Christos C. Chamis

Glenn Research Center, Cleveland, Ohio

Probabilistic Simulation for Combined 

Cycle Fatigue in Composites

NASA/TM—2010-216104

March 2010
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Christos C. Chamis

Glenn Research Center, Cleveland, Ohio

Probabilistic Simulation for Combined 

Cycle Fatigue in Composites

NASA/TM—2010-216104

March 2010

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
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NASA/TM—2010-216104 1 

Probabilistic Simulation for Combined Cycle Fatigue in Composites 

Christos C. Chamis 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

A methodology to compute probabilistic fatigue life of polymer matrix laminated composites has 

been developed and demonstrated. Matrix degradation effects caused by long term environmental 

exposure and mechanical/thermal cyclic loads are accounted for in the simulation process. A unified timetemperature-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 reliabilitybased 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. 

Introduction 

High Speed Research (HSR) Programs, technological solutions which will ensure the economic 

viability and environmental compatibility of a future High Speed Civil Transport (HSCT) plane are 

currently being sought. Lighter structural concepts for both airframe primary structures and engine 

structure components are being investigated. It is envisioned that such objectives can be achieved through 

the use of composites as well as other conventional lighter weight alloys. One of the prime issues for 

these structural components is assured long term behavior with a specified reliability. For example, the 

engine structure components are required to last 18 000 hr, and the airframe structural components are 

required to last 60 000 hr of flight time. These objectives are being pursued with the cooperation of 

industry and university participants through developments in accelerated testing strategies as well as 

computational simulation methodologies for predicting the long term durability of HSCT structural 

components. The present article addresses issues pertaining to probabilistic fatigue life of polymer matrix 

composites under combined thermal/mechanical cyclic loads through 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 induce variations, inherent 

uncertainties in the constituent properties, and 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 resistant composites in 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 to components. Furthermore, it is 

difficult, if not impossible, to isolate the effects of the composite multi-scale make-up experimentally. 

However, it is time consuming and expensive to conduct a large number of long-term tests to capture the 
NASA/TM—2010-216104 2 

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 

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. 4) under 

development at the NASA Glenn Research Center (Refs. 5 to 11) 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 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 is computed for a reliability of 0.999. The reader should note that this type of probabilistic 

evaluation on composites has not appeared anywhere in the literature as of this writing. 

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 
NASA/TM—2010-216104 3 

Sf fiber strength 

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 

αm 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 

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 (Ref. 12). 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

a

OF i

F

PO

P

VV

VV

M

M

1

 (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. 

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 
NASA/TM—2010-216104 4 

Figure 1.—Schematic of ICAN computer code. 

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 first ply 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 reliability 

based 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 thermal and 

mechanical cyclic loads. 

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 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. 4). 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 is 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ σ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ σ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ σ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ σ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

− = Tff f

TT

P

mff

mM

n

ff

T

m

ogd f

gw

PO

P

t

t

NS

N

NS

N

tSSTT

TT

M

M 11 1 1 1

1

 (2) 
NASA/TM—2010-216104 5 

The value of the exponents is assumed to be 0.5 for each one. The ratios in each factor are evaluated 

from the respective conditions. For example, the ratio in the first term is about 0.387 and its effect will 

be 0.147. 

Each term in parenthesis accounts for a specific physical effect which starts at 1 at ambient condition 

and degrades progressively to zero with load. 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 degradation physics of behavior. It can 

describe all 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 13. 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 14. 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 it 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). 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 
NASA/TM—2010-216104 6 

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. 

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, νf12 ............................................................................ 0.20 

Poisson’s ratio, νf23 ............................................................................ 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, νm12 .......................................................................... 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 

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 
NASA/TM—2010-216104 7 

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

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, μ, 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 
NASA/TM—2010-216104 8 

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. 

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. 
NASA/TM—2010-216104 9 

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. 

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. 

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 

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. 
NASA/TM—2010-216104 10 
NASA/TM—2010-216104 11 
NASA/TM—2010-216104 12 

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 15. 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. 

It is important to note that no experimental data has appeared in the literature to compare with the 

predictions included herein. However, the methodology has been for a long time. 

Conclusion 

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 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. A Review of Research on Aeronautical Fatigue in the United States-2003-2005. Compiled by James 

L. Rudd, Air Force Research Lab., Wright-Patterson Air Force Base, OH. Presented at the Meeting 

of the International Committee on Aeronautical Fatigue, Hamburg, Germany, June 6-10, 2005 (CD). 

4. 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. 
NASA/TM—2010-216104 13 

5. Minnetyan, L. and Chamic, C.C.: Computational Simulation of Composite Fatigue Life. 41st AIAA 

Structures, Structural Dynamics and Materials Conference, Atlanta, GA. April 3-6, 2000, No. AIAA 

2000-1721. (CD Rom). 

6. Minnetyan, L., Abdi, F. and Chamis, C.C., Cyclic Fatigue Damage Evolution in Airfoil Structures, 

46th International SAMPE Symposium, Vol. 46, Book 1 of 2 Books, pp 1037-1048, May 6-10, 2001. 

7. Chamis, C.C. and Minnetyan, L.,: A Multi-Factor Interaction Model (MFIM) for Damage Initiation 

and Progressions. Presented in the Session on Damage Initiation and Prediction in Composites at the 

2001 ASME International Mechanical Engineering Congress and Exposition (IMECE), New York, 

NY, November 11-16, 2001. (CD Rom) 

8. Minnetyan, L. and Chamis, C.C.: Cyclic Fatigue Degradation Response of Composite Structures. 47th

International SAMPE Symposium, May 12-16, 2002, Long Beach, CA, Vol. 47, Book 2 of 2 Books, 

pp. 1632-1646. 

9. Minnetyan, L. and Chamis, C.C.: Probabilistic Fatigue of Integrally Stiffened Composite Structures, 

44th AIAA Structures, Structural Dynamics and Materials Conference, April 7-10, 2003, 

Norfolk,VA, No. AIAA 2003-1839. (CD Rom) 

10. Minnetyan, L. and Chamis, C.C.: Acoustic Fatigue of Composite Engine Structures, 48th

International SAMPE Symposium, May 11-15, 2003, Long Beach, CA, Vol 48, Book 2 of 2 Books, 

pp. 1786-1798. 

11. Chamis, C.C. and Pai, S.S.: Probabilistic Fatigue: Computational Simulation. Journal of ASTM 

International, Vol. 1, No. 8, Paper ID: JA111560, pp. 15-29, September 1004. 

12. 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. 

13. 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. 

14. 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. 

15. 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 for Combined Cycle Fatigue in Composites 

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Chamis, Christos, C. 

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14. ABSTRACT

A methodology to compute probabilistic fatigue life of polymer matrix laminated composites has been developed and 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; Multifactor model; Fast probability integrator; Constituent materials 

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