Dynamic Experiments and Constitutive Model Performance 

for Polycarbonate

by Stephan R. Bilyk

ARL-TR-6937 July 2014

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Army Research Laboratory

Aberdeen Proving Ground, MD 21005-5069

ARL-TR-6937 July 2014

Dynamic Experiments and Constitutive Model Performance 

for Polycarbonate

Stephan R. Bilyk

Weapons and Materials Research Directorate, ARL

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Dynamic Experiments and Constitutive Model Performance for Polycarbonate

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14. ABSTRACT

An experimental and numerical investigation has been performed to examine material deformation of polycarbonate (PC) from 

low (quasi-static) to high (dynamic) strain rates. In 2012, a revised physics-based constitutive model for thermoplastics was 

implemented into several U.S. Department of Energy (DOE) hydrocodes at the U.S. Army Research Laboratory. This report 

briefly describes the Mulliken-Boyce (M-B) amorphous polymer model and reviews significant modifications in 

implementation for DOE hydrocodes. The original material model parameters obtained for the M-B model only considered 

isothermal deformations of PC, while a subsequent implementation of M-B into the finite element code LS-DYNA obtained 

material parameters for adiabatic heating. Observed results are used in this report for validation and to examine performance

differences in the mentioned model parameter sets. Dynamic-Tensile-Extrusion is an integrated experimental technique that 

allows the study of material deformation at high strain rates (greater than10,000/s) and large strains (greater than 1) when 

subjected to dynamic tension loading conditions. This is an important complement to the more traditional Taylor cylinder 

impact experiment, which achieves large strain and high-strain-rate deformation but under hydrostatic compression. 

Additionally, model parameter sensitivity is evaluated herein by considering pressure dependence of flow strength and its 

influence for amorphous polymer materials subjected to hydrostatic tension or compression.

15. SUBJECT TERMS

amorphous polymers, Mulliken-Boyce model, rate dependent, pressure dependent, parameter sensitivity

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iii

Contents

List of Figures iv

List of Tables vi

Acknowledgments vii

1. Introduction 1

2. Materials and Experimental Methods 3

3. Constitutive Model 7

4. Numerical Considerations 12

5. Results 14

5.1 Dynamic Compression ..................................................................................................16

5.2 Dynamic Tension ..........................................................................................................23

6. Summary, Conclusions and Recommendations 30

7. References 32

Distribution List 34
iv

List of Figures

Figure 1. In situ x-ray absorption characterization of shaped charge jet/target interaction. 

Performed at ARL EF-7; courtesy of M. Zellner, ARL.............................................................2

Figure 2. Dramatic change in mechanical properties for amorphous polymers as the 

temperature passes through glass transition, Tg.........................................................................3

Figure 3. Original schematic of a Taylor impact experiment. .........................................................4

Figure 4. (a) Schematic of the Dyn-Ten-Ext experiment and (b) sequenced high-speed video 

frames of the sample event. .......................................................................................................4

Figure 5. Schematic of the single stage gas gun used for the Taylor cylinder impact 

experiments at MIT ISN. The assembly shown is for an impact-penetration experiment 

using a finite thickness target.....................................................................................................5

Figure 6. Hemishere-ended cylinder samples and directionality considered for polycarbonate 

as used for DynTen-Ext experiments.........................................................................................7

Figure 7. Storage and loss tangent moduli for PC; DMA experiments performed at 1 Hz and 

shift at 100 Hz showing the  and transition regions using the Deconstruct-ShiftReconstruct analytical method...................................................................................................8

Figure 8. (a) Observed uniaxial compression response of PC over a range of strain rates 

performed at room temperature; (b) strain rate sensitivity of flow stress..................................9

Figure 9. One-dimensional (1-D) depiction of the M-B constitutive model. The spring in 

segment B is a strain hardening Langevin (nonlinear) spring representing chain stretching 

and alignment; the springs in segment A are rate- and temperature-dependent elastic 

(linear) springs and the dashpots represent rate-, temperature-, and pressure-dependent 

viscoplastic terms for intermolecular resistance. .....................................................................10

Figure 10. Comparison of PC material response in uniaxial compression using the M-B 

model for parameters obtained assuming isothermal conditions (labeled Mulliken and 

Boyce in plot) and modified parameters for plastic work (labeled Varghese and Batra in 

plot)..........................................................................................................................................12

Figure 11. Schematic illustrating the reference configuration, initial conditions, and boundary 

conditions used for the ALE3D Taylor cylinder impact simulations. .....................................13

Figure 12. Schematic illustrating the reference configuration, initial conditions, and boundary 

conditions used for the ALE3D Dyn-Ten-Ext simulations......................................................13

Figure 13. High-speed video frames recorded at various stages during a Taylor cylinder 

impact experiment on a 76.2-mm PC rod at 187 m/s. Note elastic recovery after 100 s 

and rebound (detachment from rigid surface) at 198 s..........................................................17
v

Figure 14. Parameter sensitivity showing numerical contours of axial stress using the M-B 

model for a Taylor cylinder impact experiment at a striking velocity of 187 m/s (labeled 

D2) at sequential times after rigid impact. (a) Full original parameters used in M-B, (b) 

full VBB parameters used in M-B, (c) VBB parameters used in M-B with the 

viscoplastic phase disabled, (d) VBB parameters used in M-B with the viscoplastic

phase disabled, (e) VBB parameters used in M-B with pressure dependence of strength 

disabled, (f) comparison of temperature rise due to plastic work-to-heat conversion near 

end of impact event using (i) full VBB parameters in M-B, (ii) phase disabled, (iii) 

phase disabled, (iv) pressure dependence disabled...............................................................18

Figure 15. Parameter sensitivity showing numerical contours of strain rate using the M-B 

model for a Taylor cylinder impact experiment at a striking velocity of 187 m/s (labeled 

S2) at sequential times after rigid impact. (a) Full original parameters used in M-B, (b) 

full VBB parameters used in M-B, (c) VBB parameters used in M-B with the 

viscoplastic phase disabled, (d) VBB parameters used in M-B with the viscoplastic

phase disabled, and (e) VBB parameters used in M-B with pressure dependence of 

strength disabled. .....................................................................................................................20

Figure 16. Parameter sensitivity showing pressure contours using the M-B model for a Taylor 

cylinder impact experiment at a striking velocity of 187 m/s (labeled S2) at sequential 

times after rigid impact. (a) Full original parameters used in M-B, (b) full VBB 

parameters used in M-B, (c) VBB parameters used in M-B with the viscoplastic phase 

disabled, (d) VBB parameters used in M-B with the viscoplastic phase disabled, and (e) 

VBB parameters used in M-B with pressure dependence of strength disabled.......................21

Figure 17. Comparison of strain hardening due to chain stretching for PC striking a rigid 

anvil at 187 m/s. Deformed shape and contours shown at the end of impact event for (a) 

experiment, (b) full VBB parameters, (c) VBB parameters with pressure dependence of 

strength disabled, (d) VBB parameters used in M-B with the viscoplastic phase 

disabled, and (e) VBB parameters used in M-B with the viscoplastic phase disabled.........22

Figure 18. Recovered .30-caliber specimens from Dyn-Ten-Ext experiments for PC. (a) 

Arrested intact specimen from low velocity impact (less than 350 m/s), showing necking, 

void nucleation behind the neck, and circumferential fracture behind the void. (b) 

Comparison of arrested intact specimen and failed specimen (upper is low velocity, lower 

is high velocity), where central voiding and peripheral fracture have proceeded nearly to 

complete failure. There is a small ligament of unfractured material remaining in the failed 

part. ..........................................................................................................................................24

Figure 19. Dyn-Ten-Ext experiment for PC as captured at the die exit by a Shimadzu highspeed framing camera. The time in microseconds from camera trigger is noted in the 

lower-right corner of the images (18, 40, 98, and 198 s, respectively). The test 

conditions were striking velocity of 523 m/s with a die exit of 0.36 cm. PC failure is in 

the corkscrew fracture/tearing mode. While it appears to behave in a molten fashion, the 

recovered pieces were found to have retracted almost to their original shape, indicating 

solid behavior...........................................................................................................................25

Figure 20. Recovered .30-caliber specimens from Dyn-Ten-Ext experiments for PC showing 

nearly complete elastic recovery (rebound) at 363 m/s; deformation near the exiting tip at 

451 m/s; and failure with fracture above 600 m/s....................................................................25

Figure 21. Parameter sensitivity showing numerical contours of velocity using the M-B 

model for a Dyn-Ten-Ext experiment at a striking velocity of 363 m/s at sequential times 
vi

after die impact. (a) Full VBB parameters used in M-B, (b) VBB parameters used in M-B 

with pressure dependence of strength disabled, (c) VBB parameters used in M-B with the 

viscoplastic phase disabled, and (d) VBB parameters used in M-B with the 

viscoplastic phase disabled...................................................................................................27

Figure 22. Parameter sensitivity showing numerical contours of radial stress using the M-B 

model for a Dyn-Ten-Ext experiment at a striking velocity of 363 m/s at sequential times 

after die impact. (a) Full VBB parameters used in M-B, (b) VBB parameters used in M-B 

with pressure dependence of strength disabled, (c) VBB parameters used in M-B with the 

viscoplastic phase disabled, and (d) VBB parameters used in M-B with the 

viscoplastic phase disabled. Note, positive stress is tensile and negative is compressive....28

Figure 23. Parameter sensitivity showing numerical contours of axial stress using the M-B 

model for a Dyn-Ten-Ext experiment at a striking velocity of 363 m/s at sequential times 

after die impact. (a) Full VBB parameters used in M-B, (b) VBB parameters used in M-B 

with pressure dependence of strength disabled, (c) VBB parameters used in M-B with the 

viscoplastic phase disabled, and (d) VBB parameters used in M-B with the 

viscoplastic phase disabled. Note, positive stress is tensile and negative is compressive....29

List of Tables

Table 1. M-B model parameters for PC.........................................................................................16
vii

Acknowledgments

The author is indebted to invaluable technical conversations and advice provided by Dr. Richard 

C. Becker. The author would also like to thank Dr. Adam D. Mulliken for courteously providing 

the experimental results and the Abaqus version of the model and Professor Romesh C. Batra, 

Virginia Polytechnic Institute and State University, for the revised model and parameters. The 

experiments would not have been possible without the expertise provided by Dr. Jevan 

Furmanski, Exxon, United States Military Academy Cadet James Tyler, Dr. G. “Rusty” Gray, 

Dr. Carl Trujillo, and Daniel T. Martinez, LANL MST-8. It is my honor to work and collaborate 

with such talented and highly intelligent individuals.
viii

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1

1. Introduction

Materials often exhibit phenomena on a broad range of spatial and temporal scales that combine 

together to dictate deformation response to subsequent failure. The physical processes and 

evolution are complicated and may be difficult to access experimentally. Because of the complex 

processes present during a ballistic impact event, it is often difficult to identify the armor or 

material parameters that significantly influence performance. These controlling parameters can 

be identified using a canonical approach by performing carefully instrumented impact 

experiments, examination of dominant parameters in analytical and numerical penetration 

mechanics solutions, and/or computational modeling of the ballistic impact event (1, 2). Once 

limiting performance parameters and defeat mechanisms are identified, the armor designer can 

seek to control these aspects for optimized and enhanced armor performance. 

Survivability of the U.S. Army’s fighting forces relies on lightweight armor systems capable of 

sustaining multi-hit impacts. Emerging armor systems utilize multiple material components 

(e.g., metals, ceramics, polymers) that achieve mass and space efficiencies with improved energy 

absorption and durability. Further dramatic armor improvements including fast response to 

adaptive threats require implementing relevant material response and deformation mechanisms

into hydrocodes, providing researchers the ability to identify key interactions (e.g., properties, 

wave propagation, interfaces, geometry, composition, processing) that pilot valued protection 

performance functions (e.g., stop threat, modular, rapid install and remove). For polymer 

selection and formulation, information relating key material properties to performance is limited. 

In this work, in situ dynamic behavior for polycarbonate (PC) is investigated using experimental 

and numerical capabilities developed at the U.S. Army Research Laboratory (ARL), Los Alamos 

National Laboratory (LANL), Massachusetts Institute of Technology (MIT) and Virginia 

Polytechnic Institute and State University (VA Tech). Properties evaluated and tested include 

pressure sensitivity, strain rate sensitivity, and temperature dependence. The observed results and 

parameter sensitivity are presented to validate a new physics-based polymer material model

implemented into the U.S. Department of Energy (DOE) hydrocode ALE3D (3), developed at 

Lawrence Livermore National Laboratory (LLNL).

Amorphous glassy polymers are versatile materials that can be easily worked, molded, and 

thermoformed. They are effective materials for use in numerous applications requiring 

mechanical performance subjected to high-strain-rate loading conditions, ranging from golf balls 

to laminated blast/bullet-proof transparent windshields. Thermoplastic polymers have also been 

evaluated against high velocity chemical energy shaped charge jets, as shown in figure 1. 

Information relating polymer properties to performance at these extreme environments is limited. 

Our canonical strategy at ARL for polymers is to identify key mechanisms for protection 
2

performance, develop and validate physics-based constitutive models, and then transfer the 

numerical capabilities to protection design engineers. 

Figure 1. In situ x-ray absorption characterization of shaped charge jet/target interaction. 

Performed at ARL EF-7; courtesy of M. Zellner, ARL.

Mechanical properties for amorphous glassy polymers are highly influenced by the glass 

transition temperature, as shown in figure 2. Glassy polymers are generally soft and ductile 

above Tg (rubber-like state), and hard and brittle below Tg (glass-like state). The ductility under 

pressure is the primary reason for high impact resistance, which allows structures made of these 

materials to undergo large deformation before failure (4). Additionally, the nonlinear viscoelastic 

and viscoplastic behavior of amorphous polymers permit the material to transform impact energy 

into heat or internal energy. Glassy polymers exhibit pressure dependence of modulus and flow 

strength where strength increases with increasing strain rate and pressure. Because of these 

properties, PCs find many applications. Therefore, selection and formulation of PC for 

performance in specific protection technologies can be guided by identifying key material 

properties (pressure sensitivity, rate sensitivity, and temperature dependence). 

This research effort will briefly review the thermo-elasto-viscoplastic Mulliken-Boyce (M-B)

constitutive model for amorphous glassy polymers, significant modifications for model 

implementation into ALE3D, and an evaluation of PC material model parameters including 

parameter sensitivity of pressure and strain rate dependence on flow strength as compared with 

observed results. It provides insight into the inner-workings of the M-B model that will guide 

engineers and scientists for application of the constitutive model toward innovative lightweight 

hybrid protection design technologies for the U.S. Army. 
3

Figure 2. Dramatic change in mechanical properties for amorphous polymers as the 

temperature passes through glass transition, Tg.

2. Materials and Experimental Methods

Few experimental approaches are available for unambiguously assessing the deformation 

response of materials under extreme mechanical loading conditions, i.e., simultaneous large 

strains, ε greater than 1, and high strain rates, dε/dt greater than 10,000/ s (5). Ultra-high strain

rates can be achieved during plate impact experiments, but these require elaborate experimental 

and specimen preparation techniques. 

For the Taylor cylinder impact experimental method, a right circular cylinder is fired at high 

velocity onto a hard (rigid) surface, as shown in figure 3. This classic ballistic experiment was 

named after Sir G. I. Taylor, who developed the experiment in 1948 to screen materials for 

ballistic applications during World War II. Taylor recognized that this experimental procedure 

permitted an estimate of a dynamic yield stress by measuring the overall length of the reference 

or original state of the cylinder and the final deformed specimen length (6). Currently, the Taylor 

cylinder impact experiment is primarily used as a valuable tool to validate constitutive models 

for various ductile materials subjected to dynamic compressive loading conditions. Although the 

experiment is relatively simple in theory, great attention must be given to maintaining planar 

impact of the cylinder on the rigid anvil and accurate post-mortem analysis (7). 
4

Figure 3. Original schematic of a Taylor impact 

experiment (1).

The Dynamic-Tensile-Extrusion (Dyn-Ten-Ext) experiment was developed by G. T. Gray III and 

coworkers at LANL to examine extreme tensile conditions in materials (8, 9). The apparatus 

utilizes the same hardware as a Taylor cylinder impact experiment, except that a conical 

extrusion die is fixed to the end of the gun barrel, forcing the specimen to extrude through it at a 

high velocity (see figure 4 [a]), and the anvil is not employed. The leading edge velocity of the 

specimen is relatively unaffected by the extrusion process, but the trailing portion rapidly 

decelerates inside the die, and thus the extruded ligament at the die exit is pulled in high strain

rate tension, typically to large strains and, ultimately, to failure (see figure 4 [b]). As an 

integrated experiment that generates extreme dynamic tensile deformation, Dyn-Ten-Ext 

provides a complementary means to challenge the quantitative accuracy of constitutive models 

parameterized under less extreme conditions. It also reveals tensile and shear instabilities that are 

not active in dynamic compression. 

Figure 4. (a) Schematic of the Dyn-Ten-Ext experiment and (b) sequenced high-speed video frames of the sample 

event.

 

(a) (b)
5

The Taylor cylinder impact experiments were conducted at the MIT Institute for Soldier 

Nanotechnologies’ (ISN’s) high strain rate laboratory under Contract No. DAAD-19-02-D0002 

with the U.S. Army Research Office. A 12.7-mm diameter bore single-stage gas gun was used to 

perform the Taylor cylinder impact experiments, as shown in figure 5 (10, 11). A double 

diaphragm assembly was burst using pressurized nitrogen gas. Typical impact velocities reached 

up to 300 m/s when the breech pressure was of the order of 70 bar. Experiments were performed 

using PC cylinders that were 12.67-mm diameter and either 25.4-mm or 76.2-mm length (L/D 

ratios of 2:1 and 6:1, respectively) at impact velocities varying from approximately180–280 m/s. 

The flats of the samples were polished prior to experiments. A hardened stainless steel tile 100 × 

100 mm wide × 12.7 mm thick and machine-ground and lubricated, was mounted on a steel 

frame and acted as the rigid surface. Impact velocity was measured by triggering two parallel 

laser ribbons monitored with photodiodes. Impacted specimens were recovered after rebound and 

sectioned for final deformed geometry. A Cordin high-speed charge-coupled device camera with 

a Nikon 70–300 mm lens, capable of acquiring images at a frame rate of 2 million frames per 

second, was used to record the event (10, 11). A machine-grade PC procured from GE Plastics 

was used for all the MIT Taylor cylinder impact experiments. Although the material was 

extruded, compression Kolsky bar experiments (approximately 4000/s) showed that the material 

was fairly isotropic and the dynamic response was identical to Lexan PC 9034 (10).

Figure 5. Schematic of the single stage gas gun used for the Taylor cylinder impact experiments at MIT ISN (11).

The assembly shown is for an impact-penetration experiment using a finite thickness target. 
6

Large sheets (1.22 × 2.44 m wide × 12.7 mm thick) of PC were obtained from Bayer Sheffield 

Plastics Makrolon. The sheets as received by ARL were extruded along the 2.44-m width. As 

previously mentioned, the Dyn-Ten-Ext apparatus in use at LANL is a slight modification to the 

Taylor cylinder impact configuration (12). The barrel has a smooth 7.62-mm (0.300-in) bore.

The barrel discharges into a target chamber that is evacuated with a roughing pump to prevent 

interaction of the projectile with an atmosphere. While the chamber contains a polished hardened 

steel anvil during the Taylor cylinder impact experiment, for Dyn-Ten-Ext this is replaced with a 

catch tank loosely filled with cloth for soft recovery of any specimen fragments. The main data 

collected during a Dyn-Ten-Ext experiment consist of initial velocity from the bore prior to 

interaction with the die and high-speed multiframe photography of the extruded material for 

obtaining deformed shape and speed, which is then compared to numerical simulations to 

understand the material behavior during the experiment. The specimen’s initial velocity is 

measured via two pressure transducers on the barrel near the target chamber and captured with 

an oscilloscope. The images of the extruded material are captured with an Imacon 200 or 

Shimadzu HPV2 framing camera (Hadland, Santa Cruz CA), typically with a 3–5 μs interframe 

time, and illumination is provided opposite the camera by a Powerlight 2599 flash lamp 

(Photogenic, Bartlett, IL). The velocity of extruded material is collected from the captured 

images and is calibrated with a transparent scale near the die exit. 

The Dyn-Ten-Ext specimen geometry can be a sphere or a hemisphere-ended cylinder, both of 

which use a diameter and length of 7.60 mm; hemisphere-ended cylinders are shown in figure 6.

The die uses a conical reducing section inclined 9° (half-angle) to the extrusion axis. The die 

exit diameter is specified depending on the desired draw ratio to be imposed prior to the onset of 

dynamic self-extrusion after the die exit. The nominal design of the die uses an exit diameter of 

2.80 mm (0.110 in), which imposes an axial true strain ε~2 at the die exit. All samples were 

machined at ARL and sent to LANL as part of a collaborative agreement between the author and 

G. T. Gray. The experiments were performed by J. Furmanski, C. Trujillo, D. Martinez, and 

CDT J. Tyler under the Joint Munitions Program, Technology Coordinating Group-I

collaborative research efforts between DOE and U.S. Department of Defense agencies. To 

examine the influence of anisotropy due to rolling, hemisphere specimens were prepared in all 

three directions as shown in figure 5. Unfortunately, a faulty high-speed camera prevented 

completion of the parametric study but will be reported in the future upon completion. Results of 

the successful experiments provided ample data for the objectives of this report.
7

Figure 6. Hemishere-ended cylinder samples and directionality considered for polycarbonate as used for 

DynTen-Ext experiments.

3. Constitutive Model 

It is well known that amorphous glassy polymers exhibit a strong dependence on strain rate and 

temperature as well as pressure dependence of modulus and strength. For numerous polymers, it 

has also been observed that there exists a transitional threshold in strain rate and temperature 

beyond which the strain rate sensitivity dramatically increases. The most widely accepted theory 

of rate dependent yield for glassy polymers connects macromolecular mechanisms of 

deformation resistance with macroscopic mechanical behavior (13, 14). The Eyring cooperative 

model assumed an existence of an internal stress as a structural parameter where defects are only 

inherited from thermal evolution. As an example, the glass transition temperature, Tg, is a useful 

parameter supporting this theory by differentiating the solid state from the rubbery state. Eyring 

considered plastic flow originated as a rate-activated process where molecules flowed from a 

particular potential well to another by overcoming an energy barrier (13). The shear strain rate, 

 , as a function of the shear stress, , in the polymer was written as,

 

 

 

 

 (1)

where H is the activation energy; R is the universal gas constant; T is temperature;  is the 

activation volume; and is a pre-exponential factor. In 1955, the Ree-Eyring model further

accounted for microstructural mechanisms by relating molecular motions to yield behavior, 

denoted as phases and written as, 









































 



RT

Q

A C

RT

Q

A C

T

y 

 



 

 



ln(2 ) sinh exp 1

 

(2)

where y is the yield stress under uniaxial loading; is the strain rate; Qi (i =  are activation 

energies for the two processes; and Ai and Ci are activation parameters (14, 15). For PC, the 

results of Dynamic Mechanical Analysis (DMA) plotting storage and loss moduli as functions of
8

temperature are shown in figure 7. Mulliken and Boyce (16) developed an expression describing 

the  mechanism for low strain rates and high temperatures, whereas the mechanism refers to 

high strain rates and low temperatures. Taken together, the andphases will be denoted as 

segment A.

To determine strain rate sensitivity of the yield stress, a set of quasi-static Instron and dynamic 

Kolsky bar compression experiments were conducted for PC, as shown in figure 8 (a). Rate 

sensitivity was then reduced from the observations in figure 8 (b) as two distinct regions for 

increasing flow stress with strain rate. It was then determined that this transition 

(approximately1500/s) was in close agreement with the transition observed from DMA data (16). 

This insight of rate-sensitive transitions formed the foundation of the M-B constitutive model for 

amorphous polymers. 

Figure 7. Storage and loss tangent moduli for PC; 

DMA experiments performed at 1 Hz and 

shift at 100 Hz showing the  and

transition regions using the DeconstructShift-Reconstruct analytical method (16). 
9

Figure 8. (a) Observed uniaxial compression response of PC over a range of strain rates performed at room 

temperature; (b) strain rate sensitivity of flow stress (16). 

The M-B model is built on the foundation of the Ree-Eyring and Eyring cooperative model as 

previously described, and is an extension of the Arruda-Boyce model by capturing glassy 

polymer behavior at high rates of deformation (16). Similar to a mixture theory, material 

response in the M-B model are described using three mechanisms (two segment A’s and a 

segment B) acting in parallel and simultaneously co-existing at a material point, as shown in 

figure 9. The deformation is decomposed into elastic and plastic parts for each phase and each 

phase sees the same total deformation. The constitutive equations do not assume a yield surface; 

therefore, plasticity occurs at all times during deformation. The total Cauchy stress tensor, , at a 

material point is written as the sum,

 (3)

where and are the Cauchy stress tensors for the and phases. Referring back to 

figure 9, , the segment B mechanism represents macromolecular resistance to chain stretching 

and alignment; thus, it is a strain hardening term with no dependence on strain rate and 

temperature. 

(a) (b)
10

Figure 9. One-dimensional (1-D) depiction of the M-B constitutive model (16). The spring in segment B is a 

strain hardening Langevin (nonlinear) spring representing chain stretching and alignment; the springs in 

segment A are rate- and temperature-dependent elastic (linear) springs and the dashpots represent rate-, 

temperature-, and pressure-dependent viscoplastic terms for intermolecular resistance.

The stress in the nonlinear hardening segment B, , uses the original description by Arruda and 

Boyce (17) and written as,

 

 

 

 

 

 

 

 

 

 

 

 

(4)

where 

 

 

 measures chain stretch in an 8-chain network; is the Langevin 

function defined by 

 

 

; 

 

is the deviatoric part of the isochoric left CauchyGreen tensor, 

 

 

 

 

; Nl is the limiting chain stretch; and CR = nkT is the rubbery 

modulus (where n is the number of chains per unit volume, k is Boltzmann’s constant, and T is 

the absolute temperature).

Segment A represents the intermolecular resistance to chain-segment rotation; thus, the two 

parallel subsegments are thermal softening terms that are strain rate and temperature dependent. 

For PC, the -subsegment represents rotations of the entire monomer chain and the 

-subsegment represents rotations of the phenyl group. The viscoplastic behavior connecting 

stress to effective plastic strain rate for each intermolecular resistance subsegment can be written 

as,

 

 

  

 

 

 

 

 

 (5)
11

where i= for the two component phases, is the athermal shear strength (related to shear 

modulus and Poisson ratio), a pressure coefficient, and is the pressure. Note the 

similarity to equation 1. Also, the viscoelastic behavior as observed from the DMA experiments 

is incorporated into the M-B model through the shear modulus in segment A. The internal history 

variable in equation 5 accounts for strain softening in glassy polymers. The evolution equation 

for the internal variables is written as,

 

 

 

 

 

 

 

 

 

 

 

(6)

and the shear strength is,

 

 

 

(7)

where are softening slopes, 

 

are the “preferred state” of the shear strengths, 

 are material 

parameters used to limit softening due to plastic work, are shear moduli as functions of 

temperature, and strain rate and are the Poissons ratios for each phase, which are assumed to 

be constant. The effective plastic strain rate, 

 

 

, as given by equation 5 is equal to zero when 

deformations are elastic only. Also, 

 is a steady state value for 

, which sets a lower limit on 

strength.

The M-B constitutive equation as described using equations 5–7 include amendments by 

Varghese and Batra (18) to account for the temperature rise due to 100% plastic work-to-heat 

conversion. In 2012, the M-B model was implemented by R. Becker, ARL, into the LLNL 

hydrocode suite ALE3D (3). The implementation followed the kinematic approach used by 

Varghese and Batra (18); however, some modifications were required. Specifically, Becker did 

not use multiplicative decomposition of the deformation gradient into elastic and plastic parts but 

used an updated Lagrangian form that would be less problematic with advection. Also, the 

initialization of the temperature and rate dependent shear modulus was set using prevailing 

conditions instead of initialization to the anticipated experimental conditions used by Varghese 

and Batra. Finally, the pressure-volume response is given by a nonlinear equation-of-state which 

also depends on deformation energy, whereas the earlier work used a constant bulk modulus. 

Further details on the implementation of M-B into ALE3D and other DOE hydrocodes will be 

made available in a future report. 

Model parameters for the M-B model were optimally fit using two types of experiments: DMA 

experiments for the viscoelastic behavior of the loss storage modulus and quasi-static/dynamic 

compression experiments for rate sensitive viscoplastic response. The original parameter fits for 

PC obtained by Mulliken and Boyce (16) considered only isothermal conditions and are shown 

in figure 8 (a). The contribution of the -component is clearly seen in figure 8 (b) with the ability 

to describe the increase in flow stress at high strain rates. Varghese and Batra (18) assumed that 

all the energy dissipated due to plastic work was converted into heat (adiabatic heating) and re-fit 

the model by adjusting the chain stretch coefficients. They also split the original internal variable
12

 into two components as now given in equation 5; one accounting for the rate and temperature 

dependent modulus and the other a pure evolution, as written in equations 6 and 7, respectively. 

As a result, they obtained another set of model parameters for PC that considered the temperature 

rise due to plastic deformation and compared it with the isothermal parameters, shown in 

figure 10. 

Figure 10. Comparison of PC material response in uniaxial compression using 

the M-B model for parameters obtained assuming isothermal 

conditions (labeled Mulliken and Boyce in plot) and modified 

parameters for plastic work (labeled Varghese and Batra in plot) 

(18).

4. Numerical Considerations

A performance evaluation using all three model parameter sets discussed in the previous section 

was completed numerically for dynamic compression and dynamic tension loading conditions. 

All computational modeling was performed on a twelve-processor Linux computer using the 

hydrocode ALE3D (3), developed at LLNL. The boundary conditions for the Taylor cylinder 

impact experiments are shown in figure 11. An ALE3D input file was created describing the 

geometry, initial conditions, boundary conditions, and material response models. Analyses were 

conducted using a two-dimensional axisymmetric geometry. Symmetry along the centerline was 

described by constraining the centerline nodes with no displacement in the radial direction. The
13

boundary of the cylinder was Lagrangian, and the interior nodes were free to advect. Also, the 

rigid wall was described by constraining the boundary nodes along the impacted end to not cross 

the impact plane. This input allowed the cylinder to rebound in the negative axial direction after 

the reflected tensile wave arrived at the impacted end and effectively stop any further loading 

due to impact. 

Figure 11. Schematic illustrating the reference configuration, initial conditions, and boundary conditions 

used for the ALE3D Taylor cylinder impact simulations.

The computational domain for the Dyn-Ten-Ext experiments is shown in figure 12. The 

simulation was run in an Eulerian mode with a fixed mesh. Axisymmetry along the centerline 

was described by constraining the centerline nodes with no displacement in the radial direction. 

A frictionless surface was used to describe the rigid extruding die. The options used in these 

Eulerian simulations do not allow the material to separate from the die. This drives the timestep 

down and eventually stops problem execution during elastic recoil. The issue during rebound can 

be avoided by alternatively applying a slide surface that detaches from the deforming cylinder. 

Results for model performance will only be discussed as the cylinder begins to exit the die and 

before the stress on the die surface becomes tensile. 

Figure 12. Schematic illustrating the reference configuration, initial conditions, and boundary 

conditions used for the ALE3D Dyn-Ten-Ext simulations.
14

For both numerical simulations, the initial velocity of the projectile was the observed impact 

velocity. The governing equations for spatial and temporal deformation of the cylinder were 

solved using an Eulerian formulation. Elements were approximately square with a mesh density 

of approximately 0.2 mm/element in the r- and z-directions. The consistent units used for all 

numerical simulations were cm, g, s, Mbar, and K. 

ALE3D is a large deformation continuum mechanics code that contains a material strength 

formulation. Typically, the constitutive formulation can be subdivided into five parts: (1) a 

stress/strain relationship for elastic material response, (2) a yield surface that specifies the 

multiaxial stress state corresponding to the start of plastic flow, (3) a flow rule that describes 

plastic behavior after yielding by relating strain increments to stress increments, (4) a 

consistency statement that constrains the stress state to remain on the yield surface during plastic 

flow, and (5) a hardening rule that describes the evolution of the flow strength during plastic 

deformation. The constitutive relation or flow rule physically describes material response by 

relating stresses to deformations. The finite element framework mathematically connects 

kinematics and field equations with boundary conditions. Most generally, a constitutive relation 

for the material flow strength, , can be written as some function, 

 (8) 

is the plastic strain in the current configuration, is the plastic strain rate, T, the temperature, 

p, the pressure, and  is some internal state variable. Most empirical constitutive models (e.g., 

Johnson-Cook strength model) are constructed on macroscopic material response experiments 

and do not track a state variable; they are written as,

 (9)

Several phenomenological constitutive models including an M-B attempt to provide a better 

representation of the underlying physics using an internal state variable to describe structure 

evolution. The flow stress can now be written as,

 (10)

5. Results

A parameter sensitivity analysis was performed with the M-B model at extreme loading 

environments that approach the ballistic regime. The two experiments previously described 

provide ‟isolated” loading conditions of dynamic compression and dynamic tension. The 

purpose of this numerical analysis was not to match or validate model parameters to the 

observed. Instead, the given model parameters where used as a baseline for a given loading 

condition, and particular mechanisms were carefully disabled then compared to the baseline case. 

This is an example of one method required for sensitivity analysis. Another would require
15

changing the functional form of physical descriptions in model and comparing to a baseline 

ballistic impact experiment. The method of changing mathematical forms will be considered in 

future research endeavors.

The M-B model in its most general form describes three intermolecular activation processes that 

simultaneously influence material deformation at the continuum level. The first describes chain 

stretch as related to strain hardening and the others describe strain softening for low strain rates, 

high temperatures and high strain rates, low temperatures, respectively. This was described 

mathematically using the 1-D spring-dashpot system, as shown in figure 9, using springs for the 

nonlinear elastic behavior and dashpots for viscoplasticity. A model parameter sensitivity 

analysis was performed for a better understanding on the influence of these simultaneously 

competing mechanisms subjected at high rate loading conditions. The dependence of flow 

strength on strain rate sensitivity and pressure was examined as function of these competing 

deformation mechanisms. 

Specifically, three mechanisms were individually turned off or minimized in this parametric 

study:

(1) No low strain rate activation mechanism; disabled dashpot- of subsegment A.

(2) No high strain rate activation mechanism; disabled dashpot- of subsegment A.

(3) No pressure dependence in the chain rotation activation mechanism; disabled pressure in 

the effective plastic strain rates terms of segment A. 

Values of M-B material parameters for PC are given in table 1 and subdivided according to the 

1-D spring-dashpot system, as previously described. To obtain M-B model parameters requires 

reduced DMA data for defining the elastic springs. These are piecewise cubic splines that are not 

given in table 1 but are available in the ALE3D implementation. A set of uniaxial compression 

experiments (quasi-static and dynamic) are required for a piecewise breakdown of yield behavior 

as defined using the viscoplastic dashpots. Note that the values listed in table 1 were obtained by 

Varghese and Batra (18) and accounted for the temperature rise from plastic work to heat (i.e., 

locally adiabatic deformations) as encountered during high rate loading events. 
16

Table 1. M-B model parameters for PC (18).

MB Parameter Parameter Description Nominal Value

Rate-dependent elastic springs

 

 Process-specific shear moduli as functions of temperature 

and strain rate

Obtained from DMA 

data

 

 Process-specific bulk moduli as functions of temperature 

and strain rate

Obtained from DMA 

data

Viscoplastic dashpots

 

 

 Pre-exponential factors 2.94e+16 , 3.39e+05

 Activation energies 3.744e-18, 3.769e-20

 

 

 Pressure coefficients 0.168, 0.245

 Softening slopes 125 MPa, 400 MPa

 

 Preferred states of strain softening; added by (18) 0.33, 2.00

Langevin springs

CR at 300K Rubbery modulus 35 MPa

 

Limiting chain extensibility 3.5

5.1 Dynamic Compression

Taylor cylinder impact experiments were conducted using PC cylinders with a 12.7-mm 

diameter and 76.2-mm length at normal impact velocities ranging from 180 to 200 m/s. Results 

are numerically predicted for the representative case with no fracture (only deformation) at 

187 m/s (18). 

High-speed video frames of the impact event are shown in figure 13. After the PC cylinder has 

made contact with the rigid anvil (frame 2 in figure 13), the deformation at the impacted end 

flows radially and results in the formation of a mushroom head (frames 3–5 in figure 13). At 

72 s (frame 6 in figure 13), radial barreling forms above the mushroom head at approximately 

20% above the foot of the original length. This spreading of the deformation zone is primarily

due to strain hardening from chain alignment. The cylinder continues to deform until it rebounds 

at approximately 198 s (frame 11 in figure 13). It is during and after this rebound stage that the 

cylinder displays elastic recovery until the loading event stops at 230s. Measurements of the 

recovered specimen revealed plastic radial expansion over approximately one-half the length 

(11).

Considering the differences in implementation and parameters, numerical simulations of the 

Taylor cylinder impact experiment using the Varghese, Batra, Becker (VBB) model parameters 

favorably agreed with the original M-B model results, as shown in figures 14, 15, 16 (a), and 

16 (b). Figure 14 shows the evolution of axial stress, figure 15 plastic strain rate, and 

figure 16 pressure. The montage of predicted shapes matched well with the observed including 

the mushroomed head, radial barreling, and elastic recoil. Subtle differences compared with the 

predictions using the original M-B model parameters are attributed to the following features 

using the VBB model parameters:
17

(1) Temperature rise from plastic work-to-heat conversion (original M-B parameters 

considered only isothermal deformations).

(2) Equation-of-state used with VBB parameters in ALE3D where the shear modulus is split 

into two parts (Young’s modulus and Poisson’s ratio used to obtain a shear modulus with 

the original M-B parameters). 

(3) Modifications to ALE3D VBB model parameters for segment A terms.

(4) Modulus and history variables initialized to predefined strain rate in M-B model.

These differences are expected to be more pronounced at higher strain rates and larger 

deformations.

Figure 13. High-speed video frames recorded at various stages during a Taylor cylinder impact experiment on a

76.2-mm PC rod at 187 m/s. Note elastic recovery after 100 s and rebound (detachment from rigid 

surface) at 198 s (11).
18

Figure 14. Parameter sensitivity showing numerical contours of axial 

stress using the M-B model for a Taylor cylinder impact 

experiment at a striking velocity of 187 m/s (labeled D2) at 

sequential times after rigid impact. (a) Full original parameters 

used in M-B courtesy of (11), (b) full VBB parameters used in 

M-B, (c) VBB parameters used in M-B with the viscoplastic

phase disabled, (d) VBB parameters used in M-B with the 

viscoplastic phase disabled, (e) VBB parameters used in M-B 

with pressure dependence of strength disabled, (f) comparison 

of temperature rise due to plastic work-to-heat conversion near 

end of impact event using (i) full VBB parameters in M-B, (ii) 

phase disabled, (iii) phase disabled, (iv) pressure 

dependence disabled. 
19

Figure 14. Parameter sensitivity showing numerical contours of axial 

stress using the M-B model for a Taylor cylinder impact 

experiment at a striking velocity of 187 m/s (labeled D2) at 

sequential times after rigid impact. (a) Full original parameters 

used in M-B courtesy of (11), (b) full VBB parameters used in 

M-B, (c) VBB parameters used in M-B with the viscoplastic

phase disabled, (d) VBB parameters used in M-B with the 

viscoplastic phase disabled, (e) VBB parameters used in M-B 

with pressure dependence of strength disabled, (f) comparison 

of temperature rise due plastic work-to-heat conversion near 

end of impact event using (i) full VBB parameters in M-B, (ii) 

phase disabled, (iii) phase disabled, (iv) pressure 

dependence disabled (continued). 
20

Figure 15. Parameter sensitivity showing numerical contours of strain rate using the M-B model for a Taylor 

cylinder impact experiment at a striking velocity of 187 m/s (labeled S2) at sequential times after rigid 

impact. (a) Full original parameters used in M-B courtesy of (11), (b) full VBB parameters used in 

M-B, (c) VBB parameters used in M-B with the viscoplastic phase disabled, (d) VBB parameters 

used in M-B with the viscoplastic phase disabled, and (e) VBB parameters used in M-B with pressure 

dependence of strength disabled.
21

Figure 16. Parameter sensitivity showing pressure contours using the M-B model for a Taylor cylinder impact 

experiment at a striking velocity of 187 m/s (labeled S2) at sequential times after rigid impact. (a) Full 

original parameters used in M-B courtesy of (11), (b) full VBB parameters used in M-B, (c) VBB 

parameters used in M-B with the viscoplastic phase disabled, (d) VBB parameters used in M-B 

with the viscoplastic phase disabled, and (e) VBB parameters used in M-B with pressure 

dependence of strength disabled.

The numerical results provided insight into the evolving stress state within the material as related 

to deformation. High strain rates and compressive stress (on order of 5e+04/s and 400 MPa) are 

induced in PC immediately after impact within the first 10 s of the event (figures 14 [b] and 

15 [b]). Afterwards, the magnitude of stress decreases until the transient nature of stress begins 

to stabilize (after 30 s) as the compressive front travels to the rear of the cylinder (figures 14 [b]

and 15 [(b]). The elastic compressive wave front reached the rear end at approximately 45 s and 

is followed by a slower moving plastic deformation wave front. It is the plastic front that induced 

the resulting mushroomed head on the impacted (foot) end. Upon reaching the non-impacted end 

of the cylinder, the elastic compressive front reflects as a tensile front to maintain the zero stress 

state on the free surface. Afterwards, the reflected elastic tensile front interacts with the plastic 

front as seen by the profiles at 100 s and 150 s of figure 14 (b). The cylinder rebounded at 

approximately 195 s followed by significant elastic recoil, as seen by comparing the deformed 

profiles for 150 s and 230 s of figure 15 (b). 
22

For the same instant of time, regions of high strain rate are directly related to zones of high 

plastic deformation. Peak strain rates of about 4e+04/s were achieved between 5 and 30 s after 

impact, as shown in figure 15 (a) and (b) at 5 s and 30 s, and subsequently diminish after 

45 s, as shown in figure 15 (a) and (b) at 150 s. Most of the plastic deformation including 

mushrooming occurred in the initial 40 s of the impact event and radial barreling occurred in 

the range of 50–100 s. Negligible plastic deformation was predicted after rebound from the 

rigid surface. A review of the results shown in figures 15–17 (a) and (b) clearly indicated that the 

high rate dynamic loading conditions subjected PC to high-deformation gradients during a 

Taylor cylinder impact experiment. 

The parameter sensitivity analysis examined the influence of the viscoplastic dashpots on the 

deformed state of the PC cylinder. Recall the rate dependence of the dashpots on flow strength, 

as shown in figure 8 (b). By disabling the -dashpot, more strain hardening spread axially within 

the cylinder; the zone of moderate strain rate expanded at early stages (before 40 s) and resulted 

in a larger barreled zone, as shown by comparing figures 15, 16, and 17 (c) with (b). By 

disabling the high-rate  dashpot and maintaining the low-rate  dashpot, more localized high 

strain rates were generated above the mushroomed head and resulted in wider radial barreling 

between 30 and 100 s, as shown by comparing figures 15, 16, and 17 (d) with (b) and (c). 

Finally, disabling pressure dependence in both viscoplastic and dashpots resulted in 

negligible differences as compared with full VBB parameter set for the M-B model, as shown by 

comparing figures 15, 16, and 17 (e) with (b). The pressure dependent results are not surprising 

since hydrostatic pressures generated during Taylor cylinder impact are small when compared 

with axial stress. This is not expected to be the case for more extreme ballistic environments 

where pressure dependence of flow strength has a greater influence on material deformation.

Figure 17. Comparison of strain hardening due to chain stretching for PC striking a 

rigid anvil at 187 m/s. Deformed shape and contours shown at the end of 

impact event for (a) experiment courtesy of (11), (b) full VBB parameters, 

(c) VBB parameters with pressure dependence of strength disabled, (d)

VBB parameters used in M-B with the viscoplastic phase disabled, and 

(e) VBB parameters used in M-B with the viscoplastic phase disabled.

(a) (b) (c) (d) (e)
23

Figure 17 shows the final deformed lengths of the full VBB parameter M-B model compared 

favorably with observed. Additionally for figure 17 (b–e), chain stretch in the deformed cylinder 

as calculated from the Langevin spring segment B is shown for the three parameter sensitivity 

sets. There are negligible differences in hardening by disabling pressure dependence at these 

strain rates, as seen by comparing figure 17 (c) with (b). By disabling the low-rate  dashpot, an 

increased zone of chain alignment indicated that large deformation gradients existed along the 

axial direction that resulted with increased strain hardening within the cylinder as shown in

figure 17 (d). By disabling the high-rate  dashpot and maintaining the low-rate  dashpot, 

resistance to high-rate deformation decreased resulting in a softer or more rubbery-like material 

that barreled in the radial direction, as shown in figure 17 (e). 

Finally, an important advantage for using the Varghese and Batra (18) modifications with 

hydrocode adjustments by Becker, is shown in figure 14 (f) for the temperature rise in the 

cylinder. The cylinder temperature increase an amount of 30 K by the end of the impact event 

using the modifications for 100% plastic work-to-heat conversion. This increase in temperature 

was not accounted for in the original M-B implementation (considered only isothermal 

deformations). Therefore, to model ballistic events for amorphous polymers at extreme loading 

conditions it is strongly recommended to use the modified M-B model with VBB parameters as 

given in table 1. 

5.2 Dynamic Tension

A range of experimental results was obtained as the striking velocity of the .30-caliber specimen 

was increased, progressing from intact arrest with incipient cracking and void nucleation, to 

terminal damage progression through a circumferential tearing and central void progression 

process. Although damage and failure of PC is not considered for evaluating parameter 

performance in the M-B model, the Dyn-Ten-Ext results merit a brief sidebar on this topic. 

The experiments performed to date captured damage evolution within the specimen subjected to

extreme tensile conditions and also revealed that two separate mechanisms (or at least pathways) 

of damage were active during deformation. At lower velocities, the specimen did not extrude but 

was observed to form a neck at the hemisphere tip, as shown in figure 18 (a). Behind this neck,

spherical voids nucleated in a number of cases. Further behind the neck, circumferential fracture 

was observed to initiate. These arrested damage features are believed to dominate failure and 

resulted in rupture of the specimen under more extreme loading conditions, as shown in 

figure 18 (b). Alternative experiments using notched specimens observed PC cavitations and 

failure in a brittle manner when subjected to moderate strain rates and experiencing tensile 

pressures with magnitude on the order of 80 MPa (19, 20, 21).
24

Figure 18. Recovered .30-caliber specimens from Dyn-Ten-Ext experiments for PC. (a) Arrested intact specimen

from low velocity impact (less than 350 m/s), showing necking, void nucleation behind the neck, and 

circumferential fracture behind the void. (b) Comparison of arrested intact specimen and failed specimen

(upper is low velocity, lower is high velocity), where central voiding and peripheral fracture have 

proceeded nearly to complete failure. There is a small ligament of unfractured material remaining in the 

failed part.

High-speed video frames of the failure process during extrusion experiments revealed the 

dynamics, though they appeared potentially quite complicated. In an experiment at an 

intermediate velocity (523 m/s [figure 19]), the extruded ligament was tearing while extending, 

and ultimately the fracture was in a helical or corkscrew geometry. Much of the behavior during 

extrusion and exit appeared fluid-like. However, recovered failed specimens (figure 20) showed 

that the torn ligament recovered approximately to its original configuration. This indicated that 

the behavior was in fact solid-like. At high rates it appeared fluid-like due to very low stiffness.

The final deformed PC shapes for the experiments performed at different velocities are shown in 

figure 20. Below impact velocities of 500 m/s, PC exhibited no fracture, little plastic

deformation, and did not completely pass through the die exit. The simultaneous activity of void 

nucleation and coalescence combined with circumferential fracture was an intriguing result that 

deserves further study. Indeed, for ballistic applications, if two such mechanisms interact during 

penetration, then the mechanics need to be systematically investigated in a controlled manner 

like that provided by Dyn-Ten-Ex. 

 

(a) (b)
25

Figure 19. Dyn-Ten-Ext experiment for PC as captured at the die exit by a Shimadzu high-speed framing camera. 

The time in microseconds from camera trigger is noted in the lower-right corner of the images (18, 40, 

98, and 198 s, respectively). The test conditions were striking velocity of 523 m/s with a die exit of 

0.36 cm. PC failure is in the corkscrew fracture/tearing mode. While it appears to behave in a molten 

fashion, the recovered pieces were found to have retracted almost to their original shape, indicating solid 

behavior.

Figure 20. Recovered .30-caliber specimens from Dyn-Ten-Ext experiments for PC 

showing nearly complete elastic recovery (rebound) at 363 m/s; deformation 

near the exiting tip at 451 m/s; and failure with fracture above 600 m/s. 
26

The experiment for the specimen striking at 363 m/s was chosen as the baseline case because it 

exhibited no failure and fracture. The complete VBB parameter set was used to activate all the 

competing mechanisms in the M-B model (called ‘VBB parameters’ in figures 21–23). As shown 

by the axial velocity plots in figure 21 for the VBB parameters column, the hemisphere ended 

PC cylinder made contact with the rigid die at approximately 5 s and began to exit the die after 

30 s. At 40 s the .30-caliber PC bullet stopped stretching and deforming in the axial direction 

and began to rebound back into the die (elastic recovery). Also, the highest axial velocity began 

to localize near the tip after 20 s. A peak strain rate of approximately 3.5e+05/s was predicted 

near the contact region at early times (less than 20 s) and then dropped to 10/s near the exit. For 

the ‘no pressure’ results shown in figure 21, similar trends were observed, but the bullet 

stretched further past the exit as compared with VBB. This is because the pressure dependent 

strength term for both viscoplastic dashpots was disabled which decreased strength or increased 

thermally activated deformation, thus predicting more deformation. When the alpha viscoplastic 

dashpot was disabled, as shown in figure 21 ‘no alpha,’ the flow strength dropped to values 

between 10 and 40 MPa and was dramatically under-predicted at low strain rates. For the ‘no 

beta’ case shown in figure 21, the beta dashpot was disabled and the flow strength at high strain

rates reached values 20–25 MPa below the baseline value. In general, differences in deformation 

encountered for the Dyn-Ten-Ext predictions by disabling either viscoplastic dashpot follow the 

path for rate dependence of strength, as shown in figure 8 (b) for each viscoplastic component. 

Disabling the viscoplastic deformation mechanisms lowered the strength and resulted in higher 

deformation than the baseline case (VBB parameters) as well as for the no pressure case. The

flow began to separate after exiting the die for the no alpha and no beta cases and resembled 

fracture even though no failure model was described in the input deck.
27

(a) (b) (c) (d)

Figure 21. Parameter sensitivity showing numerical contours of velocity using the M-B model for a Dyn-Ten-Ext 

experiment at a striking velocity of 363 m/s at sequential times after die impact. (a) Full VBB 

parameters used in M-B, (b) VBB parameters used in M-B with pressure dependence of strength 

disabled, (c) VBB parameters used in M-B with the viscoplastic phase disabled, and (d) VBB 

parameters used in M-B with the viscoplastic phase disabled.
28

(a) (b) (c) (d)

Figure 22. Parameter sensitivity showing numerical contours of radial stress using the M-B model for a Dyn-TenExt experiment at a striking velocity of 363 m/s at sequential times after die impact. (a) Full VBB 

parameters used in M-B, (b) VBB parameters used in M-B with pressure dependence of strength 

disabled, (c) VBB parameters used in M-B with the viscoplastic phase disabled, and (d) VBB 

parameters used in M-B with the viscoplastic phase disabled. Note, positive stress is tensile and 

negative is compressive.

Figures 22 and 23 show the evolution of stress tensor components as a function of the parametric 

parameter sensitivity analysis. Radial stress for the PC bullet began to emanate at the region of 

contact with the angled die, as shown in figure 22 VBB parameters at 8 s. By 20 s the tip of 

the bullet exhibited negligible stress because it was a free end while the center region was being 

squeezed from the sides in a compressive state, as shown in figures 22 and 23. As it began to exit 

the die at 30 s, the tip remained stress free but now a highly compressed region formed 

immediately behind it, as shown in figure 22 VBB parameters at 30 s. This compressive wave 

eventually moved down toward the back end and reflected in tension as the tip began to rebound 

(elastic recovery). It was during this stage that having the slide line for the rigid die tied to the 

bullet, caused the timestep to decrease. 
29

(a) (b) (c) (d)

Figure 23. Parameter sensitivity showing numerical contours of axial stress using the M-B model for a 

Dyn-Ten-Ext experiment at a striking velocity of 363 m/s at sequential times after die impact. (a) Full 

VBB parameters used in M-B, (b) VBB parameters used in M-B with pressure dependence of strength 

disabled, (c) VBB parameters used in M-B with the viscoplastic phase disabled, and (d) VBB 

parameters used in M-B with the viscoplastic phase disabled. Note, positive stress is tensile and 

negative is compressive.

For the no alpha and no beta cases shown in the axial stress plots of figure 23 at 40 s, an 

increase in radial compression as compared with the no pressure case evolved in the die contact 

region during rebound (elastic recovery) of the PC bullet. By disabling either viscoplastic term, 

the region of increased flow stress was due to high compressive strains from the chain stretching 

term in segment B. To verify this, numerical output was data reduced to calculate the stress due 

to chain stretching for three locations of the bullet at 40 s. The highest stress due to chain 

stretch was located directly behind the tip at approximately 220 MPa, dropping to moderate 

values of approximately 130 MPa at the midsection and the smallest values of approximately 

5 MPa at the rear of the hemisphere bullet. 

Additionally, the modified M-B model accounted for plastic work-to-heat conversion resulted in 

a temperature increase of approximately 32 K for all cases examined. The increase in 

temperature due to plastic deformation was similar in dynamic tension and dynamic compression 

and is an important feature for simulations of any ballistic event. 
30

6. Summary, Conclusions and Recommendations

The M-B constitutive model for amorphous glassy polymers includes three competing 

intermolecular deformation mechanisms that together describe material response from low to 

high strain rates. One mechanism describes chain alignment or stretching using an elastic only 

hardening condition that is not dependent on strain rate or temperature. The remaining two 

mechanisms are thermally activated energy equations for chain rotation at low and high strain

rates, respectively. The chain rotation mechanisms describe thermal softening that is rate and 

temperature dependent including the temperature rise due to energy dissipation from the 

conversion of plastic work to heat. 

A parameter sensitivity analysis was completed using a modified form of the M-B model to 

determine the influence of competing, multiple mechanisms of deformation for PC when 

subjected to extreme thermo-mechanical states. Isolated loading conditions for dynamic 

compression and dynamic tension traditionally applied for model validation, were used for 

numerical comparisons and analysis. A parametric study was completed by individually 

disabling three modes of deformation: pressure dependent flow strength, thermal softening at 

low strain rates (-chain rotation viscoplastic term), and thermal softening at high strain rates 

(-chain rotation viscoplastic term). 

For dynamic compression, the complete VBB parameter set was compared with the original 

parameters used in the M-B model. Small differences in the final deformed shape between the 

two parameter sets as compared with the Taylor cylinder impact experiments were attributed to 

using modified viscoplastic VBB model parameters that accounted for the temperature rise from 

the conversion of plastic work to heat (deformation heating) and the shear modulus being split 

into two parts using an energy-based equation-of-state. The influence of pressure on flow 

strength was examined by disabling pressure dependence in M-B. PC exhibited no change in 

deformation with pressure dependence disabled when subjected to dynamic compression for 

strain rates at or below 105/s. For events occurring at higher strain rates, pressure dependence is 

expected to exhibit greater influence on deformation. When the -chain rotation term was 

disabled the flow stress was nearly zero below the uniaxial stress data at 102/s and increased 

above the 102/s level due to the -chain rotation term (but still below the observed rate sensitive 

yield stress). For this case, the Langevin spring had the greatest influence on deformation 

resulting with an increase in strain hardening along the axis of the cylinder (as evidenced by an 

equivalent increased zone of chain stretching along the axis). When the -chain rotation term 

was disabled the flow stress was 10–20 MPa below the observed uniaxial results for strain rates 

above 102/s. For this case, thermal softening due to -chain rotation had the greatest influence 

resulting with increased radial barreling.
31

For behavior in dynamic tension using the Dyn-Ten-Ext experiment, disabling pressure 

dependence or either viscoplastic dashpots numerically lowered the dynamic yield strength of 

PC, which resulted in larger displacement through the die exit. When the viscoplastic dashpots 

were disabled, it revealed the largest deformations and the formation of a large compressive zone 

with increasing flow stress due to chain stretching directly behind the tip of the bullet during 

elastic recovery (rebounding stage). 

Recommendations for future work include parameter sensitivity analysis using dynamic shear 

and obtaining a new set of VBB parameters for M-B by considering a 50–60% conversion of 

plastic work to heat (22, 23). Model predictions were also examined when the initial temperature 

of PC was at or above the glass transition temperature, Tg. When the initial temperature was 

20 K below Tg, the code had difficulties as specimen temperature increased due to energy 

dissipation from plastic work. Essentially, it could not decide if the amorphous polymer was in 

the soft or solid phase and localized high strain rates appeared temporally in random elements. 

An extension of the M-B model through the glass transition region is recommended for improved 

effectiveness toward predicting deformation states at extreme environments. Additionally, the 

Langevin spring describing molecular chain stretching assumed elastic hardening that is 

completely recovered. Experiments that closely monitor recovery may uncover a nonreversible 

portion for chain stretching that would promote modifications to segment A in M-B. 

The most general product of this research effort was to offer an alternative purpose for 

experiments conventionally used to validate constitutive models. Although model validation is 

necessary, investing time “adjusting” model parameters to match observed results can always be 

accomplished, as long as the observed comes first (24). But the value of this investment will 

have small returns when applied at extreme ballistic environments. Instead, it is recommended to 

use model validation of an experiment as the baseline case for performing model parameter 

sensitivity, as illustrated in this report. Far more physical understanding of competing 

deformation mechanisms as described within physics-based constitutive models can be achieved 

that become relevant for uncovering protection defeat mechanisms at ballistic regimes. An 

additional relevant sensitivity analysis would involve altering the functional forms of terms 

described in a constitutive model for uncovering protection effectiveness as related to property 

performance. Together, this approach is an important ingredient toward achieving goals for 

protection-by-design involving material processing and fabrication to performance. 
32

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34

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