Christos C. Chamis

Glenn Research Center, Cleveland, Ohio

Probabilistic Simulation of Combined

Thermo-Mechanical Cyclic Fatigue

in Composites

NASA/TM—2011-217094

August 2011
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Christos C. Chamis

Glenn Research Center, Cleveland, Ohio

Probabilistic Simulation of Combined

Thermo-Mechanical Cyclic Fatigue

in Composites

NASA/TM—2011-217094

August 2011

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

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NASA/TM—2011-217094 1 

Probabilistic Simulation of Combined Thermo-Mechanical 

Cyclic Fatigue in Composites 

Christos C. Chamis 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135

Abstract 

A methodology to compute probabilistically-combined thermo-mechanical fatigue life of polymer 

matrix laminated composites has been developed and is demonstrated. Matrix degradation effects caused 

by long-term environmental exposure and mechanical/thermal cyclic loads are accounted for in the 

simulation process. A unified time-temperature-stress-dependent multifactor-interaction relationship 

developed at NASA Glenn Research Center has been used to model the degradation/aging of material 

properties due to cyclic loads. The fast probability-integration method is used to compute probabilistic 

distribution of response. Sensitivities of fatigue life reliability to uncertainties in the primitive random 

variables (e.g., constituent properties, fiber volume ratio, void volume ratio, ply thickness, etc.) computed 

and their significance in the reliability-based design for maximum life is discussed. The effect of variation 

in the thermal cyclic loads on the fatigue reliability for a (0/45/90)s graphite/epoxy laminate with a ply 

thickness of 0.127 mm, with respect to impending failure modes has been studied. The results show that, 

at low mechanical-cyclic loads and low thermal-cyclic amplitudes, fatigue life for 0.999 reliability is most 

sensitive to matrix compressive strength, matrix modulus, thermal expansion coefficient, and ply 

thickness. Whereas at high mechanical-cyclic loads and high thermal-cyclic amplitudes, fatigue life at 

0.999 reliability is more sensitive to the shear strength of matrix, longitudinal fiber modulus, matrix 

modulus, and ply thickness. 

Nomenclature 

Ef11,22,23 fiber normal modulus 

Em matrix normal modulus 

Gf12,13,23 fiber shear modulus 

kf actual fiber volume ratio 

kv apparent void volume ratio 

l,m,n,p,q exponents for multifactor interaction equation 

MP material property 

MPO reference material property 

N number of effects causing material degradation 

Nm number of mechanical cycles 

Nmf final number of mechanical cycles 

NT number of thermal cycles 

NTf final number of thermal cycles 

p coefficient of variation 

r applied stress to laminate ratio 

S final strength 

Sf fiber strength 
NASA/TM—2011-217094 2 

SfC fiber compressive strength 

SfT fiber tensile strength 

SmC matrix compressive strength 

SmS matrix shear strength 

SmT matrix tensile strength 

T temperature 

Tgd dry glass transition temperature 

Tgw wet glass transition temperature 

To initial temperature 

t current time 

tf final time 

t1 ply thickness 

V primitive variable for material or load (e.g., temperature, stress, mechanical cycles, etc.) 

VF value of primitive variable at final condition 

VO value of primitive variable at reference condition 

m matrix thermal expansion coefficient 

μ mean value 

νf12,13,23 fiber Poisson’s ratio 

νm12,13,23 matrix Poisson’s ratio 

σ stress 

σM mechanical cyclic stress amplitude 

σT thermal cyclic stress amplitude 

Introduction 

The aeronautics industry is in continuing pursuit of lightweight structural concepts for both airframe 

primary structures and engine structural components. One of the prime issues for these structural 

components is assured long-term behavior with a specified reliability. These objectives are being 

addressed through the use of accelerated testing strategies as well as computational simulation 

methodologies for predicting long-term durability of structural components. The present article addresses 

issues pertaining to probabilistic fatigue life of polymer matrix composites under a combined thermaland mechanical-cyclic loading through the use of some typical examples. 

The complexity of predicting composite behavior is compounded by the presence of multiple scales 

that must be considered in the simulation. These are the micro-level scale where fiber, matrix, and 

interface participate, the macro-level where individual plies take part, and the laminate level where the 

ply configuration becomes important. Additionally, the fabrication process can induce variations, there 

are inherent uncertainties in the constituent properties, and the cyclic variations in an aggressive loading 

environment all contribute to the ever increasing complexity in the computational simulation. Successful 

inclusion of the lighter high-temperature composites for future aircraft hinges largely upon the ability to 

predict and assure their long-term behavior. Current practices depend largely on a building block 

approach of conducting long-term testing of materials in the form of coupons up through components. 

However, this is time consuming and expensive to conduct a large number of long-term tests to capture 

the effects of all the design variables and loads. Furthermore, accounting for the previously discussed 

uncertainties in the experiments is a monetarily unfeasible proposition. Therefore in view of these facts, 

realization of the full potential of lightweight composites is more likely to be achieved with the real 

innovative computational approaches capable of handling these aspects in an integrated manner together, 

with some accelerated and strategically-chosen experiments to minimize the testing costs. Also such 
NASA/TM—2011-217094 3 

computational procedures provide a priori insight to identify critical experiments to be performed as well 

as reduce the number of tests to be conducted to an optimum number. 

Traditional computational approaches are deterministic in nature; they do not account for 

uncertainties associated with composite structures and materials. The focus of ongoing research at the 

NASA Glenn Research Center has been to develop advanced integrated computational methods and 

related computer codes to perform a complete probabilistic assessment of composite structures. These 

methods account for uncertainties in all the constituent properties, fabrication process variables, and loads 

to predict probabilistic micro, ply, laminate, and structural responses. These methods have already been 

implemented in the Integrated Probabilistic Assessment of Composite Structures (IPACS) (Ref. 1) 

computer code. 

The objective of this article is to describe a computational simulation methodology to predict the fatigue 

life reliability and probabilistic long-term behavior in polymer matrix composites. Also, the implementation 

of the methodology in the ICAN computer code (Ref. 2) has been described. A unified time, stress, and 

load-dependent multifactor-interaction equation (MFIE) model (Ref. 3) developed at the NASA Glenn 

Research Center has been used to simulate the long term behavior of polymer matrix composites. 

To illustrate the application of the methodology, a typical composite system consisting of graphite 

fibers in epoxy matrix is chosen. The methodology can be applied to other types of polymer matrix 

composites as well. The layup of the composite system is chosen as (0/45/90)s. The cumulative 

probability distribution functions (CDF’s) for combined thermal mechanical fatigue is computed. The 

fatigue life cycles are computed for different applied stress to laminate strength ratios, r. The laminate 

strength is evaluated based on first ply failure criteria (hereinafter referred to as laminate strength). First 

ply failure criteria assumes that the laminate has failed when any stress component in a ply exceeds its 

respective allowable limit. By using these CDF’s a fatigue life cycle curve for a reliability of 0.999 is 

obtained in order to demonstrate how this methodology can be used to aid the design process. Also, the 

sensitivity of fatigue life reliability to the primitive random variables are computed for a reliability of 

0.999. It is noted that the computer codes are continuously updated, but the concepts remain the same as 

described in the initial references. 

Computational Simulation 

Simulation of uncertainties in the primitive random variables throughout the load history requires 

fundamental governing field equations in time variable. The probabilistic simulation process in this article 

uses time dependent MFIE model, micromechanics theory, and the fast probability integration (FPI) 

theory 9 (Ref. 4). Uncertainties in the primitive variables related to physical process fabrication, 

constituent properties, and ply geometry are included in the simulation. Micromechanics theory available 

in the ICAN computer code has been used together with the MFIE to include the degradation of 

constituent properties resulting from aggressive load effects for the simulation of long term effects. A 

generic form of the previous model is given by 







 







 

N

i 1

a

F O i

F

PO

P

V V

V V

M

M (1)

The terms in parenthesis represent a physical effect that causes the degradation of material property. The 

exponent accounts for the amount of degradation due to a particular effect. More details on this model are 

given later in this article. The following sections describe the developed/modified computer codes and 

procedures used in the computational simulation of probabilistic long-term behavior in polymer matrix 

composites subjected to combined thermo-mechanical fatigue. 
NASA/TM—2011-217094 4 

Integrated Composite Analyzer (ICAN) 

The ICAN code computationally simulates the material behavior of polymer matrix composites from 

fiber/matrix constituents to the laminate scale including fabrication effects (Fig. 1). ICAN uses composite 

micromechanics and laminate analysis based on linear elastic theory to compute constituent, ply, and 

laminate level properties required for the global structural analysis (left side of Fig. 1). ICAN also 

decomposes the global structural response to laminate, ply, and constituent response levels (stress and 

strains) which helps the user to evaluate failure (right side of Fig. 1). Details of the ICAN computer code are 

given in Reference 2. ICAN also performs the failure analysis based on different failure criteria such as firstply failure criteria, fiber break criteria, modified distortion energy criterion, and Hoffman’s criteria. 

The ICAN computer code was modified to implement the time-dependent MFIE and perform 

sensitivity evaluation for primitive random variables. The time variable in the MFIE is divided into a 

smaller number of time steps to simulate long-term effects. At every time step the degradation magnitude 

of the matrix properties, which are used in micro-mechanics and laminate analysis, is evaluated. Response 

variable sensitivity to the random variables at every time step are also performed to compute the 

respective scatter in response variables. Also, at every time step a failure analysis based on first-ply 

failure and fiber break criteria is performed to determine whether the laminate can take any further load. 

Failure analysis determines the possible failure modes and maximum load capacity in the respective 

failure mode. The analysis stops when the laminate is incapable of carrying any more load. In addition, a 

data base is created to store results required to perform probabilistic life analysis and develop reliabilitybased life assessment of polymer matrix composites. Thus, the current version of ICAN is capable of 

performing probabilistic time variable analysis to incorporate material degradation due to combined 

thermal and mechanical cyclic loads. 

Figure 1.—Schematic of ICAN computer code. 
NASA/TM—2011-217094 5 

Time Dependent Multi-Factor Interaction Equation (MFIE) Model 

It is known that predicting the behavior of composite materials is a complex task. Accounting for all 

the physical effects and how they affect the material properties in the time domain is even more complex. 

Over the years, research in developing a unified law describing the material behavior driven by primitive 

variables has been an ongoing activity at NASA Glenn Research Center. The result of this research is the 

development of a unified MFIE model (Ref. 3).Concepts used in Reference 3 have been expanded to 

include the time-dependent degradation effect on material behavior because of environmental, fabrication, 

and load effects (Ref. 5). The modified form of the equation including time effect in expanded form is 

r

f

q

f Tf

T T

p

f mf

M m

n

f f

T

m

f

1

gd o

gw

PO

P

t

t 1

S N

N 1

S N

N 1

S t

1

S

1

T T

T T

M

M















  













   













   













   













   

















  (2) 

Each term in parenthesis accounts for a specific physical effect. Any number of effects can be included in 

one single equation as seen by the nature of the equation. The exponents are determined from the 

available experimental data or estimated from the anticipated material behavior due to a particular 

primitive variable. Each primitive variable and the exponent in the previous equation can be random with 

a statistical distribution. The insufficiency of a set of experimental data can be taken into account by 

means of uncertainties in the exponent. 

An important part of the previous model is the fact that only one equation can include all the effects 

with any nonlinearity in the material behavior and follow the physics of behavior. It can describe all of 

the interacting effects of different variables (thermal, metallurgical, mechanical, chemical, and load). 

Since the variables used are at a primitive level, it simulates the in situ degradation in material properties 

caused by applied cyclic and environmental effects. 

Probabilistic Simulation 

An advanced first-order second-moment fast probability integration (FPI) technique (Ref. 4) is used 

to compute cumulative-probability distribution function of the fatigue life. Results of the random variable 

perturbations are used to compute the CDF of responses. The FPI technique is very efficient compared to 

the Monte Carlo simulation technique. FPI does not generate random samples; however, it does use a 

numerical integration technique to compute the joint probability density functions and probability of 

failures. It transforms the physical random variable space system into unit normal space to perform 

probability integration easily and more accurately. On the other hand, the Monte Carlo method generates 

a large number of random samples to compute the CDF of response. Hence, FPI is computationally more 

economical than Monte Carlo. The sensitivity of responses is also computed by FPI. Sensitivity 

information helps improve the design and quality. 

Simulation Cases, Results, and Discussion 

Probabilistic simulation of long-term behavior under mechanical cyclic loads only was demonstrated 

in Reference 5. In this report the methodology has been extended to simulate probabilistic long-term 

behavior and fatigue reliability under combined thermo-mechanical cyclic loads. Probabilistic behavior of 

a graphite/epoxy laminate as an illustrative example is studied. The deterministic long-term behavioral 

aspects for polymer matrix composites are discussed in Reference 6. A (0/45/90)s laminate made of 

graphite fibers and epoxy matrix is subjected to uniaxial mechanical tensile and thermal cyclic load as 

shown in Figure 2. The load shown in Figure 2 is sinusoidal. However, any cyclic load shape such as 

triangular, square, and so forth can be used by accounting for this through the exponent value in MFIE. 

Each ply has a thickness of 0.127 mm. Typical mean values, scatter, and probability distribution of all the 

fiber, matrix properties, and fabrication variables are taken as those given in Table 1 (Ref. 2). 
NASA/TM—2011-217094 6 

Figure 2.—Description of thermo-mechanical cyclic load. 

TABLE 1.—UNCERTAINTIES IN THE CONSTITUENT PROPERTIES 

[Scatter, 5 percent; nominal distribution.] 

Property Mean 

Fiber 

Normal modulus, Ef11, GPa 213.7 

Normal modulus, Ef22, GPa 13.8 

Poisson’s ratio, f12 0.20 

Poisson’s ratio, f23 0.25 

Shear modulus, Gf12, GPa 13.8 

Shear modulus, Gf23, GPa 6.9 

Tensile strength, SfT, GPa 2.8 

Compressive strength, SfC, GPa 2.8 

Matrix 

Normal modulus, Em, GPa 3.4 

Poisson’s ratio, m12 0.35 

Tensile strength, SmT, GPa 0.1 

Compressive strength, SmC, GPa 0.24 

Shear strength, SmS, GPa 0.09 

Fabrication variables 

Fiber volume ratio, kf, percent 60 

Void volume ratio, kv, percent 2 

Ply thickness, t1, mm 0.127 

Normal distribution for the primitive variables has been assumed in the present study. However, any other 

type of distribution for these variables can be used in the described methodology. Initially, the laminate 

was subjected to a mean static load, and failure analysis was performed to evaluate static laminate 

strength. It was found to be 1184.0 kg per unit cm width based on the first ply failure criteria. Several 

simulation cases, with different combinations of mechanical and thermal load magnitudes, given in 

Table 2 were analyzed, and the CDF’s and sensitivity evaluations were performed. The mechanical load 

is applied as a ratio, r, defined as applied mechanical stress over static laminate strength. These load cases 

are divided into three categories. The first category includes only the mechanical cyclic load (load cases 1 

to 3 in Table 2). However, the category 2 load cases are only for thermal cyclic loads (load cases 4 to 7 in 

Table 2). In the category 3 load cases, the mechanical cyclic loads are combined with the thermal cyclic 

loads of different amplitudes such as 0, 15.6, 23.9, 37.8, and 51.7 °C (load cases 8 to 22 in Table 2). The 

purpose of having these categories is to illustrate the combined effect of thermal and mechanical cyclic 

loads on the fatigue life reliability. 

 
NASA/TM—2011-217094 7 

TABLE 2.—COMPUTATIONAL LOAD CASES AND RELIABILITY-BASED FATIGUE LIFE 

Case Mechanical 

cyclic load 

ratio, 

r 

Mean 

temperature, 

C 

Cyclic 

temperature 

amplitude, 

C 

Mean fatigue 

lifeNmf

Scatter, 

percent 

0.999 

Reliability 

lifeNmf

1 0.6 21.1 0.0 0.8792 14.40 0.4909 

2 0.7 21.1 0.0 0.3956 25.65 0.0934 

3 0.8 21.1 0.0 0.1609 32.3 0.0068 

4 0.0 65.6 15.6 0.9080 6.47 0.7318 

5 0.0 65.6 23.9 0.8623 6.91 0.6788 

6 0.0 65.6 37.8 0.7909 6.57 0.6287 

7 0.0 65.6 51.7 0.7254 6.73 0.5758 

8 0.6 65.6 0.0 0.9045 5.1 0.7616 

9 0.7 65.6 0.0 0.7785 5.42 0.6486 

10 0.8 65.6 0.0 0.6423 6.5 0.5118 

11 0.6 65.6 15.6 0.9278 5.6 0.7529 

12 0.7 65.6 15.6 0.8076 5.26 0.6744 

13 0.8 65.6 15.6 0.6736 4.03 0.5516 

14 0.6 65.6 23.9 0.8980 6.63 0.7126 

15 0.7 65.6 23.9 0.8231 5.51 0.6764 

16 0.8 65.6 23.9 0.6898 6.57 0.5433 

17 0.7 65.6 37.8 0.8186 5.45 0.6610 

18 0.8 65.6 37.8 0.6949 6.0 0.5590 

19 0.9 65.6 37.8 0.4009 44.0 –0.1154 

20 0.7 65.6 51.7 0.7709 6.72 0.6143 

21 0.8 65.6 51.7 0.7047 6.15 0.5710 

22 0.9 65.6 51.7 0.3984 44.0 –0.1159 

In each load case simulation, deterministic sensitivity evaluation of the fatigue life to each random 

variable was performed using perturbation analysis (Ref. 7). Perturbation analysis means evaluating the 

change in the responses if the value of a particular random variable is changed by a given magnitude of 

standard deviation of that variable. From each perturbation analysis, the fatigue life for each ply was 

computed based upon the first-ply failure criteria. The shortest life of a ply in the laminate was considered 

to be the life of a laminate for that perturbation. Perturbation analysis is performed to develop a response 

surface which is required for the probabilistic analysis. By using these perturbation analysis results and 

probability distributions of random variables, the reliability analysis was performed using the FPI method. 

The CDF’s and sensitivity factors for the first three load cases are summarized in Figure 3. It shows the 

CDF of fatigue life caused by mechanical cyclic loads only. The fatigue life in these three cases was 

primarily governed by failure in 90 ply because the transverse tensile stress was greater than the 

corresponding strength. Thus, the life under uniaxial tensile cyclic loads of that laminate is governed by 

the transverse tension failure in 90 plies. Figure 3 also shows that the fatigue life CDF shift to the left 

means that the life under mechanical cyclic loads drops rapidly as the amplitude of applied cyclic stress 

increases. At load ratios, r, equal to 60, 70, and 80 percent, the respective mean fatigue life, , was 

computed to be 88, 40, and 16 percent of the “endurance limit, Nmf.” The corresponding scatter, p, with 

respect to mean value is 14.4, 26, and 32 percent. Also, the respective fatigue life for 0.999 reliability is 

49, 9.3, and 0.6 percent of Nmf as shown in Figure 4 (S-N curve) and Table 2. Thus, the fatigue life drops 

rapidly and the p increases. Since ply transverse tension is the governing failure mode, the fatigue life for 

0.999 reliability is sensitive to the SmT, ply thickness, E, Ef11, kf, and k. Note that the magnitude of 

sensitivity for a particular random variable almost remains the same with respect to different values of r. 

This means that the scatter in these variables should be minimized or controlled in order to enhance the 

fatigue life reliability under mechanical loads. 
NASA/TM—2011-217094 8 

Figure 3.—Cumulative distribution function for and sensitivity of fatigue life for 0.999 reliability of (0/±45/90)s 

graphite/epoxy laminate. Ply thickness, 0.127 mm under mechanical cyclic load; mean mechanical load = 

0.5×static strength. (a) Cumulative distribution function. (b) Sensitivity of fatigue life for 0.999 reliability. 

Figure 4.—Fatigue life variation for 0.999 reliability of (0/±45/90)s 

graphite/epoxy laminate. Ply thickness, 0.127 mm under 

mechanical cyclic load. Mean mechanical load = 0.5×static 

strength. 

Figure 5 depicts the CDF’s and random variable sensitivity for the fatigue life caused by variation in the 

cyclic temperature amplitude (load cases 4 to 7 and Table 2). Fatigue life caused by thermal cyclic loads in 

the laminate under consideration is mainly governed by the transverse compression failure in 90 and 45

plies. It is seen from this figure that the fatigue life variation is not as severe due to variation in cyclic 

temperature amplitude as was the case for mechanical load amplitude. The computed mean fatigue life, m, 

for temperature amplitudes of 15.6, 23.9, 37.8, and 51.7 C is 91, 86, 79, and 73 percent of Nmf, 

respectively, whereas the scatter, p, is 6.47, 6.91, 6.57, and 6.73 percent, respectively. Also, the respective 

fatigue life for 0.999 reliability is 73, 68, 63, and 58 percent (Fig. 6 and Table 2). Thermal loads induce 

compressive stresses in the transverse direction of both 45 and 90 plies. Since allowable ply compressive 

strength is higher than the tensile strength, a higher life is expected under thermal loads. Also, as seen from 

Figure 5, E, α, SmC, kf, SmS, Tgd, ply thickness, and k govern the fatigue life for 0.999 reliability. 

Again, the magnitude of sensitivity for different thermal cyclic load amplitudes does not change 

much. The sensitivity trend of fatigue life to SmC is downward with the increase in the thermal load 

amplitude which is indicative of other eminent failure modes at very high thermal amplitudes. It is 

interesting to note that the scatter, p, remains the same at all amplitude levels. 
NASA/TM—2011-217094 9 

Figure 5.—Cumulative distribution function for and sensitivity of fatigue life for 0.999 reliability of (0/±45/90)s 

graphite/epoxy laminate. Ply thickness, 0.127 mm under thermal cyclic load; mean mechanical load = 

0.5×static strength. (a) Cumulative distribution function. (b) Sensitivity of fatigue life for 0.999 reliability. 

Figure 6.—Fatigue life variation for 0.999 reliability of 

(0/±45/90)s graphite/epoxy laminate. Ply thickness, 0.127 

mm under thermal cyclic load. Mean mechanical load = 

0.5×static strength. 

Figures 7 to 11 are CDF’s and sensitivity factors for the fatigue life under combined thermomechanical loading with cyclic temperature amplitudes of 0, 15.6, 33.9, 37.8, and 51.7 °C, respectively. 

Figure 12 shows the fatigue life design curves for 0.999 reliability. The mean fatigue life, scatter, and the 

life for 0.999 reliability for these load cases are given in Table 2. It is seen from Figures 7 and 8 that the 

fatigue life CDF shifts to the left in proportion to the applied load ratio, r, at low thermal amplitudes of up 

to 23.8 °C which means that the probability distribution of fatigue life does not change much. This is due 

to the fact that the failure mode up to 23.9 °C cyclic amplitude remains the same. This results in linear 

variation of the mean, scatter, and 0.999 reliability value of fatigue life (Fig. 12). The variables that 

control the fatigue life magnitude at 0.999 reliability are ply thickness, SmC, SmS, E, α, Tgd, kf, k, and SmT. 

Ply thickness is the important variable in these cases caused by the mechanical load effects. 
NASA/TM—2011-217094 10 

Figure 7.—Cumulative distribution function for and sensitivity of fatigue life for 0.999 reliability of (0/±45/90)s 

graphite/epoxy laminate. Ply thickness, 0.127 mm under thermo-mechanical cyclic load; mean temperature, 

65.6 °C; amplitude temperature, 0 °C. (a) Cumulative distribution function. (b) Sensitivity of fatigue life for 

0.999 reliability. 

Figure 8.—Cumulative distribution function for and sensitivity of fatigue life for 0.999 reliability of (0/±45/90)s 

graphite/epoxy laminate. Ply thickness, 0.127 mm under thermo-mechanical cyclic load; mean temperature, 

65.6 °C; amplitude temperature, 15.6 °C. (a) Cumulative distribution function. (b) Sensitivity of fatigue life for 

0.999 reliability. 

Figure 9.—Cumulative distribution function for and sensitivity of fatigue life for 0.999 reliability of (0/±45/90)s 

graphite/epoxy laminate. Ply thickness, 0.127 mm under thermo-mechanical cyclic load; mean temperature, 

65.6 °C; amplitude temperature, 23.9 °C. (a) Cumulative distribution function. (b) Sensitivity of fatigue life for 

0.999 reliability. 
NASA/TM—2011-217094 11 

Figure 10.—Cumulative distribution function for and sensitivity of fatigue life for 0.999 reliability of (0/±45/90)s 

graphite/epoxy laminate. Ply thickness, 0.127 mm under thermo-mechanical cyclic load; mean temperature, 

65.6 °C; amplitude temperature, 37.8 °C. (a) Cumulative distribution function. (b) Sensitivity of fatigue life for 

0.999 reliability. 

Figure 11.—Cumulative distribution function for and sensitivity of fatigue life for 0.999 reliability of (0/±45/90)s 

graphite/epoxy laminate. Ply thickness, 0.127 mm under thermo-mechanical cyclic load; mean temperature, 

65.6 °C; amplitude temperature, 51.7 °C. (a) Cumulative distribution function. (b) Sensitivity of fatigue life for 

0.999 reliability. 

As the cyclic thermal amplitude increases, the shift in fatigue life CDF’s is not in proportion to r as 

seen from Figures 9 to 11. Thus, the trend in variation of m, p, and 0.999 reliability value of fatigue life 

gradually becomes nonlinear implying that the nature of probability distribution of fatigue life is not same 

at all load ratios. For amplitudes of 37.8 and 51.7 °C it becomes highly nonlinear (refer to Fig. 12 and 

Table 2). This is due to the fact that, at low-temperature amplitudes, the failure is mostly governed by the 

longitudinal compressive strength in a 90° ply whereas, at higher thermal amplitudes and low-load ratio, 

the governing failure mode shifts from 90° ply to 45° ply with compressive strength in the transverse 

direction being the limiting one. Also, at high-load ratio and high-temperature amplitudes, the failure 

mode is controlled by the shear strength in a 45° ply. Thus, when the shear mode failure controls, the drop 

in fatigue life is also steep and the scatter in fatigue life increases dramatically. 

As evidenced by sensitivity factors depicted in Figures 9 to 11, the fatigue life at 0.999 reliability at 

low-load ratio is dominated by SmC, E, α, kf, and SmS. However, at higher load ratios and thermal 

amplitudes, the fatigue life sensitivity to SmS, Ef11, and ply thickness increases whereas the others’ effect 

decreases. Thus, it is seen that the nature of failure mode governs the reliability and corresponding 

variables become sensitive to the life. Also, it is observed that the shear failure mode causes a large 
NASA/TM—2011-217094 12 

Figure 12.—Fatigue life variation for 0.999 reliability of 

(0/±45/90)s graphite/epoxy laminate. Ply thickness, 

0.127 mm under thermo-mechanical cyclic load; mean 

temperature, 65.6 °C. 

scatter in the fatigue life as well as reduction in the life for 0.999 reliability. It can be seen from Figure 12 

that for the laminate under consideration, the 0.999 reliability cannot be achieved for load ratio, r, of 0.9 

and thermal amplitudes of 37.8 and 51.7 °C. At this magnitude of reliability the fatigue life is negative in 

Figure 12, meaning the laminate has already failed at this reliability. 

The information in Figures 4, 6, and 12 can be used as a design aid to determine the component life for 

0.999 reliability as well as to make decisions on its continuing use. Similar design curves can be 

developed readily for any polymer matrix composite with any laminate configuration and for any load 

conditions. It is also important to mention here that, when the failure mode changes for different 

perturbations under same load conditions, one should iterate the probabilistic analysis to ensure the 

convergence of reliability using advanced first-order mean value analysis given in Reference 7. This is 

necessary because a small perturbation in the random variables could change the entire failure mechanism 

and the response surface becomes highly nonlinear which in turn affects the probability of failure. A 

reliability-based approach provides the level of reliability of a given design as well as identifies the 

controlling variables. Thus, it helps improve the manufacturing process, design experiments, and develop 

inspection guidelines. 

Conclusions 

A methodology to simulate the probabilistic fatigue life of polymer matrix composites under thermal 

and mechanical loads has been developed, implemented in the in-house computer code Integrated 

Composite Analyzer (ICAN), and demonstrated by the examples. The methodology incorporates the 

ICAN computer code, a generic time dependent multi-factor interaction equation model, and fast 

probability integration technique. Cumulative distribution functions of fatigue life cycles for a (0/45/90)s

graphite/epoxy laminate under uniaxial tensile cyclic loads and thermal cyclic loads were computed. 

Sensitivities of fatigue life to the random variables for 0.999 reliability were plotted. Fatigue life design 

curves for 0.999 reliability were generated for different combinations of thermal cyclic load amplitude 

and mechanical cyclic load. The importance of the failure mode on fatigue life reliability, scatter, and 

sensitivity were discussed. The fatigue life for (0/45/90)s graphite/epoxy laminate under mechanical

cyclic loads only was most sensitive to thickness and matrix tensile strength at 0.999 reliability since 

tension failure in 90° ply controls the life. However, at low mechanical cyclic loads and low thermal 

cyclic amplitudes, the matrix compressive strength, matrix modulus, thermal expansion coefficient, and 

ply thickness were most sensitive to the fatigue life for 0.999 reliability since the transverse compression 

failure in 45° and 90° plies controls the life. Also, at high mechanical-cyclic loads and high thermal-
NASA/TM—2011-217094 13 

cyclic amplitudes, shear failure in 45° ply controlled the life. Therefore, at these load magnitudes, the 

fatigue life at 0.999 reliability was more sensitive to the shear strength of matrix, longitudinal fiber 

modulus, matrix modulus, and ply thickness. The reliability-based fatigue curves are useful in 

determining the allowable fatigue load on the structure or assessing the life of a given component. 

Sensitivity information provides guidelines to the designer to improve the reliability of the structure’s 

fatigue life. 

References 

1. Integrated Probabilistic Assessment of Composite Structures (IPACS) User’s Manual. Ver. 1.0, 

NASA Contract No. NAS3–25266, Task Order No. 5209–02, Dec. 1991. 

2. Murthy, P.L.N.; and Chamis, C.C.: Integrated Composite Analyzer (ICAN): Users and Programmers 

Manual, NASA TP–2515, March 1986. 

3. Boyce, L.; and Chamis, C.C.: Probabilistic Constitutive Relationships for Cyclic Material Strength 

Models. Proceedings of the 29th AIAA Structures, Structural Dynamics, and Materials Conference, 

pt. 3, J. Propul. P., 1988, pp. 1299–1306. 

4. Wu, Y.T.: Demonstration of a New Fast Probability Integration Method for Reliability Analysis. 

Advances in Aerospace Structural Analysis, O.H. Burnside and C.H. Parr, ed., ASME, New York, 

pp. 63–73, 1985. 

5. Shah, A.R.; Singhal, S.N.; Murthy, P.L.N.; and Chamis, C.C.: Probabilistic Simulation of Long Term 

Behavior in Polymer Matrix Composites. Proceedings of the 35th AIAA/ASME/AHS/ASCE 

Structures, Structural Dynamics, and Materials Conference, April 1994. 

6. Shah, A.R; and Chamis, C.C.: Cyclic Load Effects on Long Term Behavior of Polymer Matrix 

Composites. Proceedings of the 40th International SAMPE Symposium and Exhibition, May 1995. 

7. Probabilistic Structural Analysis Methods (PSAM) for Select Space Propulsion System Components 

User’s Manual, South West Research Institute, NASA Contract NAS3–29389, Nov. 1989. 
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Probabilistic Simulation of Combined Thermo-Mechanical Cyclic Fatigue in Composites 

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14. ABSTRACT

A methodology to compute probabilistically-combined thermo-mechanical fatigue life of polymer matrix laminated composites has been 

developed and is demonstrated. Matrix degradation effects caused by long-term environmental exposure and mechanical/thermal cyclic 

loads are accounted for in the simulation process. A unified time-temperature-stress-dependent multifactor-interaction relationship 

developed at NASA Glenn Research Center has been used to model the degradation/aging of material properties due to cyclic loads. The 

fast probability-integration method is used to compute probabilistic distribution of response. Sensitivities of fatigue life reliability to 

uncertainties in the primitive random variables (e.g., constituent properties, fiber volume ratio, void volume ratio, ply thickness, etc.) 

computed and their significance in the reliability-based design for maximum life is discussed. The effect of variation in the thermal cyclic 

loads on the fatigue reliability for a (0/±45/90)s graphite/epoxy laminate with a ply thickness of 0.127 mm, with respect to impending failure 

modes has been studied. The results show that, at low mechanical-cyclic loads and low thermal-cyclic amplitudes, fatigue life for 0.999 

reliability is most sensitive to matrix compressive strength, matrix modulus, thermal expansion coefficient, and ply thickness. Whereas at 

high mechanical-cyclic loads and high thermal-cyclic amplitudes, fatigue life at 0.999 reliability is more sensitive to the shear strength of 

matrix, longitudinal fiber modulus, matrix modulus, and ply thickness.

15. SUBJECT TERMS

Fiber composites; Polymer; Cumulative distributions; Sensitivity factors 

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