Distance (mm)

162 163 164 165 166 167 168 169 170 171 172 173

Vertical Dimension (mm)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

40

60

100

80

100

80

120

40

140

60

Propagation

Direction
88 

(a) 

(b) 

Figure 62. (a) Thermal detector output (after calibration) for shear-dominated fracture of 

PMMA with side grooves to suppress crack kinking. ∆T denotes temperature 

change from ambient and t = 0 corresponds to time of impact. (b) Initial 

portion of thermal signal from shear-dominated fracture of PMMA with side 

grooves. Temperature field surrounding crack tip is shown. ∆T denotes 

temperature change from ambient and t = 0 corresponds to time of impact. 

Time (μs)

50 75 100 125 150 175 200 225 250 275 300

∆T (K)

0

25

50

75

100

125

150

175

200

225

250

8

7

9 10

11

12

13

14

Time (μs)

50 55 60 65 70 75 80 85 90 95 100

∆T (K)

0

10

20

30

40

50

60

70

80

90

100

8

7

9
89

To examine the spatial extent of heating, temperature contour plots for PMMA 

were generated from the infrared detector output in a manner identical to that 

used in the previous section. Figure 63 shows such a spatial temperature contour 

plot for PMMA. The temperature contours depict a very narrow zone of thermal 

dissipation, indicative of a craze-dominant means of dissipation as opposed to 

plasticity. The lateral span of the heat dissipation region at the location of peak 

temperature was 140 μm or less. The exact span could not be determined 

because only one detector element responded at the time of peak temperature. 

The crack tip speed in PMMA was slightly supersonic relative to the shear wave 

speed, indicating the possible presence of a shear shock. Both Rosakis et al. 

(1999) and Ravi-Chandar et al. (2000) observed the presence of stress field 

discontinuities during their shear-dominated fracture experiments. The 

temperature contours do not show any discontinuities that would confirm the 

presence of the shock, but do illustrate the limited extent of temperature increase 

ahead of the crack tip. 

Figure 63. Temperature contours at crack tip region for shear-dominated fracture of 

PMMA. Temperature increase from room temperature (K). Steady-state 

crack growth has been assumed. 

Recall that the temperature contours for high-speed opening mode fracture of 

PMMA had a lateral span of ~300 μm and significant microcracking was present 

on the fracture surface for higher crack growth speeds. The contours for 

shear-dominated fracture of PMMA are much more localized, and as will be seen 

in section 4.5, microcracking similar to what was observed for the mode I 

dynamic fracture of PMMA was not present on the fracture surface. The 

Length Along Propagation Distance (mm)

86 87 88 89 90 91 92 93 94 95 96 97

Vertical Dimension (mm)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

20

80

60

40

80

60 40 20

Propagation

Direction
90 

temperature signals are highly localized, similar to the low-speed PMMA 

experiments where a single craze was formed. This suggests the possibility that 

highly localized “shear crazing” is the mechanism responsible for 

shear-dominated fracture of PMMA. 

4.4 Thermal Dissipation of Mechanical Work 

As shown in section 1, the heat energy equation for an adiabatic event with 

negligible thermoelastic cooling reduces to the form 

q c pT p

ij ij & βσ ε& + & = ρ , (36) 

with the parameters having been defined in section 1.2.2. Contours of plastic 

work rate ( p

ij ij σ ε& ) and internal heating rate ( q& ) were generated from the 

temperature measurements presented in the previous section using a technique 

identical to that used for the opening mode experiments discussed in section 3. 

The large lateral span of the temperature contours for PC (shown in Figure 61) 

suggest thermal dissipation will be composed primarily of the rate of plastic 

work term. Figure 64 shows contours of plastic work rate at the crack tip region 

assuming that this is the only dissipative term in equation (36). A value of β 

= 0.5 was assumed. The length and width of the plastic work rate zone is ~2 mm 

and ~0.5 mm, respectively. The maximum plastic work rate was 

100 × 1012 J/m3

s. These values are similar to those found for opening mode 

fracture of PC. 

Figure 64. Plastic work rate contours at crack tip region for shear-dominated fracture of 

PC. β = 0.5 and steady-state crack growth have been assumed. 

Length Along Propagation Direction (mm)

164.0 164.5 165.0 165.5 166.0 166.5 167.0

Vertical Dimension (mm)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

20

40

60 80 80

100 80

20

6080 40 20

x 1012 J/m3s

Propagation

Direction
91

Internal heating was assumed to be the sole heat source for PMMA based on the 

highly localized temperature contours (see section 3.5 for additional discussion). 

Figure 65 shows contours of internal heat generation rate for PMMA assuming q&

to be the sole coupling term in equation (36). Note that the contours are for a 

region in the immediate vicinity of the propagating crack tip, the subsequent 

frictional heating is not included. The length of the heat generation rate region is 

~3 mm, and the width is less than 200 μm. The maximum value of dissipation 

for PMMA is 50 × 1012 J/m3

s, half the maximum for PC. 

Figure 65. Rate of internal heat generation contours at crack tip region for 

shear-dominated fracture of PMMA. Negligible plastic work rate and 

steady-state crack growth have been assumed. 

As discussed earlier, low-speed opening mode fracture of PMMA produced a 

highly localized thermal field. The maximum rate of internal heat generation 

was 11 × 1012J/m3

s for the low-speed opening mode experiment, considerably 

less than the maximum for shear-dominated fracture. The thermal dissipative 

mechanisms for low-speed opening mode and high-speed shear-mode fracture of 

PMMA are both highly localized. However, the energy dissipation rate for shear 

fracture is approximately four times larger than low-speed opening-mode 

fracture. Normalizing the energy dissipation rate by crack-speed yields an 

energy dissipation per unit crack area. If this is done for the shear- and lowspeed opening mode fracture cases, the energy dissipation values become much 

closer, giving a value of 35 × 109

J/m2 for shear-dominated fracture and 36 

× 109

J/m2 for low-speed opening mode. This further suggests the possibility of 

crazing as the dissipation mechanism for shear-dominated fracture. 

Length Along Propagation Direction (mm)

87.0 87.5 88.0 88.5 89.0 89.5 90.0 90.5

Vertical Dimension (mm)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

20 30 50 40 30 20

x 1012 J/m3s

Propagation

Direction
92 

4.5 Model Friction Experiments 

The experiments of Rosakis et al. (1999) showed evidence of crack face contact aft 

of the advancing crack tip. The infrared detector results for PMMA presented in 

Figure 62a show two discrete thermal events, one associated with the passage of 

the crack tip and a subsequent one speculated to be associated with frictional 

crack face contact. This section discusses a cursory examination aimed at 

determining if crack face sliding could produce thermal signatures similar to 

those observed in Figure 62a. The frictional heating study was not a 

comprehensive effort, it was by design a preliminary study to provide an 

indication of the magnitude of heating possible from high-speed frictional 

sliding of PMMA. 

Frictional heating of polymers has been the subject of very few studies. The 

maximum sliding speed for these studies has generally been limited to about 

30 m/s (Ettles 1986; Ettles and Shen 1988; Kong and Ashby 1992). Rosakis et al. 

(1999) found the propagation speed of the trailing crack face contact zone for 

shear-dominated fracture to be approximately the same as the crack tip, i.e., 

s 2c . The relative sliding speed between the two crack faces was not 

determined by Rosakis et al. (1999). However, given that the mechanism that 

causes shear-dominated fracture originates from an applied material speed 

mismatch between the top and bottom half of the specimen panel (resulting from 

projectile impact), it seems reasonable to expect the sliding speed to be on the 

order of the projectile speed. 

Frictional heating of generic materials was first modeled by Blok (1937). For a 

heat source of intensity Q& (per unit area) moving over a half space of uniform 

thermal properties, the surface temperature rise ∆T is 

, πkρc

t ∆T 2Q

p

& = (37) 

where t is the time the source had been over the point of interest, and k, ρ, and cp

are thermal properties defined in section 2 (see Table 2). The heat source for 

frictional contact is given as 

Q = μPV , & (38) 

where μ is the coefficient of friction, P is the interface pressure, and V is the 

sliding speed. The work of Ettles (1986) and Ettles and Shen (1988) determined 

that the frictional sliding of polymers produced a viscous melt layer between the 

sliding bodies. The friction coefficient in equation (38) was found to decrease 

with increasing sliding speed, effectively limiting the maximum temperature 

increase predicted from equation (37). Kong and Ashby (1992), who studied the 

speed dependence on the wear behavior of polymers, observed a thin layer of 

, 

, 
93

hot polymeric material deposited on the sliding surfaces. This layer corresponds 

with the viscous melt layer noted by Ettles (1986) and Ettles and Shen (1988). If 

the heating observed aft of the shear-dominated crack tip in Figure 62(a) is due to 

frictional sliding, a thin melt layer would likely be present. The fracture surfaces 

from the shear-dominated experiments of this study are examined in detail in 

section 4.6. To help in identifying the features on the fracture surface, melt layers 

resulting from high-speed friction of PMMA are needed. Additionally, the 

thermal signature from a high-speed polymer friction event is desired for 

comparison to the temperature field measurements shown in Figure 62(a). These 

are the goals of the friction experiments discussed next. 

High-speed friction experiments were performed using the configuration 

illustrated in Figure 66. Pieces of PMMA from the same lot used for the fracture 

experiments were cut to size and stacked as shown in Figure 66. The middle 

piece was impacted by a steel projectile, after which it then began to slide relative 

to the top and bottom pieces. The sliding surfaces were polished prior to the 

experiments with a 5-μm polishing slurry on a polishing wheel to achieve a 

glassy, smooth finish. The infrared detector monitored the interface between the 

top and middle pieces. Normal pressure was applied to the interfaces with four 

equispaced clamps. The magnitude of the pressure was not measured. The 

intent of these experiments was not to provide a comprehensive study of 

polymer frictional heating, but rather to give an indication of the thermal 

signature associated with high-speed polymer sliding and to obtain sliding 

surfaces for comparison to the fracture experiments. A much more detailed 

analysis could form the basis of a future project. 

Typical temperature field measurements from the friction experiments are 

shown in Figure 67. The projectile-impact speed was 40 m/s, identical to the 

impact speed for the shear-dominated fracture experiments. A peak temperature 

increase of 43 K was measured by detector element 14, which is about half the 

temperature increase needed to reach Tg. Flanking elements detected a 

temperature increase of 15 K and 10 K. The specific variation of temperature 

with time is likely due to the reflections of the stress waves in the three polymer 

pieces that produce compressive tractions on the sliding interfaces. The wave 

reflection pattern will not be similar to the fracture experiments because the 

overall size of the test specimens differ considerably. The significance of the 

temperatures shown in Figure 67 is that appreciable temperatures can be 

generated from dynamic frictional sliding of PMMA. Unique features were 

found on the sliding surfaces indicating the presence of a friction surface. This 

will be examined in the next section. 

4.6 Fracture Surface Micrography 

The fracture surfaces from the shear-dominated fracture experiments were 

examined to provide additional insight into the dissipation mechanisms in the 
94 

Figure 66. Configuration for high-speed friction experiments. Infrared detector 

monitored the sliding interface between the top and middle PMMA pieces. 

Figure 67. Typical thermal detector output (after calibration) for high-speed friction 

experiments. ∆T denotes temperature change from ambient and t = 0 

corresponds to time of impact. 

Steel Projectile

8.5 mm

127 mm

117 mm

Confinement Pressure

63.5 mm

19 mm

IR Detector

Coverage

Time (μs)

180 200 220 240 260 280 300 320 340 360 380

∆T (K)

0

5

10

15

20

25

30

35

40

45

50

10

14

16

12
95

crack tip process zone. Photographs of the PC and PMMA fracture surfaces 

generated from shear-dominated crack growth are shown in Figures 68 and 69, 

respectively. The surface for PC has a rippled appearance with the ripples 

oriented perpendicular to the crack propagation direction. The fracture surface 

for PMMA has a white-colored coating and small cracks extending below the 

crack surface. 

Figure 68. Photograph of fracture surface from shear-dominated dynamic fracture of PC 

with side grooves to suppress crack kinking. Fracture surface ripples are 

visible on the crack face. 

SEM images of the fracture surfaces were obtained to examine the surface 

features in greater detail. A fracture surface image for PC is shown in Figure 70 

along with a schematic depiction of the location it was taken from. The crack 

propagated from left to right in the figure. Ripples covering what was once 

molten material are evident in the figure. Fine strands of material oriented along 

the crack growth direction are present at a few locations. It is not possible to 

determine if the features are attributable to the shear band, the shear crack, or 

late time friction. The features on the fracture surface are considerably different 

from those observed on the opening mode PC fracture surface, where crazing 

and ductile tearing was observed (see Figure 53). The formation of ripples 

observed in Figure 70 is unique to this mode of failure for PC. 

The high-temperature increase recorded during the PMMA shear-dominated 

experiments indicates that evidence of polymer melting should exist on the 

fracture surfaces. SEM images of the fracture surfaces from this experiment are 

shown in Figure 71. Two images are shown, one from the specimen interior, and 

the other from a location near the groove surface. As in the work of 

Initial Notch

Tip

Painted

Lines

Crack Propagation

Direction
96 

Figure 69. Photograph of fracture surface from shear-dominated dynamic fracture of 

PMMA with side grooves to suppress crack kinking. White-colored wear 

layer is visible on the crack face. 

Figure 70. SEM image of fracture surface from shear-dominated dynamic fracture 

experiment using PC with side grooves. The crack propagated from left to 

right leaving surface ripples covered with a molten coating. 

Crack Propagation

Initial Notch Direction

Tip

100 μm

Top View of

Fracture Surface

Propagating

Crack
97

Figure 71. SEM image of fracture surface from shear-dominated dynamic fracture using 

PMMA with side grooves. Left-side image was taken in the interior of the 

panel. Right-side image is near location where infrared detector monitored. 

The crack propagated from left to right. 

Ravi-Chandar et al. (2000), the image taken from the interior of the specimen 

shows signs of considerable polymer melting, suggesting that the 

shear-dominated fracture event did generate high temperatures. The image 

taken near the surface of the groove also shows melting, but to a much lesser 

degree. Figure 71 clearly shows the three-dimensional nature of the 

thermomechanical phenomenon. Because the infrared detector system collects 

heat emitted from the specimen surface, the data are expected to be a lower 

bound of the actual temperature of the propagating crack tip in the interior of the 

specimen. Variations of the surface features throughout the specimen thickness 

were not observed for PC. Also evident on the fracture surface is a ripple feature 

below the melt region. Whether this occurred during the passage of the shear 

crack or the frictional motion of the two-panel halves remains unknown. 

The temperature field measurements for PMMA showed indications that 

frictional heating of the crack faces occurs aft of the crack tip. An SEM image of 

the sliding surface from the friction experiment (described in section 4.5) is 

shown in Figure 72. The friction surface from the experiment is shown on the left 

and the original surface condition of the surface prior to the experiment is shown 

on the right. The sliding direction of the mating surface was from left to right. A 

thin layer of molten material is spread along the surface with thin fibrils of 

IImage taken in interior of panel. Image taken near surface of panel.

25 m 25 m

Crack Propagation

Direction

Crack Propagation

Direction

Crack

Propagation Top View of Fracture Surface
98 

Figure 72. SEM image of PMMA high-speed friction surface. The mating surface slid 

over the surface shown from left to right. Deposits are visible in the friction 

surface. The original surface condition prior to the experiment is shown on 

the right. 

material stretched along the sliding direction. However, the temperature 

increase for the friction experiment was only 45 K, well below the PMMA glass 

transition temperature. The recorded temperature may be underestimated due 

to the relative size of the heat source compared to the detector element. This is 

discussed in section 4.3 and pursued further in section 6. Comparing this image 

with the PMMA fracture surface in Figure 71 shows some similarity. The 

fracture surface image shows a thin layer of material that has some resemblance 

to the resulting surface from the friction experiment. The implication of this is 

that the melting observed in the SEM image of the PMMA fracture surface may 

not be entirely due to thermal dissipation from shear-dominated cracking, but 

may also be the result of subsequent frictional heating that accompanies it. 

4.7 Additional Comments on Shear-Dominated Crack Tip Behavior 

The interpretation of the infrared emissions during shear-dominated fracture 

yielded insight into the coupling of mechanical work and thermal dissipation. 

Substantial heating was observed during shear-dynamic fracture of both PC and 

PMMA. The temperature signatures from the shear-dominated fracture 

experiments showed the two polymers to behave very differently to the same 

loading condition. An adiabatic shear band followed by a fracture event 

occurred in PC. The maximum temperature increase produced by the 

100 μm

Top View of

Friction Surface

50 μm

Surface Condition 

Prior to Experiment

Direction of

Sliding
99

propagating adiabatic shear band was 45 K, similar to that reported by Rittel 

(2000) who used a single thermocouple. The fracture event, which for the case of 

PC was subsonic, increased the temperature an additional 100 K. The 

temperature field of the propagating crack tip indicated the dissipation of 

mechanical work is primarily through plastic deformation. PMMA, on the other 

hand, showed an abrupt 85 K increase in temperature resulting from a crack 

propagating faster than the shear wave speed of the material. Thermal evidence 

of crack face contact and frictional heating were observed behind the advancing 

crack tip in PMMA. The frictional heating increased the temperature beyond the 

glass transition temperature of PMMA. SEM images of fractured PMMA 

surfaces confirmed that polymer melting did occur during the shear-dominated 

fracture event. 

Comparing the results from the shear-dominated fracture experiments with 

those of opening mode reveal several similarities and differences. Significant 

ductile deformation appears to contribute to material heating for PC regardless 

of fracture mode. The shear loading that causes shear-dominated fracture in PC 

also induces the formation of a shear band propagating ahead of the crack. The 

presence of the shear band is believed to decrease the energy flux to the crack tip 

in PC, resulting in a decrease in terminal crack tip speed. No dissipation 

mechanisms were found ahead of the shear crack in PMMA, which accounts for 

the higher terminal crack speed. The highly localized thermal field surrounding 

the shear-dominated crack tip was somewhat similar in size and dissipation 

strength to the single craze dissipation event observed for the low-speed PMMA 

opening fracture experiments. The fracture surfaces resulting from 

shear-dominated fracture of both materials contained a ripple feature that was 

not found on the opening-mode fracture surfaces, as well as on deposits of 

material that were molten. Given that shear-dominated fracture of these 

materials consists of multiple distinct events (adiabatic shear band for PC, shear 

fracture, and frictional sliding), the modeling of the shear-dominated dynamic 

fracture response of these materials will need to address multiple dissipative 

events. 

The results of the opening-mode and shear-dominated dynamic fracture 

experiments indicate that material heating is due to either thermoplastic or 

thermofracture heat sources, depending on the nature of the polymeric material. 

The next two sections provide a detailed examination of the relative contribution 

of each of these mechanisms. The next section discusses a computational study 

that attempts to identify the amount of heating attributable to plastic 

deformation for opening mode fracture of PC. Section 6 presents a dissipative 

cohesive zone model derived for dynamic opening-mode fracture of PMMA, 

where the thermofracture mechanism appears to dominate material heating. 

This begins the sizeable task of modeling the various dissipative phenomena 

observed during dynamic fracture of these materials. 
100 

5. Effect of Plasticity in Thermomechanical Heating of 

Ductile Polymers During Opening-Mode Dynamic 

Fracture 

5.1 Introduction 

Rapid plastic deformation of a material can produce an increase in temperature. 

A common example of this is the back-and-forth bending of a thin metal strip. 

After flexing the metal several times, it will feel warm when touched. This 

occurs because of the coupling that exists between heat generation and 

mechanical fields. The coupled heat energy relation was given in section 1 as 

, (3 2 ) , kT T0 q cpT p

ij ij

e

ii kk & −α λ + μ ε& + βσ ε& + & = ρ (39) 

where the parameters were defined in section 1.2.2. The linear thermoelastic 

constitutive relation used in deriving the heat energy equation was also given in 

section 1 as 

2 (3 2 ) ( ) , ij = ij kk + μ ij − λ + μ δ ijα T −T0 σ λδ ε ε (40) 

where T0 is the initial temperature of the material. Although the linear 

thermoelastic constitutive equation is fully coupled, the heat energy equation 

relation given by equation (39) is not because the dissipative term associated 

with plastic deformation is not fully coupled. As such, equation (39) can be used 

to calculate the temperature increase in a material resulting from mechanical 

fields and internal heating, but the mechanical fields and internal heating are not 

affected by the temperature increase. 

It was shown in sections 2 and 3 that nearly adiabatic conditions exist during 

dynamic opening mode fracture of PC and PMMA. Additionally, the 

thermoelastic term was shown in section 3 to be negligible compared to the 

temperature increases recorded during fracture. Neglecting the conduction and 

thermoelastic terms reduces equation (39) to the following: 

βW q ρc T , p

p & & & + = (41) 

with p W& , the plastic work rate, defined as 

. p

ij ij

p W σ ε& & ≡ (42) 

The quantity β is the fraction of plastic work rate that is converted to heat. As 

discussed in section 1, Rittel (1999) found β for PC to vary from 0.4 to 1.0 

, 

, 

, 
101

depending on strain rate, and Li and Lambros (2001) found β for PC to range 

from 0.5 to 0.6 during high-rate compression. 

The thermal fields surrounding propagating crack tips were shown in sections 3 

and 4 to produce significant temperature increase. The sources for dissipation 

were attributed to the work from plastic deformation and internal heating 

resulting from microstructure breakdown and molecular chain scission. For 

PMMA, evidence suggested the dissipation source was solely from internal heat 

generation associated with dynamic fracture. The data for PC, on the other hand, 

indicated that plastic deformation contributed a significant portion to the overall 

crack tip temperature. 

This chapter examines the contribution of plastic deformation heating to the 

thermal field measured for PC during opening mode dynamic fracture, 

presented in section 3 and shown in Figure 73. Finite-element simulations of the 

dynamic opening mode experiments are performed with an incremental 

plasticity model that uses a rate-dependent Mises yield surface and isotropic 

hardening to describe the behavior of PC. The only source of material heating 

permitted in the simulation is that resulting from plastic deformation. A 

comparison of temperature increase between experimental and numerical results 

is made to identify the plasticity contribution to the overall heating of the 

material. This allows the relative contribution of p W& in equation (41) to be 

determined for PC during opening mode dynamic fracture. The results of this 

section provide a better understanding of the heating mechanics in ductile 

polymers during dynamic fracture. This approach has been successfully used in 

the past in modeling the thermoplastic coupling in ductile metals (Zhou et al. 

1996). 

5.2 Finite-Element Model 

A Lagrangian-based finite-element technique was used in this study for 

simulating the PC opening-mode fracture experiments presented in section 3. A 

Lagrangian approach was selected because of the need to accurately track crack 

tip motion. The finite-element technique was used because it offered a 

convenient way to propagate the crack tip with time, as discussed later. Of the 

many Lagrangian finite-element codes available, the ABAQUS/Explicit finiteelement code (Hibbitt et al. 1998) was chosen primarily because of its ease of use 

and availability. The ABAQUS finite-element code is available with two 

different time integration techniques, implicit and explicit. In an implicit time 

integration scheme, the displacements at any time cannot be obtained without 

knowledge of the accelerations at the same time. The relationships among 

velocity, displacement, and acceleration must be combined with the equations of 

motion and the resulting set of equations, which are then solved simultaneously 

for each time-step. For problems with large numbers of elements, the 
102 

Figure 73. Measured temperature field surrounding a crack tip propagating in PC at 550 

m/s under mode I loading. 

simultaneous solution of the large set of equations can be a significant 

computational burden. The explicit time integration scheme does not require the 

accelerations at a given time to calculate the nodal displacements. An explicit 

expression for node displacement results from the equation of motion when the 

node velocities and accelerations are expressed by a central difference relation. 

Explicit time integration eliminates the need to solve large sets of equations 

simultaneously. However, explicit integration can cause the computed response 

to grow without bound (i.e., become computationally unstable) if the time step is 

too large. Most codes, including the ABAQUS/Explicit finite-element code, that 

was used in this study, have built-in routines to automatically select stable timesteps during the simulation. User intervention is not required. 

The modeling procedure in the ABAQUS finite-element code used is to construct 

a finite-element mesh of the material in the initial configuration, assign 

appropriate boundary and initial conditions to the model, supply a description 

of the material behavior (i.e., a constitutive material model), and then perform 

the finite-element solution. These various aspects are discussed next. 

The finite-element technique was used to simulate dynamic opening mode 

fracture in PC. The experiment used for comparison with the simulation is 

described in section 3.4.1, and a typical crack tip temperature field is shown in 

Figure 73. The projectile-impact speed for this particular experiment was 

25 m/s. The finite-element model duplicating the experimental configuration 

consisted of the polymer fracture specimen and the steel projectile. Because 

Time (μs)

200 205 210 215 220 225 230 235 240 245 250

∆Τ (Κ)

0

20

40

60

80

100

120

3

4

11

5

6

7

8

14

13

9 12

10
103

mode I fracture is symmetric about the crack propagation path, only the top half 

of the polymer specimen and projectile were modeled, with a plane of symmetry 

defined along the crack path. Introducing a plane of symmetry enabled a 

convenient method for advancing the crack tip, and is discussed later. The single 

edge notched PC specimen was discretized using an expanding mesh of four 

node-reduced integration quadrilateral elements and three node triangular 

elements. Quadrilateral elements were used near the region where plastic 

deformation was expected to occur (i.e., the crack tip region). This is also the 

location where the temperature fields of the numerical simulation will be 

compared to those measured during the experiments. Because the infrared 

detector element size was 80 μm, the cell size in the plastic deformation zone 

should also be ~80 μm for a valid comparison of temperature results. With such 

a small cell size in the crack tip region, great flexibility was needed to expand the 

mesh at locations distant from the crack tip region. This was achieved by 

transitioning from quadrilateral to triangular elements far from the plastic 

deformation zone. The triangular elements permitted greater flexibility in 

expanding the mesh. The steel projectile was also modeled with four nodereduced integration quadrilateral elements. The element size of the projectile 

was 3 mm. The final mesh used for the dynamic fracture simulation is shown in 

Figure 74. 

Plane stress elements were used throughout. The out-of-plane thickness of the 

PC plane stress elements was 8.5 mm. An element out-of-plane thickness of 

32 mm was used for the projectile to preserve the mass of the projectile. As 

mentioned earlier, the geometry is symmetric about the crack propagation path 

so only the top half of the polymer specimen and projectile were modeled. A 

boundary condition was imposed along the plane of symmetry that prevented 

nodes along the plane of symmetry from moving in the vertical direction. A 

mathematically sharp crack in the PC specimen was modeled by excluding from 

this restriction the nodes along the plane of symmetry that belonged to the crack. 

The finite-element model had a total of 77,600° of freedom. 

Crack growth in the numerical simulation was achieved by manually releasing 

nodes along the axis of symmetry at a prescribed time to duplicate the 

experimentally measured crack tip position history obtained from the crack 

initiation and propagation gages. This manual node release technique has been 

successfully used in the past (e.g., Lo et al. [1994]). The timing data from the PC 

experiment used for comparison indicated that crack growth began 165 μs after 

projectile impact for a 25 m/s impact speed. It is important to determine if there 

is significant variability in the initiation time because the stress state at the crack 

tip changes with time as stress waves propagate throughout the polymer 

specimen. Figure 75 shows the measured crack initiation time for mode I PC 

dynamic fracture experiments as a function of projectile impact speed. The 
104 

Figure 74. Finite-element mesh used for the numerical simulations. Element size in the 

crack tip region is 80 μm. The plane-stress model has 77,600° of freedom. 

impact speed of interest is 25 m/s. At this speed, there is some data scatter. The 

initiation time for the experiment corresponding to the temperature field 

presented in Figure 73 (that will be used for comparison with the numerical 

simulation) is within the data scatter. As such, the node release scheme used for 

crack growth was programmed to commence at 165 μs after projectile impact. 

Timing data from the crack propagation gage indicated the crack accelerated to 

terminal speed (550 m/s) within 3 mm of initiation. The position history of the 

crack tip from the experiment of interest is shown in Figure 76. The position vs. 

time data shown in Figure 76 was used to indicate the proper time for the 

sequential release of nodes along the crack path to simulate crack growth. Each 

node was released individually from its vertical displacement restriction at the 

appropriate time to cause the crack tip position history in the simulation to 

conform to Figure 76. Because there were more nodes released during the time 

interval of interest than there are data points in Figure 76, linear interpolation 

was used for prescribing the release time for intervals between data points in 

Figure 76. The manual node release technique has the effect of indirectly 

applying an empirical fracture toughness vs. crack tip speed criterion and 

eliminated the need to introduce and calibrate a failure model for elements along 

the axis of symmetry. Due to computer memory restrictions, only the first 4 mm 

Steel

Projectile

Polycarbonate

Panel

Crack Tip

Symmertry

Plane

Crack Tip

Steel

Projectile

Polycarbonate

Panel

Crack Tip

Symmertry

Plane

Crack Tip
105

Figure 75. Dynamic crack initiation time measured during the opening-mode fracture 

experiments using PC for various projectile-impact speed. 

Figure 76. Crack tip position history measured during dynamic opening-mode crack 

growth in PC. Position history was used for timing the sequential release of 

nodes along the plane of symmetry to simulate crack tip propagation in the 

numerical simulation. 

Impact Speed (m/s)

20 22 24 26 28 30 32 34 36 38 40 42 44 46

Crack Initiation Time (μs)

135

140

145

150

155

160

165

170

175

Time after Crack Initiation (μs)

0 10 20 30 40 50 60 70

Crack Tip Position (mm)

0

5

10

15

20

25

30

35
106 

of crack propagation were simulated, corresponding to 50 nodes being released. 

The crack tip speed was 550 m/s for the final 1 mm of crack growth (last 12 

nodes released). 

The initial conditions specified for the simulation consisted of prescribing an 

initial horizontal translation speed of 25 m/s for the nodes that comprised the 

steel projectile. The nodes from the projectile were prevented from penetrating 

into the PC finite elements at the impact interface by assigning a frictionless 

contact interface between the projectile and polymer specimen. The contact 

interface is a standard algorithm available in the ABAQUS/Explicit 

finite-element code. 

A mesh convergence study using square elements near the crack tip with 

dimensions of 2400, 400, and 80 μm was performed to ensure that the desired 

element size of 80 μm provided adequate resolution in the plastic deformation 

region. The material models for PC and steel described in the next section were 

used for each of the convergence study simulations. The temperatures of 

elements surrounding the crack tip were compared to evaluate convergence. The 

temperature fields for the grids with 400-μm and 80-μm cells were identical. The 

temperature field for the 2400 μm element grid showed essentially no 

temperature increase, primarily because the element size was about twice as big 

as the plastic deformation zone. Consistency of the computed temperature fields 

for the two smaller element size grids indicated that both the 400 and 80 μm 

element size was adequate. However, the 80-μm size was chosen so that a direct 

comparison with the infrared detector measurements would be possible. 

5.3 Material Model 

The heating that occurs during high-rate deformation of ductile materials was 

shown in section 5.1 to result from plastic deformation through the coupling of 

plastic strain and temperature given by equations (41) and (42). Because the goal 

of this section is to establish the relative contribution of ductile deformation 

heating to the crack tip temperature field measured for PC during dynamic 

opening-mode fracture, an accurate description of high-rate deformation 

behavior of PC is needed. It was beyond the scope of this study to develop a 

rate-sensitive constitutive model for PC. Instead, high-rate behavior from 

published studies was used. In particular, Li and Lambros (2001) determined the 

high rate compressive behavior of PC specimens that were taken from the same 

lot of material used in this study. A split Hopkinson pressure bar technique was 

used to generate strain rates as high as 2200 s-1. All tests were performed with 

the polymer material initially at room temperature. The stress-strain data 

measured by Li and Lambros (2001) for PC at compressive strain rates of 1200, 

1700, and 2200 s-1 are shown in Figure 77. Note that the data reported by Li and 
107

Figure 77. Nominal stress-strain data for dynamic compression of PC reported by Li and 

Lambros (2001). Results from three different strain rates are shown. 

Lambros (2001) and which was shown in Figure 77 is nominal (engineering) 

stress and strain. The yield stress was observed by Li and Lambros (2001) to be 

dependent upon strain rate, varying from 45 to 70 MPa for the strain rates shown 

in Figure 77. The fraction of plastic work rate converted to heat, β, was also 

measured by Li and Lambros (2001) with the same infrared detector system used 

in this study. A β value of 0.5 was found for all strain rates examined. Table 3 

shows the yield stress and fraction of plastic work converted to heat obtained 

from the experiments of Li and Lambros (2001). Rittel (1999) observed that β can 

increase to a value of 1 for strain rates of ~8000 s-1. 

Table 3. Uniaxial high strain-rate yield stresses and fraction of plastic work converted to 

heat (β) from Li and Lambros (2001). 

Strain Rate 

(s-1) 

Yield Stress 

(MPa) 

β 

1200 

1700 

2200 

45 

50 

70 

0.5 

0.5 

0.5 

The material model used for PC was a rate-dependent incremental plasticity 

model that is standard in the ABAQUS/Explicit finite-element code and is 

commonly used for metals. The model decomposes the strain rate into elastic 

and plastic components and uses a Mises yield surface with associated plastic 

flow. Isotropic hardening is permitted. Equivalent compressive and tensile 

Nominal Strain

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Nominal Stress (MPa)

0

20

40

60

80

100

120

140

160

2200 s-1

1700 s-1

1200 s-1
108 

behavior and the same Young’s modulus for all strain rates are assumed in the 

model. The inputs required by the code user for this material model are the 

Young’s modulus and a table of stress-strain pairs used to define the plastic 

portion of deformation for each strain rate. The code uses linear interpolation 

between the discrete number of points for strains not included in the table. To 

begin this process, the three nominal stress-strain curves shown in Figure 77 

were approximated by a number of points connected with straight line segments. 

The data points approximating the data of Li and Lambros (2001) are shown in 

Figures 78–80 for the three strain rates. The actual data input to the material 

model need to be in the form of Cauchy stress, σCauchy, and logarithmic plastic 

strain, p

ln ε . Assuming plastic deformation of PC to be incompressible (as is found 

to be the case for most metals), the conversion of nominal stress and strain to 

Cauchy stress and logarithmic plastic strain is 

(1 ) , Cauchy nom nom σ = σ + ε (43) 

ln(1 ) ,

int

ln

yield po

Cauchy

nom

p

E 













 = + − σ

ε ε (44) 

where E is Young’s modulus and the second term in equation (44) is evaluated at 

the material yield point. Equations (43) and (44) were used to convert the data 

points shown in Figures 78–80. The resulting “true” stress-strain data that were 

used in the code are shown in Figure 81. Note that the logarithmic strain in 

Figure 81 is plastic strain only. The elastic portion has been removed. 

Figure 1. Approximation of the plastic part of the nominal stress-strain curve for PC at a 

strain rate of 1200 s-1 as used in the numerical simulation material model.

Nominal Strain

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Nominal Stress (MPa)

0

20

40

60

80

100

120

140

160

, 

, 
109

Figure 2. Approximation of the plastic part of the nominal stress-strain curve for PC at a strain 

rate of 1700 s-1 as used in the numerical simulation material model. 

Figure 3. Approximation of the plastic part of the nominal stress-strain curve for PC at a strain 

rate of 2200 s-1 as used in the numerical simulation material model. 

Nominal Strain

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Nominal Stress (MPa)

0

20

40

60

80

100

120

140

160

Nominal Strain

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Nominal Stress (MPa)

0

20

40

60

80

100

120

140

160
110 

Figure 81. Cauchy (true) stress as a function of plastic logarithmic strain for three 

different strain rates of dynamic compression for PC. 

The density, specific heat capacity, and Young’s modulus used for PC were 

discussed in section 2 and are shown in Tables 1 and 2. It was mentioned in 

section 2 that Theocaris and Andrianopoulos (1982) found the modulus of PC to 

be relatively insensitive to strain rate at higher rates. This was also noted by Li 

and Lambros (2001) and is consistent with the assumptions of the rate-dependent 

plasticity model that is used here (i.e., constant Young’s modulus). 

Adiabatic heating of the material was permitted to occur only from plastic 

deformation (i.e., the internal heat generation term q& in equation (41) was not 

included in the thermomechanical coupling). The relationship that the ABAQUS 

finite-element code uses for thermomechanical heating is thus, 

c T . p

p

ij ij & βσ ε& = ρ (45) 

In all cases, β in equation (45) was taken to be 0.5. This was the experimentally 

obtained value from Li and Lambros (2001). If β = 1 was used (as suggested by 

Rittel 1999) the temperature results presented subsequently would scale 

accordingly. Either way, this does not affect the final conclusion as will be seen 

in the next section. 

5.4 Simulation Results and Discussion 

Because of computer memory restrictions, the finite-element simulation was of 

the first 4 mm of crack growth. After the crack had grown 4 mm according to the 

Plastic Logarithmic Strain

0.00 0.05 0.10 0.15 0.20 0.25

True Stress (MPa)

0

20

40

60

80

100

120

140

160

180

200

2200 s-1

1700 s-1

1200 s-1
111

crack tip position history given by Figure 76, the crack tip had reached the 

terminal speed of 550 m/s, and 50 nodes had been released along the plane of 

symmetry. The temperature field produced by the plastic deformation 

surrounding the propagating crack tip is shown in Figure 82. The finite-element 

grid is also shown in Figure 82. The elements along the plane of symmetry 

became sufficiently strained so much that the small strain assumptions built into 

the governing equations of the finite-element code were no longer valid. Hence, 

the temperatures calculated for these elements are likely not valid. For this 

reason, the results from the numerical simulation cannot be used to determine 

the temperature at the tip of a propagating crack, but rather the surrounding 

field should be used for comparison to the experimental results. In addition, the 

rigid body motion of the entire specimen in front of the detectors, which affects 

the experimental results, has not been accounted for in the numerical simulation. 

The effects of this motion are that the infrared detectors in the experiment view 

the passage of a crack propagating with an apparent speed slower than actual. 

Hence, some caution must be exercised when directly comparing the time scale 

of the numerical and experimental results. 

Figure 82. Temperature increase contours at the tip of a crack tip propagating in PC at a 

speed of 550 m/s predicted from the numerical simulation. Adiabatic heating 

was due solely to plastic work. 

The temperature contours from the numerical simulation show a region of 

elevated temperature ahead of and lateral to the crack tip. The boundary of this 

heat affected zone extends ~0.5 mm ahead of the crack tip and 

0.6 mm transversely. The temperature contours from the experiment, shown in 

Figure 83, shows the heat affected zone to extend several millimeters ahead of 

Original

Crack Tip Current

Crack tip

1 mm

∆T (K)
112 

Figure 83. Temperature contours surrounding a crack tip propagating in PC at 550 m/s 

under mode I loading. 

the peak in temperature. This clearly is not consistent with the numerical 

simulation. However, the lateral span of the measured temperature field is in 

reasonable agreement with the simulation. The peak temperature increase in the 

region near the crack tip for the experiment was 105 K. For the simulation, a 

temperature increase of 8 K was calculated. The simulation produced a 

maximum temperature increase only 8% of what was measured during the 

experiment. The simulation contour also shows the temperature to increase aft 

of the crack tip. This may be artificial because of the overstrained elements 

discussed earlier. 

A column of elements located 3.2 mm (40 nodes) ahead of the initial crack tip 

was monitored for comparison with the experimental results. The column of 

nodes was oriented much like the infrared detector elements in the experiments, 

except the detector array was located 20 mm from the notch, whereas the column 

of elements in the simulation was 3.2 mm from the notch. The temperature of 

the first seven vertical elements closest to the crack propagation plane, excluding 

the element on the symmetry plane, is shown in Figure 84. This figure should be 

compared to Figure 73, which shows the temperatures measured during the 

Length along Propagation Direction (mm)

109 110 111 112 113 114 115 116

Vertical Dimension (mm)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

10

20

50

40 30

80

70 60

90

100

100

90

80

70 60 50 40 30 20 10

40 20 30

10

Propagation

Direction
113

Figure 84. Temperature increase of a vertical array of finite elements located 3.2 mm 

from the initial notch tip for comparison to infrared detector measurements. 

experiment in a similar format. Recall that in the experiments, the detectors are 

stationary and the crack propagates through their field of view. Thus, the 

infrared detector array viewed multiple material points passing by the field of 

view as the polymer specimen fractured and translated. This led to the decrease 

in recorded thermal signals after crack passage in Figure 73, reasons for which 

were discussed in section 3. The column of finite elements is tied to the moving 

specimen, thus the temperature will rise to a maximum and then plateau, as it 

does in Figure 84. The temperature-time curves for the simulation presented in 

Figure 84 show the temperature to slowly increase for about 2 μs, and then 

abruptly increase to a plateau value during 1 μs. For the experiment, the 

temperature rise time was about 8 μs, considerably longer than for the 

simulation. There are several possible reasons for the discrepancy in 

temperature rise time. The node release technique that was used results in an 

abrupt release of nodes regardless of stress state in the material that it is 

associated with. In a real material, the failure of the material is likely to be less 

abrupt, and a gradual transition from fully constrained to fully released may be a 

more appropriate node release scheme. This could change the stress state in 

material surrounding the node being released. The recoil of suddenly severed 

polymer fibrils in the failed craze could potentially introduce a heat source aft of 

the actual crack tip, increasing the temperature rise time of the experiments. 

There is also the possibility that the opening of the heated crack faces past the 

Time (μs)

178 179 180 181 182 183 184 185 186 187 188

∆T (K)

0

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

1

2

3

4

5

6

7

Crack Propagation

Plane of

Symmetry

Elements
114 

stationary infrared detector elements could introduce an increased rise time of 

the temperature signal recorded by the detector system. This is examined in 

detail in section 6. 

The dynamic fracture experiments subjected PC to loading conditions different 

from those for which the material was characterized, i.e., the compressive split 

Hopkinson pressure bar experiments of Li and Lambros (2001). The polymer 

material in the vicinity of the propagating crack tip is predominantly in tension, 

and the tensile stresses are high enough to cause polymer chain scission and 

fracture at the tip of the crack. Significant temperature increase was observed 

during the experiments, and the heat-affected zone was relatively broad. Thus, 

under the loading conditions of the dynamic fracture experiments, classical metal 

plasticity behavior might provide an adequate description of PC. 

The numerical simulations of the fracture event indicate that classical metal 

plasticity behavior does not adequately describe PC during opening-mode 

dynamic fracture. The numerical results, shown in Figures 82 and 84, show the 

temperature field surrounding the propagating crack to be significantly lower 

than that recorded during the experiments. There are several possible reasons 

for the discrepancy in the magnitude of the temperatures surrounding the 

propagating crack tip. The fracture event in a ductile, amorphous glassy 

polymer involves triaxial stress states, plastic deformation, and the motion, 

disentanglement, and scission of long chain molecules (Kambour 1964). The 

material model that was used to specify the behavior of PC in the finite-element 

code accounted only for ductile deformation, and the parameters used to 

describe this deformation were obtained only from uniaxial compression 

experiments. The mode I fracture surface of PC showed evidence of a highly 

localized single craze failure at the region where the infrared detector was 

focused. Given that the temperature contours from the experiments were 

relatively large in size (compared to those observed in brittle PMMA where a 

single craze also occurred), it seems likely that an improved ductility model for 

PC is needed. It stands to reason that adding a significant thermofracture 

contribution ( q& ) to the coupled heat energy equation given by equation (41) will 

increase the total crack tip temperature, but only in a highly localized manner. 

The thermofracture contribution to crack tip heating is explored in the next 

section. Because the thermofracture process is highly complex, crack tip heating 

in PMMA is considered to avoid introducing plasticity. This will permit a 

thermofracture model to be developed without needing to include the 

complexities of coupled plastic deformation. 
115

6. Dissipative Cohesive Zone Model 

6.1 Introduction 

When a polymer well below the glass transition temperature is subjected to an 

imposed strain, a competition between classical shear yielding and craze 

formation occurs. As was discussed in section 1, crazing dominates this 

competition during opening mode crack growth of brittle polymers. The process 

of polymer crazing is quite well understood. There are three stages of polymer 

crazing (Kambour 1973; Kramer 1983), namely craze initiation, widening, and 

breakdown. These stages are depicted schematically in Figure 85. Using x-ray 

diffraction techniques, Kawabe et al. (1992) observed polymer chain alignment to 

occur ahead of the craze in PMMA and PC during quasi-static opening-mode 

loading. The region of molecular chain alignment is followed by areas of 

microvoid nucleation. Additional drawing of the material causes the 

micro-voids to enlarge, leading to the formation of oriented fibrils. The fibrils 

stretch until they reach a stretch ratio of ~4 (Donald and Kramer 1982; Wool 

1995). The craze widens by a process of surface drawing (Kramer 1983): new 

polymer material is drawn into the craze fibrils from the bulk-fibril interface, 

referred to as the active layer (see Figure 85). The fibrils that do not elongate 

much beyond a stretch ratio of 4 grow in length by adding more material to their 

ends from the active layer. Kramer (1983) and Kramer and Berger (1990) assume 

the material in the active layer to behave as a non-Newtonian fluid. Craze 

breakdown occurs by the eventual failure of the fibrils. Kramer and Berger 

(1990) observed fibril failure to occur at the fibril-bulk interface, and not at the 

center of the craze width, and both noted that the fibril failure event is triggered 

by impurities in the polymer, which served as failure nucleation sites at the fibrilbulk interface. 

The craze breakdown process results in the formation of two new surfaces, 

adding to the outer surface atoms a greater potential energy than those in the 

bulk of the material. During the creation of the two new surfaces, energy must 

be supplied to the material for the elevated potential energy state of the surface 

atoms. The amount of energy required is 2γ, where γ is the surface energy of the 

material (a material property). The craze formation event discussed above 

involves energy consuming processes additional to the surface energy 

requirements. The energy available for the dissipative crazing mechanisms is 

(G-2γ), where G is the dynamic energy release rate supplied by the far field 

loading. The relative amount of energy available for additional dissipative 

processes is evident by comparing the energy needed to fracture a polymer, G, 

with the surface energy for the polymer. The dynamic energy release rate for 
116 

Figure 85. The initiation, widening, and breakdown of a polymer craze during 

opening-mode fracture. Aligned molecular chains form thick fibrils that 

resist crack opening. 

opening mode fracture in PMMA was found in section 3 to vary from 2800 to 

5600 J/m2, depending on the crack tip speed measured. These measurements are 

in line with other previously quoted values (Döll 1976). However, in sharp 

contrast, the surface energy for polymers has been found to be several orders of 

magnitude lower than these measured G values. Wool (1995) cites a surface 

energy of 0.04 J/m2 for polystyrene. Berry (1960) estimates the surface energy for 

PMMA to be ~0.1 J/m2. These values for γ are quite small compared to the 

measured G values solely by added fracture surface area. For example, Sharon et 

al. (1996) found the amount of fracture surface area to increase six fold during 

high-speed dynamic fracture of PMMA, as discussed in section 3. Increasing the 

surface energy requirements by a factor of six still leaves the surface energy 

requirements for fracture very small compared to measured values of G. As 

such, approximately all of the energy used for polymer crazing during fracture 

can be assumed to be dissipated and likely results in the generation of heat. This 

assumption will be made in the remainder of this section. 

Voids

Aligned

Molecules

Crack

Face

Recoiling

Fibril

Stubs

Fibrils Microvoids

Aligned

Molecules

Crack Propagation

Direction

Amorphous

Region

Active

Layer

Voids

Craze Length

Open

Space
117

The craze formation and breakdown process discussed earlier and shown in 

Figure 85 includes many mechanisms that likely generate heat. As mentioned 

earlier, Kawabe et al. (1992) observed molecular chain alignment ahead of a craze 

in PMMA and PC. Movement of the chains to the oriented configuration (to 

include sliding and rotation) would lead to heat generation. As craze fibrils form 

and lengthen, significant molecular chain motion occurs within the active layer 

(Kramer 1983; Kramer and Berger 1990). The chain motion within the active 

layer is also a likely source of heat generation. The ultimate failure of the craze 

fibrils involves molecular chain scission (Kramer and Berger 1990), molecular 

chain pull-out from surrounding material (Wool 1995), and recoil of the fibril 

stubs (Kambour 1964; Wool 1995). Each of these events can also occur in a 

dissipative manner. Although the mechanisms associated with polymer craze 

formation, growth, and failure have been studied, the amount of heating 

associated with the mechanisms is unknown. 

The anatomy of a craze includes fibrils distributed along the length of the craze 

that resist craze opening. As mentioned above, this opening resistance occurs in 

a dissipative manner. The theoretical notion of a cohesive zone, introduced in 

section 1 and shown in Figure 86, is of similar character. Dugdale (1960) used the 

cohesive zone concept to account for material inelasticity in the vicinity of the 

crack tip and to analytically eliminate the stress singularity at the tip of a crack 

under mode I loading. The cohesive zone consists of a region ahead of the crack 

tip where the opening of the zone, δ, is resisted by a cohesive stress, σ, 

distributed along the cohesive zone, L. This is conceptually similar to craze 

formation ahead of a crack tip in brittle polymers, where the fibrils distributed 

along the craze length resist craze opening. As a consequence, various studies 

have successfully modeled the quasi-static crazing process during fracture of 

polymers with cohesive zones (Brown and Ward 1973; Peterson et al. 1974; Fraser 

and Ward 1978; Tijssens et al. 2000; Panda and Williams 2000; Allen and Searcy 

2001). A critical element of the cohesive zone concept is the cohesive traction law 

that establishes the cohesion stress as a function of opening, i.e., σ =σ (δ). 

Tijssens et al. (2000) developed a craze cohesive model with a traction-separation 

law motivated by micromechanical considerations. Their model includes the 

initiation, growth, and breakdown of polymer crazes, but unfortunately yields a 

set of equations that must be solved using a linear incremental analysis within a 

finite-element framework. Their application was limited to quasi-static loading 

of a PMMA plate with a central hole. A slow growth crack model for 

polyethylene using a cohesive zone methodology was developed by Panda and 

Williams (2000). A traction-separation law was measured using a tensile test 
118 

Figure 86. Cohesive zone concept at the tip of a propagating crack. Crack face opening, 

δ, is resisted by a cohesive traction σ. 

specimen with a pre-cut notch to induce a craze. However, in that study the 

highest displacement rate of the load frame grips was limited to 50 mm/min, far 

below the rates of interest in this study. Allen and Searcy (2001) presented a 

viscoelastic cohesive zone model that incorporates dynamic effects. The cohesive 

traction law for this model was based on micromechanical observations of 

polymer craze fibril behavior. The model of Allen and Searcy (2001) was 

presented as an analytical tool and was not evaluated against dynamic fracture 

experiments. Kubair et al. (2001) derived a rate-dependent cohesive traction law, 

but it also was not compared to experimental data. 

The opening profile of the craze at the tip of a quasi-statically growing crack was 

measured by Brown and Ward (1973) and Fraser and Ward (1978) for PMMA, 

and by Morgan and Ward (1977) for PC. Each of these studies concluded that the 

constant stress cohesive traction law of Dugdale (1960) provided reasonable 

agreement with the optical measurements of craze opening. However, the 

studies were limited to quasi-static crack growth. Peterson et al. (1974) used 

holographic interferometry to deduce the stress distribution along a stationary 

craze in PC as well as the craze opening profile. In contrast to Morgan and Ward 

(1977), they concluded that the stress profile was nonlinear, and the peak stress 

occurred at a position aft of the craze tip. Suzuki et al. (1997) measured the crack 

face opening profile for cracks propagating at 200 m/s in PMMA. 

Unfortunately, measurements of crack opening in the vicinity of the craze could 

not be made in that work. Thus, although some (conflicting) information on the 

nature of the traction-separation law, σ = σ (δ), for quasi-static crack growth in 

PMMA exists, little is known in the dynamic crack growth case. 

This section presents a dissipative cohesive zone model in which the craze 

energy is assumed to be converted to heat, applicable to brittle polymers prone 

to single craze failure, such as PMMA impacted at low-speeds. Several cohesive 

σc

L

δc

Crack Faces Crack

Tip

Cohesive

Zone

Tip

σ(δ)
119

traction laws are evaluated against experiment, and the model is used to make 

predictions of crack tip heating for each of the traction laws at various crack tip 

speeds. The predictions are then directly compared to the temperature 

measurements presented in section 3 for opening-mode dynamic fracture of 

PMMA. To the author’s knowledge, this effort represents the first attempt to use 

the cohesive zone concept for modeling thermal dissipation in brittle polymers 

during high-speed fracture. 

6.2 Cohesive Zone Model 

The notion of a cohesive zone at the tip of a crack was first introduced by 

Barenblatt (1959) and Dugdale (1960) to analytically eliminate the stress 

singularity at the tip of a crack under mode I loading. A cohesive zone is 

introduced as a line that extends from the crack tip to model the inelastic 

response of the material ahead of the crack, depicted in Figure 86. In this model, 

the material opening, δ, ahead of the crack tip is opposed by a cohesive stress, σ, 

acting along the cohesive zone line of length L. The opening displacement is zero 

at the front tip of the cohesive zone and reaches a maximum, denoted δc, at the 

rear of the cohesive zone. At the same time the cohesive traction σ, starts from a 

maximum value of σc at the forward tip of the cohesive zone and decreases to 

zero as the end of the cohesive zone is approached (i.e., at δ =δc). The end of the 

cohesive zone, which is the first point of separation of the crack faces, can be 

considered the crack tip. Therefore, the location of maximum opening 

displacement in the cohesive zone is the location where the physical crack tip 

begins. Barenblatt (1959) considered the case of pure material cleavage at the tip 

of a crack. For the crack to advance, atomic bonds must be broken to permit the 

creation of new surfaces. Barenblatt (1959) assumed a unique relationship 

between atomic force and the separation between atoms as the basis of a cohesive 

traction law. Dugdale (1960) considered the cohesive zone for an elasticperfectly plastic material. The transition from elastic-to-plastic behavior 

occurred at the leading edge of the cohesive zone. The cohesive traction law 

used by Dugdale (1960) was of constant stress and will be one of several cohesive 

laws explored in this chapter. 

Freund (1990) extended the cohesive zone idea to the dynamic crack growth case. 

Because the cohesive zone modeling performed in the present study builds upon 

the results of Freund (1990), a summary of the Freund (1990) analysis is 

presented here for completeness. Consider a mode I crack tip propagating with 

speed a& under plane stress conditions. To obtain a relationship between an 

arbitrary cohesive traction law and the length of the cohesive zone, L, the flux of 

energy into the crack tip is considered from two different contours surrounding 

the crack tip, as shown in Figure 87. The contour Γappl is circular in shape with a 

center at the leading edge of the cohesive zone and has a large radius compared 

to the length of the cohesive zone. If this contour is located within the region 
120 

where the stresses are accurately described by the asymptotic solution given by 

equation (1), the energy flux F through Γappl is given by Freund (1990) as 

( ) ( ) , 2

appl

d

I

I

appl K

E

aA F & Γ = (46) 

where E is the Young’s modulus for the material, AI is the mode I universal 

constant defined in section 2 which is a function of a& , and ( )appl

d KI is the 

dynamic mode I stress intensity factor resulting from applied far-field tractions. 

Figure 87. Far field and near tip contours used for calculating energy flux into a crack tip 

with a cohesive zone of length L (Freund 1990). 

A second contour, Γcoh, surrounds the actual cohesive zone itself. The energy flux 

through this contour (Freund 1990) is 

∫

−

Γ =

0

2 ( ) 2 ,

L

F coh σ ju& jdζ (47) 

L

ξ

y

Γcoh

Γappl

1

2

, 

, 
121

where uj is displacement, repeated indices imply summation, and the 

superposed dot denotes a material time derivative. Within the cohesive zone, 

δ (ξ, t) = 2u2(ξ, 0, t), σ12(ξ, 0, t) = 0, and σ22(ξ, 0, t) = σ[δ(ξ)]. Thus, equation (77) 

reduces to 

∫

−

Γ =

0

( ) ( ) ( , ) ,

L

F coh σ δ δ ζ t dζ & (48) 

where the superposed dot again denotes a material time derivative. The material 

time derivative of δ(ξ, t) is 

( , ) . ξ

δ δ δ ξ ∂∂ − ∂∂ = a

t

t & & (49) 

Using equation (49) in equation (48) gives the following for the energy flux 

through Γcoh: 

( ) ∫ ∫

−

+

∂

∂ Γ =

0

0

( ) ( ) .

L

coh d a d

t

F

c

ζ σ δ δ δσ δ

δ

& (50) 

Under steady-state crack growth conditions, the first term on the right hand side 

of equation (50) is zero. Additionally, steady state crack propagation ensures 

that ( ) ( ) F Γcoh = F Γappl , yielding 

( ) ( ) .

0

2

∫ =

c

K d

E

A

appl

d

I

I

δ

σ δ δ (51) 

The dynamic energy release rate G is introduced using equation (18) 

( ) , 2

appl

d

I

I K

E

A G = (52) 

which gives 

( ) .

0

∫ =

c

G d

δ

σ δ δ (53) 

Equation (53) relates the cohesive traction law to the dynamic energy release rate. 

It is easy to see from this relation that G is the area under the σ-δ curve. This 

makes sense physically as well because the cohesive tractions, σ, do exactly this 

amount of work over the cohesive opening δ, as the crack extends. 

The next step in obtaining the relationship between an arbitrary cohesive traction 

law and the length of the cohesive zone, L, is to obtain an expression for ( )appl

d KI

in equation (51). Freund (1990) derived an expression for the stress intensity 

factor resulting from the cohesive traction, ( ) KI coh , as 

, 

, 
122 

( ) ( ) ∫ = −

L

I coh dl

l

l K

0

, 2 σ

π

 (54) 

with the negative sign in equation (54) giving a negative stress intensity because 

the cohesive tractions are resisting crack opening. The total stress intensity factor 

for the crack is the sum of ( ) KI coh and ( )appl

d KI . To eliminate the stress 

singularity at the crack tip (the reason for introducing a cohesive zone in the first 

place), the total stress intensity factor must equal zero. Thus, 

( ) ( )appl

d KI coh = − KI , giving 

( ) ( ) ∫ =

L

appl

d

I dl

l

l K

0

. 2 σ

π

 (55) 

Substituting equation (55) into equation (52) gives the final expression relating 

the cohesive traction to the length of the cohesive zone (Freund 1990) 

( ) . 2 ( )

2

0 0

∫ ∫ 



 



 =

c L

I dl

l

l

E

A d

δ

σ

π

σ δ δ (56) 

If equation (53) is used, G can be related to the cohesive zone length through the 

relationship 

( ) . 2

2

0





 



 = ∫

L

I dl

l

l

E

A G σ

π

 (57) 

To solve for the cohesive zone length, L, a particular form for the cohesive 

traction law must be assumed. However, as can be seen in equation (56), the 

cohesive tractions can be expressed either as the cohesive traction law, σ ( ) δ , or 

as a distribution along the length of the cohesive zone, σ (ξ ). The σ ( ) ξ relation 

will be referred to herein as the cohesive zone stress distribution. Freund (1990) 

provides a relationship between the opening displacement and position along 

the cohesive zone for an arbitrary cohesive stress distribution as follows: 

( ) ( )

( ) ∫ −

− = ∂∂ L

d

D 0

2

1 2 , 2 1

η ξ η

σ η

η

ξ

πμ

λ λ

ξ

δ (58) 

where λ1, λ2, and D were defined in section 2, and μ is the material shear 

modulus. Closed form solution of equation (56), which requires solution of the 

Cauchy singular integral equation given by equation (58), is then possible only 

for simple cohesive traction laws, such as the constant stress law used by 

Dugdale (1960). For more complicated σ (δ ) traction laws, an iterative 

numerical procedure is used to solve equations (56) and (58). An easy 

, 

, 
123

alternative, which has a wider variety of analytical solutions, is to assume a σ ( ) ξ

relation. This effectively assumes the spatial variation of stresses ahead of the 

crack tip. Equation (58) can then be used to analytically solve for a number of 

σ ( ) ξ relations. Because there is a unique one-to-one relation between σ ( ) δ

relations and resulting σ ( ) ξ relations, it is possible to inversely derive the 

behavior of various σ ( ) δ laws. This approach has been used in the present 

work. Alternatively, a numerical solution of equation (58) for σ ( ) δ is possible, 

as discussed in Kubair et al. (2001). This point will be further discussed later. 

6.2.1 Thermal Dissipation in a Cohesive Zone 

For a specific cohesive stress distribution, equations (56) and (58) give the 

cohesive zone length and opening profile along the length. The aim of this study 

is to extend the previous set of equations, derived in Freund (1990), to yield the 

thermal dissipation within the cohesive zone. For a material that exhibits 

negligible plastic deformation, the heat energy equation given by equation (8) 

reduces to 

c T q , p & ρ & = (59) 

where q& is the volumetric internal heating associated with brittle fracture. 

Integrating equation (59) gives an explicit expression for the temperature 

increase of the material from an initially traction free state as 

( ) ( ) . 1

0

∫ ∆ =

t

p

q d

c

T t τ τ

ρ & (60) 

The primary task in evaluating equation (60) is to relate the internal heating 

rate, q&( )t , to the cohesive zone. 

Sun and Hsu (1996) obtained an analytic expression for q& of the form 

( 2 ) ( ) ( ) 0 0 G − γ a&δ x − x δ y − y , under the conditions of stable crack growth, 

where δ is the Dirac-delta function, x and y are coordinates, and the zero 

subscript denotes the current crack tip coordinates. Physically the term (G-2γ) 

represents energy that has flowed through the crack tip region (G), but is in 

excess of that needed for crack surface formation (2γ), as discussed in the 

previous section. The analytic expression for q& by Sun and Hsu (1996) 

conceptually makes sense, but is dimensionally incorrect. The expression has 

units of J/m·s, whereas the governing equation for which it was derived requires 

it to have the units of J/m3 s. The expression for q& obtained in this study differs 

significantly from that of Sun and Hsu (1996) in that the internal heating rate is 

related to the mechanics of the cohesive zone. 

, 
124 

The rate of work W& done by the cohesive traction at any point inside the 

cohesive zone is 

(ξ ) σ (ξ )δ (ξ ). & & W = (61) 

Equation (61) represents the rate of energy dissipated at a location ξ within the 

cohesive zone, and is equated to the rate of internal heat generation q& . 

However, the work rate given by equation (61) is per unit area, whereas the 

internal heating rate used in equation (60) is per unit volume. What is needed to 

remedy this situation is to include in equation (61) the amount of material above 

and below the fracture surface associated with heat generation. Adding a 

volumetric aspect to the cohesive zone concept deviates from the development of 

the cohesive zone model because the cohesive zone is idealized as a line of zero 

thickness along the crack propagation path. Defining the thickness of material 

within the cohesive zone associated with material heating as H(ξ), the expression 

for the internal heating rate is 

( ) ( ) ( )

( ) . ξ

σ ξ δ ξ ξ H q & & = (62) 

The thickness H(ξ) is not necessarily the same as the cohesive opening 

displacement δ(ξ) because, as discussed in section 6.1, polymer material feeds 

into the fibril from an active layer at the bulk-fibril interface (see Figure 85). The 

thickness of this active layer is unknown. Substituting equation (62) into 

equation (60) and noting that under steady-state conditions time and position 

within the cohesive zone can be interchanged, the expression for the temperature 

increase within the cohesive zone is 

( ) ( ) ( )

( ) . 1

0

∫ ∆ =

t

p

d

c H

T t τ

τ

σ τ δ τ

ρ

& (63) 

Because the temperature predicted by the model derived in this section will be 

compared to the temperature measurements made by the infrared detector 

presented in section 3, an expression for the temperature recorded by the 

infrared detector elements is derived here. As stipulated above, the thickness of 

material associated with heating is H(ξ). The infrared detector has a dimension 

of 80 μm and, as discussed in section 3, will provide an output proportional to 

the radiation incident on the entire detector element area. Figure 88 illustrates 

the case of H less than the detector size which is most likely the case based on 

quasi-static results for H. The temperature sensed by the detector element in 

Figure 88 is 

, actual

det

det T

y

H ∆T = ∆ (64) , 
125

Figure 88. Heat source smaller than the infrared detector element passing over the 

region monitored by the detector. The element responds to the average 

radiation incident on the entire area monitored. 

where ∆Tdet is the temperature increase sensed by the detector element, ∆Tact is 

the true temperature of the heat-affected region, and ydet is the height of the 

detector element (80 μm for this study). Substituting equation (64) into equation 

(63) gives the following as the temperature increase that the infrared detector 

element would sense from a dissipative cohesive zone, 

[ ] ( ) () () . 1

0

∫ ∆ =

t

p det

det d

c y

T t σ τ δ τ τ

ρ

& (65) 

Calculating the temperature increase for the infrared detector element eliminates 

the need to determine the actual thickness of the heat affected region. The 

disadvantage to this is that the true temperature increase cannot be determined 

unless an additional relationship is introduced to specify the thickness H. For 

heat sources with a thickness less than the detector element size, the infrared 

detector system underpredicts the true temperature. 

An algorithm based on the previous model development was constructed for 

predicting the temperature increase at the tip of a dynamically propagating 

mode I crack in PMMA. The following procedure was used. 

• A cohesive stress distribution is specified as σ(ξ). 

• The dynamic energy release rate G is obtained for a particular crack tip 

 speed. 

• The length of the cohesive zone is obtained using equation (57). 

Stationary

Detector

Element

Crack Propagation

Direction

Heat

Source

δ ydet = 80 μm
126 

• Equation (58) is used to obtain an expression for ∂δ ∂ξ . 

• The temperature increase sensed by the infrared detector element is 

calculated using equation (65) assuming steady-state crack growth 

and the result of step 4. 

The results prior to step 4 of the algorithm permit the cohesive traction law, 

σ ( ) δ , to be determined for a given cohesive stress distribution σ (ξ ), specified in 

step 1. 

It was pointed out in the section 6.1 that the total amount of energy available for 

dissipation is (G-2γ), not simply G as assumed in equation (57). However, it was 

also discussed that 2γ is several orders of magnitude less than G. In the results 

that follow, all of the measured G will be assumed available for dissipation 

within the cohesive zone given the relatively insignificant surface energy 

requirements. 

6.2.2 Cohesive Traction Laws and Stress Distributions 

The first step in the algorithm listed above is to select a suitable cohesive zone 

stress distribution. In doing so, a unique cohesive traction law is implicitly 

selected given the one-to-one relation between σ (ξ ) and σ (δ ) discussed earlier. 

Numerous traction laws have been examined in the past, and they generally fall 

into one of two categories, intrinsic and extrinsic (Kubair and Geubelle 2001). 

Intrinsic cohesive traction laws, depicted on the left of Figure 89, have a material 

failure criterion built into the traction law. Material within the cohesive zone 

strengthens as the zone opens until a critical stress, σc, is achieved. Physically, σc, 

represents the material strength and must be obtained from experiments, i.e., it is 

a material property. Further opening of the cohesive zone results in a weakening 

of the material to eventually zero stress. The area under the σ-δ curve is equal to 

the dynamic energy release rate, G. Needleman (1987), Suo et al. (1992), and Xu 

and Needleman (1994) used an exponential law as an intrinsic cohesive traction. 

A bilinear intrinsic cohesive traction law was used by Geubelle and Baylor (1998) 

and Lin et al. (2001), and a trapezoidal-shaped law was used by Tvergaard and 

Hutchinson (1992). The other category of cohesive traction law is extrinsic, 

where the strengthening portion of the traction curve is not included, as shown 

on the right side of Figure 89. Extrinsic laws were used by Barenblatt (1959), 

Dugdale (1960), Geubelle and Rice (1995), and Camacho and Ortiz (1996). In this 

study, an extrinsic cohesive traction law will be used; however, as discussed, it is 

derived from an initial cohesive stress distribution σ(ξ). This is done to permit a 

closed form solution of equation (58). The area under the traction law curve, G, 

was measured during the experiments discussed in section 3. The material 

critical stress, σc, however, was not directly measured. In fact, it is not very clear 

how to measure the critical stress of a material. As such, the only parameter that 

is not known a priori is the critical stress, σc. However, as will be discussed later, 
127

Figure 89. Illustration of intrinsic and extrinsic cohesive traction laws (Kubair and 

Geubelle 2001). 

a reasonable estimate of σc can be made from the quasi-static experiments shown 

in Figure 5. All other parameters needed for the algorithm previously listed are 

measured quantities. 

The two extrinsic rate-independent cohesive stress distributions initially 

examined are the constant stress distribution and a linear stress decay 

distribution. These distributions were selected for their simplicity and because 

an analytic solution of equation (58) could be obtained. The constant stress 

distribution is considered first. Dugdale (1960) proposed a constant stress 

cohesive traction law of the form 

( ) , σ δ = σ c (66) 

where σc is a constant, critical stress. Dugdale (1960) assumed the material to 

deform plastically when stressed beyond σc. Because the cohesive stress is 

constant and independent of the cohesive opening, the cohesive stress 

distribution σ(ξ) is also constant and equal to σc, as shown in Figure 90. With 

this in mind, the length of the cohesive zone, L, is determined from equation (57). 

Details of the solution procedure are provided in Appendix A. The resulting 

analytic expression for the cohesive zone length is 

. 8 2 AI c

EG L

σ

π = (67) 

With the length of the cohesive zone determined, the rate of change of the 

cohesive opening displacement, i.e., equation (58), can be obtained. An analytic 

expression was obtained for equation (58). The solution technique for solving 

the resulting Cauchy singular integral equation is provided in Appendix B. The 

final expression is 

Intrinsic Cohesive Law

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

δ / δc

σ / 

σc

Extrinsic Cohesive Law

strengthening weakening weakening

δ / δc

σ / 

σc

, 
128 

Figure 90. Form of constant and linear decay stress distributions used in the dissipative 

cohesive zone model. 

ln , 















+

− = − ∂

∂

ξ

ξ

πμ

σ

ξ

δ

L

C L c (68) 

where 

( ) . 2 1 2

1 2

D

C λ − λ = (69) 

This completes step (4) from the algorithm outlined above. Calculation of the 

temperature increase from the widening cohesive zone (solution of equation [65]) 

requires a change of variables in equation (60) from ξ to t. Assuming steady state 

crack propagation speed, the change of variables is ξ = a&t . The integration of 

equation (6.20) to yield ∆Tdet could not be performed analytically. Instead, 

numerical integration was performed using a trapezoidal area technique that is 

standard in SigmaPlot (SPSS Inc. 2000). The high-rate baseline properties for 

PMMA listed in Tables 1 and 2 were used, along with a G value of 3000 J/m2 and 

a critical stress σc of 120 MPa. The value for G corresponds to a crack tip speed of 

300 m/s (see Figure 36), which was the low-speed crack growth case in section 3 

where a single craze formed along the crack path. It is for this case that the 

dissipative cohesive model would be expected to have best agreement. The 

critical stress was estimated based on the tensile stress-strain data presented in 

section 2 for PMMA, assuming additional rate effects. The critical stress value 

used is discussed in greater detail in the next section. The predicted temperature 

ξ / L

0.0 0.2 0.4 0.6 0.8 1.0

σ / 

σc

0.0

0.2

0.4

0.6

0.8

1.0

Constant Stress

Distribution

Linear Decay

Distribution

, 
129

increase (normalized by ambient temperature) for the constant stress cohesive 

traction law is shown in Figure 91. The normalized temperature increases with 

normalized time until a final value of 0.062 is reached at the crack tip. A 

thorough comparison of these results with the dynamic mode I fracture 

experiments of PMMA will be made in the next section. 

Figure 91. Temperature increase predictions made with the dissipative cohesive zone 

model using the constant stress and linear decay stress distributions. 

Baseline material properties for PMMA and a crack propagation speed of 

300 m/s were used. Ambient temperature T0 is 295 K. 

The second cohesive stress distribution considered is one where the stress decays 

linearly. It will be shown that this stress distribution does not result in a linear 

decay cohesive traction law, σ(δ), as was used by Camacho and Ortiz (1996). The 

particular form of the stress distribution is shown in Figure 90, and has the 

following analytic form: 

( ) 1  ,



 



 = − L c

ξ σ ξ σ (70) 

where once again σc is the critical stress and L is the yet to be determined 

cohesive zone length. Solution of equation (57) for L using the linear stress 

distribution results in the following analytic expression: 

. 32

9

2 AI c

EG L

σ

π = (71) 

t a / L

0.0 0.2 0.4 0.6 0.8 1.0

∆Tdet / T0

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Linear Decay

Stress Distribution

Constant Stress

Distribution

.

, 
130 

Details of the solution procedure for obtaining equation (71) are provided in 

Appendix A. For equivalent crack growth conditions, the cohesive zone length 

of the linear stress distribution model is 2.25 times greater than that of the 

constant stress case. The closed form expression of ∂δ ∂ξ for the linear decay 

cohesive stress distribution is derived in Appendix B. The final expression is 

ln , 2

2

































+

− − = +

∂

∂

ξ

ξ ξ ξ

πμ

σ

ξ

δ

L

L

L

L

L

C L c (72) 

where C is given by equation (69). The quantity ∆Tdet from equation (65) was 

obtained for the linear distribution case in the same manner as was used for the 

constant stress case. The results are also shown in Figure 91, along with the 

results from the constant stress case. The temperature increase is approximately 

the same as was calculated for the constant stress case. However, as will be seen 

later, the rise time for the temperature increase is considerably longer for the 

linear case because the cohesive zone length, L is longer (by a factor of 2.25 as 

previously discussed). A significant difference in the qualitative behavior of the 

two different stress distributions is that the temperature increase for the constant 

stress case terminates with a high slope, whereas the linear decay case has a rolloff near the peak temperature. The temperature roll-off will be shown in the next 

section to be consistent with the general behavior of the temperatures recorded 

during the fracture experiments presented in section 3. 

During the solution procedure previously mentioned, the stress and opening 

displacement are both calculated as a function of position within the cohesive 

zone. These two quantities can be cross-plotted to depict the cohesive traction 

law σ(δ). Figure 92 shows this for the two σ(ξ) distributions considered thus far. 

As should occur, the constant stress distribution is constant in both σ-ξ and σ-δ

space. The linear decay case, however, is not of the same form for the two 

spaces. Slight deviations from linearity are observed for this traction law. Most 

of the deviation is for small cohesive zone openings. The remainder of the curve, 

however, is nearly linear. 

Before attempting a comparison with experimental data, a parametric study was 

performed to examine the sensitivity of the cohesive zone model to parameter 

variations and thus provide a better understanding of the effects important to the 

generation of heat within the cohesive zone. The parameters varied were E, G, 

σc, and a& . The linear decay cohesive stress distribution derived above was used 

for the parametric study. The cohesive zone algorithm was used to obtain the 

variations of temperature (∆Tdet) and internal heating rate ( q& ) with time. A 

baseline value for each parameter was selected, and one value higher and one 

value lower than the baseline were investigated in each case. The baseline value 

for each parameter was based on the nominal value for PMMA and a crack tip 

, 
131

Figure 92. Calculated form of the constant stress and linear decay traction laws obtained 

from matching cohesive zone stress distributions. 

speed of 300 m/s. Table 4 shows the baseline and variation of each parameter 

examined. Values for the other parameters were taken from Tables 1 and 2. 

Although the results are not shown here, the remaining parameters used in the 

model (ν, ρ, and cp) were systematically varied to determine their influence on 

calculated temperatures and heating rates. Poisson’s ratio, ν, was found to have 

negligible influence on calculated results for values ranging from 0.3 to 0.4. The 

other two parameters, ρ and cp, are present in the algorithm only through 

equation (65), and have a direct inverse proportionality effect on temperature. 

The coupling between temperature ρ and cp for PMMA is discussed in 

section 6.4. 

Table 4. Parameter values used for the parametric study. 

Parameter Value Parameter Low Baseline High 

E (GPa) 4.5 5.5 6.5 

G (J/m2) 2000 3000 4000 

σc (MPa) 100 120 140 

a& (m/s) 250 300 350 

The results from the parametric study are shown in Figures 93–100. Figures 93 

and 94 show Young’s modulus influence on predicted temperature ∆Tdet and 

internal heat rate q& , respectively. Increasing E does not change the maximum 

temperature increase, but it does increase the rise time of the temperature 

δ / δc

0.0 0.2 0.4 0.6 0.8 1.0

σ /σc

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Constant Stress

Distribution

Linear Decay

Stress Distribution
132 

Figure 93. Temperature increase predictions using a linear decay σ(ξ) relation from 

parametric study with E varied. 

increase. The magnitude of the internal heat rate (Figure 94) decreases with 

increasing E and occurs over a longer time duration. In contrast to the effect of E, 

the dynamic energy release rate, G, is observed to have a significant influence on 

the maximum temperature increase. Figure 95 shows that doubling G doubles 

the maximum temperature. This is caused by the increased area under the q& -t

curve, shown in Figure 96. The critical stress, σc, in Figure 97 is observed to have 

a significant influence on the rise time of the temperature increase, but no 

influence on the peak temperature. Increasing σc decreases the rise time of the 

temperature gain. An increase in σc also increases the magnitude of q& , as shown 

in Figure 98. But, because G was held constant, the resulting cohesive zone 

length decreased to keep the area under the q& -t curve constant for the different 

critical stress values. Thus, in some sense, the effects of σc and G on temperature 

evolution are complimentary: σc increases the rise time and G increases the 

maximum ∆Tdet. The crack tip speed is shown in Figures 99 and 100 to have a 

modest influence on the temperature rise time and magnitude of q& . Increasing 

the crack tip speed shortens the temperature rise time and increases q& . 

This parametric study provides a means for bounding the model predictions if 

there is uncertainty in the experimentally measured parameters. The baseline 

value of the various parameters correspond to approximately those appropriate 

for low-speed crack growth in PMMA. There are uncertainties in these values 

Time (μs)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

∆T (K)

0

5

10

15

20

25

E = 4.5 GPa

E = 5.5 GPa

E = 6.5 GPa
133

Figure 94. Internal heating rate predictions using a linear decay σ(ξ) relation from 

parametric study with E varied. 

Figure 95. Temperature increase predictions using a linear decay σ(ξ) relation from 

parametric study with G varied. 

Time (μs)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

q

(J/m3

s)

0.0

5.0e+12

1.0e+13

1.5e+13

2.0e+13

2.5e+13

E = 4.5 GPa

E = 5.5 GPa

E = 6.5 GPa

.

Time (μs)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

∆T (K)

0

5

10

15

20

25

G = 2000 J/m2

G = 3000 J/m2

G = 4000 J/m2
134 

Figure 96. Internal heating rate predictions using a linear decay σ(ξ) relation from 

parametric study with G varied. 

Figure 97. Temperature increase predictions using a linear decay σ(ξ) relation from 

parametric study with σc varied. 

Time (μs)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

5.0e+12

1.0e+13

1.5e+13

2.0e+13

2.5e+13

G = 2000 J/m2

G = 3000 J/m2

G = 4000 J/m2

q

(J/m3

s)

.

Time (μs)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

∆T (K)

0

5

10

15

20

25

σc = 100 MPa

σc = 120 MPa

σc = 140 MPa
135

Figure 98. Internal heating rate predictions using a linear decay σ(ξ) relation from 

parametric study with σc varied. 

Figure 99. Temperature increase predictions using a linear decay σ(ξ) relation from 

parametric study with a& varied. 

Time (μs)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

∆T (K)

0

5

10

15

20

25

a = 250 m/s

a = 300 m/s

a = 350 m/s

.

.

.

Time (μs)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

5.0e+12

1.0e+13

1.5e+13

2.0e+13

2.5e+13

σc = 100 MPa

σc = 120 MPa

σc = 140 MPa

q

(J/m3

s)

.
136 

Figure 100. Internal heating rate predictions using a linear decay σ(ξ) relation from 

parametric study with a& varied. 

however. All parameters except σc were either measured directly in this study or 

were obtained from the literature. Uncertainties resulting from measurement 

technique or even the appropriate load rate locally within the material (for the 

case of rate sensitive E) prevent the exact values of the parameters from being 

known. For the parameter σc, the exact meaning and the appropriate 

measurement technique of this parameter is not very clear. With this in mind, 

the variations of the parameters used in the parametric study are believed to be 

in excess of the experimental uncertainty. 

The parametric study was performed to examine the influence of a single 

parameter while the other parameters were kept constant. In a real material, it 

stands to reason that E, G, and σc are not independent parameters. Equation (46) 

shows that the energy flux into the crack tip region is inversely proportional to E. 

The cohesive model, as formulated in this study, is stress limited, meaning 

inelastic behavior initiates once a critical stress is exceeded. Increasing only E

will result in a decrease in the energy flowing into the crack tip region because 

the strain energy density in the material resulting from far field tractions will be 

less. However, the energy flux in equation (46) is simply (G a& ), so increasing E

will likely decrease G in a real material. Changes in σc will also influence G

because of strain energy density considerations. Given the possible 

interdependencies of the various model parameters, a true evaluation of the 

model will by necessity require comparison with experimental data where many 

of the needed parameters were measured. In this way, the model can be used 

Time (μs)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

5.0e+12

1.0e+13

1.5e+13

2.0e+13

2.5e+13

a = 250 m/s

a = 300 m/s

a = 350 m/s

q

(J/m3

s)

.

.

.

.
137

with actual parameter values appropriate for the dynamic fracture event being 

modeled. This is done in the next section. 

6.3 Comparison of Dissipative Cohesive Zone Model With 

Experiments 

6.3.1 Single Craze Failure Mode 

The dissipative cohesive zone model derived in section 6.2 was based primarily 

on first principles. To determine if the model correlates with actual material 

behavior requires comparison with experimental data. The temperature fields 

surrounding high-speed opening-mode crack tips in PMMA were presented in 

section 3. As discussed previously, the lateral span of the temperature fields 

indicated that there was little, if any, plasticity surrounding the crack tip. Thus, 

the temperature increase for PMMA was by thermofracture mechanisms rather 

than thermoplasticity. The dissipative cohesive zone model derived in the 

previous section is applicable for the thermofracture portion of crack tip heating. 

The thermoplasticity contribution, present only for ductile materials, was 

examined in section 5. To examine the validity of the dissipative cohesive zone 

model for brittle materials, comparisons are made in this section with the 

temperature measurements of dynamic opening-mode fracture for high MW 

PMMA. The fracture surfaces from the low-speed PMMA experiments 

presented in section 3 indicated that a single craze formed along the crack plane 

(see Figure 55). As discussed earlier, the cohesive zone model is expected to 

have best agreement with crack growth involving a single craze. Given this, the 

model is evaluated against the low-speed PMMA dynamic fracture data. 

The temperature field measurements of two nonidentical low-speed 

opening-mode high molecular weight PMMA fracture experiments are shown in 

Figure 101. The crack tip propagation speed was 300 m/s for both experiments. 

Recall from section 3 that the thermal signal sensed by the infrared detector 

system was highly localized for the low-speed case, with only one detector 

element registering a temperature increase during the early part of the 

measurement interval. Because of this, the temperature signals from flanking 

detector elements are not included in Figure 101. The first 4 μs of the 

temperature signals are shown, corresponding to the initial temperature increase 

to peak value. This is the time regime appropriate for model comparison. The 

inset in Figure 101 shows a longer portion of the temperature signals for 

perspective. There is little difference between the two temperature signals, 

indicating good repeatability from test to test. 

The cohesive zone model will be compared to one of the low-speed temperature 

signals shown in Figure 101 to assess the validity of the model, specifically, the 

signal depicted by the solid line in Figure 101. To further assess model validity, 
138 

Figure 101. Leading portion of experimentally measured temperature signal from two 

different PMMA opening-mode fracture experiments with crack tip speeds 

of 300 m/s. Inset at top left shows a longer time span of the signals. 

q& calculated from the model using equation (62) will be compared to the rate of 

internal heating obtained from the experimental data using equation (59). This 

requires the derivative of the experimental temperature signals to be calculated 

numerically. The procedure used for obtaining the derivative was presented in 

section 3. The calculated q& for the low-speed experiments shown in Figure 101 

are presented in Figure 102. Again, the two signals are in good agreement. The 

signal depicted with the solid line in Figure 102 will be used for comparison with 

the model. Comparing temperature and q& between the model and experiments 

is equivalent to comparing an analytic solution and its first derivative to data. 

This gives a clearer indication of the model’s behavior. This is in contrast to 

earlier cohesive model studies (Tvergaard and Hutchinson 1992; Needleman 

1992; Xu and Needleman 1994; Tvergaard and Hutchinson 1996; Camacho and 

Ortiz 1996) where the traction-separation relation, σ(δ), was based on theoretical 

arguments or post-mortem fracture surface observations and not on 

experimental measurements taken during the decohesion process. 

Time (μs)

-1 0 1 2 3 4

∆T (K)

0

10

20

30

0 10 20 30 40

0

10

20

30
139

Figure 102. Rate of internal heating derived from dynamic opening-mode fracture of 

PMMA. The crack tip speeds were 300 m/s, and the experimental signals 

are from the peak detector. 

The dissipative cohesive zone model parameters corresponding to the low-speed 

experiment in Figure 102 are the baseline values listed in Tables 2 and 4. Recall 

that G, E, a& , ρ, cp, and ν are independently measured. They are not fitted to 

experimental data. The critical stress, σc, is a parameter that was not directly 

measured. On the basis of the low and moderate rate PMMA tension tests 

presented in section 2 (see Figure 5), a high-rate estimate of a maximum stress 

value of 120 MPa was made. This value was used for all of the calculations in 

this chapter; it was not varied. The constant stress and linear decay stress 

distributions were used in the dissipative cohesive zone model with the baseline 

parameter values. The model predictions are compared with the low-speed 

temperature measurements in Figure 103. Both models predict a peak 

temperature that is slightly less than the peak temperature measured for the 

experiment. However, the constant stress distribution has a rise time that is too 

short, and the linear decay distribution is too slow in reaching peak temperature. 

With this in mind, a third cohesive stress distribution was used that qualitatively 

is in between the two distributions previously presented. This stress 

distribution, referred to as the quad-high stress distribution, is a quadratic 

function of the form 

( ) 1 .

2















 



 



 = − L c

ξ σ ξ σ (73) 

Time (μs)

-1 0 1 2 3 4

q

(J/m3

s)

0.0

5.0e+12

1.0e+13

1.5e+13

2.0e+13

2.5e+13

3.0e+13

.
140 

Figure 103. Comparison of the temperature increase predicted by the dissipative 

cohesive zone model with experimental data for opening-mode fracture of 

PMMA with a crack speed 300 m/s. Constant stress and linear decay 

stress distributions were used in the model. 

This stress distribution is shown in Figure 104 along with the two previously 

presented cohesive stress distributions for comparison. This stress distribution 

was used in the cohesive zone model in the same way as for the other two stress 

distributions. The solution for the cohesive zone length, L, and ∂δ ∂ξ was 

obtained in a manner similar to that described in Appendices A and B for the 

linear stress distribution. The resulting expressions are 

, 128

25

2 AI c

EG L

σ

π = (74) 

ln . 3 2

2

2

2 2

2

































+

− − = + +

∂

∂

ξ

ξ ξ ξ ξ ξ

πμ

σ

ξ

δ

L

L

L

L

L

L

L

C L c (75) 

Although the stress distribution σ(ξ) has the desired shape, as shown in Figure 

104, the shape of the cohesive traction law σ(δ) is not guaranteed to be the same. 

The resulting σ(δ) traction law calculated from the cohesive zone algorithm is 

shown in Figure 105. The baseline values for the various parameters were used. 

The shape is observed to change slightly from the shape of the stress distribution 

σ(ξ), but it still remains between the two previous traction laws. 

Time (μs)

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

∆T (K)

-5

0

5

10

15

20

25

30

Linear Decay

Stress Distribution

Experimental

Data

Constant Stress

Distribution

;
141

Figure 104. Form of the quad-high stress distribution compared to the constant stress 

and linear decay stress distribution. 

Figure 105. Calculated shape of the quad-high traction law. The constant stress and 

linear decay traction laws are shown for comparison. The baseline 

parameters for 300 m/s crack growth in PMMA were used. 

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

σ / 

σc

ξ / L

Constant Stress

Distribution

Quad-High

Stress Distribution

Linear Decay

Stress Distribution

δ / δc

0.0 0.2 0.4 0.6 0.8 1.0

σ / 

σc

0.0

0.2

0.4

0.6

0.8

1.0

Constant Stress

Distribution

Linear Decay

Stress Distribution

Quad-High

Stress Distribution
142 

The predicted temperature for the quad-high stress distribution is compared to 

the low-speed PMMA experiment in Figure 106. The maximum temperature 

increase is about the same as was predicted for the constant and linear decay 

stress distributions; however, the rise time is in better agreement with the 

experimental data. To further compare the model with the low-speed PMMA 

data, the internal heating rate from the experiment was compared with the q&

calculated for the three stress distributions. The comparison is shown in Figure 

107. The quad-high distribution closely matches the experimental data, whereas 

the linear decay distribution underpredicts q& and the constant stress distribution 

has a form that deviates significantly from the experimental data. Once again it 

should be emphasized that the parameter values used in Figures 106 and 107 

were not fitted to the experimental data, but rather were independently 

measured (G, E, a& , ρ, cp, ν) or estimated (σc). Yet, the agreement for the quadhigh stress distribution is remarkably good, both in ∆Tdet and q& . 

Figure 106. Temperature increase predicted by the dissipative cohesive zone model 

compared to experimental data for 300 m/s crack growth in PMMA. 

6.3.2 Distributed Damage Failure Mode 

In addition to the low-speed single craze experiments, the cohesive zone model 

was used to predict the temperature increase for cracks propagating at higher 

speeds with the full knowledge that there is considerably more spatial 

distribution of damage around the crack tip. Higher speed temperature data 

used for model comparison are shown in Figure 108, along with the results from 

the low-speed tests. The data are from experiments with cracks propagating at 

Time (μs)

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

∆T (K)

-5

0

5

10

15

20

25

30

Experimental

Quad-High Data

Stress Distribution

Linear Decay

Stress Distribution

Constant Stress

Distribution
143

Figure 107. Internal heating predicted by the dissipative cohesive zone model compared 

to experimental data for 300 m/s crack growth in PMMA. 

400 and 600 m/s in high molecular weight PMMA. As before, the temperature 

signal from the leading detector for each of the two experiments is shown. The 

inset in the figure shows the temperature data for a longer time duration for 

perspective. The internal heating rate q& of the three different speed experiments 

is shown in Figure 109. The rate of internal heating was calculated in the same 

way as for the low-speed case. A comparison of predicted q& with the 

high-speed data will aid in assessing the model behavior for high-speed crack 

growth. 

The only dissipative cohesive zone model parameters that were changed for the 

higher speed cases were G and a& . The G values used were 3550 J/m2 for the 400 

m/s case and 5600 J/m2 for the 600 m/s case. These values were taken from the 

measured G vs. a& plot shown in Figure 36. Comparison of the model with the 

400 m/s crack speed experimental data is shown in Figure 110. The temperature 

increases predicted by all three stress distributions were about half the increase 

measured during the experiment. The three distributions all resulted in a 

temperature increase of about 22 K, compared to a measured increase during the 

experiment of 35-45 K. Of the three distributions, the quad-high distribution best 

matches the data despite the underprediction in temperature. Model comparison 

for the 600 m/s case is shown in Figure 111. The predicted temperature 

increases from the three different stress distributions ranged from 31 to 35 K, 

with the constant stress distribution giving the lowest increase. The 

experimental data show a temperature increase of ~72 K, roughly twice the 

model predictions. The quad-high distribution gave the best qualitative 

agreement. 

Time (μs)

-1 0 1 2 3 4

0.0

1.0e+13

2.0e+13

3.0e+13

4.0e+13

Constant Stress Distribution

Linear Decay Stress Distribution

Quad-High Stress Distribution

q

.

(J/m3

s)

Experimental

Data
144 

Figure 108. Leading portion of experimentally measured temperature signal from three 

different PMMA opening mode fracture experiments with crack tip speeds 

of 300, 400, and 600 m/s. Inset at top left shows a longer time span of the 

signals. Peak detector only shown for 400 and 600 m/s experiments. 

Figure 109. Rate of internal heating derived from dynamic opening mode fracture of 

PMMA. The crack tip speeds were 300, 400, and 600 m/s, and the 

experimental signals are from the peak detector. 

Time (μs)

-1 0 1 2 3 4

∆T (K)

0

20

40

60

80

0 10 20 30 40

0

20

40

60

80

300 m/s

400 m/s

600 m/s 600 m/s

400 m/s

300 m/s

Time (μs)

-1 0 1 2 3 4

q

0

2e+13

4e+13

6e+13

8e+13

1e+14

.

(J/m3s)

300 m/s

400 m/s

600 m/s
145

Figure 110. Temperature increase predicted by the dissipative cohesive zone model 

compared to experimental data for 400 m/s crack growth in PMMA. 

Figure 111. Temperature increase predicted by the dissipative cohesive zone model 

compared to experimental data for 600 m/s crack growth in PMMA. 

Time (μs)

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

∆T (K)

0

10

20

30

40

50

Constant Stress Distribution

Linear Decay Distribution

Quad-High Distribution

Experimental

Data

Time (μs)

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

∆T (K)

0

10

20

30

40

50

60

70

80

Constant Stress Distribution

Linear Decay Distribution

Quad-High Distribution
146 

To better examine the model behavior relative to the experimental data, the 

internal heating rates predicted by the quad-high stress distribution for all three 

crack tip speeds are compared to q& derived from the experimental data. The 

comparison is shown in Figure 112. As was shown earlier, good agreement 

exists for the 300 m/s case. The quad-high model follows the experimental data 

during most of the active heating phase (i.e., the first 2 to 3 μs). The lack of 

agreement between model and experiment for the higher speed temperature data 

(shown in Figures 110 and 111) is evident in the internal heating comparison. 

The q& predicted by the model initially agrees with the experiments. However, 

q& from the model reaches a maximum and decreases to zero much earlier than is 

observed for the experiments, leading to less overall heating and less 

temperature increase. 

Figure 112. Comparison of the internal heat rate predicted by using the quad-high stress 

distribution in the dissipative cohesive zone model with experimental data 

for low, mid, and high crack tip speed fracture of PMMA. 

The reason for the discrepancy between model and experiment at the higher 

crack tip speeds is the microcracking phenomenon discussed in section 3 that 

occurs during high-speed crack propagation in PMMA. As the crack tip speed 

increases in PMMA, the fracture surface changes from a smooth “glassy” surface 

to a rough surface with many microcracks present. The smooth fracture surface 

is due to the passage of a single polymer craze, and is produced when the crack 

tip speed is low, typically 300 m/s or lower. For higher crack tip speeds, 

multiple craze fronts occur above and below the primary fracture plane, 

resulting in surface roughness (see Figure 54). Thus, whereas a single craze can 

Time (μs)

-1 0 1 2 3 4

0

2e+13

4e+13

6e+13

8e+13

1e+14

q

.

(J/m3

s) Experimental Data

600 m/s

400 m/s

300 m/s

Model

600 m/s

400 m/s

300 m/s
147

be approximated as failure occurring along the fracture plane through a cohesive 

zone, the multiple, layered crazing that exists during high speed fracture in 

PMMA is more of a volumetric failure event. The cohesive zone concept 

assumes failure to occur on a single plane. It was not devised to model 

distributed damage. Nilsson and Hutchinson (1994) have devised a modified 

constant stress cohesive law to account for distributed damage; however, it was 

derived for ductile materials. 

To illustrate the volumetric aspect of damage during high-speed fracture of 

PMMA, the distribution of temperature at a single time was recorded for a 

representative experiment. The temperature distribution, shown in Figure 113, is 

the temperature sensed by each detector element at the time of peak 

temperature. The crack tip speed for the experiment shown was 600 m/s. As 

observed in Figure 113, three detector elements sensed material heating, 

implying the thickness of the damage zone was equivalent to the spacing of three 

elements in the infrared detector array. The finite size of the detector element 

prevents an exact thickness from being determined, but the damage zone 

thickness is estimated to be between 110 nm and 190 μm. A more precise 

measurement would require a detector system with higher spatial resolution. 

The estimated thickness is consistent with measurements of the surface 

roughness shown in Figure 54. The damage distribution was fit to a 

transcendental function of the form 

, 2cos 2 



 



 ∆ = b

y Tdet d π (76) 

where d and b are fitting constants and y corresponds to the vertical position of 

the elements, i.e., the x axis of the plot. The good agreement suggests a possible 

empirical damage distribution in the material. Although not shown here, other 

high-speed experiments showed a similar degree of agreement with equation 

(76). The modeling of distributed damage is not pursued further in this study 

and remains a topic of future work. 

6.4 Further Analysis of Temperature Field Measurements 

The particular stress distribution used in the dissipative cohesive zone model 

determined the temporal form of the predicted temperature increase. 

Comparison of the model results with experimental data was used to indicate the 

validity of the various stress distributions considered. For this comparison to be 

meaningful, correct interpretation and understanding of the temperature signals 

from the infrared detector system is crucial. In this section, a detailed analysis of 

the low-speed experimental data is performed. The aspect of crack tip 

propagation past a stationary detector array along with crack face opening is 

further explored. The results of this analysis will place a limit on the precision 

that can be attributed to the initial portion of the temperature measurements. 

, 
148 

Figure 113. Distribution of temperature among the vertically oriented infrared detector 

elements at the time of peak temperature. 

6.4.1 Flanking Thermal Signal Decay 

During the low crack tip speed fracture experiments of PMMA presented in 

section 3, the stationary infrared detector array monitored the passage of a single 

craze and the subsequent opening of the newly formed crack faces. The 

temperature curves from the experiments show a rise in temperature followed by 

a slower decrease. As was discussed in section 3, Mason and Rosakis (1992) 

suggest crack opening as the likely reason for the decrease in temperature but do 

not pursue this further. If the crack is assumed not to open and adiabatic 

conditions exist (due to the short times involved), the temperature curves would 

be expected to increase to a peak value and remain constant in the trailing 

thermal wake. This is precisely what happens during shear-dominated fracture 

of PMMA, discussed in section 4 (see Figures [62a] and [62b]) If, on the other 

hand, crack opening occurs, the temperature recorded would decrease as the 

detector began to view the air gap between separated fracture surfaces as 

conjectured by Mason and Rosakis (1992). The opening of the crack would move 

the heated crack faces to adjacent detector elements later in time. A contour plot 

of the temperature increase measured by the detector system for a crack tip 

propagating at 300 m/s in PMMA and the trailing thermal wake is shown in 

Figure 114. The temperature contour plot was constructed in a manner similar to 

those discussed in section 3, except the time scale was not transformed to length. 

The displacement of the heated crack surfaces is apparent, consistent with the 

explanation offered by Mason and Rosakis (1992). 

Vertical Position (μm)

-200 -100 0 100 200

∆T (K)

0

20

40

60

80

Experimental Data
149

Figure 114. Contours of temperature increase in PMMA for a propagating crack tip and 

trailing thermal wake. Crack tip propagation speed is ~300 m/s. The crack 

opening displacement is evident by the shape of the trailing wake region. 

To better understand the effect of crack face opening on the measured 

temperature signals, three possible configurations involving the opening of a 

single craze in the region monitored by the infrared detector elements are 

considered. Along with each of the configurations considered, the resulting 

temperature signals from a representative low-speed opening-mode PMMA 

experiment corresponding to each configuration will be shown. The first 

configuration, depicted in Figure 115, has the craze passing directly through the 

region monitored by a single detector element. Experimental temperature 

signals corresponding to such a situation are shown at the bottom of Figure 115. 

For a crack tip propagating at 300 m/s, the time required for the tip to move the 

entire distance of one detector element (80 μm) is 0.26 μs. This is a small fraction 

of the total rise time actually recorded, indicating that the rise time of the 

temperature signal is not significantly influenced by crack tip motion in the 

direction of propagation. However, as the craze widens, more of the area 

monitored by the detector element is covered, increasing the overall output for 

that element. Eventually, the craze splits and the heated material moves 

transversely to the crack propagation direction at the speed of crack face 

opening. This situation is shown in the middle sketch of Figure 115. Because all 

the actively heated region is still within the 80-μm detector space, the detector 

output remains constant until portions of the heated material move outside of 

Time (μs)

250 300 350 400 450

Vertical Position (mm)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

5

5

5

5

5

5

5

5

5

5

5

5

5

5

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5

5

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5

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5

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5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

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5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

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5

5

5

5

5

5

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5

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5

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5

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5

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10

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5

1010

5

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10

10

10

5

10

10

10

10

5 5

5

10

10

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10

10

10

10

15

10

10 10

15

10

15

15

20

5

15

20

20

5

5

5

Crack Tip

Location
150 

Figure 115. Depiction of craze thickening, splitting, and crack opening relative to the 

stationary infrared detector elements for a craze passing through the region 

monitored by an element. Bottom of figure is representative data for the 

configuration illustrated. 

Craze

Thickening

Crack Face

Motion

Craze

Splitting

Detector

Element

Crack Propagation

Direction

Craze

Time (μs)

250 275 300 325 350 375 400 425 450 475

∆T (K)

-5

0

5

10

15

20

25
151

the region viewed by the detector, at which time the output decreases. 

Eventually the heated material moves to areas monitored by flanking detector 

elements (right side of Figure 115). The output from these neighboring elements 

will increase as the heated material moves into the coverage zone, plateau once 

fully inside the area and then decrease as the heated material exits the coverage 

zone. However, the thickness of the heated material viewed by neighboring 

detectors is half the maximum craze thickness, so the detector output should be 

half that of the center detector. The temperature signals for the flanking detector 

elements will coincide in time only if the original craze passed exactly through 

the middle of the central detector element. The signals will not coincide in time 

if the craze passed slightly high or low through the central detector. This 

mechanism corresponds to the actual signals recorded during the experiment 

shown at the bottom of Figure 115. 

The second possible configuration of crack opening is shown in Figure 116. For 

this configuration, the craze passes between two adjacent detector elements, 

where it splits and separates. This time, both primary detector elements view 

only half of the original craze thickness. An example of this for low-speed 

fracture of PMMA is shown at the bottom of Figure 116. The time of crack 

passage through the detector measurement region cannot be determined solely 

from the temperature signals for this case. 

The third configuration, shown in Figure 117, is for the case where the crack tip 

passed above or below the entire detector array. The thermal event sensed by 

the detector elements then corresponds to the vertical propagation of the split 

craze through the detector field of coverage. A low-speed PMMA experiment 

illustrating this is shown at the bottom of Figure 117. The crack opening speed 

can be experimentally determined from an experiment such as the one shown at 

the bottom of Figure 117, but the crack tip position cannot be ascertained (and 

hence the position along the crack face corresponding to the transverse speed 

measurement is unknown). 

The temperature signals from the low-speed experiment presented in this 

section and section 3 show a plateau for each detector, and the elements flanking 

the peak detector sense a temperature increase half that of the peak. This 

suggests that configurations similar to Figures 115–117 occurred. In particular, 

the low-speed PMMA data used for comparison with the dissipative cohesive 

zone model (Figure 101) show trends that are consistent with the sequence of 

events depicted in Figure 115. An analytic approximation of the crack face 

opening speed, 2 u& , is given by Freund (1990) as 

[( ) 1 2], 2

2

2

1

2 ≈ + λ − μ π

λ

D r

a K

u

d

I & & (77) , 
152 

Figure 116. Depiction of craze thickening, splitting, and crack opening relative to the 

stationary infrared detector elements for a craze passing between two 

detector elements. Bottom of figure is representative data for the 

configuration illustrated. 

Craze

Thickening

Crack Face

Motion

Craze

Splitting

Detector

Element

Crack Propagation

Direction

Craze

Time (μs)

300 320 340 360 380 400 420 440

∆T (K)

-5

0

5

10

15

20

25

30
153

Figure 117. Depiction of crack opening relative to the stationary infrared detector 

elements for a craze passing below the detector element array. Bottom of 

figure is representative data for the configuration illustrated. 

Detector

Element

Crack Propagation

Direction

Half

Craze

Time (μs)

200 220 240 260 280 300

∆T (K)

-5

0

5

10

15

20
154 

where r is the distance from the crack tip and the other parameters were defined 

in section 2. The crack face opening speed predicted using equation (77) for a 

crack propagating at 300 m/s in PMMA is shown in Figure 118 using d KI

derived from equation (18) and the measured value of G. Also shown in Figure 

6.34 are data points obtained from the slow-speed PMMA experiment used for 

comparison with the cohesive zone model (i.e., the low-speed curve shown in 

Figure 108). The experimental values of crack face opening speed were obtained 

by noting the time at which the flanking detector elements in the detector array 

sensed an increase in temperature. The speed was calculated by dividing the 

distance between detector elements by the time increment of sensed temperature 

between adjacent elements. The data are in reasonable agreement with the 

analytic expression, corroborating the notion of crack face motion as being 

responsible for the temperature sensed by the detector elements flanking the 

peak detector. 

Figure 118. Crack face opening speed for fracture of PMMA with a crack tip speed of 

300 m/s. Experimental data derived from temperature measurements are 

compared to analytical expression from Freund (1990). 

Unfortunately, the configuration depicted in Figure 115 indicates that there is 

some ambiguity in interpreting the initial rise time of the temperature signal. It 

is possible that the increase in temperature is due to the expanding craze 

thickness and not an increase in craze temperature. It also suggests that the 

roll-off in temperature near the peak could be artificial, resulting from the craze 

halves exiting the detector coverage area. This affects not only the temperature 

signal, but also the calculated q& . The quad-high cohesive stress distribution was 

Time (μs)

0 20 40 60 80 100

u2

(m/s)

0

5

10

15

20

25

30

35

Theory

Experiment

.
155

identified as best matching the experimental data. If the roll-off portion of the 

experimental temperature signal is artificial, the constant stress distribution may 

in fact provide the best match with the data. These issues can only be addressed 

with a temperature measurement system having a spatial resolution significantly 

better than the 80 μm resolution used in this study. However, as is, the final 

resulting difference between the quad-high and linear stress distributions is not 

large. 

6.4.2 Cohesive Zone Length 

The length of the cohesive zones predicted by the model for the low-speed case 

were 375, 585, and 840 μm for the constant, quad-high, and linear stress 

distributions, respectively. The typical length for a single craze in PMMA grown 

under quasi-static loading is less than 100 μm (Kambour 1964). Since the 

cohesive zone concept was introduced in this study to model polymer craze 

formation and breakdown, the predicted cohesive zone length should compare 

with typical craze lengths. However, as mentioned above, crazes from quasistatic loading are significantly shorter than the predicted cohesive zone length. If 

the actual cohesive zone had a length of 100 μm and a speed of 300 m/s, the rise 

time of temperature for a stationary observer would be ~0.3 μs. This is on the 

order of the system response time. Yet, the actual thermal signals are several 

microseconds long. A more sophisticated rate-dependent traction law (Kubair et 

al. 2001) could be used to shorten the predicted cohesive zone length, but this 

does not change the fact that the measured temperature rise lasted microseconds 

in duration. One possible explanation for this is that the thermally active zone 

ahead of a crack propagating at high speed in PMMA is considerably longer than 

the quasi-static craze length. As mentioned in section 6.1, Kawabe et al. (1992) 

observed polymer chain alignment to occur ahead of the craze in PMMA and PC 

during opening mode loading. This extended zone of molecular motion would 

likely result in heat dissipation and could explain the discrepancy between the 

predicted cohesive zone length and observed craze lengths. 

There is an additional phenomenon associated with polymer crazing that may 

increase the duration of the thermal event sensed by the infrared detector 

elements. Following rupture of the stretched craze fibrils, molecular chain recoil 

will occur (Kambour 1964; Wool 1995). During recoil, internal heating of the 

fibril stubs may be sensed by the detector elements. The heating would occur aft 

of the crack tip, increasing the length of the thermally active region and thus the 

duration of the thermal event recorded by the detectors. The internal heating 

mechanism associated with molecular chain recoil is not pursued in this study, 

and the magnitude and duration of fibril stub recoil heating is unknown. 
156 

6.4.3 Effect of Temperature on μ and cp

As the material in the vicinity of the crack tip increases in temperature, several 

material properties may change from their baseline value. A parametric study 

was performed in section 6.2 to examine the sensitivity of material property 

variation on the model. Here, the actual variation of two material properties 

with temperature, namely ρ and cp, is examined for PMMA (Domininghaus 

1988). The temperature dependence noted by Domininghaus (1988) is shown in 

Figures 119 and 120 for ρ and cp, respectively. A 70 K increase from room 

temperature decreases ρ from 1190 to 1168 kg/m3. The same increase in 

temperature changes cp from 1470 to 1900 J/kg-K. Note that the variations of ρ

and cp with temperature are inverse to each other. Interpretation of the 

temperature signals measured during the experiment was done assuming 

constant values of these parameters (i.e., one-way coupling), despite a measured 

temperature dependence. However, strictly speaking, the temperature 

dependence should be taken into account and could potentially form another 

source of uncertainty in the analysis. These two parameters (ρ and cp) are used in 

the cohesive zone algorithm during the final step through equation (65). If the 

higher temperature values of ρ and cp are used instead of the nominal room 

temperature values, the predicted temperature from the dissipative cohesive 

zone model would be reduced by about 21% for the case of a 75 K temperature 

increase (the high-speed PMMA case). For the lower speed case where the 

experiments showed a 25 K temperature increase, the temperature predicted by 

the model would be lowered by only 7%. Thus, although the model does not 

include fully coupled material properties, the error in not accounting for the 

thermal coupling of ρ and cp is minimal for the low-speed case. 

7. Conclusions 

7.1 Conclusions 

The main goal of this study was to develop a better understanding of the 

dissipative processes present during opening and shear-dominated dynamic 

fracture of nominally brittle and ductile amorphous polymers. This was 

achieved through an experimental, numerical, and analytical investigation of the 

thermal dissipation of mechanical work in PMMA and PC. The main results of 

this work are summarized below. 

• Opening mode dynamic fracture of ductile PC and brittle PMMA was 

found to be a highly dissipative event with significant temperature 

increases observed surrounding the propagating crack tip. 
157

Figure 119. Variation of density with absolute material temperature for PMMA 

(Domininghaus 1988). 

Figure 120. Variation of specific heat with material absolute temperature for PMMA 

(Domininghaus 1988). 

Temperature (K)

250 275 300 325 350 375 400 425 450

ρ

(kg/m3)

1.15

1.16

1.17

1.18

1.19

1.20

Temperature (K)

250 275 300 325 350 375 400 425 450

cp

(J/kg-K)

1000

1250

1500

1750

2000

2250

2500
158 

The noncontact method of infrared radiometry was used to measure in high 

spatial and temporal resolution the temperature fields surrounding the 

advancing crack tips. This represents the first time dynamic crack tip 

temperature fields have been measured in polymeric materials. The maximum 

temperature increase recorded for cracks propagating near the limiting speed of 

0.55 cR was 105 K for PC and 72 K for PMMA. Dynamic fracture of PC was found 

to show dissipation due to plastic deformation as well as polymer craze 

formation and breakdown. A single craze formed along the surface of the 

material, where the temperature field was measured. Near the center of the 

fracture specimen thickness, ductile tearing was found to accompany polymer 

crazing. The multiple infrared detector thermal signals along with fracture 

surface microscopy indicated that polymer crazing is the primary energy 

dissipation mechanism during opening mode dynamic fracture of PMMA (i.e., 

no plasticity present). For crack speeds in PMMA around 0.24 cR, dynamic 

fracture occurred by the formation of a single craze. As the crack speed 

increased, multiple layered craze fronts formed along the main fracture surface, 

leading to increased crack tip heating. 

• Shear-dominated dynamic fracture of PC and PMMA was also found to be 

a highly dissipative event. Such shear-dominated thermal measurements 

have never before been performed. Interpretation of the temperature 

signatures from the shear-dominated fracture experiments showed PC and 

PMMA to behave very differently to the same loading condition. An 

adiabatic shear band followed by a fracture event occurred in PC. The 

maximum temperature increase produced by the propagating adiabatic 

shear band was 45 K. The fracture event, which for the case of PC was 

subsonic, increased the temperature an additional 100 K. PMMA, on the 

other hand, showed an abrupt 85 K increase in temperature resulting from 

a shear crack propagating ~40% faster than the shear wave speed of the 

material. No dissipation mechanisms were found ahead of the shear crack 

in PMMA. The highly localized thermal signature indicated the crack tip in 

PMMA was formed from a shear crazing mechanism. Thermal evidence of 

crack face contact and frictional heating were observed behind the 

advancing crack tip in PMMA. The frictional heating increased the 

temperature beyond the melting point of PMMA. SEM images of fractured 

PMMA surfaces confirm that polymer melting did occur during the sheardominated fracture event. 

• The dynamic energy release rate was measured for both materials during 

the opening-mode experiments using a strain gage technique. The fracture 

energy was measured simultaneously with crack tip speed and 

temperature for the very first time, offering the ability to directly correlate 

the crack tip thermal field with crack tip mechanical fields. The 

measurements indicated that the energy needed to form new surfaces is 
159

relatively insignificant compared to the energy required to propagate the 

crack dynamically. Thus, nearly all of the energy needed for fracture is 

dissipated in the form of heat. A numerical simulation of the 

opening-mode fracture of PC that considered only the energy dissipation 

from plasticity indicated that less than 10% of crack tip heating was due to 

plastic deformation of the ductile material. For PMMA, the measured crack 

tip heating along with the large amount of energy available for dissipation 

show the material to have a highly dissipative process zone during 

dynamic fracture. This was somewhat surprising given the nominally 

brittle nature of PMMA. 

• The molecular weight of PMMA was found to influence the crack tip 

temperature field and the onset of crack branching for opening-mode 

dynamic fracture. The recorded temperature for the low molecular weight 

PMMA was only 35% that of the high molecular weight PMMA under 

similar impact loading conditions. The crack in low molecular weight 

PMMA was observed to branch much earlier than the high molecular 

weight PMMA. These measurements and observations suggest the 

physical mechanisms associated with brittle fracture of PMMA are less 

dissipative when the molecular weight is reduced. This furthermore 

suggests that molecular weight may be a significant material parameter for 

modeling thermal dissipation in brittle polymers. 

• A dissipative cohesive zone model was developed by using a dynamic 

formulation of a cohesive zone and coupling it with an adiabatic form of 

the heat-energy equation. Several cohesive stress distributions were used 

to obtain unique cohesive traction laws. Predictions of crack tip heating 

were made from each of the cohesive stress distributions considered. The 

predictions were compared to the crack tip temperature field 

measurements made during opening mode fracture of high molecular 

weight PMMA. Good agreement was found between experimental data 

and the model for low crack tip speeds using one of the considered 

cohesive stress distributions. For high crack tip speeds, the dissipative 

cohesive zone model predicted only half the temperature increase observed 

during experiment. The cohesive zone modeling showed remarkable 

agreement with experiments where dynamic crack growth occurred by the 

formation of a single polymer craze (i.e., slow-speed crack propagation). 

For higher speed crack propagation, the single craze is replaced by a region 

of distributed damage. The dissipative cohesive zone model, as formulated 

in this study, was unable to account for the high crack tip temperatures 

observed within the distributed damage zone. 

7.2 Future Work 

The modeling that was performed for the mode I situation was limited to a 

rate-independent formulation. The modeling should be expanded to a 
160 

rate-dependent cohesive traction law (Kubair et al. 2001), particularly for 

polymeric materials because they exhibit considerable rate-dependent behavior. 

Additionally, a means of accounting for the distributed damage observed for 

high-speed crack propagation in PMMA should be developed within the 

cohesive zone methodology. Doing so will extend the applicability of the 

dissipative cohesive zone model to fracture with crack tips propagating near the 

terminal speed. 

The shear-dominated fracture portion of this study included a cursory 

examination of high-speed frictional heating of PMMA. This is a topic where 

additional experimental and analytical work is obviously needed. The effort 

presented in section 4 was not intended to be comprehensive. A more 

comprehensive study will need to examine the influence of contact pressure, 

sliding speed, and the non-Newtonian aspect of high-temperature polymer flow. 

In addition to frictional heating, the entire kinematic behavior of the fracture 

surfaces aft of the crack tip in shear-dominated fracture of PMMA needs further 

exploration. 

The modeling performed in section 6 was for opening mode fracture of the 

nominally brittle PMMA. The same techniques used for the dissipative mode I 

cohesive model should be applied to the shear-dominated case. This will require 

measurement of the dynamic energy release rate for this loading configuration, 

which may entail additional theoretical study. However, the cohesive zone 

approach for the shear-dominated fracture of PMMA will likely be useful 

because the highly localized thermal signatures of the shear crack suggest a 

shear-crazing phenomenon. 

The ultimate conclusion of this effort will occur when fully coupled 

thermoplastic and thermofracture relationships are included in numerical 

algorithms to accurately predict the failure of amorphous polymers. The fully 

coupled relationships will need the thermal dependence of all material and 

fracture parameters to be established. The parametric study performed in 

section 6 to examine the influence of certain parameters on the cohesive zone 

model showed that material property variation can have a significant impact on 

resultant heating. However, as was pointed out in section 6, the parameters that 

were varied are likely not independent of each other. The interdependencies of 

the material parameters with temperature and each other will need to be 

established. This is an effort that will likely take many years to complete. 

Finally, it was shown in the parametric study that the critical stress, σc, had 

significant influence on amount of crack tip heating. The physical meaning of 

this parameter in the cohesive zone methodology and a means of reliably 

measuring it will be needed to cast all of the parameters in the dissipative 

cohesive zone model as material properties.
161

8. References 

Adams, G. W., and R. J. Farris. “Latent Energy of Deformation of Bisphenol A 

Polycarbonate. Journal of Polymer Science: Part B: Polymer Physics, vol. 26, 

no. 2, pp. 433–445, 1998. 

Allen, D. H., and C. R. Searcy. “A Micromechanically-Based Model for 

Predicting Dynamic Damage Evolution in Ductile Polymers.” Mechanics of 

Materials, vol. 33, pp. 177–184, 2001. 

American Society for Testing and Materials. “Tensile Properties of Plastics.” 

ASTM D638, West Conshohocken, PA. 

Anderson, T. L. Fracture Mechanics: Fundamentals and Applications. Vol. 325, Boca 

Raton: CRC Press, 1995. 

Andrews, D. J. “Rupture Velocity of Plane Strain Shear Cracks.” Journal of 

Geophysical Research, vol. 81, no. 32, pp. 5679–5687, 1976. 

Anthony, S. R., J. P. Chubb, and J. Congleton. “The Crack Branching Velocity.” 

Philosophical Magazine, vol. 22, pp. 1201–1206, 1970. 

Barenblatt, G. I. “The Formation of Equilibrium Cracks During Brittle Fracture: 

General Ideas and Hypotheses, Axially Symmetric Cracks.” Applied 

Mathematics and Mechanics, vol. 23, pp. 622–636, 1959. 

Beahan, P., M. Bevis, and D. Hull. “Electron-Microscopy Studies of Fracture 

Processes in Amorphous Thermoplastics.” Polymer, vol. 14, no. 3, pp. 96–102, 

1973. 

Berger, L. L. “On the Mechanism of Craze Fibril Breakdown in Glassy 

Polymers.” Macromolecules, vol. 23, no. 11, pp. 2926–2934, 1990. 

Berry, J. P. “Surface Characteristics of Fractured Poly(methyl methacrylate).” 

Nature, vol. 185, no. 4706, pp. 91–92, 1960. 

Blok, H. “Theoretical Study of Temperature Rise at Surfaces of Actual Contact 

Under Oiliness Lubricating Conditions.” Proceedings of the General Discussions 

on Lubrications and Lubricants, vol. 2, Institute of Mechanical Engineers, 

pp. 222–235, 1937. 

Boley, B. A., and J. H. Weiner. Theory of Thermal Stresses. New York: Dover 

Publications, pp. 32–37, 1997. 

Broberg, K. B. “The Propagation of a Brittle Crack.” Arkiv for F ysik, vol. 18, 

pp. 159–192, 1960. 
162 

Broberg, K. B. “On Crack Paths.” Engineering Fracture Mechanics, vol. 28, 

pp. 663–679, 1987. 

Brown, H. R., and I. M. Ward. “Craze Shape and Fracture in Poly(methyl 

methacrylate).” Polymer, vol. 14, pp. 469–475, 1973. 

Burridge, R., G. Conn, and L. B. Freund. The Stability of a Rapid Mode II Shear 

Crack With Finite Cohesive Traction.” Journal of Geophysical Research, vol. 85, 

no. B5, pp. 2210–2222, 1979. 

Camacho, G. T., and M. Ortiz. “Computational Modeling of Impact Damage in 

Brittle Materials.” International Journal of Solids and Structures, vol. 33, 

no. 20–22, pp. 2899–2938, 1996. 

Chona, R. “The Stress Field Surrounding the Tip of a Crack Propagating in a 

Finite Body.” Ph.D. dissertation, University of Maryland, College Park, MD, 

1987. 

Cotterell, B. “Velocity Effects in Fracture Propagation.” Applied Materials 

Research, vol. 4, pp. 227–232, 1965. 

Döll, W. “Application of an Energy Balance and an Energy Method to Dynamic 

Crack Propagation.” International Journal of Fracture, vol. 12, no. 4, 

pp. 595–605, 1976. 

Döll, W. “Fractography and Failure Mechanisms of Amorphous 

Thermoplastics.” Fractography and Failure Mechanisms of Polymers and 

Composites. New York: Elsevier Science, pp. 387–436, 1989. 

Domininghaus, H. Plastics for Engineers – Materials, Properties, Applications. 

New York: Hanser Publishers, pp. 268–269, 428–429, 1988. 

Donald, A. M., and E. J. Kramer. “Craze Initiation and Growth in High-Impact 

Polystyrene.” Journal of Polymer Science, vol. 27, no. 10, pp. 3729–3741, 1982. 

Duffy, J. “Temperature Measurements During the Formation of Shear Bands in a 

Structural Steel.” Mechanics of Material Behavior, edited by G. J. Dvorak and 

R. T. Shield, Amsterdam: Elsevier Science Publishers B. V., pp. 75–86, 1984. 

Dugdale, D. S. “Yielding of Steel Sheets Containing Slits.” Journal of the 

Mechanics and Physics of Solids, vol. 8, no. 2, pp. 100–104, 1960. 

Dulaney, E. N., and W. F. Brace. “Velocity Behavior of a Growing Crack.” 

Journal of Applied Physics, vol. 31, pp. 2233–2236, 1960. 

Erdogan, F., and G. C. Sih. “On the Crack Extension in Plates Under Plane 

Loading and Transverse Shear.” Journal of Basic Engineering, vol. 85, no. 1, 

pp. 519–527, 1963. 
163

Ettles, C. C. M. “The Thermal Control of Friction at High Sliding Speeds.” 

Journal of Tribology—Transactions of the ASME, vol. 108, no. 1, pp. 98–104, 

1986. 

Ettles, C. C. M., and J. H. Shen. “The Influence of Frictional Heating on the 

Sliding Friction of Elastomers and Polymers.” Rubber Chemistry and 

Technology, vol. 61, no. 1, pp. 119–136, 1988. 

Farren, W. S., and G. I. Taylor. “The Heat Developed During Plastic Extension of 

Metals.” Proceedings of the Royal Society of London, Series A, vol. 107, pp. 422–

451, 1925. 

Fineberg, J., S. P. Gross, M. Marder, and H. L. Swinney. “Instability in Dynamic 

Fracture.” Physical Review Letters, vol. 67, p. 457, 1992. 

Fraser, R. A. W., and I. M. Ward. “Temperature Dependence of Craze Shape and 

Fracture in Polycarbonate.” Polymer, vol. 19, pp. 220–224, 1978. 

Freund, L. B. Dynamic Fracture Mechanics. New York: Cambridge University 

Press, 1990. 

Fuller, K. N. G., P. G. Fox, and J. E. Field. “The Temperature Rise at the Tip of 

fast Moving Cracks in Glassy Polymers.” Proceedings of the Royal Society of 

London, Series A, vol. 341, no. 1627, pp. 537–557, 1975. 

Geubelle, P. H., and J. Baylor. “Impact-Induced Delamination of Laminated 

Composites: A 2-D Simulation.” Composites B, vol. 29, pp. 589–602, 1998. 

Geubelle, P. H., and J. R. Rice. “A Spectral Method for 3-D Elastodynamic 

Fracture Problems.” Journal of the Mechanics and Physics of Solids, vol. 43, pp. 

1791–1824, 1995. 

Henkee, C. S., and E. J. Kramer. “Crazing and Shear Deformation in Crosslinked 

Polystyrene.” Journal of Polymer Science Part B: Polymer Physics, vol. 22, no. 4, 

pp. 721–737, 1984. 

Hibbitt, Karlsson, and Sorensen. ABAQUS/Explicit User’s Manual. HKS 

Incorporated, Providence, RI, 1998. 

Hodowany, J., Ravichandran, G., Rosakis, A. J., and Rosakis, P. “Partition of 

Plastic Work Into Heat and Stored Energy in Metals.” Experimental Mechanics

vol. 40, no. 2, pp. 113–123, 2000. 

Incropera, F. P., and D. F. DeWitt. Fundamentals of Heat Transfer. New York: 

John Wiley & Sons, pp. 582–589, 1980. 

Irwin, G. R., 1980. Series Representation of the Stress Field Around Constant 

Speed Cracks. University of Maryland Lecture Notes, College Park, MD, 

unpublished data. 
164 

Jandel Scientific. TableCurve 2D User’s Manual. Jandel Scientific, San Rafael, CA, 

1994. 

Kallivayalil, J. A., and A. T. Zehnder. “Measurement of the Temperature Field 

Induced by Dynamic Crack Growth in Beta-C Titanium.” International Journal 

of Fracture, vol. 66, no. 2, pp. 99–120, 1994. 

Kalthoff, J. F., and S. Winkler. “Failure Mode Transition at High Rates of Shear 

Loading.” Impact Loading and Dynamic Behaviour of Materials, edited by 

C. Y. Kunze, H. D. Chiem, and L. W. Meyer, Berlin: Springer Verlag, 1987. 

Kambour, R. P. “Direct Observation of the Craze Ahead of the Propagating 

Crack in PMMA and Polystyrene.” Journal of Polymer Science: Part A-2, vol. 4, 

pp. 349–358, 1964. 

Kambour, R. P. “Mechanism of Fracture in Glassy Polymers: I. Fracture 

Surfaces in Polymethyl Methacrylate.” Journal of Polymer Science: Part A, 

vol. 3, pp. 1713–1724, 1965. 

Kambour, R. P. “Mechanism of Fracture in Glassy Polymers: II. Survey of 

Crazing Response During Crack Propagation in Several Polymers.” Journal of 

Polymer Science: Part A-2, vol. 4, pp. 17–24, 1966a. 

Kambour, R. P. “Mechanism of Fracture in Glassy Polymers. III. Direct 

Observation of the Craze Ahead of the Propagating Crack in Poly(methyl 

methacrylate) and Polystyrene.” Journal of Polymer Science: Part A-2, vol. 4, 

pp. 349–358, 1966b. 

Kambour, R. P. “A Review of Crazing and Fracture in Thermoplastics.” Journal 

of Polymer Science Part D: Macromolecular Reviews, vol. 7, pp. 1–154, 1973. 

Kambour, R. P., A. S. Holik, and S. Miller. “Plane-Strain Fracture Energy 

Instability and Mixed Mode Crack Propagation in Polycarbonate at Room 

Temperature.” Journal of Polymer Science Part B: Polymer Physics Edition, vol. 

16, pp. 91–104, 1978. 

Kapoor, R., and S. Nemat-Nasser. “Determination of Temperature Rise During 

High Strain Rate Deformation.” Mechanics of Materials, vol. 27, no. 1, pp. 1–

12, 1998. 

Kawabe, H., Y. Natsume, Y. Higo, and S. Nunomura. “Micromechanism of a 

Deformation Process Before Crazing in a Polymer During Tensile Testing.” 

Journal of Material Science, vol. 27, no. 20, pp. 5547–5552, 1992. 

Khanna, S. K., and A. Shukla. “On the Use of Strain Gages in Dynamic Fracture 

Mechanics.” Engineering Fracture Mechanics, vol. 51, no. 6, pp. 933–948, 1995. 

Kong, H., and M. F. Ashby. “Wear Mechanisms in Brittle Solids.” Acta 

Metallurgica et Materialia, vol. 40, no. 11, pp. 2907–2920, 1992. 
165

Kramer, E. J. “Microscopic and Molecular Fundamentals of Crazing.” Journal of 

Advanced Polymer Science, vol. 52–53, pp. 1–56, 1983. 

Kramer, E. J. “Craze Fibril Formation and Breakdown.” Journal of Polymer 

Engineering Science, vol. 24, no. 10, pp. 761–769, 1984. 

Kramer, E. J., and L. L. Berger. “Fundamental Processes of Craze Growth and 

Fracture.” Journal of Advanced Polymer Science, vol. 91/92, pp. 1–68, 1990. 

Krishmaswamy, S., A. J. Rosakis, and G. Ravichandran. “On the Extent of 

Dominance of Asymptotic Elastodynamic Crack-Tip Fields: Pt. 2—Numerical 

Investigation of Three-Dimensional and Transient Effects.” Journal of Applied 

Mechanics, vol. 58, no. 1, pp. 95–103, 1991. 

Kubair, D. V., and P. H. Geubelle. “Comparative Analysis of Extrinsic and 

Intrinsic Cohesive Models of Dynamic Fracture.” International Journal of 

Solids and Structures, 2001. 

Kubair, D. V., P. H. Geubelle, and Y. Y. Huang. “Analysis of a Rate-dependent 

Cohesive Model for Dynamic Crack Propagation.” Engineering Fracture 

Mechanics, 2001. 

Lambros, J., and A. J. Rosakis. “Development of a Dynamic Decohesion 

Criterion for Subsonic Fracture of the Interface Between Two Dissimilar 

Materials.” Proceedings of the Royal Society of London, Series A, vol. 451, pp. 

711–736, 1995. 

Li, Z., and J. Lambros. “Strain Rate Effects on the Thermomechanical Behavior of 

Polymers.” International Journal of Solids and Structures, vol. 38, no. 20, 

pp. 3549–3562, 2001. 

Lin, G., P. H. Geubelle, and N. R. Sotos. “Simulation of Fiber Debonding with 

Friction in a Model Composite Pushout Test.” International Journal of Solids 

and Structures. vol. 38, no. 46-47, pp. 8547–8562, 2001. 

Lo, C. Y., T. A. Nakamura, and A. Kushner. “Computational Analysis of 

Dynamic Crack Propagation Along a Bimaterial Interface.” International 

Journal of Solids and Structures, vol. 31, no. 2, pp. 145–168, 1994. 

Mason, J. J., and A. J. Rosakis. “On the Dependence of the Dynamic Crack Tip 

Temperature Fields in Metals Upon Crack Tip Velocity and Material 

Parameters.” Mechanics of Materials, vol. 16, no. 4, pp. 337–350, 1992. 

Mason, J. J., A. J. Rosakis, and G. Ravichandran. “On the Strain and Strain Rate 

Dependence of the Fraction of Plastic Work Converted to Heat: An 

Experimental Study Using High-Speed Infrared Detectors and the Kolsky 

Bar.” Mechanics of Materials, vol. 17, no. 2, pp. 135–145, 1994. 

Measurements Group Inc. “Errors Due to Wheatstone Bridge Nonlinearity.” 

Measurements Group Tech Note TN-507, Raleigh, NC, 1982. 
166 

Morgan, G. P., and I. M. Ward. “Temperature Dependence of Craze Shape and 

Fracture in Poly(methyl methacrylate).” Polymer, vol. 18, pp. 87–91, 1977. 

Murray, J., and D. Hull. “Dependence on Strain Rate of Nucleation of Cracks in 

Polystyrene at 293 Degrees K.” Journal of Polymer Science: Part A-2, vol. 8, 

no. 9, pp. 1521–1540, 1970. 

Needleman, A. “A Continuum Model for Void Nucleation by Inclusion 

Debonding.” Journal of Applied Mechanics, vol. 54, no. 3, pp. 525–531, 1987. 

Needleman, A. “Micromechanical Modeling of Interfacial Decohesion.” 

Ultramicroscopy, vol. 40, no. 3, pp. 203–214, 1992. 

Nilsson, K. F., and J. W. Hutchinson. “Interraction Between a Major Crack and 

Small Crack Damage in Aircraft Sheet Material.” International Journal of Solids 

and Structures, vol. 31, no. 17, pp. 2331–2346, 1994. 

O’Shea, D. C. Elements of Modern Optical Design. New York: John Wiley & Sons, 

1985. 

Panda, K. C., and J. G. Williams. “Cohesive Zone Modeling of Crack Growth in 

Polymers. Part 1—Experimental Measurement of Cohesive Law.” Plastics, 

Rubber and Composites, vol. 29, no. 9, pp. 439–446, 2000. 

Paxson, S. L., and R. A. Lucas. “An Investigation of the Velocity Characteristics 

of a Fixed Boundary Fracture Model.” Dynamic Crack Propagation, edited by 

G. C. Sih, Leyden: Noordhoff International Publishing, pp. 415–426, 1973. 

Peterson, T. L., D. G. Ast, and E. J. Kramer. “Holographic Interferometry of 

Crazes in Polycarbonate.” Journal of Applied Physics, vol. 45, no. 10, 

pp. 4220–4228, 1974. 

Planck, M. The Theory of Heat Radiation. New York: Dover Publications, 1959. 

Poulikakos, D. Conduction Heat Transfer. New Jersey: Prentice Hall, pp. 146–150, 

1994. 

Ravi-Chandar, K. “On the Failure Mode Transitions in Polycarbonate Under 

Dynamic Mixed Mode Loading.” International Journal of Solids and Structures, 

vol. 32, no. 6, pp. 925–938, 1995. 

Ravi-Chandar, K., J. Lu, B. Yang, and Z. Zhu. “Failure Mode Transitions in 

Polymers Under High Strain Rate Loading.” International Journal of Fracture, 

vol. 101, no. 1-2, pp. 33–72, 2000. 

Rittel, D. “Experimental Investigation of Transient Thermoelastic Effects in 

Dynamic Fracture.” International Journal of Solids and Structures, vol. 35, 

no. 22, pp. 2959–2973, 1998. 
167

Rittel, D. “On the Conversion of Plastic Work to Heat During High Strain Rate 

Deformation of Glassy Polymers.” Mechanics of Materials, vol. 31, no. 2, 

pp. 131–139, 1999. 

Rittel, D. “Experimental Investigation of Transient Thermoplastic Effects in 

Dynamic Fracture.” International Journal of Solids and Structures, vol. 37, no. 

21, pp. 2901–2913, 2000. 

Rosakis, A. J. “Application of Coherent Gradient Sensing (CGS) to the 

Investigation of Dynamic Fracture Problems.” Optics and Lasers in 

Engineering, vol. 19, no. 1–3, pp. 3–41, 1993. 

Rosakis, A. J., O. Samudrala, and D. Coker. “Cracks Faster Than the Shear Wave 

Speed.” Science, vol. 284, no. 5418, pp. 1337–1340, 1999. 

Sanford, R. J., and J. W. Dally. “Strain-Gage Methods for Measuring the 

Opening-Mode Stress-Intensity Factor.” Experimental Mechanics, vol. 27, pp. 

381–388, 1987. 

Sanford, R. J., J. W. Dally, and J. R. Berger. “An Improved Strain Gage Method 

for Measuring KID for a Propagating Crack.” Journal of Strain Analysis, vol. 25, 

no. 3, pp. 177–183, 1990. 

Sharon, E., S. P. Gross, and J. Fineberg. “Energy Dissipation in Dynamic 

Fracture.” Physical Review Letters, vol. 76, no. 12, pp. 2117–2120, 1996. 

Singh, R. P., J. Lambros, A. Shukla, and A. Rosakis. “Investigation of the 

Mechanics of Intersonic Crack Propagation Along a Bimaterial Interface 

Using Coherent Gradient Sensing and Photoelasticity.” Proceedings of the 

Royal Society of London, Series A, vol. 453, pp. 2649–2667, 1997. 

SPSS Inc. SigmaPlot 2000 Programming Guide. Chicago, IL, 2000. 

Stalder, B., P. Beguelin, and H. H. Kausch. “A Simple Velocity Gauge for 

Measuring Crack Growth.” International Journal of Fracture, vol. 22, R47-R54, 

1983. 

Sun, N. S., and T. R. Hsu. “Thermomechanical Coupling Effects on Fractured 

Solids.” International Journal of Fracture, vol. 78, no. 1, pp. 67–87, 1996. 

Suo, Z., M. Ortiz, and A. Needleman. “Stability of Solids With Interfaces.” 

Journal of the Mechanics and Physics of Solids, vol. 40, pp. 613–640, 1992. 

Suzuki, S., Y. Nozaki, and H. Kimura. “High-Speed Holographic Microscopy for 

Fast-Propagating Cracks in Transparent Materials.” Applied Optics, vol. 36, 

no. 28, pp. 7224–7233, 1997. 

Taylor, G. I., and H. Quinney. “The Latent Energy Remaining in a Metal After 

Cold Working.” Proceedings of the Royal Society, Series A, vol. 143, pp. 307–326, 

1934. 
168 

Theocaris, P. S., and N. P. Andrianopoulos. “Strain Rate Effects on the 

Mechanical Properties of Polymers.” Proceedings of the VIIth SESA 

International Conference, edited by A. Bester, Haifa, Israel, pp. 473–482, 1982. 

Tijssens, M. G. A., E. van der Giessen, and L. J. Sluys. “Modeling of Crazing 

Using a Cohesive Surface Methodology.” Mechanics of Materials, vol. 32, 

no. 1, pp. 19–35, 2000. 

Tvergaard, V., and J. W. Hutchinson. “The Relation Between Crack Growth 

Resistance and Fracture Process Parameters in Elastic-Plastic Solids.” Journal 

of the Mechanics and Physics of Solids, vol. 40, no. 6, pp. 1377–1397, 1992. 

Tvergaard, V., and J. W. Hutchinson. “Effect of Strain-Dependent Cohesive Zone 

Model on Predictions of Crack Growth Resistance.” International Journal of 

Solids and Structures, vol. 33, no. 20–22, pp. 3297–3308, 1996. 

Wool, R. P. Polymer Interfaces: Structure and Strength. New York: Hanser 

Publishers, 1995. 

Xu, X. P., and A. Needleman. “Numerical Simulations of Fast Crack Growth in 

Brittle Solids. Journal of the Mechanics and Physics of Solids, vol. 42, no. 9, 

pp. 1397–1434, 1994. 

Yang, A. C., E. J. Kramer, C. C. Kuo, and S. L. Phoenix. “Craze Fibril Stability 

and Breakdown in Polystyrene.” Macromolecules, vol. 19, no. 7, pp. 2010–

2019, 1986. 

Yoffe, E. H. “The Moving Griffith Crack.” Philosophical Magazine, vol. 42, 

no. 330, pp. 739–750, 1951. 

Zehnder, A. T., and A. J. Rosakis. “Dynamic Fracture Initiation and Propagation 

in 4340 Steel Under Impact Loading. International Journal of Fracture, vol. 43, 

no. 4, pp. 271–285, 1990. 

Zehnder, A. T., and A. J. Rosakis. “On the Temperature Distribution at the 

Vicinity of Dynamically Propagating Cracks in 4340 Steel: Experimental 

Measurements Using High-Speed Infrared Detectors.” Journal of the 

Mechanics and Physics of Solids, vol. 39, no. 3, pp. 385–417, 1991. 

Zehnder, A. T., and A. J. Rosakis. “Temperature Rise at the Tip of Dynamically 

Propagating Cracks: Measurements Using High-Speed Infrared Detectors.” 

Experimental Techniques in Fracture III, Society for Experimental Mechanics, 

Bethel, CT, pp. 125–169, 1993. 

Zehnder, A. T., P. R. Guduru, A. J. Rosakis, and G. Ravichandran. “Million 

Frames per Second Infrared Imaging System.” Review of Scientific Instruments, 

vol. 71, no. 10, pp. 3762–3768, 2000. 
169

Zhou, M., A. J. Rosakis, and G. Ravichandran. “Dynamically Propagating Shear 

Bands in Impact-Loaded Prenotched Plates—I. Experimental Investigations 

of Temperature Signatures and Propagation Speed.” Journal of the Mechanics 

and Physics of Solids, vol. 44, no. 6, pp. 981–1006, 1996a. 

Zhou, M., G. Ravichandran, and A. J. Rosakis. “Dynamically Propagating Shear 

Bands in Impact-Loaded Prenotched Plates—II. Numerical Simulations.” 

Journal of the Mechanics and Physics of Solids, vol. 44, no. 6, pp. 1007–1032, 

1996b.
170 

INTENTIONALLY LEFT BLANK. 
171

Appendix A. Closed Form Solution of Cohesive Zone Length 

The analytical expression for the cohesive zone length, L, was given in section 6 

as (Freund 1990), 

( ) , 2

2

0





 



 = ∫

L

I dl

l

l

E

A G σ

π

 (A-1) 

where the parameters were defined in section 6. An explicit analytic solution of 

equation (A-1) for L is provided here for two cohesive stress distributions, 

namely the constant stress and linear decay distributions discussed in section 6. 

The constant stress cohesive distribution has the form 

( ) , σ ξ = σ c (A-2) 

where σc is a constant and ξ is the length along the cohesive zone as shown in 

Figure 87. Substituting equation (A-2) into (A-1) gives 

. 2 1

2

0

2





 



 = ∫

L

I c dl

E l

A G

π

σ (A-3) 

The integral is solved by using the substitution η = l , which gives 

2 . 2

2

0

2















 = ∫

L

I c d

E

A G η

π

σ (A-4) 

Solution of the integral in equation (A-4) yields 

[2 ] . 2 2 2

L

E

A G I c

π

σ = (A-5) 

Solving equation (A-5) for L gives the length of the cohesive zone for the constant 

stress case, 

. 8 2 AI c

EG L

σ

π = (A-6) 

The analytical expression for the linear decay cohesive stress distribution was 

given in section 6 as 

( ) 1  .



 



 = − L c

ξ σ ξ σ (A-7) 

, 

, 
172 

Substituting the linear decay distribution into equation (A-1) gives the following: 

( ) . 2

2

0

2

2





 



 − = ∫

L

I c dl

l

L l

EL

A G

π

σ (A-8) 

Using the same substitution as was used for the constant stress case (i.e., η = l ) 

in equation (A-8) yields 

2( ) . 2

2

0

2

2

2















 = − ∫

L

I c L d

EL

A G η η

π

σ (A-9) 

Solution of the integral in equation (A-9) results in the following expression: 

. 3

8 2 2

2

2





 



 = L L

EL

A G I c

π

σ (A-10) 

The solution of the cohesive zone length for the linear decay distribution thus 

becomes 

. 32

9

2 AI c

EG L

σ

π = (A-11) 
173

Appendix B. Closed Form Solution of Cohesive Zone 

Opening Rate 

Closed form expressions for the rate of cohesive zone opening, ∂δ ∂ξ , are next 

derived in this Appendix for the constant stress and linear decay cohesive stress 

distributions. The Cauchy singular integral equation for ∂δ ∂ξ was given in 

section 6 as (Freund 1990): 

( )

( ) ∫ − = ∂

∂ L

d C

0

η , ξ η

σ η

η

ξ

ξ πμ

δ

 (B-1) 

where 

( ) , 2 1 2

1 2

D

C λ − λ = (B-2) 

and the parameters were defined in section 6. The constant stress cohesive 

distribution has the form 

( ) , σ ξ = σ c (B-3) 

where σc is a constant and ξ is the length along the cohesive zone as shown in 

Figure 87. Substituting the constant stress distribution given by equation (B-3) 

into equation (B-1) yields 

( ) ∫ − = ∂

∂ L

c d C

0

. 1η

η ξ η

ξ

πμ

σ

ξ

δ (B-4) 

Performing the substitution l = η in equation (B-4) gives 

( ) ∫ −

− = ∂

∂ L

c dl

l

C

0

2 . 2 1

πμ ξ

σ ξ

ξ

δ (B-5) 

Using partial fractions, the integrand can be simplified to the following 

equivalent expression: 

∫ 















+ − −

− = ∂

∂ L

c dl

l l

C

0

. 1 1

πμ ξ ξ

σ

ξ

δ (B-6) 

The integral in equation (B-6) contains two terms, which are solved individually. 

Starting with the second term, the substitution u = l + ξ gives 

, 

, 

, 
174 

. 1 1

0

∫ ∫

+

= − + −

L L

du

u

dl

l

ξ

ξ ξ (B-7) 

Evaluating the integral on the right hand side of equation (B-7) yields 

ln . 1

0

∫ + = + −

L

L

dl

l ξ

ξ

ξ (B-8) 

The first term in the integral of equation (A-6) is now evaluated. The path of 

integration is from zero to L ; however, the expression has a singularity along 

the integration path at ξ . Because the integrand is not analytic at the 

singularity, the integral cannot be evaluated by simply “passing through” the 

singular point. Instead, the integral will be evaluated along the interval ξ − ε

to ξ − ε , where ε is a very small number, and then along the remaining 

interval ξ + ε to L . The resulting expression is then evaluated in the limiting 

case of ε → 0 . Proceeding in this manner: 

lim . 1

0

0 















− + − = − ∫ ∫ ∫

−

+ →

ξ ε

ε ξ ε ξ ε ξ ξ

L L

l

dl

l

dl dl

l (B-9) 

The integrals are evaluated to give 

lim{ } [ ] ln( ) ln( ) [ ] ln( ) ln( ) . 1

0

0

ε ε ξ ξ ε

ξ ε = − − − + − − − ∫ → dl Ll

L

 (B-10) 

This simplifies to 

lim ln ln . 1

0

0 





























 − +















−

− = ∫ − → ε

ξ

ε ξ

ε

ξ ε

L dl

l

L

 (B-11) 

Noting the following: 

ln ln , 















− =















−

−

ξ ε

ε

ε ξ

ε (B-12) 

equation (B-11) becomes 

lim ln ln . 1

0

0 





























 − +















− = ∫ − → ε

ξ

ξ ε

ε

ξ ε

L dl

l

L

 (B-13) 

Equation (B-13) simplifies to 

, 
175

lim ln , 1

0

0 































−

− = ∫ − → ξ ε

ξ

ξ ε

L dl

l

L

 (B-14) 

which, in the limit, becomes: 

ln . 1

0















 − = − ∫ ξ

ξ

ξ

L

dl

l

L

 (B-15) 

Equation (B-15) provides the expression for the integral of the first term in the 

integrand of equation (B-6). With the integral of equation (B-15) now completely 

solved, the expression for ∂δ ∂ξ is 

ln ln ,

































+

+













 − = − ∂

∂

ξ

ξ

ξ

ξ

πμ

σ

ξ

δ

L

C L c (B-16) 

which simplifies to the expression presented in section 6 for the constant stress 

distribution case, 

ln . 















+

− = − ∂

∂

ξ

ξ

πμ

σ

ξ

δ

L

C L c (B-17) 

A similar procedure is used to derive an expression of ∂δ ∂ξ for the linear 

decay cohesive stress distribution. The analytical expression for the linear decay 

stress distribution was given in section 6 as 

( ) 1  .



 



 = − L c

ξ σ ξ σ (B-18) 

Substituting equation (B-18) into the Cauchy singular equation (B-1) gives 

( )

( ) ∫ −

− = − ∂

∂ L

c d C L

0

. 1η

η ξ

η

η

ξ

πμ

σ

ξ

δ (B-19) 

As before, the substitution l = η is used in the integrand. The resulting 

expression is 

∫ −

− = ∂

∂ L

c dl

l

l L

L

C

0

2

2

. 2

πμ ξ

σ ξ

ξ

δ (B-20) 

The integrand can be expressed in a more tractable form for integration by 

performing polynomial division on the integrand and then simplifying with 

partial fractions to give 

, 

, 
176 

∫ 































+ 













 − −















− 













 − = +

∂

∂ L

c dl

l

L

l

L

L

C

0

. 1

2

1

2

1

2

ξ ξ

ξ

ξ ξ

ξ

πμ

σ ξ

ξ

δ (B-21) 

The second and third terms in the integrand have a form similar to the integrand 

in equation (B-6) that was obtained during the derivation for the constant stress 

distribution. Using the same procedure to evaluate these two terms as was used 

before, i.e., equations (B-7) through (B-15), and integrating the first term in the 

integrand results in the following expression: 

ln . 2

2

































+

− − = +

∂

∂

ξ

ξ

ξ

ξ

πμ

σ ξ

ξ

δ

L

L L

L

L

C c (B-22) 

This expression simplifies to 

ln , 2

2

































+

− − = +

∂

∂

ξ

ξ ξ ξ

πμ

σ

ξ

δ

L

L

L

L

L

C L c (B-23) 

which is identical to equation (111) given in section 6. 

, 
dynamic fracture, polymer fracture, craze, thermomechanical heating 192

The dissipative processes that occur during opening and shear-dominated dynamic fracture of amorphous polymers were

examined in a combined experimental, computational, and analytical investigation. Experiments were performed using

two materials, nominally brittle polymethyl methacrylate and nominally ductile polycarbonate to quantify crack tip

heating and identify dominant dissipative mechanisms. Shear-dominated dynamic fracture of polymethyl methacrylate

was found to exhibit heating from craze formation and frictional sliding of the fracture surfaces aft of the propagating

crack tip. Heating in polycarbonate during shear-dominated dynamic fracture was from two dissipative processes, the

formation of an adiabatic shear band and plastic deformation surrounding the propagating crack. Plastic deformation

heating was noted for opening mode fracture of polycarbonate. Finite-element simulations of dynamic crack growth in

polycarbonate were performed to isolate the heating from thermoplasticity. The simulations indicated that although

thermoplastic heating does occur, thermofracture heating may be significant. Heating from craze formation was

observed during opening mode fracture of polymethyl methacrylate. A dissipative cohesive zone model was developed

to predict heating from thermofracture mechanisms associated with polymer crazing. The model predictions were

consistent with measurements of single craze heating during opening mode fracture of polymethyl methacrylate.

ARL-TR-2793

U.S. Army Research Laboratory

ATTN: AMSRL-WM-TC

Aberdeen Proving Ground, MD 21005-5066

Todd W. Bjerke

Thermomechanical Behavior of Amorphous Polymers During High-Speed Crack 1L1601102AH42

Propagation

August 2002 Final, October 1997–December 2001

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