Trenton M. Ricks and Thomas E. Lacy, Jr. 

Mississippi State University, Mississippi State, Mississippi 

Brett A. Bednarcyk and Steven M. Arnold 

Glenn Research Center, Cleveland, Ohio

John W. Hutchins 

Mississippi State University, Mississippi State, Mississippi 

A Multiscale Progressive Failure Modeling 

Methodology for Composites That Includes 

Fiber Strength Stochastics

NASA/TM—2014-215990

May 2014
	
	
	








	



	

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Trenton M. Ricks and Thomas E. Lacy, Jr. 

Mississippi State University, Mississippi State, Mississippi 

Brett A. Bednarcyk and Steven M. Arnold 

Glenn Research Center, Cleveland, Ohio

John W. Hutchins 

Mississippi State University, Mississippi State, Mississippi 

A Multiscale Progressive Failure Modeling 

Methodology for Composites That Includes 

Fiber Strength Stochastics

NASA/TM—2014-215990

May 2014


	



	


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NASA/TM—2014-215990 1 

A Multiscale Progressive Failure Modeling Methodology for 

Composites That Includes Fiber Strength Stochastics 

Trenton M. Ricks and Thomas E. Lacy, Jr. 

Mississippi State University 

Mississippi State, Mississippi 39762 

Brett A. Bednarcyk and Steven M. Arnold 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

John W. Hutchins 

Mississippi State University 

Mississippi State, Mississippi 39762 

Abstract 

A multiscale modeling methodology was developed for continuous fiber composites that incorporates 

a statistical distribution of fiber strengths into coupled multiscale micromechanics/finite element (FE) 

analyses. A modified two-parameter Weibull cumulative distribution function, which accounts for the 

effect of fiber length on the probability of failure, was used to characterize the statistical distribution of 

fiber strengths. A parametric study using the NASA Micromechanics Analysis Code with the Generalized 

Method of Cells (MAC/GMC) was performed to assess the effect of variable fiber strengths on local 

composite failure within a repeating unit cell (RUC) and subsequent global failure. The NASA code 

FEAMAC and the ABAQUS finite element solver were used to analyze the progressive failure of a 

unidirectional SCS-6/TIMETAL 21S metal matrix composite tensile dogbone specimen at 650 °C. 

Multiscale progressive failure analyses were performed to quantify the effect of spatially varying fiber 

strengths on the RUC-averaged and global stress-strain responses and failure. The ultimate composite 

strengths and distribution of failure locations (predominately within the gage section) reasonably matched 

the experimentally observed failure behavior. The predicted composite failure behavior suggests that use 

of macroscale models that exploit global geometric symmetries are inappropriate for cases where the 

actual distribution of local fiber strengths displays no such symmetries. This issue has not received much 

attention in the literature. Moreover, the model discretization at a specific length scale can have a 

profound effect on the computational costs associated with multiscale simulations. 

1.0 Introduction 

As a result of recent increases in computational capabilities, numerous models have been developed 

to simulate material behavior across multiple length scales (Sullivan and Arnold (2011)). While most 

material models are deterministic in character, real materials exhibit statistical variations in properties and 

features over a range of different length scales. When performing multiscale analyses, a number of 

challenges arise when accounting for statistically varying material characteristics (Graham-Brady, 

Arwade, Corr, Gutierrez, Breysse, Grigoriu, and Zabaras (2006); Sriramula and Chryssanthopoulos 

(2009)). For instance, how does statistical variability at one length scale affect the predicted material 

response over a hierarchy of scales including the macroscale (Graham-Brady, Arwade, Corr, Gutierrez, 

Breysse, Grigoriu, and Zabaras (2006))? To answer this question, multiscale modeling strategies have 

been developed to account for variations in properties and morphologies at different scales (cf., 

(Sriramula and Chryssanthopoulos (2009)) for a summary of different multiscale methods for fiberreinforced polymer matrix composites). 
NASA/TM—2014-215990 2 

Many researchers have recognized the need to include a distribution of fiber segment strengths in 

analytical/numerical models rather than using a single deterministic value in order to better estimate the 

composite strength and characterize the failure behavior. Early models of unidirectional composites 

developed by Rosen (1964) and Zweben (1968) first considered a distribution of flaws along the fiber 

length. The Equal (or Global) Load Sharing model of Rosen (1964) assumes that upon the failure of an 

individual fiber, the load is equally distributed to intact fibers. In contrast, the Local Load Sharing model 

developed by Zweben (1968) assumes that a greater fraction of the load is redistributed to intact fibers 

near the fiber break. Various other models have since been developed to predict the strength of 

unidirectional composites based upon a distribution of fiber strengths (Harlow and Phoenix (1978); Oh 

(1979); Curtin (1991), (1993); Foster, Ibnabdeljalil, and Curtin (1998); Curtin (2000); Landis, Beyerlein, 

and McMeeking (2000); Bednarcyk and Arnold (2001); Okabe, Takeda, Kamoshida, Shimizu, and Curtin 

(2001); Okabe and Takeda (2002); Lekou and Philippidis (2008)). For example, Bednarcyk and Arnold 

(2001) developed the Evolving Compliant Interface model. This model was incorporated into the 

Micromechanics Analysis Code with Generalized Method of Cells (MAC/GMC) (Bednarcyk and Arnold 

(2002)) where the interface between fiber subcells in adjacent mating triply periodic repeating unit cells 

(RUCs) was given a fiber tensile strength consistent with fiber vendor data. This method was used to 

simulate the longitudinal failure of a unidirectional metal matrix composite at the microscale (i.e., RUC 

behavior) and compared with the Curtin fiber breakage model (Curtin (1991), (1993)) that combined a 

statistical probability of fiber failure based upon a shear lag approach. However, these models have not 

been generally applied within a multiscale analysis framework. 

Various techniques have been proposed in order to characterize the effects of microscale material 

uncertainty on the macroscale response. The Multiscale Stochastic Finite Element Method (MSFEM) was 

developed to simulate random heterogeneous materials from the micro- to meso- to macroscales (Xu, 

Chen, and Shen (2009); Shen and Xu (2010)). Uncertainties can be introduced in the form of random 

coefficients in partial differential equations (e.g., moduli) or boundary conditions (e.g., load, 

displacement). A homogenization technique can be implemented at the microscale to account for different 

material phases (Shen and Xu (2010)). This method has yet to be extended to simulate the multiscale 

progressive failure of composite materials. A multilevel finite element (FE) method (i.e., FE2

) (Feyel and 

Chaboche (2000); Feyel (2003)) was also developed in which a combination of localization and 

homogenization techniques is employed between two or more length scales through a series of FE 

calculations. For example, macroscale integration point strains from a global FE model (e.g., a coupon 

specimen or structure) can be mapped onto a microscale FE model containing representations of 

individual constituents. The ensuing microscale stress field can be used to predict evolution of structure 

within the RUC and homogenized. The RUC-averaged response can then be used to update the 

macroscale integration point stresses. A similar approach was adopted by Blassiau, Thionnet, and Bunsell 

(2006a), (2006b), (2008). Multiple FE models of individual RUCs were developed to characterize the 

interaction of a single broken fiber with varying numbers of unbroken fibers (Blassiau, Thionnet, and 

Bunsell (2006a), (2006b)). Multiscale FE simulations, which accounted for statistical variation in fiber 

strengths, were performed to predict the effect of local fiber breakage, stress redistribution in the vicinity 

of broken fibers, and interfacial debonding around fiber breaks on bulk damage accumulation and failure 

for unidirectional and filament wound carbon fiber composites with elastic or viscoelastic epoxy matrices 

(Blassiau, Thionnet, and Bunsell (2008)). 

An alternative multiscale method has been implemented by Bednarcyk and Arnold (2007) where the 

reformulated generalized method of cells (GMC) (Pindera and Bednarcyk (1999)) was used to perform 

microscale calculations; the homogenized microscale material response was integrated into macroscale 

calculations performed using a hybrid FE approach. The reformulated GMC as implemented within the 

ABAQUS Standard or Explicit FE solver (2013) is referred to herein as FEAMAC (cf., Figure 1). Since 

the RUC-averaged response obtained using the GMC is more computationally efficient than traditional 

FE approaches (Pindera and Bednarcyk (1999)), this multiscale approach is considered to be more 

computationally efficient than FE2

. Bednarcyk and Arnold (2007), (2011) demonstrated the feasibility of 
NASA/TM—2014-215990 3 

Figure 1.—Schematic showing the coupling of MAC/GMC 

with ABAQUS via FEAMAC. 

using FEAMAC to simulate the progressive multiscale failure of composite materials. A traditional 

maximum stress failure criterion for fiber failure and the Curtin fiber breakage model were each 

implemented within RUCs at the microscale and used to simulate progressive failure of a metal matrix 

composite (MMC) tensile dogbone specimen at the macroscale using global-to-local-to-global analyses. 

In the current study, MAC/GMC and FEAMAC are used to investigate the effect of statistical variations 

in fiber strengths on failure progression within RUCs at the microscale and subsequent macroscale 

failure. Future work will investigate the effects of statistical variations in constitutive moduli, strengths, 

volume fractions, and orientations on local and global composite failures. 

When performing multiscale analyses involving damage and failure of fibrous composites, the 

stochastic variations in fiber strengths are commonly characterized using a Weibull cumulative 

distribution function (CDF) (Weibull (1951)), although other distribution functions (e.g., sigmoidal 

functions (Baxevanakis, Jeulin, and Valentin (1993))) may also be used. The classic two-parameter 

Weibull CDF, however, does not account for the effect of fiber segment length on the measured strengths 

and, as a consequence, has been shown to yield inaccurate strength predictions (Curtin (2000)). Early 

modifications to the classic two-parameter Weibull CDF to incorporate the effect of fiber segment lengths 

also led to inaccurate predictions in the strength distribution of fiber segment strengths particularly at 

shorter segment lengths (cf., (Hitchon and Phillips (1979))). Accordingly, a modified two-parameter 

Weibull CDF was proposed by Watson and Smith (1985) to better account for the effect of fiber segment 

length on the probability of failure, i.e., 

 













	













































   

 

0 0

1 exp L

L Pf (1) 

where Pf 		
	
"
	<	L0)

!	
	
"	
	


			 L0 represents the 

reference fiber length (i.e.*0 were determined) and L represents the 

characteristic fiber length of interest. 			
		
	



experimental strength data in which the tested fiber lengths are varied (e.g., 0    1). For example, 

Beyerlein and Phoenix (1996) determined this value to be equal to 0.6 for an AS4 carbon fiber.  = 0, 

then it is easily seen that Equation (1) reduces to the classic two-parameter Weibull CDF. A number of 
NASA/TM—2014-215990 4 

researchers have used the modified two-parameter Weibull CDF (Eq. (1)) to characterize the effect of 

fiber segment length on failure for a wide variety of composites with silicon carbide monofilaments as 

well as carbon, glass, and flax fibers (Padgett, Durham, and Mason (1995); Beyerlein and Phoenix 

(1996); Foster, Ibnabdeljalil, and Curtin (1998); Curtin (2000); Landis, Beyerlein, and McMeeking 

(2000); Andersons, Joffe, Hojo, and Ochiai (2002); Okabe and Takeda (2002); Andersons, Spamins, 

Joffe, and Wallstrom (2005); Naito, Yang, Tanaka, and Kagawa (2012)). As will be shown in this study, 

accounting for the effect of fiber lengths on the probability of failure is of critical importance in 

multiscale progressive failure analyses of continuous fiber-reinforced composites. In order to simulate 

progressive failure within RUCs at the microscale, the modified two-parameter Weibull CDF (Eq. (1)) 

was implemented within the framework of MAC/GMC (Bednarcyk and Arnold (2002)). 

MAC/GMC provides a computationally efficient means of modeling composites based on Aboudi’s 

method of cells micromechanics theories (Aboudi (1991); Paley and Aboudi (1992); Aboudi (1995), 

(1996); Pindera and Bednarcyk (1999); Aboudi (2004); Aboudi, Arnold, and Bednarcyk (2012)). Using 

the method of cells, a doubly or triply periodic RUC is discretized into an arbitrary number of subcells. 

Each subcell is then assigned material properties and a constitutive law to describe the local material 

behavior. Continuity of displacements and tractions are then enforced along the subcell boundaries in an 

average sense, and all field quantities are evaluated at the subcell centroids. An illustration of this scheme 

for a unidirectional composite is shown in Figure 2. Using this model, a doubly-periodic RUC may be 

defined in the x2-x3 plane and is discretized into an arbitrary number of subcells along the x2-direction 

(height) and the x3-direction (width), respectively, while the fibers extend in the x1-direction (length). If a 

triply periodic RUC is selected, the RUC can be discretized along the x1-direction as well. 

Figure 2.—(a) Representation of a unidirectional composite with fibers aligned in the x1-direction and (b) 

RUC representation of the unidirectional composite. Figure from Aboudi, Arnold, and Bednarcyk (2012). 
NASA/TM—2014-215990 5 

When performing global analyses using FEs, geometric symmetries (e.g., one-half, one-quarter, oneeighth symmetry) are often exploited to reduce the number of model degrees-of-freedom and decrease the 

computational time. In fact, such symmetry assumptions were employed by Bednarcyk and Arnold 

(2007), (2011) and Blassiau, Thionnet, and Bunsell (2008). However, it will be shown that use of 

geometric symmetries is inappropriate in global (structural) progressive failure analyses when the actual 

spatial distribution of fiber strengths displays no such symmetries. This issue has not received much 

attention in the literature. One central goal of this work is to present a method for systematically assigning 

an experimentally determined spatial distribution of fiber strengths to individual fibers within RUCs. 

These RUCs can then be analyzed separately or implemented within a multiscale framework (e.g., 

FEAMAC). 

The first part of this study analyzes the effect that a stochastic distribution of fiber strengths has on 

the RUC-averaged stress-strain response and failure using MAC/GMC. While this initial 

micromechanics-based study provides insight into local pointwise material behavior, the true stochastic 

nature of the problem is only revealed when integrated multiscale micromechanical/FE analyses are 

performed. This is demonstrated herein using an MMC dogbone specimen as the representative structure. 

2.0 Material System 

The effect of a statistical variation in fiber strengths on the failure behavior of a 25 percent fiber 

volume fraction SCS-6/ TIMETAL 21S MMC at 650 °C was analyzed in this study. The SCS-6 fiber is a 

high-stiffness, high-strength silicon carbide monofilament with a diameter of approximately 142 μm. In a 

previous work (Bednarcyk and Arnold (2011)), a two-parameter Weibull probability density function 

(PDF) was fit to room temperature fiber strength data (0 = 4198.9 MPa,  = 10). These Weibull 

parameters were determined based on monofilament tensile tests at a fiber length of L0 = 25.4 mm 

(Bednarcyk and Arnold (2011)). Additionally, the Weibull scale factor (0) was reduced by 5.3 percent 

for simulations performed at 650 °C to account for the fiber strength dependence on temperature (Mall, 

Fecke, and Foringer (1998); Bednarcyk and Arnold (2007), (2011)). In the current study, the SCS-6 fiber 

was assumed to be linearly elastic and isotropic. TIMETAL 21S is a metastable beta strip titanium alloy 

possessing a high strength and good creep and oxidation resistance. The titanium matrix was considered 

to be viscoplastic and was simulated using the Generalized Viscoplasticity with Potential Structures 

(GVIPS) constitutive model (Arnold and Saleeb (1994); Arnold, Saleeb, and Castelli (1996)). Table I 

contains a summary of the thermoelastic material properties for both the SCS-6 fiber and the TIMETAL 

21S matrix (Bednarcyk and Arnold (2011)). The viscoplastic material properties employed in this study 

can be found in Arnold, Saleeb, and Castelli (1996) and Bednarcyk and Arnold (2011). 

TABLE I.—MATERIAL PROPERTIES FOR SCS-6 AND TIMETAL 21S 

Material Temperature, 

°C 

Young’s modulus, 

GPa 

Poisson’s 

ratio 

Coefficient of 

thermal expansion, 

1010–6/°C 

SCS-6 

21 393 0.25 3.56 

316 382 0.25 3.72 

427 378 0.25 3.91 

538 374 0.25 4.07 

860 368 0.25 4.57 

TIMETAL 21S 

23 114.1 0.365 7.717 

300 107.9 0.365 9.209 

500 95.1 0.365 10.7 

650 80.7 0.365 12.13 

704 59.7 0.365 14.09 
NASA/TM—2014-215990 6 

3.0 RUC Analyses 

A parametric study was performed using MAC/GMC to assess the effect of variable fiber strengths on 

microscale composite failure at 650 °C. Such information is crucial for predicting the progressive 

composite failure at the macroscale (structural level). Five different doubly-periodic RUCs were analyzed 

in this study: single-, four-, nine-, 25-, and 49-fiber RUCs. These RUCs are comprised of 2x2, 4x4, 6x6, 

10x10, and 14x14 subcells, respectively, while maintaining a constant fiber volume fraction of 25 percent. 

Figure 3 shows the single-fiber and 25-fiber doubly-periodic RUCs considered in this study. Note that a 

25-fiber RUC can be subdivided into 25 individual single-fiber RUCs. The MMC material system 

considered in this work was specifically fabricated to provide uniform fiber volume fractions throughout 

the composite and a square arrangement of fibers (Bowman (1999)). Hence, these RUC architectures are 

representative of the as-fabricated material. For each individual fiber subcell in the RUC, a fiber tensile 

strength value obtained from an experimentally determined Weibull CDF was assigned. Essentially, a 

random number was generated (i.e., [0-1]) and used to solve Equation (1) 
			

strength values were then assigned to the individual fibers contained within the RUCs in the microscale 

analyses. To assess the effect of fiber length on the predicted RUC response, MAC/GMC analyses were 

performed where the fiber length-dependent strength was obtained from Equation (1). For example, for 

the single-fiber RUC, an individual fiber tensile strength was selected using the modified Weibull CDF 

(Eq. (1)) and assigned to the fiber. Similarly, four, nine, 25, and 49 fiber tensile strengths were selected 

using the CDF and assigned to the four-, nine-, 25-, and 49-fiber RUCs, respectively. 

In these MAC/GMC analyses

L	
	 = 1, L = 0.64 mm, and 

L0 = 25.4 mm (L/L0 = 0.025) were chosen. This value of L corresponds to one-half of a key macroscale 

fiber segment length *	

*	
	
	
*	 = 0 in order to 

bound the effect of fiber length on the fiber strength distribution. Figure 4 contains plots of the associated 

PDFs and CDFs for Equation (1) for the case where 0    1. Since the fiber segment length is much 

smaller than the reference length (i.e., L/L0 = 0.025), the length-dependent strength distribution is shifted 

to higher stresses increases. The modified Weibull CDF (Eq. (1)) is based on “weakest link” theory; 

a distribution of shorter fibers will typically have higher strengths than an analogous distribution of longer 

fibers due to the decreased likelihood of severe flaws. Additionally, for L/L0  1, the mean fiber strength 

increases and there is more scatter 		
	 (cf., Figure 4). This 

underscores the importance of accounting for fiber lengths in an appropriate manner. 

Figure 3.—Microstructural representation of a unidirectional SiC/Ti 

composite for a) a single-fiber RUC and b) a 25-fiber RUC. 
NASA/TM—2014-215990 7 

Figure 4.—Modified Weibull (a) PDF and (b) CDF as a function of 

the fiber strength 	
 

Similar to the work in Bednarcyk and Arnold (2011), residual stresses in the fiber and matrix were 

accounted for by simulating a 16 hr stress-free cooldown from the heat treatment temperature to room 

temperature, followed by a stress-free temperature rise to 650 °C over 5 min. A uniform axial strain in the 

x1-direction (cf., Figure 2) was then applied at a rate of 110-4/s. The maximum stress failure criterion 

was used to dictate axial fiber failures within an RUC where individual fiber strength values were 

obtained for each fiber in the RUC by generating a random number to represent a probability of fiber 

failure, which correlates to a given strength value via Equation (1). While the matrix was permitted to 

yield in accordance with the GVIPS model, ultimate failure (fracture) of the matrix was not considered in 

the analyses. Five hundred microscale simulations (i.e., using MAC/GMC) were performed for each of 

the five RUC architectures to estimate the RUC-averaged mean tensile strengths associated with the 

stochastic distribution of fiber strengths. As p	
"

*


Equation 

(1) were considered (i.e., 0, 1) in order to bound the effect of fiber segment length on the RUC-averaged 

tensile strength. This resulted in a total of 5000 distinct microscale simulations performed in this study. 

Unbiased estimates for the mean and variance of the RUC-averaged tensile strengths were calculated 

based on a random sampling methodology (McKay and Beckman (1979)). An estimate of the standard 

deviation was obtained by taking the square root of the variance estimate for a given response. Future 

studies will investigate the use of more complex sampling techniques (e.g., Latin Hypercube (Helton and 

Davis (2003))), particularly if additional material parameters (e.g., fiber volume fraction, fiber/matrix 

moduli) are allowed to vary. 
NASA/TM—2014-215990 8 

Figure 5.—Mean RUC-averaged ultimate strength compared against the 

number of fibers (at a constant fiber volume fraction) after 500 simulation runs 

per RUC where the error bars correspond to ± one standard deviation from 

the mean ultimate strength. Equation (1) was used to generate the fiber 


	 = 	 = 1 (L/L0 = 0.025). 

Figure 5 shows a plot of the RUC-averaged composite tensile strengths resulting from a statistical 

distribution of individual fiber strengths in RUCs with one, four, nine, 25, and 49 simulated fibers, 

respectively, where two different fiber length-dependent strength distributions were employed in the 

microscale simulations (i.e. = J = 1, L/L0 = 0.025). As would be expected, the simulations that account 

for the effect of fiber length on composite strength  = 1) yield a higher mean RUC-averaged strength and 

more variation in the calculated strengths 

*	 = 0 (i.e., no fiber length dependence). 

For both cases, when a single-fiber RUC was simulated, a higher mean RUC-averaged strength results. As 

the number of simulated fibers contained in an RUC was increased, however, the composite strength 

decreased. Similarly, the variability in the RUC-averaged strength decreased as the number of fibers within 

the RUC was increased. For the single-fiber RUC, the RUC-averaged strength became highly dependent on 

the selected fiber strength. This resulted in a higher mean strength value and more variation among the 

individual RUC-averaged strengths. As the number of simulated fibers was increased, the RUC-averaged 

strengths became less dependent on the individual fiber strengths. Essentially, the load carried by one fiber 

at failure was redistributed to the remaining fibers. As the number of simulated fibers was increased, the 

likelihood of encountering a weaker fiber increased leading to a decrease in the mean RUC-averaged 

strength. Additionally, if the number of fibers is large, the distributions of fiber strengths within given RUCs 

become more similar, resulting in less variation in predicted ultimate strengths. For fiber strength 

distributions *	 = J = 1, as the number of simulated fibers exceeded 25, the mean value and 

variation in the RUC-averaged strength asymptotically approached those of the representative volume 

element (RVE) averaged response. While the use of a 25-fiber RUC led to approximately the same RUCaveraged strength as for a 49-fiber RUC, roughly one-half the computational time was needed. This savings 

in computational time becomes crucial when performing global-to-local-to-global multiscale simulations. In 

order to assess the effect of the number of simulations on the calculated mean RUC-averaged strength, an 

additional 5000 progressive failure simulations were performed for a single-	)8$* = 1. The 

estimated mean fiber strength and associated standard deviation differed by less than 1 percent from the 

values obtained using 500 simulations. Hence, a relatively small number of simulations (i.e., 500 per RUC 

architecture) was sufficient in providing a reasonable approximation of the mean and variance for each 

RUC. Of course, if additional material parameters (e.g., fiber volume fraction, fiber/matrix moduli) are 

varied, a larger number of simulations would be needed as well as a more efficient sampling methodology; 

this will be the topic of future studies. 
NASA/TM—2014-215990 9 

Figure 6.—A random sampling of ten stress-strain curves from a batch of 500 simulations for 

(a) single-fiber (b) four-fiber (c) nine-fiber (d) 25-fiber and (e) 49-fiber RUCs using Equation 

(1) 	 = 0 case. 

Figure 6(a) to (e) contain ten typical (out of 500) RUC-averaged uniaxial stress-strain curves in the 

x1-(fiber) direction for the single-fiber, four-fiber, nine-fiber, 25-fiber, and 49-fiber RUCs, respectively. 

Similar results were obtained for simulations that accounted for the effect of fiber length on composite 

	 = 1). For all of the RUC architectures, the RUC-averaged stress-strain response increased 

monotonically up until the onset of fiber failure. For RUCs containing only one fiber, once the fiber 

failed, the RUC-averaged axial stress-strain response displayed a discrete sudden load drop and the 

remaining stress was carried by the matrix (cf., Figure 6(a)). The strain at failure, of course, was a strong 
NASA/TM—2014-215990 10 

function of the assumed fiber strength. A comparison of Figure 6(a) to (e) suggests that as the number of 

fibers is increased, a gradual softening behavior is observed. Furthermore, the difference in the global 

strain between the first and last fiber failures for a given analysis increased as the number of simulated 

fibers increased. This makes sense because an increase in the number of fibers raises the probability of 

encountering both weaker and stronger fibers. Additionally, the variation in the RUC-averaged stressstrain response diminished as the number of fibers increased since the local RUC fiber strength 

distribution better approximated that for the actual Weibull distribution. By adding more simulated fibers, 

a more gradual, continuum-like local stress-strain response was observed as each fiber failure occurred 

over a smaller fraction of the RUC volume. Clearly, understanding the progressive failure behavior 

associated with a statistical fiber strength distribution at the microscale is crucial to establishing and 

accurately capturing length scale effects in a robust, computationally efficient multiscale modeling 

methodology. Because of the computational efficiency of the MAC/GMC code, parametric studies can be 

performed quickly to assess the influence of RUC architecture and variations in material parameters on 

the RUC-averaged response. These results can then be used to guide the implementation of such lower 

length scale calculations within a multiscale analysis framework. 

4.0 Coupled FE/ Micromechanics Analyses 

While the preceding parametric study investigated the effect of RUC architecture and fiber strength 

distributions on the RUC-averaged local failure, the ultimate goal of this work is to use RUC deformation 

and damage evolution at a given integration point within an FE analysis (i.e., FEAMAC/ABAQUS) to 

perform global progressive failure analyses of a 25 percent fiber volume fraction SCS-6/TIMETAL 21S 

dogbone specimen under a monotonic tensile load at 650°C. Figure 7 contains a schematic of the NASA 

GRC dogbone specimen (Worthem (1994)). Such specimens were specifically designed to reduce the 

magnitude of the stress concentration associated with a reduction in cross-sectional area commonly 

observed in dogbone tensile test specimens. Two FE models were constructed using relatively coarse 

(2400 elements) or fine (19,200 elements) meshes comprised of eight-noded linear isoparametric brick 

elements with eight integration points per element. Initial analyses showed that the use of higher-order 

quadratic elements had a negligible impact on the predicted stress-strain response and failure for this 

problem. Since FEAMAC assigns an RUC to each FE, MAC/GMC is called over 150,000 times per time 

step for the fine FE mesh. Hence, the computational efficiency of the GMC (Pindera and Bednarcyk 

(1999)) is a crucial element in the multiscale FE analyses performed here and is essential to the analysis 

of more complex structures. 

Although an RUC can be used to approximate the RVE-averaged response, an RUC alone is not 

necessarily an RVE. An individual RUC with distinct constituent morphologies and properties can be 

regarded as an RVE subvolume (sub-RVE). This notion is similar to that developed by Lacy, McDowell, 

and Talreja (1999), who related the homogenized response over distinct RVE-subvolumes of varying 

sizes to the RVE-averaged (continuum) material behavior. In the current study, the homogenized material 

responses at multiple adjacent FE integration points (i.e., RUCs) can be used to replicate the RVEaveraged behavior. 

Similar to the previous local RUC analyses using MAC/GMC, thermal residual stresses in the tensile 

dogbone specimen were determined from FEAMAC/ ABAQUS analyses involving a 16 hr assumed 

stress-free cooldown from the heat treatment temperature to room temperature. This was followed by a 

stress-free temperature rise to 650 °C over 5 min. Multiscale progressive failure analyses of the dogbone 

specimen (cf. Figure 7) were then performed with a constant temperature distribution at 650 °C. The 

surface nodes corresponding to the machine grips were fixed at one end of the specimen while the surface 

nodes in the grip region at the opposite end of the specimen were given a longitudinal tensile 

displacement consistent with an initial elastic strain rate of 110-4/s in the gage section. 
NASA/TM—2014-215990 11 

Figure 7.—NASA GRC MMC tensile dogbone specimen: (a) Specimen geometry with 

top and side views of the (b) coarse 2400 FE mesh and (c) fine 19,200 FE mesh. 

The coarse (2400 element) FE mesh was initially used to study the effect of RUC architecture and 

fiber strength parameter (!
the global composite stress-strain response and failure behavior. In each 

simulation, RUCs containing single-fiber or four-fiber RUCs were generated (i.e., only one RUC 

geometry per simulation). As an aside, special care should be taken to ensure that the actual material 

volume associated with an RUC does not exceed that of the FEs used in the analysis, i.e., the RUCaveraged continuum response would occur over a domain larger than the typical element size. Recall in 

the previous local MAC/GMC analyses, large variations in the RUC-averaged composite strength were 

observed for both the single- and four-fiber RUCs. Such variability in local properties will manifest itself 

at the macroscale in multiscale progressive failure analyses. One crucial consideration is to retain a 

sufficient level of model discretization at each simulation scale to provide accurate results without 

excessive computational costs. Ninety-six distinct RUCs, each with a different set of individual fiber 

strength values based upon the modified Weibull CDF (Eq. (1)), were randomly assigned in equal 

numbers to individual FEs throughout the coarse FE mesh. In the absence of experimental data to 

	
 for this material system, the fiber strength parameter was varied between 0    1 in 

increments of 0.25. The characteristic length L = 0.64 mm (L/L0 = 0.025) corresponded to one-half of the 

typical FE length in the axial direction for the coarse mesh. Since two FE integration points are present in 

the length direction, the element half-length (L) was chosen as a length scaling parameter in order to 

validate the methodology for different FE meshes. Essentially, the length of the simulated fiber segment 

used to define the continuum-averaged FE properties should not exceed the associated FE dimensions. 

Figure 8 contains an overview schematic of the methodology used in assigning local fiber properties to 

individual RUCs and then distributing RUCs throughout the FE mesh. This process was performed ten 

times for each RUC a total of 100 multiscale FE simulations using the coarse mesh. 

In order to assess the effect of macroscale FE model symmetry on the global composite stress-strain 

response and failure behavior, a 300 element one-eighth symmetry FE mesh was generated that maintains 

the same mesh density as the 2400 FE coarse mesh. Twelve distinct four-fiber RUCs (i.e., the same FE to 

RUC ratio as the coarse FE mesh) were distributed throughout this model in the same manner as 

	
"				
	*
	"
*
	"*JO

increments of 0.25, and ten simulations w		
	

	
	

GJ

additional simulations. The individual strength predictions from each group of ten analyses were averaged 

together to obtain the macroscale composite strength. 
NASA/TM—2014-215990 12 

Figure 8.—Fiber strength distribution scheme for multiscale analyses. First, 

the input parameters for the strength distribution and the number/ 

architecture of the RUC are determined. Then the fiber strengths are 

generated by using a random number generator and solving Equation (1) 






subcells within an RUC, and the process is repeated until all RUCs have 

been defined. Finally, the RUCs are randomly assigned in equal numbers 

to FEs within the ABAQUS model. 

Figure 9 shows the average macroscale composite ultimate strength as a function of the fiber strength 

parameter for multiscale analyses performed using single-fiber and four-fiber RUCs within the coarse FE 

mesh and four-fiber RUCs within the one-"

	"?3
*	the predicted 

ultimate strength increased proportionally (cf., Figure 9). This was due to the increasingly pronounced 


	
	Ocf., Figure 4(a)). In contrast to the local MAC/GMC 

calculations (cf., Figure 5), the use of a four-fiber RUC within the coarse global FE mesh led to higher 

ultimate strengths than for a single-	)8$
	

	"

without global symmetry, as the far-field global strain was increased, local fiber failures initiated in a 

distributed fashion throughout the specimen in lower length scale RUCs surrounding FE integration 

points. This process led to failure localization within individual RUCs as well as throughout the global FE 

mesh, culminating in ultimate specimen failure. If single-fiber RUCs are employed in multiscale analyses, 

some load shedding within an element and between elements is possible once initial fiber failure occurs. 

However, after the single fiber fails, the load is rapidly shed to neighboring elements, increasing the 

likelihood of global damage localization which leads to a reduction in the predicted composite ultimate 

strength. While the calculated variation in strength values was less than 5 percent 		


variation in the composite strengths was less for multiscale calculations performed using a four-fiber 

RUC than those for the single-fiber RUC. The experimental ultimate strength for this specimen is 

973 MPa (Bowman (1999)); this suggests that the actual fiber strength parameter is likely less than unity 

 < 1). Of course, further testing is required to fully establish an appropriate  
NASA/TM—2014-215990 13 

Figure 9.—Average macroscale composite ultimate strength as a 

function of fiber strength parameter (0  	  1) over ten ABAQUS/ 

FEAMAC simulations using single-fiber and four-fiber RUCs with the 

coarse FE mesh and four-fiber RUCs with a one-eighth symmetry 

coarse mesh. The error bars denote ± one standard deviation from 

the mean. Note that the experimental ultimate strength was 973 MPa 

(Bowman (1999)). 

When performing multiscale progressive failure analyses, the use of global symmetry boundary 

conditions can profoundly impact the calculated global composite stress-strain response and failure 

behavior. The use of such boundary conditions is questionable when the distribution of constituent 

morphologies and properties display no such symmetries. To illustrate this point, the use of a one-eighth 

symmetry coarse FE model led to higher calculated mean ultimate strengths than those obtained using the 

same FE mesh density where the entire dogbone specimen was simulated (cf., Figure 9). This increase in 

mean strength was also somewhat independent of the number of fibers in the RUC. Simulations 

performed using one-eighth symmetry boundary conditions led to fiber failures that occurred in a 

symmetric fashion across all symmetry planes. Hence, local bending due to an asymmetric distribution of 

fiber failures was not permitted. Such eccentric deformation can lead to an elevated local stress state. This 

serves to lower the ultimate composite strength, which explains why the one-eighth symmetry model 

yields higher ultimate strengths than the model that does not employ geometric symmetries. Additionally, 

more variability in the calculated ultimate strengths was observed for the one-eighth symmetry model. 

Hence, global FE symmetry models can lead to erroneous strength predictions in multiscale progressive 

failure analyses and should only be used with much caution. For these reasons, use of symmetry boundary 

conditions in combination with stochastic variations in material properties is not recommended. In the 

following discussion, only results obtained from full geometry models are presented. 

The variability in predicted macroscale composite strength can also be seen in the global FE stressstrain responses. For example, Figure 10(a) and (b) contain plots of the predicted gage section stressstrain 	
	 = 0.5 from multiscale analyses of the full dogbone specimen performed using both 

single-fiber and four-fiber RUCs, respectively. Here, the macroscale (continuum-averaged) response was 

determined from the family of elements comprising the gage section of the specimen. The measured 

response from Bowman (1999) is also included in the figure. Both sets of calculations reasonably 

matched the observed specimen behavior, but the variability in predicted strengths was lower for 

simulations performed using a four-fiber RUC (cf., Figure 10(b)) than for a single-fiber RUC (cf., Figure 

10(a)). As the total number of simulated fibers within a RUC increased, the homogenized composite 

strength was less dependent on the strengths of individual fibers. The variability in predicted strengths, 

therefore, decreased with increasing numbers of fibers within the RUC (cf., Figure 9 and Figure 10). 
NASA/TM—2014-215990 14 

Figure 10.—Gage section stress-strain response of a longitudinally reinforced 

SCS-6/ TIMETAL 21S MMC for simulations using (a) single-fiber RUC and 

(b) four-fiber RUC. The coarse FE mesh was used and RUC fiber subcell 




	 = 0.5. Results are compared against gage 

section experimental data obtained from Bowman (1999). 

Accounting for the distribution of constituent morphologies and properties across relevant length 

scales is crucial for predicting progressive damage accumulation and failure behavior of composite 

materials using multiscale analyses. For example, Figure 11 shows the distribution of failed elements after 

damage localization has occurred for a series of multiscale FE simulations with different microscale 

distributions of fiber strengths (including one using one-eighth symmetry boundary conditions). Images 

of three fractured test specimens are also shown in Figure 11(e). A progressive failure analysis was 

performed where a constant fiber strength value was used in each single-fiber RUC throughout the full 

dogbone specimen mesh (Figure 11(a)). For this case, although no a priori geometric symmetries were 

imposed, a symmetric distribution of local failures occurred. These failures emanated from regions with a 

slight stress concentration due to a decrease in specimen cross-section. These results were consistent with 

those obtained by Bednarcyk and Arnold (2007), (2011). Multiscale analyses were also performed using a 

one-eighth symmetry model, where a statistical distribution of microscale fiber strengths was employed 

within four-fiber RUCs. Multiple failures initiated at points of local material weakness that varied from 

simulation to simulation. Damage progression consisted of a series of distributed fiber failures that were 

mirrored about the eight planes of model symmetry; failed elements from three representative analyses 

are shown in Figure 11(b). In contrast, Figure 11(c) and (d) show the distribution of failed elements for 

each of three multiscale analyses of the full dogbone specimen employing single-fiber and four-fiber 
NASA/TM—2014-215990 15 

Figure 11.—
!"	#
$%(a) Full dogbone specimen 

employing a single-fiber RUC with a constant fiber strength. (b) One-eighth symmetry dogbone specimens 

employing four-fiber RUCs with a distribution of fiber strengths. (c) Full dogbone specimens employing single-fiber 

RUCs with a distribution of fiber strengths. (d) Full dogbone specimens employing four-fiber RUCs with a 

distribution of fiber strengths; blue represents no failure and red indicates complete fiber subcell failure for a given 

element. (e) Experimentally observed failure behavior. 

RUCs, respectively, where a statistical distribution of fiber strengths was employed. Here, fiber failure 

localization was distributed throughout the gage section (cf., Figure 11(c) and (d)) consistent with the 

experimentally observed fracture behavior (cf., Figure 11(e)). Note that damage/ failure was much more 

widespread for the four-fiber RUC simulations (cf., Figure 11(d)) than for the single-fiber RUC 

simulations (cf., Figure 11(c)). Using a single-fiber RUC, the rapid onset of localized failures also led to 

fewer total fiber failures across the model compared to simulations using four-fiber RUCs. These results 

demonstrate that the use of global symmetry boundary conditions is inappropriate for progressive failure 

analyses when the actual spatial distribution of fiber strengths display no such symmetries. Such global 

symmetry models can lead to higher strength predictions as well as unrealistic failure patterns. How this 

result impacts findings of Blassiau, Thionnet, and Bunsell (2008) is unclear. 

For illustration purposes, ten additional multiscale progressive failure analyses using a coarse FE 

mesh were performed in which a 	

		 = 0.5) were employed within 25-fiber 

RUCs. This RUC was used since it led to RUC-averaged ultimate strengths that were somewhat 

independent of the number of simulated fibers (cf., Figure 5). Predictions of the macroscale composite 

stress-strain response and ultimate strength from these analyses were compared to a similar analysis 

where the local fiber strength was held constant throughout the mesh (cf., Figure 11(a)). In the constant 

strength analysis, a fiber strength corresponding to the mean value from the Weibull fiber strength 

distribution was used. The predicted macroscale composite strength obtained using a constant local fiber 

strength was markedly different from those from analyses where a spatial distribution of local fiber 

strengths was simulated. Figure 12 contains the predicted uniaxial stress-strain responses obtained using a 

constant local fiber strength and statistically varying local fiber strength models. As can be seen from the 

figure, use of a constant local fiber strength led to a roughly 25.5 percent higher estimate of the 

macroscale composite strength than the mean strength from the locally varying fiber strength simulations. 

Moreover, the constant local fiber strength simulation resulted in a series of element failures emanating 
NASA/TM—2014-215990 16 

Figure 12.—Gage section stress-strain response of a longitudinally reinforced SCS-6/ 

TIMETAL 21S MMC for simulations using 25-fiber RUCs with the coarse FE mesh 

	 = 0.5. 

from mild stress concentrations in the transition region between the grips and gage section (cf., Figure 

11(a)). These results were inconsistent with experimental observations where the actual specimens tended 

to fail within the gage section (cf., Figure 11(e)). In contrast, the predicted failure locations obtained using 

spatially varying local fiber strengths predominately occurred within the gage section. These results 

underscore the importance of accounting for spatial variations in the distributions of fiber strengths when 

performing multiscale analyses. 

5.0 Local/ Global Discretization Considerations 

The preceding multiscale calculations investigated the effect of variations in the local distribution of 

fiber strengths on global MMC composite failure based upon a fixed level of macroscale discretization, 

i.e., a relatively coarse FE mesh was used to simulate the global response. One key element in the 

development of robust, computationally-efficient multiscale materials models is to assess the appropriate 

level of model discretization at each relevant length scale that leads to tractable yet accurate macroscale 

solutions. To illustrate this point, consider an arbitrary SCS-6/ TIMETAL 21S MMC subvolume 

containing 16 uniformly distributed aligned fibers. The continuum-level material response of this 

subvolume may be determined from multiscale models that employ different levels of discretization at the 

macroscale and microscale, respectively. For example, using eight-noded linear isoparametric elements 

(each with eight integration points), the 16-fiber subvolume could be idealized using one FE to simulate 

the global response and a four-fiber RUC at each integration point to simulate the local axial response 

using MAC/GMC (cf., Figure 13). Alternatively, the material subvolume could be idealized using eight

FEs at the macroscale and a single-fiber RUC surrounding each FE integration point (cf., Figure 13). Of 

course, the latter multiscale discretization results in an eight-fold increase in the number of FEs, and a 

corresponding increase in model degrees-of-freedom at the macroscale. Both multiscale discretizations 

were used to perform multiscale progressive failure analyses of the NASA SCS-6/ TIMETAL 21S 

dogbone specimen, where relatively coarse (2400 element) and fine (19,200 element) global FE models 

(cf., Figure 7) were used in conjunction with four- and single-fiber RUCs, respectively. The FE to RUC 

ratio was held constant for each of these simulations; 96 RUCs were used for the coarse FE mesh while 

768 RUCs were used for the fine FE mesh. 
NASA/TM—2014-215990 17 

Figure 13.—Comparison of mesh discretization at different scales where a constant 

simulation volume is maintained. The first case combines a coarse (2400 FE) mesh 

with a four-fiber RUC while the second case combines a fine (19,200 FE) mesh 

(i.e., double the mesh density) with a single-fiber RUC. 

Analyses were performed using each discretization where the local fiber strengths were assigned to 

individual subcells in the same manner as before using the modified Weibull CDF (Eq. (1)) with the fiber 

		
	 = 0.5. The characteristic fiber length (L) in Equation (1) corresponded to half of the 

average FE dimension in the axial direction for the coarse (L = 0.64 mm) and fine (L = 0.32 mm) meshes, 

respectively. Note that as the characteristic length (L) was reduced for the fine FE mesh, and the mean 

fiber strength and variation in the associated distribution of fiber strengths increased slightly (cf., Figure 

14). Ten multiscale simulations were performed for each multiscale discretization/local fiber strength 

parameter combination, and the average macroscale ultimate composite strength was determined for each 

case. 

Figure 15 contains plots of the macroscale stress-strain responses from each of ten representative 

multiscale analyses performed using the coarse mesh/ four-fiber RUC and fine mesh/single-fiber RUC 

discretizations. The analyses performed using the fine mesh/ single-fiber RUC yielded an average 

ultimate composite strength of 1059.9 MPa (standard deviation, 5.3 MPa). The coarse mesh/four-fiber 

RUC analyses produced an average ultimate composite strength of 1022.3 MPa (standard deviation, 

8.0 MPa). 
NASA/TM—2014-215990 18 

Figure 14.—Modified Weibull PDFs that illustrate the effect of the length scale 

(L/L0%!
&	 = 0 case, 

the fiber strength distribution no longer depends on length; but when 

	 = 0.5 and 1, the length scale has a noticeable effect on the strength 

distribution. 

Figure 15.—Gage section stress-strain response of a longitudinally reinforced 

SCS-'*+<!>?@HJ<<K"	 = 0.5) using the 

coarse FE mesh with a four-fiber RUC and the fine FE mesh with a singlefiber RUC. 
NASA/TM—2014-215990 19 

Figure 16.—Distribution of fiber failures within the fine mesh 

after the onset of localization for three single-fiber RUC 


"	#
$%

represents no failure and red indicates complete fiber subcell 

failure for a given element. Each illustration denotes the 

outer layer of elements in a separate simulation. 

Figure 16 shows the distribution of FEs with failed fibers from three separate analyses obtained using 

the fine mesh/ single-fiber RUC discretization for  = 0.5. The distribution of failure locations is similar 

to that found using a coarse mesh/ four-fiber RUC discretization (cf., Figure 11(d)). These results suggest 

that both multiscale discretizations lead to similar estimates of the macroscale composite material 

behavior. The solution time for the fine mesh/ single-fiber RUC analyses, however, was roughly nine 

times greater than for the coarse mesh/ four-fiber RUC analyses without a significant difference in 

calculated results. 
NASA/TM—2014-215990 20 

Clearly, the model discretization at a specific length scale can have a profound effect on the 

computational costs associated with multiscale simulations. Since the solution algorithm implemented in 

the GMC is substantially more efficient than that for traditional FE analyses (Pindera and Bednarcyk 

(1999)), use of a more highly refined microscale model in combination with a coarser global FE mesh led 

to more computationally effective solutions for this problem. For similar reasons, coupled MAC/GMCFE multiscale analyses may be much more computationally efficient for a given problem than analogous 

multiscale analyses that are purely FE based (i.e., FE2

 (Feyel and Chaboche (2000); Feyel (2003))). In 

general, the optimal discretization at each relevant length scale is likely to be problem dependent. For 

instance, problems with a global stress gradient (e.g., open-hole composites or full scale structures), may 

require a different discretization than that used here. These sorts of issues must be addressed in order to 

fully exploit the benefits of multiscale analyses. 

6.0 Summary and Conclusions 

A parametric study investigating the effect of a statistical fiber strength distribution and repeating unit 

cell (RUC) architecture on the predicted composite ultimate strength was performed using the 

computationally efficient code MAC/GMC for a SCS-6/TIMETAL 21S material system. Progressive 

failure at an elevated temperature of 650°C was simulated at the microscale by assigning strengths to 

individual fibers within an RUC based upon a modified Weibull cumulative distribution function (CDF), 

which accounts for the effect of fiber length on the probability of failure. By increasing the number of 

fibers in the RUC, a more gradual, continuum-like stress-strain behavior was observed. 

Global multiscale progressive failure analyses of a 25 percent fiber volume fraction SCS-6/ 

TIMETAL 21S tensile dogbone specimen at 650 °C were performed by implementing a modified Weibull 

CDF of fiber strengths within FEAMAC/ ABAQUS. Fiber strengths were appropriately assigned to 

individual fibers within RUCs corresponding to specific finite elements (FEs) in order to assess the effect 

of a spatial distribution of local fiber strengths on the predicted macroscale composite stress-strain 

response and failure. The composite ultimate strengths and distribution of failure locations 

(predominately within the gage section) reasonably matched the experimentally observed failure 

behavior. Moreover, these analyses suggest that the use of models that exploit global geometric 

symmetries biases the characteristics of failure and are thus inappropriate for cases where the actual 

distribution of local fiber strengths displays no such symmetries. This issue has not received much 

attention in the literature. Additionally, the discretization at a specific length scale can have a profound 

effect on the computational costs associated with multiscale simulations. Multiscale analyses were 

performed using coarse FE mesh/ our-fiber RUC and fine FE mesh/ single-fiber RUC discretizations. 

Both multiscale discretizations led to similar estimates of the macroscale composite material behavior and 

failure. The solution time for the fine mesh/ single-fiber RUC analyses, however, was roughly nine times 

greater than for the coarse mesh/four-fiber RUC analyses. Clearly, the model discretization at a specific 

length scale can have a profound effect on the computational costs associated with multiscale simulations. 

Understanding these issues is crucial to the development of robust multiscale material models that yield 

accurate yet tractable results. 

References 

ABAQUS Unified FEA, (2013): December 29, 2013. 

Aboudi, J. (1991): Mechanics of Composite Materials: A Unified Micromechanical Approach, Elsevier. 

Aboudi, J. (1995): Micromechanical Analysis of Thermo-Inelastic Multiphase Short-Fiber Composites. 

Compos. Eng., vol. 5, no. 7, pp. 839-850. 

Aboudi, J. (1996): Micromechanical Analysis of Composites by the Method of Cells - Update. App. 

Mech. Rev., vol. 49, no. 10S, pp. S83-S91. 

Aboudi, J. (2004): The Generalized Method of Cells and High-Fidelity Generalized Method of Cells 

Micromechanical Models - A Review. Mech. Adv. Mat. Struct., vol. 11, no. 4-5, pp. 329-366. 
NASA/TM—2014-215990 21 

Aboudi, J.; Arnold, S. M.; Bednarcyk, B. A. (2012): Micromechanics of Composite Materials: A 

Generalized Multiscale Analysis Approach, Butterworth-Heinemann. 

Andersons, J.; Joffe, R.; Hojo, M.; Ochiai, S. (2002): Glass Fibre Strength Distribution Determined by 

Common Experimental Methods. Compos. Sci. Tech., vol. 62, no. 1, pp. 131-145. 

Andersons, J.; Spamins, E.; Joffe, R.; Wallstrom, L. (2005): Strength Distribution of Elementary Flax 

Fibres. Compos. Sci. Tech., vol. 65, no. 3-4, pp. 693-702. 

Arnold, S. M. Saleeb, A. F. (1994): On the Thermodynamic Framework of Generalized Coupled 

Thermoelastic-Viscoplastic-Damage Modeling. Int. J. Plast., vol. 10, no. 3, pp. 263-278. 

Arnold, S. M.; Saleeb, A. F.; Castelli, M. G. (1996): A Fully Associative, Nonisothermal, Non-Linear 

Kinematic, Unified Viscoplastic Model for Titanium Based Matrices, Life Prediction Methodology 

for Titanium Matrix Composites. Johnson, W. S., Larsen, J. M., and Cox, B. N. American Society for 

Testing and Materials. 

Baxevanakis, C.; Jeulin, D.; Valentin, D. (1993): Fracture Statistics of Single-Fibre Composite 

Specimens. Compos. Sci. Tech., vol. 48, no. 1-4, pp. 47-56. 

Bednarcyk, B. A. Arnold, S. M. (2001): Micromechanics-Based Deformation and Failure Prediction for 

Longitudinally Reinforced Titanium Composites. Compos. Sci. Tech., vol. 61, no. 5, pp. 705-729. 

Bednarcyk, B. A. Arnold, S. M. (2002): MAC/GMC 4.0 User's Manual - Keywords Manual. NASA/TM 

2002-212077/VOL2. 

Bednarcyk, B. A. Arnold, S. M. (2007): A Framework for Performing Multiscale Stochastic Progressive 

Failure Analysis of Composite Structures. NASA/TM 2007-214694. 

Bednarcyk, B. A. Arnold, S. M. (2011): A Multiscale, Nonlinear, Modeling Framework Enabling the 

Design and Analysis of Composite Materials and Structures, Models, Databases, and Simulation 

Tools Needed for the Realization of Integrated Computational Materials Engineering. Arnold, S. M. 

and Wong, T. ASM International. 

Beyerlein, I. J. Phoenix, S. L. (1996): Statistics for the Strength and Size Effects of Microcomposites with 

Four Carbon Fibers in Epoxy Resin. Compos. Sci. Tech., vol. 56, no. 1, pp. 75-92. 

Blassiau, S.; Thionnet, A.; Bunsell, A. R. (2006a): Micromechanisms of Load Transfer in a 

Unidirectional Carbon Fibre-Reinforced Epoxy Composite Due to Fiber Failures. Part 1: 

Micromechanisms and 3D Analysis of Load Transfer: The Elastic Case. Compos. Struct., vol. 74, no. 

3, pp. 303-318. 

Blassiau, S.; Thionnet, A.; Bunsell, A. R. (2006b): Micromechanisms of Load Transfer in a 

Unidirectional Carbon Fibre-Reinforced Epoxy Composite Due to Fiber Failures. Part 2: Influence of 

Viscoelastic and Plastic Matrices on the Mechanisms of Load Transfer. Compos. Struct., vol. 74, no. 

3, pp. 319-331. 

Blassiau, S.; Thionnet, A.; Bunsell, A. R. (2008): Micromechanisms of Load Transfer in a Unidirectional 

Carbon Fibre-Reinforced Epoxy Composite Due to Fiber Failures. Part 3: Multiscale Reconstruction 

of Composite Behavior. Compos. Struct., vol. 83, no. 3, pp. 312-323. 

Bowman, C. L.: (1999): Experimentation and Analysis of Mechanical Behavior Modification of Titanium 

Matrix Composites Through Controlled Fiber Placement, Case Western University, Department of 

Materials Science and Engineering. 

Curtin, W. A. (1991): Theory of Mechanical Properties of Ceramic-Matrix Composites. J. Am. Cer. Soc., 

vol. 74, no. 11, pp. 2837-2845. 

Curtin, W. A. (1993): Ultimate Strengths of Fibre-Reinforced Ceramics and Metals. Compos., vol. 24, no. 

2, pp. 98-102. 

Curtin, W. A. (2000): Tensile Strength of Fibre-Reinforced Composites III. Beyond the Traditional 

Weibull Model for Fiber Strengths. J. Compos. Mat., vol. 34, no. 15, pp. 1301-1332. 

Feyel, F. Chaboche, J.-L. (2000): FE2

 Multiscale Approach for Modelling the Elastoviscoplastic Behavior 

of Long Fibre SiC/Ti Composite Materials. Comp. Meth. Appl. Mech. Eng., vol. 183, no. 3-4, pp. 

309-330. 
NASA/TM—2014-215990 22 

Feyel, F. (2003): A Multilevel Finite Element Method (FE2) to Describe the Response of Highly NonLinear Structures Using Generalized Continua. Comp. Meth. Appl. Mech. Eng., vol. 192, no. 28-30, 

pp. 3233-3244. 

Foster, G.; Ibnabdeljalil, M.; Curtin, W. A. (1998): Tensile Strength of Titanium Matrix Composites: 

Direct Numerical Simulations and Analytic Models. Int. J. Sol. Struct., vol. 35, no. 19, pp. 2523-

2536. 

Graham-Brady, L. L.; Arwade, S. R.; Corr, D. J.; Gutierrez, M. A.; Breysse, D.; Grigoriu, M.; Zabaras, N. 

(2006): Probability and Materials: From Nano- to Macro-Scale: A Summary. Prob. Eng. Mech., vol. 

21, no. 3, pp. 193-199. 

Harlow, D. G. Phoenix, S. L. (1978): The Chain-of-Bundles Probability Model for the Strength of Fibrous 

Materials I: Analysis and Conjectures. J. Compos. Mat., vol. 12, no. 2, pp. 195-214. 

Helton, J. C. Davis, F. J. (2003): Latin Hypercube Sampling and the Propagation of Uncertainty in 

Analyses of Complex Systems. Rel. Eng. Sys. Saf., vol. 81, no. 1, pp. 23-69. 

Hitchon, J. W. Phillips, D. C. (1979): The Dependence of the Strength of Carbon Fibres on Length. Fib. 

Sci. Tech., vol. 12, no. 3, pp. 217-233. 

Lacy, T. E.; McDowell, D. L.; Talreja, R. (1999): Gradient Concepts for Evolution of Damage. Mech. 

Mat., vol. 31, no. 12, pp. 831-860. 

Landis, C. M.; Beyerlein, I. J.; McMeeking, R. M. (2000): Micromechanical Simulation of the Failure of 

Fiber Reinforced Composites. J. Mech. Phys. Sol., vol. 48, no. 3, pp. 621-648. 

Lekou, D. J. Philippidis, T. P. (2008): Mechanical Property Variability in FRP Laminates and Its Effect 

on Failure Prediction. Compos. B, vol. 39, no. 7-8, pp. 1247-1256. 

Mall, S.; Fecke, T.; Foringer, M. A. (1998): Titanium Matrix Composites: Mechanical Behavior, 

Technomic Publishing Co. 

McKay, M. D. Beckman, R. J. (1979): A Comparison of Three Methods for Selecting Values of Input 

Variables in the Analysis of Ouput from a Computer Code. Technom., vol. 21, no. 2, pp. 239-245. 

Naito, K.; Yang, J. M.; Tanaka, Y.; Kagawa, Y. (2012): The Effect of Gauge Length on Tensile Strength 

and Weibull Modulus of Polyacrylonitrile (PAN) and Pitch-Based Carbon Fibers. J. Mat. Sci., vol. 

47, no. 2, pp. 632-642. 

Oh, K. P. (1979): A Monte Carlo Study of the Strength of Unidirectional Fiber-Reinforced Composites. J. 

Compos. Mat., vol. 13, no. 4, pp. 311-328. 

Okabe, T.; Takeda, N.; Kamoshida, Y.; Shimizu, M.; Curtin, W. A. (2001): A 3D Shear-Lag Model 

Considering Micro-Damage and Statistical Strength Prediction of Unidirectional Fiber-Reinforced 

Composites. Compos. Sci. Tech., vol. 61, no. 12, pp. 1773-1787. 

Okabe, T. Takeda, N. (2002): Size Effect on Tensile Strength of Unidirectional CFRP Composites - 

Experiment and Simulation. Compos. Sci. Tech., vol. 62, no. 15, pp. 2053-2064. 

Padgett, W.; Durham, S.; Mason, A. (1995): Weibull Analysis of the Strength of Carbon Fibers Using 

Linear and Power Law Models for Length Effect. J. Compos. Mat., vol. 29, no. 14, pp. 1873-1884. 

Paley, M. Aboudi, J. (1992): Micromechanical Analysis of Composites by the Generalized Cells Model. 

Mech. Mat., vol. 14, no. 2, pp. 127-139. 

Pindera, M. J. Bednarcyk, B. A. (1999): An Efficient Implementation of the Generalized Method of Cells 

for Unidirectional, Multi-Phased Composites with Complex Microstructures. Compos. B, vol. 30, no. 

1, pp. 87-105. 

Rosen, B. W. (1964): Tensile Failure of Fibrous Composites. AIAA J., vol. 2, no. 11, pp. 1985-1991. 

Shen, L. Xu, X. F. (2010): Multiscale Stochastic Finite Element Modeling of Random Elastic 

Heterogeneous Materials. Comput. Mech., vol. 45, no. 6, pp. 607-621. 

Sriramula, S. Chryssanthopoulos, M. K. (2009): Quantification of Uncertainty Modeling in Stochastic 

Analysis of FRP Composites. Compos. A, vol. 40, no. 11, pp. 1673-1684. 

Sullivan, R. W. Arnold, S. M. (2011): An Annotative Review of Multiscale Modeling and its Application 

to Scales Inherent in the Field of ICME, Models, Databases, and Simulation Tools Needed for the 

Realization of Integrated Computational Materials Engineering. Arnold, S. M. and Wong, T. ASM 

International. 
NASA/TM—2014-215990 23 

Watson, A. Smith, R. (1985): An Examination of Statistical Theories for Fibrous Materials in the Light of 

Experimental Data. J. Mat. Sci., vol. 20, no. 9, pp. 3260-3270. 

Weibull, W. (1951): A Statistical Distribution Function of Wide Applicability. J. Appl. Mech., vol. 18, no. 

pp. 293-297. 

Worthem, D. W. (1994): Flat Tensile Specimen Design for Advanced Composites. NASA Contractor 

Report 185261. 

Xu, X. F.; Chen, X.; Shen, L. (2009): A Green-Function-Based Multiscale Method for Uncertainty 

Quantification of Finite Body Random Heterogeneous Materials. Comput. Struct., vol. 87, no. 21-22, 

pp. 1416-1426. 

Zweben, C. (1968): Tensile Failure of Fiber Composites. AIAA J., vol. 6, no. 12, pp. 2325-2331.