Christos C. Chamis

Glenn Research Center, Cleveland, Ohio

Stress Analysis and Fracture in

Nanolaminate Composites

NASA/TM—2008-214928

January 2008
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Christos C. Chamis

Glenn Research Center, Cleveland, Ohio

Stress Analysis and Fracture in

Nanolaminate Composites

NASA/TM—2008-214928

January 2008

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

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NASA/TM—2008-214928 1

Stress Analysis and Fracture in Nanolaminate Composites 

Christos C. Chamis 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Summary 

A stress analysis is performed on a nanolaminate subjected to bending. A composite mechanics 

computer code that is based on constituent properties and nanoelement formulation is used to evaluate the 

nanolaminate stresses. The results indicate that the computer code is sufficient for the analysis. The 

results also show that when a stress concentration is present, the nanolaminate stresses exceed their 

corresponding matrix-dominated strengths and the nanofiber fracture strength. 

Introduction 

A nanocomposite (nanolaminate) is analyzed that is subjected to a bending moment (see fig. 1). This 

type of composite was chosen to be investigated because (1) the bottom and top plies can each be divided 

into subplies, from which the magnitudes of the stresses can be evaluated and (2) there is not a 

comparable analysis in the literature. The top and bottom plies are each assumed to be made from one 

type of nanofiber composite, and the core is made from 10 aligned nanocomposites of the same 

composition. A schematic of the nanocomposite section is shown in figure 2; the sliced subplies are 

shown in figure 3. Note that the two plies are each sliced into 10 slices to obtain a detailed stress 

distribution of the maximum bending stresses in the section. The stress analysis includes the stress 

distribution through the laminate thickness as well as the stresses in the two substructured plies. In 

addition, the corresponding strains are also computed for completeness. The objective of this paper is to 

describe the stresses in a nanoelement of a unidirectional nanolaminate subjected to a bending moment 

along the x-direction. 

Nanocomposite Simulation Properties 

The nanofibers in this study are assumed to be Thornell T300 graphite fibers with a modulus of 

6985 GPa (1×109

 psi), a tensile strength of 800 000 psi (5.52 GPa), and a fiber diameter of 4.33×10–6 in. 

(110 nm). The matrix is assumed to be an intermediate-modulus, high-tensile-strength polymer matrix. 

The properties for these two constituents are available and presented in tables I and II. The 

nanocomposite is usually made with relatively low fiber volume ratio (about 0.05) (ref. 1). Entering this 

value and the values in tables I and II in the composite mechanics computer code (ICAN/JAVA) (ref. 2), 

the following properties are predicted: 

 (1) The fiber volume ratio, void volume ratio, and matrix volume ratio, shown in figure 4 

 (2) The laminate configuration (ply thickness, fiber misalignment, and laminate thickness measured 

from the bottom ply), shown in figure 5 

 (3) The laminate thermal expansion coefficients, shown in figure 6 

 (4) The laminate fiber-dominated longitudinal modulus, shown in figure 7 

 (5) The matrix-dominated transverse moduli, shown in figure 8 

 (6) The corresponding shear moduli, shown in figure 9 

 (7) The corresponding Poisson’s ratios, shown in figure 10 
NASA/TM—2008-214928 2

The significance of the laminate properties shown in figures 7 to 10 are that (1) the horizontal spikes 

show that the two substructure plies represent the thickness averages. These two plies have 1/10 the 

average thickness and (2) they illustrate the discontinuity that exists between the core and the two end 

plies. Note that in figure 6, the matrix-dominated thermal expansion coefficients CTE22 and CTE33 at 

the spikes are about 4 times the fiber-dominated CTE11. This type of discontinuity is also apparent in 

figures 7 to 9. The discontinuity effects are somewhat different in figure 10. In this figure the fiberdominated Poisson’s ratio NUL21 has the opposite effect than the Poisson’s ratio NUL23, which is 

matrix dominated. 

The laminate fiber-dominated tensile and compressive strengths (LSC11T and LSC11C, respectively) 

are shown in figure 11. This figure shows the same trend in the two end plies for LSC11T and LSC11C at 

the spikes that was described above for the thermal expansion coefficients. The spike effects are not as 

pronounced in figure 12, which shows the matrix-dominated strengths. It is worth commenting that the 

results described thus far are independent of loading. 

The strains through the laminate thickness are shown in figure 13. In this figure, the two major 

assumptions for bending deformation through the thickness are clearly shown: (1) the line of bending 

strains due to the bending moment (orange) is linearly varying and (2) the line of Poisson strains (green) 

is also linearly varying, but having an opposite slope as it should have. However, the line of shear strains 

(blue) is “0” as it should be since there is in-plane shear loading. 

The corresponding stresses are shown in figure 14. Note the two spikes indicating that this laminate 

behaves more like a sandwich where the core stresses are negligible (fig. 15) compared to the stresses in 

the two end plies where the stresses are of opposite sign. Note also that the transverse ply stress (SIG22) 

and the in-plane shear stress (SIG12) are near “0” as they should be since there are no loads in these two 

directions. The laminate combined stress failure criteria (modified distortion energy (ref. 3) and the Hill 

criterion (ref. 4)) are plotted in figure 16. The results in this plot show no fracture on the laminate scale. 

These local stress concentration factors are shown in figure 17 around a small hole in the laminate 

(from ref. 5, see the appendix). The plot in figure 17 illustrates that the stress concentration factors reach a 

maximum at 90° and 270° from the load direction for +Kxx and Kxy. The Kyy stress reaches maximum 

for 0°, 180°, and back to 360°. Note that the maxima for Kxx and Kxy are equal in magnitude, with Kxy 

having a sharp discontinuity at 90° and 270°; that is, Kxy jumps from a negative value to an equal 

positive value within a few degrees (about 4°). Even if the stresses in figures 14 and 15 are multiplied by 

20 (the maximum stress concentration factor from fig. 17), the laminate stresses still do not cause any 

fracture based upon the corresponding strengths in figures 11 and 12. 

Stresses in a Nanoelement (Nanostresses) and Fracture 

The longitudinal and transverse flexural stresses from bending SF1L and SF1T in a nanofiber are 

plotted in figure 18. The range for these stresses is –200 000 to 200 000 psi (–1.38 to 1.38 GPa). Upon 

multiplying these stresses with 20 (the stress concentration factor), the stresses becomes 4 000 000 psi 

(27.6 GPa), which is 5 times the fiber tensile strength of 800 000 psi (5.52 GPa) and 6.7 times the 

compressive strength 600 000 psi (4.14 GPa) in table I. Therefore, the nanofibers will start breaking all 

across the bottom and top nanoplies. 

Figure 19 plots the corresponding longitudinal and transverse stresses in the matrix in a nanoelement. 

These stresses multiplied by the stress intensity factor will not cause any damage in the nanoscale matrix 

element. The plot for the nanoelement matrix-dominated stresses SM2BL and SM2BT shown in 

figure 20. As can be seen in this plot, when SM2BL is multiplied by the maximum stress concentration 

factor (20) from figure 17, it is about 800 000 psi (5.52 GPa), and the matrix will definitely crack. Even 

without the multiplication of the stress concentration factor, the nanomatrix in the nanoelement of 

figure 20 will fracture. 
NASA/TM—2008-214928 3

The normal and shear nanostresses are plotted in figure 21. As can be seen in this nanoelement, both 

the normal and the shearing stresses will fracture the matrix without the effects of stress concentration 

factors. 

The important conclusions from the above discussion are that (1) the matrix will crack in 

nanoelements even without stress concentration effects and (2) the fibers will fracture in the nanoelement 

in the presence of stress concentration. 

Concluding Remarks 

The concluding remarks from an investigation on the stresses in a unidirectional nanolaminate 

(composite) are the following: 

1. The composite mechanics incorporated in a composite mechanics computer code based on 

constituent materials and a nanoscale element is sufficient for stress analysis of a nanocomposite laminate 

in bending. 

2. The matrix-dominated stresses will fracture the matrix in the nanoelement even without stress 

concentration or residual stresses. 

3. The fibers in a nanoelement will fracture when the stress concentration is accounted for. 

4. Nanolaminate geometric and material properties are not affected by loading conditions in linear 

mechanics nanolaminate theory described herein. 

5. The bending stresses in a nanolaminate behave the same as a sandwich where the top and bottom 

nanoplies exhibit a substantial increase in stress from the core material. 

6. It is important to evaluate the stresses in a nanoelement even when the nanolaminate composites do 

not show any stresses reaching their limits based upon their corresponding strengths. 

References 

1. Chamis, Christos C.; Handler, Louis M.; and Manderscheid, Jane M.: Composite Nanomechanics: A 

Mechanistic Properties Prediction. NASA TM—2007-214673, 2007. 

2. Handler, Louis M.; and Chamis, Christos C.: Integrated Composite Analyzer (ICAN/JAVA). 

http://www.grc.nasa.gov/WWW/StructMech/ICAN/indes.html Accessed Aug. 7, 2007. 

3. Chamis, C.C.: Failure Criteria for Filamentary Composites. NASA TN−D–5367, 1969. 

4. Hoffman, Oscar: The Brittle Strength of Orthotropic Materials. J. Compos. Mater., vol. 1, 1967, 

pp. 200–206. 

5. Lekhnitskii, S.C.: Anisotropic Plates. Gordon and Breach, New York, NY, 1968. Translation. 

(Previously cited in Anizotropnye Plastinki.) 
NASA/TM—2008-214928 4

Appendix 

Stress Concentration Around a Small Hole in Composite Plates 

The stress concentration around a small hole in composite plates (see fig. 22) is evaluated by the 

equations from reference 5: 

[ ] ( ) [( ) ]

( ) ⎭

⎬

⎫ − + + φ θ

⎩

⎨

⎧ = + φ − φ θ + + φ − φ θ

σ

σ θθ θθ

1 sin 2 sin 2

4

sin cos cos 1 cos sin sin 2 2 2 2 2 2

o DI

DI

o o DI DI o

cxx

c

cxx

c

R R R

R R R R R

E

E

 (1) 

Ro Ecxx Ecyy = / (2) 

1/ 2

2

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = − ν

cxy

cxx cxy

cyy

cxx DI G

E

E

E R (3) 

= ( ) () + + { } − φ θ + [ ] + θ − φ

σ

σ θθ θθ 1 cos2 sin 2 1 cos2 1 sin 2

2 o DI DI o o

cxx

o

cxy

c R R R R R

E

E (4) 

where 

σcxx is the composite stress along the x-axis 

σcθθ is the composite stress along the θ-axis 

Ecxx is the modulus along the composite x-axis

Ecθθ is the modulus along the angle θ, which varies from 0° to 360°

Ro is defined by equation (2) 

RD1 is defined by equation (3) 

θ is the angle at which the stress concentration is evaluated 

φ is the angle that the principal plate axes are located with respect to the loading x-axis 

νcxy is Poisson’s ratio when stress is applied in the x-direction 

Gcxy is the shear modulus in the x,y-plane 

Note there are no strengths in the above equations, only stiffnesses. 
NASA/TM—2008-214928 5

TABLE I.—T300 GRAPHITE FIBER PROPERTIES 

Propertya Symbolb Value Unitsc

Number of fibers per end Nf 1.0 Number 

Filament equivalent diameter df 2.756×10−6 in. 

Weight density Rhof .064 lb/in.3

Normal modulus (11) Ef11 1.0×109 psi 

Normal modulus (22) Ef22 7.0×107 psi 

Poisson’s ratio (12) Nuf12 0.20 Nondimensional 

Poisson’s ratio (23) Nuf23 .25 Nondimensional 

Shear modulus (12) Gf12 5.0×107 psi 

Shear modulus (23) Gf23 3.5×107 psi 

Thermal expansion coefficient (11) A1faf11 −5.5×10−7

 in./in./°F 

Thermal expansion coefficient (22) Alfaf22 5.6×10−6

 in./in./°F 

Heat conductivity (11) Kf11 444.0 Btu/hr/ft2

/°F/in. 

Heat conductivity (22) Kf22 4.0 Btu/hr/ft2

/°F/in. 

Heat capacity Cf 0.22 Btu/lb/°F 

Dielectric strength (11) KeF11 0.0 V/in. 

Dielectric strength (22) Kef22 0.0 V/in. 

Dielectric constant (11) Gamma11 0.0 in./V 

Dielectric constant (22) Gamma22 0.0 in./V 

Capacitance Cef 0.0 V 

Resistivity Ref 0.0 Ω-in. 

Tensile strength SfT 8.0×105 psi 

Compressive strength SfC 6.0×105 psi 

Shear strength SfS 4.0×105 psi 

Normal damping capacity (11) psi11f 0.38 % energy 

Normal damping capacity (22) psi22f 6.3 % energy 

Shear damping capacity (12) psi12f 3.34 % energy 

Shear damping capacity (23) psi23f 6.3 % energy 

Melting temperature TMf 6000.0 °F a

Numbers in parenthesis correspond to axis directions defined in figure 1(b). 

b

Symbols used in computer code ICAN/JAVA. 

c

Conversions: 1 in. = 2.54 cm, 1 psi = 6.9 Pa, 1 in./in./°F = 9/5 cm/cm/°C, 1 btu = 1055 J, 

1 lb = 4.448 N. 
NASA/TM—2008-214928 6

TABLE II.—POLYMER MATRIX PROPERTIES 

Property Symbola Value Unitsb 

Weight density Rhom 0.044 lb/in.3

Normal modulus Em 500000.0 psi 

Poisson’s ratio Num .35 Nondimensional 

Thermal expansion coefficient Alfa m 3.6×10−5

 in./in./°F 

Heat conductivity Km 0.008681 Btu/hr/ft2

/°F/in. 

Heat capacity cm .25 Btu/lb/°F 

Dielectric strength Kem 0.0 V/in. 

Dielectric constant Gammam 0.0 in./V 

Capacitance Cem 0.0 V 

Resistivity Rem 0.0 Ω-in. 

Moisture expansion coefficient Betam .0033 in./in./% moisture 

Diffusivity Dm 2.16×10−7 in.2/hr 

Saturation Mm 0.0 % moisture 

Tensile strength SmT 15000.0 psi 

Compressive strength SmC 35000.0 psi 

Shear strength SmS 13000.0 psi 

Allowable tensile strain eps mT 0.02 in./in. 

Allowable compression strain eps mC .05 in./in. 

Allowable shear strain eps mS .035 in./in. 

Allowable torsional strain eps mTOR .035 in./in. 

Normal damping capacity psiNM 6.6 % energy 

Shear damping capacity psiSm 6.9 % energy 

Void heat conductivity Kv .0012 Btu/hr/in./°F 

Glass transition temperature Tgdr 420.0 °F 

Melting temperature TMm 0.0 °F a

Symbols used in computer code ICAN/JAVA. a

Conversions: 1 in. = 2.54 cm, 1 psi = 6.9 Pa, 1 in./in./°F = 9/5 cm/cm/°C, 1 btu = 1055 J, 

 1 lb = 4.448 N. 
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14. ABSTRACT

A stress analysis is performed on a nanolaminate subjected to bending. A composite mechanics computer code that is based on constituent 

properties and nanoelement formulation is used to evaluate the nanolaminate stresses. The results indicate that the computer code is 

sufficient for the analysis. The results also show that when a stress concentration is present, the nanolaminate stresses exceed their 

corresponding matrix-dominated strengths and the nanofiber fracture strength. 

15. SUBJECT TERMS

Nanofiber properties; Nanoelements; Nanostresses and strains; Cantilever bending 

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