Kuang C. Liu

Arizona State University, Tempe, Arizona 

Steven M. Arnold

Glenn Research Center, Cleveland, Ohio

Stochastic Nonlinear Response of Woven CMCs

NASA/TM—2013-217859

June 2013
NASA STI Program . . . in Profile

Since its founding, NASA has been dedicated to the 

advancement of aeronautics and space science. The 

NASA Scientific and Technical Information (STI) 

program plays a key part in helping NASA maintain 

this important role.

The NASA STI Program operates under the auspices 

of the Agency Chief Information Officer. It collects, 

organizes, provides for archiving, and disseminates 

NASA’s STI. The NASA STI program provides access 

to the NASA Aeronautics and Space Database and 

its public interface, the NASA Technical Reports 

Server, thus providing one of the largest collections 

of aeronautical and space science STI in the world. 

Results are published in both non-NASA channels 

and by NASA in the NASA STI Report Series, which 

includes the following report types:

• TECHNICAL PUBLICATION. Reports of 

completed research or a major significant phase 

of research that present the results of NASA 

programs and include extensive data or theoretical 

analysis. Includes compilations of significant 

scientific and technical data and information 

deemed to be of continuing reference value. 

NASA counterpart of peer-reviewed formal 

professional papers but has less stringent 

limitations on manuscript length and extent of 

graphic presentations.

• TECHNICAL MEMORANDUM. Scientific 

and technical findings that are preliminary or 

of specialized interest, e.g., quick release 

reports, working papers, and bibliographies that 

contain minimal annotation. Does not contain 

extensive analysis.

• CONTRACTOR REPORT. Scientific and 

technical findings by NASA-sponsored 

contractors and grantees.

• CONFERENCE PUBLICATION. Collected 

papers from scientific and technical 

conferences, symposia, seminars, or other 

meetings sponsored or cosponsored by NASA.

• SPECIAL PUBLICATION. Scientific, 

technical, or historical information from 

NASA programs, projects, and missions, often 

concerned with subjects having substantial 

public interest.

• TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientific and 

technical material pertinent to NASA’s mission.

Specialized services also include creating custom 

thesauri, building customized databases, organizing 

and publishing research results.

For more information about the NASA STI 

program, see the following:

• Access the NASA STI program home page at 

http://www.sti.nasa.gov

• E-mail your question to help@sti.nasa.gov

• Fax your question to the NASA STI 

Information Desk at 443–757–5803

• Phone the NASA STI Information Desk at

443–757–5802

• Write to:

 STI Information Desk

 NASA Center for AeroSpace Information

 7115 Standard Drive

 Hanover, MD 21076–1320
Kuang C. Liu

Arizona State University, Tempe, Arizona 

Steven M. Arnold

Glenn Research Center, Cleveland, Ohio

Stochastic Nonlinear Response of Woven CMCs

NASA/TM—2013-217859

June 2013

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5301 Shawnee Road

Alexandria, VA 22312

Available electronically at http://www.sti.nasa.gov

Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2013-217859 1

Stochastic Nonlinear Response of Woven CMCs

Kuang C. Liu

Arizona State University

Tempe, Arizona 85287

Steven M. Arnold

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abstract

It is well known that failure of a material is a locally driven event. In the case of ceramic matrix 

composites (CMCs), significant variations in the microstructure of the composite exist and their 

significance on both deformation and life response need to be assessed. Examples of these variations 

include changes in the fiber tow shape, tow shifting/nesting and voids within and between tows. In the 

present work, the influence of scale specific architectural features of woven ceramic composite are 

examined stochastically at both the macroscale (woven repeating unit cell (RUC)) and structural scale 

(idealized using multiple RUCs). The recently developed MultiScale Generalized Method of Cells 

methodology is used to determine the overall deformation response, proportional elastic limit (first matrix 

cracking), and failure under tensile loading conditions and associated probability distribution functions. 

Prior results showed that the most critical architectural parameter to account for is weave void shape and 

content with other parameters being less in severity. Current results show that statistically only the postelastic limit region (secondary hardening modulus and ultimate tensile strength) is impacted by local 

uncertainties both at the macro and structural level.

Introduction

Multiscale modeling has been applied to both laminated and woven composites in the past. Although 

nomenclature in the literature varies, typically a multiscale modeling analysis will follow length scales 

shown in Figure 1 for continuum-based modeling. These scales, progressing from left to right in Figure 1, 

are the microscale (constituent level; fiber, matrix, interface), the mesoscale (tow), the macroscale 

(repeating woven unit cell), and the global/structural scale. Traditionally, one traverses (transcends 

(moves right) or descends (moves left)) these scales via homogenization and localization techniques, 

respectively (Figure 1 and Figure 2(a)); where a homogenization technique provides the properties or 

response of a “structure” (higher level) given the properties or response of the structure’s “constituents” 

(lower scale). Conversely, localization techniques provide the local fields of the constituents given the 

response of the structure. Figure 2(b) illustrates the interaction of homogenization and localization 

techniques, in that during a multiscale analysis, a particular stage in the analysis procedure can function 

on both levels simultaneously. For example, during the process of homogenizing the stages represented 

by X and Y to obtain properties for the stage represented by V, X and Y should be viewed as the 

constituent level while V is on the structure level. However, during the process of homogenizing V and W 

to obtain properties for U, V is now on the constituent level (as is W). Obviously, the ability to 

homogenize and localize accurately requires a sophisticated theory that relates the geometric and material 

characteristics of structure and constituents.
NASA/TM—2013-217859 2

Figure 1.—Illustration of associated levels scales for woven/braided composite analysis.

Figure 2.—(a) Homogenization provides the ability to determine structure 

level properties from constituent level properties while localization 

provides the ability to determine constituent level responses from 

structure level results. (b) Example tree diagram.

With the recent development of the MultiScale Generalize Method of Cells (MSGMC), one can now 

ascertain the influence of architectural parameters, such as volume fraction, weave geometry, tow 

geometry, etc., at each associated length scale, for composites; particularly woven and braided 

composites. This enables the determination of which effect/parameter, at a given length scale, is 

impactful/relevant at higher length scales. For example, matrix elastic modulus is a microscale effect, 

changing this value will have a direct effect at the next largest length scale (e.g., mesoscale), but its effect 

at the macro or structural scale cannot necessarily be assumed. Similarly the tow fiber volume fraction, 

which is a mesoscale effect, should have a direct impact on the response at the macroscale, yet its effect at 

the global scale is difficult to deduce a priori. Furthermore, experimental investigations have shown that 

in typical composite (particularly woven) materials there exist significant variations in the meso and 

macroscale architectural features. Yet most analyses performed assume an idealized or pristine material 

and architecture at every length scale. Such an assumption was required, up until now, to avoid the 

(a) (b)
NASA/TM—2013-217859 3

computationally exhaustive multiscale modeling of every minute variation in architecture at every length 

scale, via the finite element method.

In prior work (Refs. 1 and 2), MSGMC was utilized to perform a multiscale1 investigation in which 

the influence of scale specific architectural features (e.g., tow fiber volume fraction, tow aspect ratio, tow 

void volume fraction, weave void distribution, void shape of woven composites, etc.) given a 

deterministic viewpoint, where examined. Prior results showed that the most critical architectural 

parameter to account for is weave void distribution and shape with other parameters being less in severity. 

Here the objective is to perform a similar multiscale investigation to References 1 and 2, but now with a 

stochastic viewpoint (i.e., assume a priori a statistical distribution for each lower length scale feature of 

interest), in order to determine the expected variance in response at both the macro and structural scale. In 

this study, as in our previous work, nesting/ply shifting will be ignored.

Multiscale Generalized Method of Cells

Overview

Analysis of woven fabric composites can be generalized into several relevant length scales (from 

largest to smallest): structural, macro, meso, and micro. The structural scale considers a relevant length 

scale in analysis, such as the gage section in a test coupon. The macroscale weave refers to the RUC of 

the weave, for a five-harness satin fabric, see Figure 3. The mesoscale refers to an RUC of the fiber 

tow; wherein this RUC represents a bundle of fibers (typically 700 to 1000 for ceramic matrix 

composites) with a given packing arrangement. The smallest length scale is the microscale, which 

represents the fundamental constituent materials, such as the monofilament fiber and matrix itself.

This multiscale analysis uses the recently developed MSGMC methodology (Refs. 1 and 2). To

simulate the structural response a representative volume element is used which contains Nα × Nβ × Nγ

macroscale RUCs. At the macroscale, each fabric composite (e.g., plain or 5HS, see Figure 6) was 

discretized into N{αβγ}α × N{αβγ}β × N{αβγ}γ subcells using the assumption of triple periodicity; wherein, for 

example, a subcell used to represent a fiber tow is further idealized at the mesoscale by N{αβγ}{αβγ}β ×

N{αβγ}{αβγ}γ subcells using the assumption of double periodicity, wherein each of these subcells are 

represented by the constitutive properties of either a fiber or matrix at the microscale. This recursive 

methodology (wherein the generalized method of cells (GMC), see Paley and Aboudi (Ref. 3) and Aboudi

(Ref. 4), is called within GMC) is shown schematically in Figure 4, and can be accounted for by attaching 

the superscript {αβγ} to each level. There are several architectural parameters at the meso, macro, and 

structural level required to fully define the discretized subcell geometries. At the mesoscale, both tow 

volume fraction and tow packing are required, while at the macroscale, weave architecture, weave volume 

fraction, tow aspect ratio and ply nesting are required. Furthermore, at the structural level, the spatial 

distribution of the macroscale RUCs are required, i.e., uniform—each subcell is associated with the same 

macroscale RUC or random—subcells are associated with a uniform distribution of macroscale RUCs. It 

has been of recent interest to study the effects of these parameters and understand what the driving factors 

for both elastic and inelastic response, see (Ref. 5).

 1

Here the term multiscale refers to an analysis in which at least three levels of scales are accounted for, wherein at 

least two homogenizations/localizations are required.
NASA/TM—2013-217859 4

Figure 3.—Five harness satin (5HS) macroscale RUC.

Figure 4.—Multiscale methodology with architectural effects being varied shown at three length 

scales considered. Actual micrographs are complements of P. Bonacuse, NASA GRC, 2010.

Microscale (Constitutive Modeling)

The MSGMC is used to represent the woven fabric composite starting with its constituent materials, 

i.e., the fiber (monofilament) and matrix and progress up the various length scales. The microscale is 

the only length scale where explicit constitutive models are applied to the various phases (e.g., fiber and 

matrix). Constitutive behavior for larger length scales are determined through the GMC triply-periodic 

homogenization procedure developed by Aboudi (Ref. 4). The monofilament fibers are modeled using a 

linear elastic relationship, i.e., Hooke’s Law, and the matrix material is represented by a scalar damage 

mechanics type relationship based on a tangent modulus relationship. Details for the damage model can 

be found in the following sections. Although, ceramics are typically stochastic herein all microscale 

constituent parameters (i.e., modulus, failure strength, etc.) were assumed to be deterministic in this 

analysis. The stresses in any subcell in the microscale can be determined from the following equation. 

The stiffness C{αβγ}{αβγ}{βγ} is determined from the given material parameters and modified by a scalar 

damage measure λ{αβγ}{αβγ}{βγ} and the strains ε

{αβγ}{αβγ}{βγ} are determined from localization from the 

mesoscale. This is possible through a concentration matrix, A, determined by GMC, which is a function 

of the subcell geometry and stiffness matrix.
NASA/TM—2013-217859 5

{ }{ }{ } { }{ }{ } ( ) αβγ αβγ βγ αβγ αβγ βγ αβγ αβγ βγ αβγ αβγ βγ { }{ }{ } { }{ }{ } σ ε = λ C (1)

{αβγ αβγ βγ αβγ αβγ βγ αβγ αβγ }{ }{ } { }{ }{ } { }{ } ε ε = A (2)

Matrix Constituent Damage Modeling

The matrix material, assumed to be the same for both the inter-weave and intra-tow, is modeled using 

linear elasticity with evolving damage, that is

σ = (1− φ)Cε (3)

The scalar damage variable, φ, varies between zero (no damage) and one (complete failure/damage) and 

its evolution is driven by the magnitude of triaxiality, i.e., the first invariant of the stress/strain tensor. 

This continuum damage model enables the capturing of progressive moduli reduction resulting from the 

initiation and propagation of micro and meso matrix cracks ultimately leading to brittle failure. Defining 

the damage rule as

(4)

which can be rewritten in incremental form with i+1 denoting the next increment (ε

i+1

= ε

i

+ ∆εi+1).

(5)

where n represents the damaged normalized secant modulus (see Figure 5), K represents the instantaneous 

tangent bulk modulus, and the first invariant stress and strain measures as

(6)

It can be shown that the instantaneous bulk modulus can be related back to the damage variable (see Ref. 2) 

(7)

where the initial value, φ0

, is zero. Note the damage rule in Equation (4) is only active once a critical 

stress criteria has been reach, i.e., it is only valid when σH > σdam.

f nK = ε −σ = 3 0 H H

1 1 3 0 ii i f nK H H + + = ∆ε − ∆σ =

( ) ( )

( ) ( )

11 22 33

1

11 22 33

1

/ 3

3

/ 3

3

H

H

I

I

σ +σ +σ

σ= σ =

ε +ε +ε

ε= ε =

( )

1 1

1 1

1 1

1

i ii

i i H H

i i

H H

n ε λε φ λ

ε ε

+ +

+ +

+ +

∆ + −= = ∆ +
NASA/TM—2013-217859 6

Figure 5.—Schematic showing bulk moduli 

change as function of triaxial strain. Note K0

represents the initial bulk modulus.

Fiber Constituent Failure Model

The fiber is also assumed to behave linearly elastic up to failure, with failure following the Hashin 

type failure criterion put forth in 1980, see Reference 6. This criterion determines the catastrophic failure 

of the fiber based on the axial and shear strengths. When the failure criterion exceeds one, the fiber 

stiffness matrix is degraded to a minimal value. A key assumption made in this analysis is that the 

compliant fiber interface is linear elastic and does not fail independently. The failure stress levels 

presented later are an in-situ failure stress considering the interface.

(8)

Mesoscale (Tow)

The mesoscale is used to represent the periodic structure of a multiphase material (e.g., fiber tow or 

matrix with voids). At the mesoscale in the case of a fiber tow, there are two significant microstructure 

parameters that govern the mesoscale subcell geometries: fiber packing arrangement and tow volume 

fraction. The response of the mesoscale is subject to these parameters as well as the material variation at 

the microscale. The continuous fiber tows are assumed to be represented by a doubly periodic RUC 

composed of N{αβγ}β×N{αβγ}γ rectangular subcells having dimensions h{αβγ}×l{αβγ} that consist of 

constituents homogenized from the microscale, this assumption is confirmed by the tow micrograph 

shown in Figure 4. An example of such an RUC discretized for GMC is shown in Figure 4, where the 

inner region (shown in grey) denotes the fiber tow and the outer region (shown in white) is the matrix. 

The RUC is discretized in such a manner that it is composed of N{αβγ}{αβγ}β × N{αβγ}{αβγ}γ rectangular 

subcells, with each subcell having dimensions h{αβγ}{αβγ}β by l{αβγ}{αβγ}γ. 

The stress within a tow subcell can be determined either through homogenization of stresses at the 

microscale or using an effective constitutive relationship (derived from microscale). The stress 

homogenization relationship, 

{ }

{ } { }

{ }{ }

{ }

{ }

{ }

{ }

{ } { }

1 1

1 N N

h l

h l

αβγ β αβγ γ

αβγ αβγ βγ

αβγ β αβγ γ

αβγ αβγ αβγ β αβγ γ = =

σ σ = ∑ ∑ (9)

is equivalent to the effective constitutive law

{ } { } { } [ ] αβγ αβγ αβγ σ = C ε . (10)

( ) 2 11 2 2

2 2 13 12

1

axial axial

f σ= + σ +σ

σ τ
NASA/TM—2013-217859 7

By using the effective constitutive law, the three terms on the right-hand side are effectively linking the 

micro-, meso-, and macroscales, see Figure 4. The effective stiffness tensor C{αβγ} is derived from the 

homogenization of the microscale stiffness tensors, represented:

{ }

{ } { }

{ } { }{ }

{ }

{ }

{ }

{ }

{ }{ }

{ } { }

1 1

1 N N

h l

h l

αβγ β αβγ γ

αβγ βγ αβγ βγ αβγ βγ

αβγ β αβγ γ

αβγ αβγ αβγ β αβγ γ

λ = =

C = ∑ ∑ C A . (11)

Lastly, the effective total strain, {αβγ} ε can be determined through localization from the macroscale as 

shown in:

{ } { } { } ε A A ε αβγ αβγ βγ = tt ip , (12)

where ε represents the globally applied strain at the macroscale.

In these equations, σ denotes the stress, A denotes strain concentration matrices, and C denotes the 

elastic stiffness matrix at a given length scale. The microscale subcell stresses and stiffness moduli that are 

needed to complete the summation are determined through the applied constitutive models for each 

constituent based on their current strain state. The mesoscale strains, which are used as the applied quantities 

for the GMC analysis, are determined from a through-thickness (tt) homogenization at the macroscale 

analogous (done on a group by group basis, see Figure 7). The subscripts tt and ip on the concentration 

matrices in Equation (13) denote the through-thickness (tt) and in-plane (ip) portion of the two-step 

homogenization process discussed later. Note that the concentration matrices Att and Aip have the same 

representative meaning as the concentration matrix A defined in the generalized method of cells (Ref. 3).

Macroscale (Weave)

At the macroscale, the RUC of the weave fabric is modeled. At this scale, the architecture is governed 

by the overall volume fraction, tow geometry (aspect ratio, width, and thickness), and overall fabric 

thickness, wherein the subcell “constituent” response is dependent on the mesoscale and microscale 

responses. The weave requires a triply periodic RUC representation, of size d×h×l that is discretized into 

Nα×Nβ×Nγ parallelepiped subcells, with each subcell having dimensions dα×hβ×lγ. At this length scale, the 

two-step homogenization procedure was employed to determine the stiffness and macroscale stresses. As 

discussed previously, this procedure was utilized to overcome the lack of shear coupling inherent in the 

GMC formulation, see (Refs. 5 and 7). The first step involves a through-thickness (tt) homogenization, 

and the second step is an in-plane (ip) homogenization. Details for the subcell geometry and RUC 

information can be found in References 5 and 8.

The resulting expressions describing the stress, strain, and stiffness at each stage are given in the 

following equations.

Through-thickness homogenization

{ } { } α

α=

βγ ∑ αβγ

α

= d

d

N

1

1

σ σ (13)

{ } { } { } α αβγ

α=

βγ αβγ ∑

α

= d

d

N

C Att C

1

1 (14)

In-plane homogenization

∑∑ { } β γ

β=

β γ

γ=

βγ =

N N

h l

hl 1 1

1

σ σ (15)
NASA/TM—2013-217859 8

{ } { } β γ

βγ

β= γ=

βγ ∑∑

β γ

= h l

hl

N N

C Aip C

1 1

1 (16)

Consequently, the key localization expression relating the microscale stresses to the global applied loads 

and specific architectural geometry are

{ }{ } { } { }{ } { }{ } { } { }

tt ip ( ) αβγ βγ βγ αβγ βγ αβγ βγ αβγ βγ λ =           σ C A A Aε . (17)

Similarly, the key homogenization equations describing the macroscale stress and stiffness matrix based 

on constituent stiffness and architectural geometry are

{ } { }

{ }{ }

{ }

{ }

{ } { }

{ }

{ }































 = ∑∑ ∑ ∑ ∑ β γ α αβγ β αβγ γ

α= αβγ β=

αβγ β αβγ γ

αβγ γ=

αβγ βγ

αβγ αβγ

α

β= γ=

β γ

N N N N N

h l

h l

d

d

h l

hl 1 1 1 1 1

1 1

σ σ (18)

and

{ } { }

{ } { }

{ } { }{ }

{ }

{ }

{ }{ } { } { }

{ }

{ }































 = ∑∑ ∑ ∑ ∑ = = = = =

β γ α αβγ β αβγ γ

α αβγ β

αβγ β αβγ γ

αβγ βγ

αβγ γ

βγ αβγ βγ

αβγ αβγ

α

αβγ

β γ

β γ

βγ λ

N N N

tt

N N

ip h l

h l

d

d

h l

hl 1 1 1 1 1

1 1 C A A C A (19)

respectively.

Structural Scale (Gage Length)

At the structural scale, a system of macroscale RUCs is represented. In this analysis, a group of Nα ×

Nβ × Nγ macroscale RUCs of size d × h × l were used to form an RUC, where each macroscale subcell is 

of size dα × hβ × lγ. Each macroscale RUC can be different in terms of architectural parameters at every 

embedded length scale and represents the scatter of a typical material. Note if each subcell within the 

structural RUC possessed the same macroscale RUC then the structural and macroscale response would 

be identical. Here a structural applied stress or strain is used to determine the overall system response as a 

function of the previous described length scales. Herein, this length scale represents a region within the 

gage section of a test specimen. Through substitution of the strains from each length scale the stresses at 

the microscale can be written in terms of a structural applied strain and the stresses at the structural scale 

can also be written in terms of the microscale stresses and stiffness.

{ }

111

1 N N N

d hl

dhl

α β γ

αβγ α βγ

α= β= γ=

σ σ = ∑∑∑ (20)

{ } { }

111

1 N N N

A d hl

dhl

α β γ

αβγ αβγ α βγ

α= β= γ=

C C = ∑∑∑ (21)

σ = Cε (22)
NASA/TM—2013-217859 9

Figure 6.—Plain and five harness satin repeating unit cell.

Modeling Ceramic Matrix Composites With MSGMC

Weave Repeating Unit Cell

For this particular study, two weaves are considered: a plain weave (PW) and five harness satin (5HS)

weave. In this idealization of the architecture, the repeating unit cell is assumed to be representative of the 

entire structure. Pictures of the fabrics with their repeating unit cell (RUC) outlined in red are shown in 

Figure 6. To create an RUC suitable for analysis, the weave is discretized into several subvolume cells at 

this scale. There are two types of materials comprising all the subcells: fiber tows and interweave matrix. 

This final three-dimensional discretization for a five harness satin is shown in Figure 7, along with example 

lower scale RUCs representing the multiscale analyses of the interweave voids, tows and intra-tow voids. In 

the figure, fiber tows are indicated through the lined subcells. The lines indicate the direction of 

orientation of the tows. The blank (white region) subcells represent the interweave matrix. Consequently, 

the 5HS macroscale RUC is subdivided into 10×10×4 subcells, each with dimensions given by

{ }

{ }

{ }

4, 4, 4, 4

,,,,

,,,,

Dtttt

H wwwww

L wwwww

=

= δ, δ, δ, δ, δ,

= δ, δ, δ, δ, δ,

(23)

For the PW an RUC of size 4x4x4 is used and details regarding this can be found in Reference 9. Within the 

context of GMC, the most important parameters are those related to volume fraction, vf, specifically the 

overall vf and the local tow vf. The aspect ratio, AR=2w/t, is the next most important parameter because it 

is responsible for the undulation and out of plane properties. The last parameter, tow spacing, is backed 

out to ensure geometric consistency by the following relationshipV = wV (w+ δ) f ftow . In this equation w is 

the tow width and δ is the spacing with the proper overall vf (herein held fixed at 0.36). Because of the 

CVI process used to manufacture the woven fabric composites, there exists high levels of porosity, as 

shown in Figure 4, that cannot be neglected; as demonstrated by our previous work, see Reference 2.

Herein voids are localized to critical areas determined from optical inspection (microscopy) in the 

case of the 5HS weave. Figure 8 illustrates the localized void regions at the macroscale for a 5HS weave, 

wherein high density (e.g., 85 percent) void regions are depicted in red and low density (e.g., 5 percent) 

void regions in blue, and a PW wherein only the high density voids are shown in red. Note in both void 

idealizations the total inter-weave void volume fraction is held constant at 12.7 percent. The voids are 

accounted for at a smaller length scale by analyzing a separate RUC and homogenizing those properties. 

This is done for two primary reasons. First, explicit modeling of voids in GMC will tend to “eliminate” an 

entire row and column due to the constant strain field assumptions within a subcell. Yet, by performing a 

separate analysis, this effect is dampened since void volume and shape merely change the resulting 
NASA/TM—2013-217859 10

anisotropic “constituent” response. Secondly, this allows for a faster, more accurate representation of void 

shape and distribution then explicitly modeling voids at this higher length scale. 

Figure 7.—Discretized five-harness satin subcell configuration, with each block representing a given group.

(a)

(b)

Figure 8.—The assumed void distributions in both 5HS and PW; white no voids, blue represent 

5 percent voids, and red represents 85 percent void content.
NASA/TM—2013-217859 11

Figure 9.—Fiber tow bundle RUC. Figure 10.—Three dimensional void RUC.

Tow Repeating Unit Cell

The fiber tow bundles are modeled using a doubly periodic (continuously reinforced) 4x4 repeating 

unit cell consisting of three materials: fiber, fiber coating/interface, and matrix. Consequently all three 

constituents influence the effective tow properties at each load step. In Figure 9, the black denotes the 

fiber, the hatched area represents the interface, and white represents the matrix. At this level there are also 

voids due to the CVI process. However, the voids at this level appear to be more evenly distributed than 

at the weave level and thus are represented by evenly distributing the void content in the tow areas (see 

tow insert in Figure 4). This is accomplished once again by calling a separate void analysis for each 

matrix subcell in the RUC, just as described in the weave RUC section. Consequently, the effective tow

properties are being influence at each increment by all three constituents, matrix damage and intra-tow 

void volume fraction. For each fiber tow bundle, the orientation is carefully computed such that the 

undulation is properly accounted for and the failure criteria can be applied in the local coordinate system.

Note no damage or failure of the interphase material will be accounted for in this study, since the BN 

coating’s stiffness is already approximately 20 times more compliant than the other two phases (see 

Table 1), thus minimizing its load carrying ability from the start.

Void Modeling

Voids are modeled through computation of a triply periodic (discontinuously reinforced) 2x2x2 RUC 

as shown in Figure 10. The hatched subcell represents the void region while the white represents the 

matrix. The relative size of the void cell is what determines the overall void content in both the fiber tow 

bundles and the weave. As mentioned previously, modeling of voids as a separate GMC analysis has 

many advantages. The overall RUC of the weave will remain constant regardless of the shape and 

distribution of the voids, i.e., no rediscretization is required. Consequently, the void location, quantity, 

and geometry can be quickly changed. Lastly, the strength and stiffness degradations and stress 

concentrations can be captured through GMC without reducing the accuracy of the analysis at the 

macroscale.

Results

Problem Description and Prior Results

In our previous study (Ref. 2) and in this current work, a five harness satin weave with a CVI-SiC 

matrix and iBN-Sylramic2 fiber were chosen, due to the availability of experimental data for correlation. An 

approximate overall fiber volume fraction of 36 percent (which was held fixed for all cases examined) was 

determined along with a tow width of 1.25 mm and total thickness of 2.5 mm (i.e., eight plies), see actual 

micrograph inserts in Figure 4 (compliments of P. Bonacuse, GRC). The properties and necessary material 

parameters are displayed in Table 1 to Table 3, elastic properties were determined from either published 

values or discussions with colleagues while the strength and damage parameters were obtain previously (see 

 2 This stands for silicon carbide fiber coated with boron nitride
NASA/TM—2013-217859 12

Ref. 2) from correlation with a macro level tensile response curve, shown in Figure 11. In Figure 11, both an 

experimental, on-axis, tensile response, taken from Morscher(see Refs. 10 and 11) is overlaid with a 

baseline correlation using the localized void model (see Figure 8(a)). The simulated response shows good 

correlation with the experimental curve, approximately capturing the deviation from proportionality (often 

referred to as “first matrix cracking” or “proportional limit stress”) and failure stress. In Liu and Arnold 

(Ref. 2) the underlying mechanisms causing nonlinearity (which are subtle in some places), that is, intra-tow 

matrix damage, inter-weave matrix damage (in the low void and also in the high void region) and then 

ultimate fiber failure were identified. As were the influence of variations in primary model parameters like 

(i.e., the initial modulus, post first matrix cracking modulus, n, and critical cracking stress, σdam) and key 

architectural parameters such as (i) tow fiber volume fraction, (ii) tow aspect ratio, and (iii) tow void volume 

fraction, (iv) weave void distribution, (v) void shape (see Table 4 for a list of specific values). These 

parameters were studied by conducting a full factorial set of numerical simulations, assuming the localized 

void model shown in Figure 8(a). Detailed monotonic tensile results are shown in Reference 2 where it was 

concluded that for the present system the localized voids (in contrast to uniform or none) must be explicitly 

modeled; and fiber volume fraction and constituent (fiber, interface, matrix) material parameters

significantly impact the simulated in-plane tensile response histories. Whereas tow fiber, void fractions and 

tow aspect ratio are less influential. Although it was shown that the tow void content has the strongest effect 

on post first matrix cracking stiffness, the tow aspect ratio has the strongest effect on failure strain and the 

tow fiber volume fraction appears to have a minimal effect relative to other parameters. Note, the tow fiber 

volume fraction and void volume fraction are both considered a mesoscale effect because their geometrical 

properties are involved in the mesoscale concentration matrix (Eq. (2)); whereas, the tow aspect ratio is 

considered a macroscale property because it is taken into account in the macroscale concentration matrices 

(Eqs. (16) and (17)). 

TABLE 1.—FIBER PROPERTIES

Modulus,

GPa

Poisson’s 

Ratio

Axial strength,

GPa

Shear strength,

MPa

σdam,

MPa

n

iBN-Sylramic 400 0.2 2.2 900 ---- -----

CVI-SiC 420 0.2 --- ---- 180 0.04

Boron nitride 22 0.22 --- ---- ---- -----

TABLE 2.—WEAVE PROPERTIES

Type............................................... 5HS

Fiber volume fraction, % .................. 36

Tow volume fraction, % ................... 78

Tow width, mm.............................. 1.25

Tow spacing, mm .......................... 0.34

Ply thickness, mm...................... 0.3125

Matrix ..................................... CVI-SiC

TABLE 3.—TOW PROPERTIES

Tow fiber volume fraction, %........................... 46

Tow packing structure ............................... Square

Fiber................................................IBN-Sylramic

Matrix ..................................................... CVI-SiC

Interface...........................................................BN
NASA/TM—2013-217859 13

TABLE 4.—VARIED PARAMETERS

Architectural 

parameter

Relevant 

length scale

Values

Tow fiber volume fraction, Vtf Meso 0.46,0.48,0.50

Tow void volume fraction Meso 0.01,0.05,0.07

Tow aspect ratio, AR Macro 8,10,12

Weave void distribution Macro None, even, localized

It was further shown (see Ref. 2) that void shape (cubic, cylindrical and flat (or sheet like)) 

significantly influenced the out-of-plane moduli (i.e., Ez = 165, 172.5, and 88.8 GPa), respectively.

However, the in-plane response (both deformation and failure) is unaffected by void shape, as one might 

expect (see Ref. 2 for details). The dramatic difference between in-plane (11-direction) and out-of-plane

(zz-direction) secant modulus as well as its evolution as a function of tensile loading is shown in 

Figure 12 for the case of sheet-like voids. The idealization of sheet like voids is consistent with the 

experimentally observed network of voids shown in Figure 13. The out of plane modulus, Ezz, initially at a 

significantly lower value (88 GPa) relative to the in-plane modulus, E11 (250 GPa), decreases as damage 

evolves. Note, the multiaxial coupling of damage is clearly evident and is a function of the multiscale 

effects. The out-of-plane modulus degrades slower than the in-plane modulus (see Figure 12), which is 

representative of micro-cracking forming perpendicular to the applied load direction (in-plane), therefore 

not significantly impacting Ezz. Although there is not experimental evidence to validate against, it is an 

important illustration of the capabilities of the present analysis technique.

Loading histories with unloading are critical for deducing mechanisms driving nonlinear response; in 

Figure 13(a) three such unload histories are illustrated schematically. The first is an unload path in which the 

unloading modulus is equal to the loading but with permanent accumulated strain (i.e., nonlinearity due to 

inelasticity), the second is a path where the unloading modulus differs from that of loading but the strain 

doesn’t return to zero (i.e., nonlinearity due to inelasticity and damage) and the third has an unload modulus 

that differs from the loading but all strain is recovered (i.e., nonlinearity due to damage only). Figure 14(b)

depicts multiple experimental tensile histories with unloading done by Morscher (see Ref. 12) for four 

different CMC materials. Clearly, for all CMC systems (in particular the Syl-iBN of interest in this study) it 

is clear that the unloading response curves return to the origins of the experiment—thus indicating that all 

nonlinearity is due to damage alone. Figure 14(c) shows the current MSGMC simulation including 

unloading; as expected the simulation, although nonlinear (due to damage accumualation), returns to the 

origin since a continuum damage model was used to represent the matrix behaivor. The blue highlighted

zone in Figure 14(c) depicts the region of comparison to the Syl-iBN experimental response in Figure 14(b).

This is a significant and key difference between the current simulations and others in the literature (e.g., Ref. 

13) which would be unable to predict such a response due to the use of plasticity instead of damage 

mechanics to represent the associated nonlinear behavior under monotonic tension. Furthermore, the good 

qualitative agreement between the two response curves are encouraging, even though the experimental 

response came from composites with 40 percent fiber volume fractions versus the 36 percent numerical 

simulation.
NASA/TM—2013-217859 14

Figure 11.—Typical experimental response curve (Refs. 10 and 11).

Figure 12.—Concurrent degradation of in plane and out 

of plane modulus as a function of tensile strain.

 (a) (b)

Figure 13.—Characterization of the porosity in a CVI SiC/SiC Woven Composite; (a) top down view and (b) 3-D 

projection of sheet like structure. Courtesy of P. Bonacuse (see Ref. 14).

0

50

100

150

200

250

300

350

400

450

500

0 0.001 0.002 0.003 0.004 0.005 0.006

Stress (MPa)

Strain

Experimental

Baseline Prediction

0 0.2 0.4 0.6 0

50

100

150

200

250

300

Strain (%)

Modulus (GPa)

E11

Ezz
NASA/TM—2013-217859 15

(a) (b)

(c)

Figure 14.—(a) Schematic of loading/unloading of a given material; (b) Experimental results (Ref. 12); and (c) 

Simulated tensile loading and unloading response for the localized void model of a 5HS weave.

.

Although our prior work (see Ref. 2) examined the sensitivity of the entire tensile response at the 

macroscale, including failure, to various architectural weave features at varying lower levels of scale; it 

did not actually address the statistical nature of these architectural features. Therefore in the next section 

attention will be focused on assessing these various architectural features statistically as well as 

understanding their influence at the structural level (see Figure 4).

Stochastic Results

In order to simulate the non-deterministic response, all the architectural input parameters in the 

section “Modeling Ceramic Matrix Composites with MSGMC” (except global volume fraction), were 

treated as random variables with Gaussian distributions. These include: tow volume fraction, thickness 

(i.e., aspect ratio), void content (high and low density, see Figure 8), and interface thickness. There are a 

total of six random variables per macroscopic weave RUC. The assumption of Gaussian distributions is 

substantiated by the work of Bonacuse et al. (Ref. 15); wherein painstaking effort was taken to determine 

the actual distribution of a few of these parameters from actual micrographs of numerous slices from 

CMC specimens. The mean values for these distributions were assumed to be identical to those in our 

prior parametric study (Ref. 2), while the variances were estimated to the best of the authors abilities 

based on a mixture of experimental and heuristic observations. The aspect ratio was controlled by 

changing the ply thickness while keeping the tow width constant at 1.25 mm. These are summarized in 

Table 5.
NASA/TM—2013-217859 16

TABLE 5.—RANDOM VARIABLES FOR STOCHASTIC ANALYSIS

Architectural 

parameter

Relevant 

length scale

Mean Standard 

deviation

Tow fiber volume fraction, Vtf Meso 0.48 0.033

Tow void volume fraction Meso 0.05 0.01

Tow thickness, mm, t Macro 0.3125 0.021

Interfacial thickness ratio Micro 0.06 0.01

Weave void low density volume fraction Macro 0.1 0.02

Weave void high density volume fraction Macro 0.75 0.05

Procedure for Incorporating Stochastics

To enable the rapid solution of multiple instances of various microstructures a systematic procedure 

was defined and automated to create the corresponding RUCs given the specified geometric and 

architectural features. Figure 15 schematically depicts the procedure, wherein step 1 samples the 

distribution curves for each architectural feature (i.e., tow volume fraction, thickness (i.e. aspect ratio), 

void content and interface thickness) described in Table 6 using Latin hypercube sampling (Ref. 16).

Given the specific values from step 1, the RUC for a given tow is constructed along with its associated 

RUC representing the intra-tow void fraction (step 2), then each unique grouping (eight for a 5HS weave 

or six for a balanced PW with linear elasticity) associated with a specific orthogonal weave type. Once 

these groups are available the actual woven RUC can be assembled and the analysis made. Obviously, 

this procedure for incorporating stochastics can require significant computational resources depending 

upon the number of subcells being analyzed in a given RUC, as illustrated in Table 6. From Table 6 one 

can immediately see almost a nine-fold reduction in cost per increment, by using a PW instead of the 5HS 

weave. Also one sees a 25 percent reduction in the total number of increments required to obtain a 

converge solution, thereby resulting in approximately a 12 times total speed increase. This fact is the 

reason why the PW architecture was used to investigate the sensitivity of local (micro and meso) 

architectural features at the structural scale, in the subsequent section. Note all previously defined

architectural feature distributions and parameters for the 5HS weave will still be used for the PW study.

Once the Monte Carlo simulation has finished evaluating all samples, the distributions of key features 

in the stress strain curve were approximated through a kernel density estimator. The accuracy of the 

results is strongly dependent on the number of samples used in the Monte Carlo analysis. Since we 

employed Latin hypercube sampling, fewer simulations were required then random sampling to achieve 

the same accuracy. However, in random sampling, additional samples can be added until the desired 

accuracy is achieved. When forming a Latin hypercube, the number of samples must be predetermined 

and all samples measured. If the desired accuracy is not achieved, supplemental runs must be carefully 

constructed, which is complex for a significant amount of random variables. While the 1x1 PW case 

contains six random variables, the 3x3 case requires 54 random variables. The number of cases required 

to achieve high accuracy estimations of the variance is large. Initially, we estimated approximately 20 per 

random variable, which provided reasonable results for the 1x1 PW case when compared to a 1000 

sample run. However, for the 3x3, running over 1000 cases was not feasible due to the high 

computational cost. A smaller run of approximately 150 cases was used to estimate the parameters;

resulting in approximately 3 samples per random variable. This implies that the results are less precise 

than the 1x1 and 2x2 cases.
NASA/TM—2013-217859 17

Figure 15.—Statistical procedure.

TABLE 6.—TYPICAL COMPUTATIONAL SPEED AS A 

FUNCTION OF WEAVE AND SIZE OF RUC

Weave type Time/increment,

sec

Typical 

increments

Number of 

subcells

5HS (1x1) 13 200 93,800

PW (1x1) 1.5 150 18,840

PW (3x3) 16 200 169,560

PW (6x6) 144 200 678,240

Macro RUC Simulation Results

Here the influence of the weave architectural features at various levels of scales given a single macro 

RUC will be examined statistically (in contrast to our previous work in Ref. 2) as described above for the 

distributions given in Table 5. In Figure 16(a) and Figure 17(a), random simulations of a single RUC at 

the macroscale (1x1) are illustrated for a 5HS weave (150 cases) and PW (638 cases), respectively, 

whereas Figure 16(b) and Figure 17(b) depict the multivariate probability density function constructed 

from a two dimensional kernel density estimator (Ref. 17) for the 5HS and PW RUCs respectively. The 

probability density functions (PDF) for the stress strain curves are a composite estimation of all the cases 

simulated. The contour line widths indicate the variance and the maximum probability regions represent 

mean values. By looking at lines of constant strain, the stress distributions can be observed. Clearly both 

weave patterns produce similar overall tensile responses, with the 5HS providing a slightly stiffer overall 

(lower strain to failure) response as expected. The distributions in stress-strain response for the 5HS and 

PW are nearly identical. Although the input parameters are Gaussian in nature, due to the nonlinear nature 

of the multiscale model (in which four levels of scale are explicitly accounted for), the output parameters

(e.g., stiffness, passion ratio, etc.) are typically non-Gaussian. For example, the out-of-plane modulus, 

which was previously shown to be strongly dependent on void and architecture, is log-normally 

distributed as evidenced by Figure 18. If the data was normally distributed, the data points would follow 

closely with the red line, however the data diverges at the higher moduli values. This indicates that the 

specific configurations to achieve a high modulus are very sparse and cannot be easily achieved, in 

contrast to the numerous possibilities for a lower modulus. 
NASA/TM—2013-217859 18

(a) (b)

Figure 16.—Stochastic simulation of a 5HS weave composite at the macroscale; (a) 150 individual cases; and 

(b) multivariate probability density function tensile curve.

(a) (b)

Figure 17.—Stochastic simulation of a PW composite at the macroscale; (a) 638 individual cases; and (b) multivariate 

probability density function tensile curve. 

Figure 18.—Normal probability plot of outof-plane modulus for 5HS composite.

30 40 50 60 70 80

0.01 

0.02 

0.05 

0.10 

0.25 

0.50 

0.75 

0.90 

0.95 

0.98 

0.99 

Data

Probability

Normal Probability Plot
NASA/TM—2013-217859 19

(a) (b)

Figure 19.—Monte Carlo simulations at the structural scale for a plain weave composite using, (a) 2x2 and (b) 3x3.

Structural Scale (Multiple RUCs are Combined) Simulation Results

Here the structural scale responses, simulated by combining several PW macroscale RUCs together 

(i.e., 2x2 and 3x3), are examined to assess the sensitivity to uncertainty of lower scale architectural 

features. Note as the structural size, i.e., the size of the structural RUC is increased from 1x1 (i.e., one 

RUC) to 3x3 (i.e., nine RUCs) the volume of material is increased and thus so is the opportunity for 

flaws. Stress-strain plots for various Monte Carlo simulations, given a 2x2 and 3x3 PW, are shown in 

Figure 19.

By evaluating the responses, it is apparent that the elastic response and first matrix cracking events 

are relatively unaffected by the number of macroscale RUCs (each potentially comprised of different 

architectural features as indicated in Figure 4 by the varying shades of grey in the upper right insert), 

whereas the nonlinear portion of the stress strain curve after damage initiation has relatively large 

variation. This wider variance is expected since the larger the structural scale (the more RUCs) the higher 

the chance to capture the measured variation; failure is driven by local events and often thought of as a 

“weakest link” phenomenon. Note the structural scale shows a wider variance for the post first matrix 

cracking region (i.e., damage active region) of the stress-strain curve, indicating the need to account for 

local random variations in the weave architecture (for this length scale) if one desires accurate predictions 

of strain to failures (or ultimate tensile strength, UTS).

Probability density functions for these four (i.e., elastic modulus, damaged modulus, first matrix 

cracking stress, and UTS) key stress-strain response measures are illustrated in Figure 20, from which 

four important conclusions can be drawn:

1. The elastic modulus mean stress and variance increases with size,

2. The damaged modulus mean stress (and variance) slightly decreases (increase) with size,

3. First matrix cracking stress mean stress slightly increases (while the variance fluctuates) with size, 

and

4. The failure stress mean (and variance) decreases (increase) with size.

Note these probability density functions are non-Gaussian. These conclusions are consistent with both the 

concept of weakest link as well as facets of the Generalized Method of Cells. Elastic modulus increases 

with size because there is larger chance of randomly sampling a higher stiffness weave when more RUCs 

are included. The damaged modulus decreases because the weakest weave will govern the response. The 

increase in first matrix cracking stress, although not intuitive, is caused by the higher elastic modulus. The 

decrease in failure stress is due to the higher probability of sampling a weaker weave. It is important to 

remember that most experimental data is obtained from specimens with dimensions that require at least a 
NASA/TM—2013-217859 20

(a) (b)

(c) (d)

Figure 20.—Distributions of salient stress-strain curve parameters; (a) elastic modulus; (b) damaged modulus; (c) first 

matrix cracking stress; and (d) failure stress.

2x2 or 2x3 RUC size or greater to be used. Therefore, caution should be used if one is attempting to 

characterize (back-out) in-situ properties from coupon data using only a single macro (1x1) RUC as the 

UTS (or strain to failure) as well as post cracking stiffness can be non-conservative. The specific impact 

on characterized constituent property values given these two approaches (single RUC versus multiple 

RUCs) will be examined in a future work. Also as the sensitivity of variance was not examined, no 

comment regarding the driving parameter(s) causing these variations can be made at this time. However, 

some insight into the order of importance and degree of influence can be obtained from our prior explicit 

deterministic study (see Ref. 2), which used the same underlying model.

Clearly, as one increases the number of RUCs, the number of random variables interacting increases 

significantly. Consequently, in an attempt to confirm that the number of random variables assessed is not 

adversely influencing the previous results; multiple analyses with a constant number of RUCs (i.e., four) 

and therefore random variables will be evaluated. This is accomplished by exploring the concept of series 

or parallel RUC configurations, by evaluating cases of a 4x1 (parallel, i.e., Nβ = 4 and Nα=1) and a 1x4 

(series, i.e., Nβ = 1 and Nα = 4)) PW as well as the combined 2x2 PW case. Given that loading is always in 

the α direction, see Figure 4, one would expect that the distributions of mechanical parameters be similar

(for the parallel case) to that of the 1x1 shown in Figure 20(a) and (d). Whereas for the series case one 
NASA/TM—2013-217859 21

(a) (b)

Figure 21.—Distributions of (a) elastic modulus (b) failure stress; given the same number of random variables but 

analyzed using a 4x1 (parallel), 1x4 (series) and combination plain weave idealization.

would expect to see behavior that is more similar to larger effective RUC sizes such as 2x2 or 3x3. From 

Figure 21, it is clear that the distributions (mean and variance) of elastic moduli are relatively unaffected; 

however the distribution of failure stress is significant impacted by the arrangement of the RUC. The 

series effect (i.e., weakest link) is evident in the narrower variance in the failure stress distribution 

compared to both the parallel and 2x2 case. 

Conclusion

This paper presents a stochastic investigation into the sensitivity of architectural features at lower 

scales on both the macroscale (single RUC) and structural (multiple RUCs) deformation responses of 

plain weave (PW) and five harness satin (5HS) woven CMCs. The recently implemented Multiscale 

Generalized Method of Cells (MSGMC) methodology was employed to model the nonlinear damage 

driven response of the woven composite fabric, where four separate material scales were considered. At 

the mesoscale, the tow fiber volume fraction and the void content within a tow were varied; whereas at 

the macroscale the influence of the tow aspect ratio and weave void content were investigated. At the 

structural scale, a group of macroscale RUCs was modeled to simulate an effective material 

representative volume. For each permutation of these effects, the tensile response to failure was analyzed; 

wherein the modulus, first matrix cracking, post damage modulus, and ultimate failure strain/stress were 

examined. Analyzing the macroscale response, it was determined (in a previous study, Ref. 2) that the 

location of the weave void content at the macroscale was the most impactful parameter for capturing 

failure related properties (FMC and UTS). Consequently, it is critical that this effect be captured and 

correctly reflected in a model to ensure accurate deformation and failure response. Further, to obtain 

accurate out-of-plane stiffness predictions the void shape should be idealized as a flat sheet. Secondly, 

tow void content had the largest effect on the initial and post stiffness with the tow aspect ratio greatly 

influencing the failure strain levels. Also, it appears that accurately modeling inter-tow failure is critical 

to predicting the deviation from proportionality. The present statistical analysis demonstrated that the 

linear elastic range was insensitive to architectural variation as compared to the damaged regime. 

Similarly, the structural scale simulations had more variance than the macroscale due to the local nature 

of failure. Therefore, suggesting that caution be used when characterizing constituent material models, 

with only a single weave RUC, as almost all experimental response curves are generated with specimens 

that have multiple RUCs within the gage section, as this can produce non-conservative results.
NASA/TM—2013-217859 22

References

1. Liu, K. and Chattopadhyay, A. and Bednarcyk, B. and Arnold, S.M.: “Efficient Multiscale Modeling 

Framework For Triaxially Braided Composites using Generalized Method of Cells,” Journal of 

Aerospace Engineering, April 2011.

2. Liu, K.C. and Arnold, S.M., “Impact of Material and Architecture Model Parameters on the Failure of 

Woven CMCs Via the Multiscale Generalized Method of Cells,” NASA/TM—2011-217011.

3. Paley, M. and Aboudi, J.: “Micromechanical Analysis of Composites by the Generalized Method of 

Cells,” Mechanics of Materials 1992; 14: 127–139.

4. Aboudi, J.: “Micromechanical Analysis of Thermo-Inelastic Multiphase Short-Fiber Composites,” 

Composite Eng. 5, 839-850, 1995.

5. Bednarcyk, B.A.: “Modeling Woven Polymer Matrix Composites With MAC/GMC,” NASA/CR—

2000-210370.

6. Hashin, Z.: “Failure Criteria for Unidirectional Fiber Composites,” J. Appl. Mech., Vol. 47, June 

1980, pp. 329–334.

7. Bednarcyk, B. and Arnold, S.M.; “Micromechanics Based Modeling of Woven Polymer Matrix 

Composites,” AIAA, Vol. 41, No. 9, pp.1788-1796, 2003.

8. Liu, K.C., Hiche, C. and Chattopadhyay, A.: “Low Speed Projectile Impact Damage Prediction and 

Propagation in Woven Composites,” 50th AIAA/ASME/ASCE/AHS/ASC Structures, Proc. 

Structural Dynamics and Materials Conference, Palm Springs, CA, May 2009.

9. Liu, K.C., Arnold, S.M., Chattopadhyay, A.; “Examination of Material and Architectural Effects at 

Various Length Scales on the Structural Deformation Response of Woven Polymeric Composites 

Determined Via Multiscale Generalized Method of Cells,” 51st AIAA/ASME/ASCE/AHS/ACS 

Structures, Structural Dynamics, Materials Conference, Orlando, Florida (April 12-15, 2010)

10. Morscher, G.: “Stress, matrix cracking, temperature, environment, and life of SiC/SiC woven 

composites” International Conference on High Temperature Ceramic Matrix Composites, Bayreuth, 

Germany, September 20–22, 2010.

11. Morscher, G. N., Singh, M., Kiser, J.D., Freedman, M., and Bhatt, R.: “Modeling stress-dependent 

matrix cracking and stress-strain behavior in 2D woven SiC fiber reinforced CVI SiC composites,” 

Composites Science and Technology, Vol. 67, 2007, pp. 1009–1017.

12. G.N. Morscher, G. Ojard, R. Miller, Y. Gowayed, U. Santhosh, J. Ahmed, and R. John, “Tensile 

Creep and Fatigue of Sylramic-iBN Melt-Infiltrated SiC Matrix Composites: Retained Properties, 

Damage Development, and Failure Mechanisms” Comp. Sci. Tech., 68 3305-3313 (2008)

13. Mital, S., Goldberg, R. and Bonacuse, P.J; “Two-Dimensional Nonlinear Finite Element Analysis Of 

CMC Microstructures,” GT2011-45930, Proceedings of ASME Turbo Expo 2011, June, Vancouver, 

Canada, 2011

14. Bonacuse, P.J., private communications, 2011.

15. Bonacuse, P.J., Mital, S., and Goldberg, R.; “Characterization of the As Manufactured Variability in a 

CVI SiC/SiC Woven Composite,” GT2011-45890, Proceedings of ASME Turbo Expo 2011, June, 

Vancouver, Canada, 2011.

16. McKay, Michael D., Richard J. Beckman, and William J. Conover. “Comparison of three methods for 

selecting values of input variables in the analysis of output from a computer code,” Technometrics,

21.2 (1979): 239-245.

17. Epanechnikov, Vassiliy A. “Non-parametric estimation of a multivariate probability density,” Theory 

of Probability & Its Applications, 14.1 (1969): 153-158.

REPORT DOCUMENTATION PAGE Form Approved 

OMB No. 0704-0188 

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the 

data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this 

burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. 

Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB 

control number. 

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 

1. REPORT DATE (DD-MM-YYYY)

01-06-2013 

2. REPORT TYPE

Technical Memorandum 

3. DATES COVERED (From - To)

4. TITLE AND SUBTITLE

Stochastic Nonlinear Response of Woven CMCs 

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Liu, Kuang, C.; Arnold, Steven, M. 

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

WBS 432938.11.01.03.02.02.16 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

John H. Glenn Research Center at Lewis Field 

Cleveland, Ohio 44135-3191 

8. PERFORMING ORGANIZATION

 REPORT NUMBER

E-18650 

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

10. SPONSORING/MONITOR'S

 ACRONYM(S)

NASA 

11. SPONSORING/MONITORING

 REPORT NUMBER

NASA/TM-2013-217859 

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited 

Subject Category: 24 

Available electronically at http://www.sti.nasa.gov 

This publication is available from the NASA Center for AeroSpace Information, 443-757-5802

13. SUPPLEMENTARY NOTES

14. ABSTRACT

It is well known that failure of a material is a locally driven event. In the case of ceramic matrix composites (CMCs), significant variations 

in the microstructure of the composite exist and their significance on both deformation and life response need to be assessed. Examples of 

these variations include changes in the fiber tow shape, tow shifting/nesting and voids within and between tows. In the present work, the 

influence of scale specific architectural features of woven ceramic composite are examined stochastically at both the macroscale (woven 

repeating unit cell (RUC)) and structural scale (idealized using multiple RUCs). The recently developed MultiScale Generalized Method of 

Cells methodology is used to determine the overall deformation response, proportional elastic limit (first matrix cracking), and failure under 

tensile loading conditions and associated probability distribution functions. Prior results showed that the most critical architectural 

parameter to account for is weave void shape and content with other parameters being less in severity. Current results show that statistically 

only the post-elastic limit region (secondary hardening modulus and ultimate tensile strength) is impacted by local uncertainties both at the 

macro and structural level. 

15. SUBJECT TERMS

Ceramic matrix composites; Micromechanics; Multiscale modeling 

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF

 ABSTRACT

UU 

18. NUMBER

 OF 

 PAGES

30 

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email:help@sti.nasa.gov) 

a. REPORT

U 

b. ABSTRACT

U 

c. THIS 

PAGE

U 

19b. TELEPHONE NUMBER (include area code)

443-757-5802 

Standard Form 298 (Rev. 8-98)

Prescribed by ANSI Std. Z39-18