Subodh K. Mital

The University of Toledo, Toledo, Ohio

Robert K. Goldberg and Peter J. Bonacuse

Glenn Research Center, Cleveland, Ohio

Two-Dimensional Nonlinear Finite Element 

Analysis of CMC Microstructures

NASA/TM—2012-217297

January 2012
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Subodh K. Mital

The University of Toledo, Toledo, Ohio

Robert K. Goldberg and Peter J. Bonacuse

Glenn Research Center, Cleveland, Ohio

Two-Dimensional Nonlinear Finite Element 

Analysis of CMC Microstructures

NASA/TM—2012-217297

January 2012

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2012-217297 1

Two-Dimensional Nonlinear Finite Element 

Analysis of CMC Microstructures

Subodh K. Mital

The University of Toledo

Toledo, Ohio 43606

Robert K. Goldberg and Peter J. Bonacuse

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abstract

A research program has been developed to quantify the effects of the microstructure of a woven 

ceramic matrix composite and its variability on the effective properties and response of the material. In 

order to characterize and quantify the variations in the microstructure of a five harness satin weave, 

chemical vapor infiltrated (CVI) SiC/SiC composite material, specimens were serially sectioned and 

polished to capture images that detailed the fiber tows, matrix, and porosity. Open source quantitative 

image analysis tools were then used to isolate the constituents, from which two dimensional finite 

element models were generated which approximated the actual specimen section geometry. A simplified 

elastic-plastic model, wherein all stress above yield is redistributed to lower stress regions, is used to 

approximate the progressive damage behavior for each of the composite constituents. Finite element 

analyses under in-plane tensile loading were performed to examine how the variability in the local 

microstructure affected the macroscopic stress-strain response of the material as well as the local 

initiation and progression of damage. The macroscopic stress-strain response appeared to be minimally 

affected by the variation in local microstructure, but the locations where damage initiated and propagated 

appeared to be linked to specific aspects of the local microstructure.

Introduction

High temperature ceramic matrix composites (CMC) are being explored as viable candidate materials 

for hot section gas turbine components, airframes and other airbreathing propulsion systems that are 

subjected to very high temperatures. However, these materials are heterogeneous and various factors 

affect their properties in a specific design environment. As shown in Figure 1, (a micrograph of a five

harness satin weave CVI SiC/SiC composite), the microstructure of a woven ceramic matrix composite 

displays a significant amount of variability and irregularity in aspects such as tow spacing, ply alignment, 

nesting of adjacent plies and matrix thickness. Furthermore, particularly for a CVI SiC/SiC composite, 

there is a significant amount of porosity that is arranged in irregular patterns. Therefore, a need exists to 

investigate the effects of this complex material microstructure on the effective properties and durability of 

these materials. A need also exists to determine how the characterized variability in the microstructure 

correlates with the known variability in CMC material properties.

As discussed in Nemeth, et al. (Ref. 1) and Mital, et al. (Ref. 2), in many analytical approaches that 

have been derived to model the response of woven ceramic matrix composites, the representative unit cell 

(the smallest repeating unit that is assumed to be representative of the composite as a whole) of the 

composite has been modeled using a “pristine”, ordered architecture with average dimensions for the tow 

geometry and spacing. In addition, the areas between the fiber tows are assumed to be composed entirely 

of matrix material. The effects of porosity are accounted for by uniformly degrading the properties of the 

matrix based on the volume fraction of porosity in the composite. Utilizing these traditional approaches, 

incorrect prediction of composite properties can occur. For example, as found by Goldberg, et al. (Ref. 3),
NASA/TM—2012-217297 2

Figure 1.—Micrograph of woven SiC/SiC Composite

the through-thickness modulus has been found to be a function not only of the amount of porosity, but the 

shape and distribution of the porosity as well. Furthermore, Goldberg, et al. (Ref. 3) found that models of 

woven ceramic matrix composites that included more realistic representations of the microstructure 

provided an improved capability to identify the sites for stress risers where damage is likely to initiate.

Traditional techniques, where the porosity is merely smeared with the matrix or the fiber tows are 

regularly spaced, can only capture the stress distribution in an average sense as shown by Mital, et al. 

(Ref. 2). The goal of an ongoing project at NASA Glenn is to investigate the effects of the complex 

microstructure of a woven ceramic matrix composite and its variability on the effective properties and the 

durability of the material. Detailed analysis of these complex microstructures may provide clues for the 

material scientists who ‘design the material’ or to structural analysts and designers who ‘design with the 

material’ regarding damage initiation and damage propagation. A representative material system, 

specifically a five-harness satin weave architecture CVI SiC/SiC composite composed of Sylramic-iBN 

fibers and a SiC matrix, is analyzed. The objective is to investigate the effects of the complex 

microstructure on the effective mechanical properties and durability. A separate effort by the authors 

involves the development of methods to characterize the distributions in as-fabricated composites of 

various microstructural parameters such as tow width and spacing, ply nesting, ply misalignment etc.

(Ref. 4) Towards that goal, a series of finite element analyses have been performed with a simple nonlinear material model. The finite element models of two-dimensional cross-sections, which were 

generated from serial sections of experimental specimens, are subjected to an in-plane tensile load. A few 

computer-generated sections (referred to as “aligned”) with a more regular spacing of the fiber tows have 

also been analyzed. The objective of this work is to perform simple progressive damage analyses on 

various sections to determine the sites for damage initiation and to relate the damage patterns to the 

unique architectural features of the given section. Determining the correlation between the local damage 

patterns and the overall stress-strain response of the material was another objective of this work. In the 

design of components using ceramic matrix composites, the first matrix cracking stress (or proportional 

limit), defined as the stress level where the macro tensile stress-strain curve deviates significantly from 

linearity, is often a key material allowable. As a result, particular emphasis was given in this study to 

determining the effects of the microstructure on the initiation and growth of damage up to the point of the 

first matrix cracking stress. The variability in constituent material properties, while likely to be a 

significant contributor to the distribution in composite properties (Refs. 5 to 7), is currently not part of 

this investigation.

Microstructural Characterization

High resolution images of the polished cross-sections of the CVI SiC/SiC composite material were 

taken for the purposes of quantifying the distributions of various composite parameters such as tow 

shapes and sizes, the distance between tows, the size and shape of the pores, relative alignment of plies 

and nesting of the plies (see Fig. 1 for an example). The images were then segmented using an automated 

process that is based on pixel intensities and statistical techniques to isolate various composite 

constituents (Ref. 3). These constituents are longitudinally sectioned tows, transverse sectioned tows, 

pores and the SiC matrix surrounding the tows (Fig. 2). The displayed section consists of longitudinal 
NASA/TM—2012-217297 3

(shown in blue) and transverse (shown in green) tows, CVI-matrix (shown in red) and porosity (shown as 

empty spaces). However, the images of the isolated composite microstructures are not amenable to the 

generation of finite element models for structural analysis due to the irregular shape and sizes of 

constituents and ply misalignment. To compensate for these irregularities, techniques were developed to 

generate simplified geometries from the segmented images while still maintaining relevant 

microstructural details. This involved several approximations including assuming a constant thickness of 

the longitudinal tows based on the average values obtained from segmented images, assuming the same 

shape and cross-sectional area for all transverse tows, assuming a sinusoidal shape for the longitudinal 

tows, growing the matrix uniformly around the fiber tows and slightly adjusting the position of the

longitudinal and transverse tows to minimize interference. A sample of a simplified cross section is 

shown in Figure 3. Some ‘aligned’ sections (see Fig. 4 for an example), where the transverse tows were 

assumed to be stacked on top of each other, were also generated to be used as a baseline to contrast with 

the more random nature of the as-manufactured composite material. As mentioned, the above technique 

involves some approximations that led to differences in segmented sections and simplified section 

idealizations. It has been observed that in actual segmented sections, the matrix deposition is nonuniform; thicker on the outside and thinner in the interior. Longitudinally sectioned tows show variation 

in the thickness due to their non-uniform cross-sectional area and slight misalignment of the plies with the 

sectioning plane. These and other assumptions lead to slight differences in the volume fractions of various 

constituents between the segmented and simplified images but the overall nature of the distribution is 

captured. Most importantly, simplified images can be used to generate finite element models for structural 

analysis as explained in the following section. The details of this technique are presented in Goldberg, et 

al. (Ref. 3). 

Figure 2.—Segmented image of SiC/SiC cross section including matrix (red), transverse tows (green) and 

longitudinal tows (blue).

Figure 3.—“Simplified” image of SiC/SiC cross section with matrix (red), transverse tow (green) and longitudinal tow 

(blue) constituents.
NASA/TM—2012-217297 4

Figure 4.—“Simplified” image of an artificially generated “aligned” cross-section.

Finite Element Analysis: Results 

A software tool called OOF2 (Object Oriented Finite Element Analysis of Material Microstructures) 

(Ref. 8) was used to generate finite element models from simplified images of the composite cross-sections.

This public domain software, running on any computer with a variant of the UNIX operating system, can 

export the finite element meshes directly into the ABAQUS general purpose finite element program (Ref. 

9). ABAQUS can then be used to analyze the meshes for effective material properties and stress 

distributions. Finite element meshes were created for many actual and idealized sections. A typical finite 

element mesh of a representative cross-section is shown in Figure 5. As before, the displayed section 

consists of longitudinal (shown in blue) and transverse (shown in green) tows, CVI-matrix (shown in red) 

and porosity (shown as empty spaces). The longitudinal and transverse tows were treated as homogenized 

materials in this model even though the tow consisted of fiber, interfacial coating, matrix and intra-tow 

porosity. Since this intra-tow porosity has been found to be relatively uniformly distributed (Ref. 3), this 

porosity was not explicitly modeled but was used in determining the homogenized properties of the tow 

material. In a previous report, the effective in-plane and through-thickness modulus of various crosssections as well as stress distributions in these sections due to an applied in-plane tensile load were 

computed (Ref. 3). The idea of these analyses, performed in the linear elastic regime, was to identify the 

patterns of stress distribution, identify the areas of stress risers and to determine if any correlation could be 

found between the stress distribution and the composite microstructure. It was observed that the in-plane 

modulus correlated very well with the amount of porosity in the section analyzed. Higher levels of porosity 

lead to lower values of in-plane composite modulus. The predicted values of the in-plane modulus only 

changed by a few percent for intra-tow porosity volume fractions ranging from 3.2 to 5.5 percent. The 

predicted values of the through-thickness modulus for the different sections show a large variation.

Furthermore, the variation of the through-thickness modulus was not related to how much intra-tow porosity 

was present in the section, but on how the porosity was arranged. If the porosity was arranged in thin sheets, 

a lower value of through-thickness modulus was predicted. If the porosity was arranged in large clumps, a 

higher value of through-thickness modulus was predicted. In addition, the average value of the throughthickness modulus correlated well with experimentally obtained values (Ref. 10). These results differed 

from results obtained using classical analysis techniques, discussed in Mital, et al. (Ref. 2), in which the 

porosity was merely smeared in the matrix. Using the classical analysis methods, the through-thickness 

modulus was generally overpredicted. Since damage is likely to initiate from locations of high stresses, 

areas of high matrix stresses were identified and they were found to be related to the locations where 

transverse tows were packed closely together. High stress (relative to the respective strength) sites were also 

found in more closely packed transverse tows, and thus they are also potential sites for damage initiation. 

Damage could initiate within the transverse tows due to matrix cracking or fiber matrix debonding. It was 

noted by Goldberg, et al. (Ref. 3) that since the large irregularly shaped inter-tow porosity is explicitly 

modeled in these analyses (as opposed to degrading the matrix properties), these analyses can more 

accurately capture locations of high stresses and thus accurately 
NASA/TM—2012-217297 5

Figure 5.—Finite element mesh of a representative SiC/SiC cross section with 17442 nodes and 19502 elements 

(14545 linear quadrilateral and 4957 linear triangular elements).

identify sites for damage initiation. Identification of the sites of stress risers can potentially have a 

significant influence on the prediction of damage initiation and durability.

Following the linear elastic analyses, efforts to model the progressive damage in several sections were 

undertaken. The primary focus for the current study was to determine, for various actual and “aligned” 

sections, where damage initiated and to relate the damage locations to features of the material 

microstructure. The effects of the material microstructure on the progression of damage, and the state of 

local damage at the first matrix cracking stress (the stress level where the stress-strain curves deviates 

from linearity) were also a focus of the current study. To simulate the progressive damage response, 

several continuum damage mechanics models provided in the ABAQUS finite element program which 

were appropriate for ceramic composites were tried. These models are based on Hashin’s damage 

initiation criteria and the damage evolution is based on energy dissipation. However, serious convergence 

issues were encountered and no credible results could be obtained. While a damage mechanics type of 

model would be more representative of the actual material behavior, as a first approximation an elasticperfectly plastic material model was used to analyze the initiation (‘yielding’) and progression of damage 

(‘inelastic flow’) in the composite. A von Mises yield surface, which allows for isotropic yield, was used 

for all of the composite constituents. In the future, Hill’s yield surface, which allows for anisotropic yield, 

will be examined along with other potential anisotropic yield surfaces. However, due to the 

unidirectionally applied loads and the two-dimensional geometries considered in the current study, the use 

of an isotropic plasticity model was assumed to be acceptable. This criteria still allows for limited 

redistribution of stresses as the stresses above the cracking stress (here idealized as a yield stress) are 

redistributed to the lower stress regions. It will be shown in the following sections that even this simple 

damage model provides results that are in good agreement with experimental data. The emphasis in the 

current study, however, is not to compare different failure models but to examine the effects of different

microstructural features on damage initiation and progression. A progressively increasing strain of 

0.3 percent in the x-direction (in-plane direction) was applied to each section analyzed at roomtemperature. Table I shows the room-temperature elastic properties of the fiber tows and matrix used in 

the analysis. In this table, for the tow properties E11 is the longitudinal modulus, E22 is the transverse 

modulus, ν12 is the in-plane Poisson’s ratio, and G12 is the in-plane shear modulus. For the matrix, E is the 

elastic modulus and ν is the Poisson’s ratio. Tow properties were computed using standard 

micromechanics techniques for unidirectional composites (Ref. 2). In the computation of the equivalent 

tow properties, the porosity within the tows was assumed to be uniformly distributed within the matrix 

and the matrix properties were degraded to account for this intra-tow porosity. For the matrix material 

located outside of the tows, since the inter-tow pores were explicitly modeled, bulk matrix properties 

were used and no calibration or in-situ effects were modeled.
NASA/TM—2012-217297 6

TABLE I.—CONSTITUENT ROOM-TEMPERATURE MATERIAL PROPERTIES

Tow Properties 

800 filament Sylramic-iBN, 

Filament diameter = 9.5 μm, 

BN coating thickness = 0.6 μm, 15% porosity (assumed 

uniformly distributed)

E11 = 37.7 Msi (260 GPa) 

E22 = 15.3 Msi (105.5 GPa) 

ν12 = 0.18 

G12 = 6.2 Msi (42.5 GPa)

CVI-SiC Matrix E = 61 Msi (420 GPa) 

ν = 0.2

The maximum allowable strain in the x-direction (loading direction) for each constituent was initially 

set as follows: 0.06 percent for matrix, 0.05 percent for the transverse tows and 0.1 percent for the 

longitudinal tows based on the judgment of the authors and the available constituent data. These strain 

values translated to a maximum allowable stress of 36.6 ksi (252 MPa) for the matrix, 37.6 ksi (260 MPa) 

for the longitudinal tows, and 7.65 Ksi (53 MPa) for the transverse tows. The yield stress for each 

constituent was assumed to be the failure stress or the maximum allowable stress. As mentioned earlier, 

on attaining the yield stress, the constituents were assumed to be perfectly plastic (i.e., the stress in the 

constituent remained constant at its yield stress even as the strain increased, thereby inducing local stress 

redistribution). In addition, the maximum allowable stresses and strains for each constituent were chosen 

to reflect the fact that experimental results show that the longitudinal tows fail well after the matrix and 

transverse tows fail. The results from the analysis of several of the actual sections (Sections 03, 10 22 and 

37 where the nomenclature is based on the sequence of polished sections) and two of the artificially 

generated “aligned” sections (Ideal-01 and Ideal-03) will be presented here. Schematics of the sections 

analyzed for this study are shown in Figure 6. The “aligned” sections have regularly spaced tows and a 

gentle undulation of longitudinal fiber tows (which is fit to the shape observed in the section 

micrographs) and, in general, a very regular structure.

Tensile stress-strain curves for Section 10, along with an experimentally obtained macro stress-strain 

curve for comparison (Ref. 6) are shown in Figure 7. It should be mentioned that the experimental data 

comes from test specimens that inherently contain multiple sections. Thus, the experimental data tends to 

homogenize the results from multiple individual sections. As can be seen in this figure, the nonlinearity of 

the stress-strain curve is captured by the model, and the stress and strain levels where the stress-strain 

curve begins to deviate from linearity coincide with the experimental results. However, the simulation 

predicts a flattening of the stress-strain response after the onset of nonlinearity, while the experimental 

results show a secondary linear region, with a steeper slope than the simulation, after the onset of 

nonlinearity. Even though, the results from only one section are shown in Figure 7, as will be discussed in 

detail later, the computed results from other sections were similar. For example, the range of proportional 

limit stress for various sections analyzed is 18 to 19 ksi.

In order to gain more insight into features of the macroscopic stress-strain response, a study was 

undertaken to evaluate the effects and sensitivity of the maximum allowable strain on the overall stressstrain behavior of the composite. The maximum allowable strain of the matrix, longitudinal tows and 

transverse tows were changed by ±25 percent from the baseline values to evaluate their sensitivity in the 

overall stress-strain behavior. The results from these simulations (along with an experimental curve for 

comparison) are shown in Figure 8. The results show that varying the matrix strength has a significant 

effect on the stress level where the overall stress-strain curve flattens out as expected. However, the slope 

of the stress-strain curve after the initial onset of nonlinearity is not significantly affected. The strength of 

the transverse tows has only a slight influence on the initiation of non-linearity as well as the secondary 

slope of the overall stress-strain curve. When the failure strain (and thus the failure strength) of the 

longitudinal tows is increased, the strain level at which the stress-strain curve makes its initial break from 

linearity does not change compared to the baseline, but the stress level at which the stress-strain curve 

flattens out increases. More significantly, as the longitudinal tow failure strain (and stress) is increased, 

the second linear region observed in the experimental results begins to appear. These results indicate that 

the longitudinal tow strength has a major influence on the secondary slope of the overall composite and, 
NASA/TM—2012-217297 7

(a) Section 03

(b) Section 10

(c) Section 22

(d) Section 37

(e) Section Ideal-01

(f) Section Ideal-03

Figure 6.—Schematics of (a) Section 03, (b) Section 10, 

(c) Section 22, (d) Section 37, (e) Section Ideal-01 and 

(f) Section Ideal-03. Longitudinal tows are shown in red, 

transverse tows in green, matrix is shown in red and porosity 

is shown as black areas.

Applied load
NASA/TM—2012-217297 8

Figure 7.—Stress-strain curve of Section 10 computed using baseline constituent 

wstrength values with comparison to experimental results.

for the most part, controls the ultimate failure. Therefore, these results indicate that the failure strain of 

the longitudinal tows should be set to the overall longitudinal failure strain of the composite, 0.5 percent, 

with a corresponding yield/failure strength of 188 Ksi (1300 MPa). In order to more realistically simulate 

the damage pattern in the composite, in the remainder of the analyses presented in this report, the 

longitudinal tow failure strain was adjusted to 0.5 percent. 

Tensile stress-strain curves of each of the actual sections were generated using the failure strain of 

0.5 percent for the longitudinal tows, and are plotted along with an experimentally obtained stress-strain 

curve (Ref. 11) for comparison in Figure 9, As can be seen from this figure, each of the simulated stressstrain curves for the actual sections were nearly identical, with deviations displayed only after the 

proportional limit. The computed stress-strain curves showed an initial linear region, and a transition to a 

second linear region, with a shallower slope. The transition began at a strain level of around 0.036 percent

and ended at a strain level of about 0.072 percent. This shows that the damage initiates well before the 

first major point of macroscopic non-linearity. This has also been substantiated experimentally by using 

acoustic emission technique to identify damage (Ref. 12). Below the proportional limit, the computed 

results somewhat underpredicted the composite stress. This discrepancy is most likely due to errors in the 

estimated constituent properties. In particular, the constituent properties used were calibrated for another 

similar woven CMC system. This approximation was deemed acceptable since the emphasis in the current 

study was not to correlate with the experimental data. However, the stress and strain levels at which the 

initial nonlinearity occurs in the stress-strain response were reasonably well predicted. After the 

proportional limit, the analytical results appear to over predict the experimental data. This phenomenon is 

most likely due to the assumption that the longitudinal tows have a linear stress-strain response until 

failure, where in reality local matrix microcracking is likely taking place in the longitudinal tows before 

they fail, leading to the tows having an elastic/plastic strain hardening response, which is not accounted 

for in the current models. The overall character of the computed stress-strain curves matches the 

experimental results, however. Note that the magnitude of local stress field at a point are highly mesh 

sensitive and therefore should not be viewed as a quantitatively precise value. In this work, no mesh 

dependence study was performed to assess the sensitivity of stress values. 
NASA/TM—2012-217297 9

Figure 8.—Sensitivity of overall stress-strain curve to maximum allowable strain of (a) 

matrix, (b) transverse tows and (c) longitudinal tows for Section 10.
NASA/TM—2012-217297 10

Figure 9.—Experimental and computed tensile stress-strain response of 

SiC/SiC composite utilizing modified failure strains for longitudinal tows. 

To better understand the local mechanisms that cause the initial nonlinearity of the macroscopic 

stress-strain curve, a detailed examination was conducted on the local stress levels within the examined 

cross-sections. The goal of the detailed examination was to obtain insights as to why sections with 

significantly different local architectures would yield nearly identical macroscopic response curves. The 

local von Mises stress levels were plotted in each of the analyzed sections at several time steps. To study 

the initiation and propagation of damage, several plots were generated where elements with von Mises 

stress levels at the specified constituent yield stress were color coded red and elements with von Mises 

stress levels below the specified constituent yield stress were color coded in blue. Figures 10 to 12 show 

the results for Section 03, Figures 13 to 15 show these results for Section 10, Figures 16 to 18 show the 

results for Section 22 and Figures 19 to 21 show the results for Section 37. In each of the figures, (a) 

shows the stresses in the transverse tows and (b) shows the stresses in the matrix. For each section, the 

first figure (Figs. 10, 13, 16 and 19 for example) show the local stresses at a global strain level of 

0.036 percent, the second figure shows the local stresses at a global strain level of 0.054 percent, and the 

third figure shows the local stresses at a global strain level of 0.072 percent.

As can be seen in Figures 10, 13, 16 and 19, for each of the analyzed sections damage initiates in the 

transverse tows at a global strain level of 0.036 percent. For each of the sections, the initial transverse tow 

damage appears to take place in regions where there is a combination of two or more transverse tows 

touching and where there is little matrix material near the transverse tows. While the damage occurs in 

different relative locations within each of the cross-sections, the overall trend appears similar in each of 

the sections. Another point of interest is that a relatively small amount of damage (‘yielding’) within each 

section is enough to initiate nonlinearity in the overall stress-strain curve. Both the experimental and 

predicted stress-strain curves begin to exhibit nonlinearity at approximately this global strain level. 

However, the transition to nonlinearity is somewhat more gradual in the predicted curve as compared to 

the experimental curve. This discrepancy is most likely due to the fact that in the plasticity based material 

model used in the current analyses, once the yield/failure stress is reached in each constituent the stress 

remains at a constant level (yield stress) and sheds all additional stress to surrounding material. In the 

actual composite, once the constituent failure stress is reached the material starts to fail and shed load to 

the nearby matrix and fiber tows. Only when the damage accumulates to a critical level does the actual 

material behavior begin to deviate from linearity. This transverse tow unloading cannot be simulated with 

the current material model. Efforts are currently underway to compare the predicted damage patterns to 

the results obtained experimentally. This transverse tow unloading is accounted for by Liu and Arnold 

(Ref. 7), by modeling the tow behavior in-situ using micromechanics.
NASA/TM—2012-217297 11

(a)

(b)

Figure 10.—Areas in (a) transverse tows and (b) matrix in Section 03 at strain level of 0.036 percent

where von Mises stress is at yield (failure) indicated by red color and below yield (failure) stress 

indicated by blue color.

(a)

(b)

Figure 11.—Areas in (a) transverse tows and (b) matrix in Section 03 at strain level of 0.054 percent

where von Mises stress is at yield (failure) indicated by red color and below yield (failure) stress 

indicated by blue color.
NASA/TM—2012-217297 12

(a)

(b)

Figure 12.—Areas in (a) transverse tows and (b) matrix in Section 03 at strain level of 0.072 percent

where von Mises stress is at yield (failure) indicated by red color and below yield (failure) stress 

indicated by blue color.

(a)

(b)

Figure 13.—Areas in (a) transverse tows and (b) matrix in Section 10 at strain level of 0.036 percent

where von Mises stress is at yield (failure) indicated by red color and below yield (failure) stress 

indicated by blue color.
NASA/TM—2012-217297 13

(a)

(b)

Figure 14.—Areas in (a) transverse tows and (b) matrix in Section 10 at strain level of 0.054 percent

where von Mises stress is at yield (failure) indicated by red color and below yield (failure) stress 

indicated by blue color.

(a)

(b)

Figure 15.—Areas in (a) transverse tows and (b) matrix in Section 10 at strain level of 0.072 percent

where von Mises stress is at yield (failure) indicated by red color and below yield (failure) stress 

indicated by blue color.
NASA/TM—2012-217297 14

(a)

(b)

Figure 16.—Areas in (a) transverse tows and (b) matrix in Section 22 at strain level of 0.036 percent

where von Mises stress is at yield (failure) indicated by red color and below yield (failure) stress 

indicated by blue color.

(a)

(b)

Figure 17.—Areas in (a) transverse tows and (b) matrix in Section 22 at strain level of 0.054 percent

where von Mises stress is at yield (failure) indicated by red color and below yield (failure) stress 

indicated by blue color.
NASA/TM—2012-217297 15

(a)

(b)

Figure 18.—Areas in (a) transverse tows and (b) matrix in Section 22 at strain level of 0.072 percent

where von Mises stress is at yield (failure) indicated by red color and below yield (failure) stress 

indicated by blue color.

(a)

(b)

Figure 19.—Areas in (a) transverse tows and (b) matrix in Section 37 at strain level of 0.036 percent

where von Mises stress is at yield (failure) indicated by red color and below yield (failure) stress 

indicated by blue color.
NASA/TM—2012-217297 16

(a)

(b)

Figure 20.—Areas in (a) transverse tows and (b) matrix in Section 37 at strain level of 0.054 percent

where von Mises stress is at yield (failure) indicated by red color and below yield (failure) stress 

indicated by blue color.

(a)

(b)

Figure 21.—Areas in (a) transverse tows and (b) matrix in Section 37 at strain level of 0.072 percent

where von Mises stress is at yield (failure) indicated by red color and below yield (failure) stress 

indicated by blue color.
NASA/TM—2012-217297 17

At a global strain level of 0.054 percent, as can be seen in Figures 11, 14, 17 and 20, damage is much 

more widespread in the transverse tows for all of the analyzed sections. The overall level of damage in the 

transverse tows in Section 10 appears to be significantly less than in the other sections, for reasons which 

are still under investigation. In addition, at this strain level a substantial amount of damage can be 

observed in the matrix. On the top and bottom surfaces of each section, the matrix damage appears to be 

concentrated in areas between transverse tows where the tows are separated by some critical distance. In 

the interior of the sections, the matrix damage (‘yielding’) appears adjacent to areas of large pores, and 

near damaged transverse tows. In addition, the areas of matrix damage appear to make vertical bands 

through the cross-section. At 0.054 percent strain, the predicted stress-strain curve displays a significant 

amount of nonlinearity. These results appear to indicate that the damage in the transverse tows is a 

significant driver of the overall material nonlinearity. However and potentially more significantly, the 

results also indicate that at this strain level the combined damage in the transverse tows and the matrix 

form vertical bands through the thickness of the cross-section. This result indicates that a key driver of 

macroscopic nonlinearity is that undamaged load paths perpendicular to the loading direction start 

becoming unavailable. As the undamaged load paths start becoming unavailable, the overall stiffness of 

the material decreases. 

At a global strain level of 0.072 percent, as can be seen in Figures 12, 15, 18 and 21, damage has 

become much more extensive. With the exception of Section 10, nearly all of the transverse tows are 

completely damaged. The level of damage in the matrix has also significantly increased, with Section 3 

showing the most extensive amount of matrix damage. In all cases, however, multiple bands of matrix 

damage through the thickness of the cross-section can be observed. In the macroscopic stress-strain curve, 

this strain level represents the point where the second linear region of the curve begins. These results 

indicate that the development of transverse tow damage and a critical density of bands of damaged matrix 

material leads to a progressive lowering of the overall modulus reflected as nonlinearity in the 

macroscopic stress-strain curve, which is commonly referred to as the “first matrix cracking stress”.

To gain further insight into the response of the material, two artificially generated “aligned” sections 

(Ideal-01 and Ideal-03) were analyzed. Figure 22 shows the tensile stress-strain curves computed for these 

sections, along with the results from Section 03 for comparison. Figure 22(a) shows the overall stressstrain curve, while Figure 22(b) shows a close-up of the stress-strain response up to approximately the 

proportional limit. The computed response of the “aligned” sections is similar in character to the response 

of the “actual” section, and the strain levels at which nonlinearity begins and levels off for the aligned 

sections are similar to the levels computed for the actual section. However, the response of Section Ideal03 is somewhat stiffer than the response of the actual sections, and the response of Section Ideal-01 is 

somewhat softer than the response of the actual sections. The causes for this discrepancy are still under 

investigation, but the discrepancies could be related to the amount and distribution of the porosity and the 

alignment of the transverse tows. The relative volume fractions of the various constituents could also play 

a role in the variation of the material response.

To further understand the local mechanisms that lead up to the initial nonlinearity of the macroscopic 

stress-strain curve in the aligned sections, a detailed examination of the local stress levels within the 

examined cross-sections was conducted. The local stress levels were again plotted in each of the analyzed 

sections at several composite strain levels, with elements having stress levels above the specified 

constituent yield stress color coded red and elements having stress levels below the specified constituent 

yield stress color coded blue. Figures 23 to 25 show the results for Section Ideal-01 and Figures 26 to 28 

show the results for Section Ideal-03. Again, in each of the figures (a) shows the von Mises stresses in the 

transverse tows and (b) shows the von Mises stresses in the matrix. For each section, the first figure 

shows the local stresses at a global strain level of 0.045 percent (since no identifiable damage was 

observed at a strain level of 0.036 percent for these sections), the second figure shows the local stresses at 

a global strain level of 0.054 percent, and the third figure shows the local stresses at a global strain level 

of 0.072 percent.
NASA/TM—2012-217297 18

(a)

(b)

Figure 22.—Computed tensile stress-strain response of SiC/SiC composite using 

idealized “aligned” cross-sections. Plot (a) shows overall response and plot (b) shows 

response up to proportional limit.

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4

Stress, Ksi

Strain, %

Ideal-01

Ideal-03

Section-03

0

10

20

30

40

50

60

0 0.02 0.04 0.06 0.08 0.1

Stress, Ksi

Strain, %

Ideal-01

Ideal-03

Section-03
NASA/TM—2012-217297 19

(a)

(b)

Figure 23.—Areas in (a) transverse tows and (b) matrix in Section Ideal-01 at strain level of 

0.045 percent where von Mises stress is at yield (failure) indicated by red color and below yield 

(failure) stress indicated by blue color.

(a)

(b)

Figure 24.—Areas in (a) transverse tows and (b) matrix in Section Ideal-01 at strain level of 

0.054 percent where von Mises stress is at yield (failure) indicated by red color and below yield 

(failure) stress indicated by blue color.
NASA/TM—2012-217297 20

(a)

(b)

Figure 25.—Areas in (a) transverse tows and (b) matrix in Section Ideal-01 at strain level of 

0.072 percent where von Mises stress is at yield (failure) indicated by red color and below yield 

(failure) stress indicated by blue color.

(a)

(b)

Figure 26.—Areas in (a) transverse tows and (b) matrix in Section Ideal-03 at strain level of 

0.045 percent where von Mises stress is at yield (failure) indicated by red color and below yield 

(failure) stress indicated by blue color.
NASA/TM—2012-217297 21

(a)

(b)

Figure 27.—Areas in (a) transverse tows and (b) matrix in Section Ideal-03 at strain level of 

0.054 percent where von Mises stress is at yield (failure) indicated by red color and below yield 

(failure) stress indicated by blue color.

(a)

(b)

Figure 28.—Areas in (a) transverse tows and (b) matrix in Section Ideal-03 at strain level of 

0.072 percent where von Mises stress is at yield (failure) indicated by red color and below yield 

(failure) stress indicated by blue color.
NASA/TM—2012-217297 22

As can be seen in Figures 23 and 26, for both of the “aligned” sections damage initiates in the 

transverse tows at a global strain level of 0.045 percent. For Section Ideal-01, damage initiates primarily 

in the transverse tows and seems concentrated in the center of the tows. Damage does not appear to be 

significant in transverse tows which are surrounded on both sides by large regions of porosity and have 

significant amounts of matrix between the transverse tows. Damage does appear to initiate in areas where 

the tows are tightly packed thereby excluding matrix infiltration. In Section Ideal-03, the damage appears 

to initiate on the edges of transverse tows near large areas of porosity, in transverse tows in regions where 

adjacent transverse tows touch. Matrix damage in Section Ideal-03 is present both within the large areas 

of porosity and the outer surfaces between transverse tows.

At a global strain level of 0.054 percent, as can be seen in Figures 24 and 27, for both of the “aligned” 

sections the damage is much more widespread in the transverse tows. The damage patterns in the matrix 

are similar to those observed in the previous strain level, with matrix damage between transverse tows 

becoming slightly more extensive in Section Ideal-01. These results once again indicate that transverse 

tow damage may be a major driving force in the nonlinearity of the macroscopic stress-strain curve.

At a global strain level of 0.072 percent, as can be seen in Figures 25 and 28, for both “aligned” 

sections the transverse tows have become almost completely damaged. The damage in the matrix has also 

become more extensive, but appears to be aligned in vertical bands across the width of the cross section.

The finding that in the aligned sections damage starts in the transverse tows and progresses to vertical 

bands of damage in the matrix is similar to what was observed in the simplified microstructure sections.

However, the detailed damage patterns in the aligned sections are different than what was observed in the 

simplified actual sections, for reasons that are still under investigation. As part of the process of 

understanding this discrepancy, detailed correlations of the initiation and progression of damage to 

specific features of the microstructure will be carried out in future work.

Summary

Detailed finite element analyses of a model material system, specifically a five-harness twodimensional satin weave architecture, CVI SiC/SiC, composite were performed. Since these composites 

display significant irregularity and variability in their microstructure, this work sought to evaluate the 

effect of this variability on overall composite properties, damage initiation and propagation of damage, 

concentrating on the factors leading to the initial nonlinearity in the macroscopic stress-strain curve. The 

results indicate that the overall macroscopic longitudinal tensile stress-strain response of various crosssections of the composite is similar despite differences in the composition of the microstructure. The 

cause of this similarity may be due to the fact that for each of the sections damage was found to start in 

the transverse tows and progress to vertical bands of matrix damage aligned with the damage in the 

transverse tows. While the specific locations where these damage patterns occurred differed between the 

various sections, the overall pattern in the initiation and development of damage were similar. The 

damage initiation and progression appears to be related to the architectural and microstructural details of 

the sections analyzed. In this analysis, a simple damage (elastic perfectly-plastic) model was able to 

produce a macroscopic tensile stress-strain response that is in reasonably good agreement with the test 

data, along with local damage patterns that also appear to be realistic. In the in-plane direction, the 

macroscopic deformation behavior is very similar (as expected) and the details of microstructure have 

little effect. However, more realistic representations of the microstructural details provide an improved 

capability to identify the locations of stress risers. However, when evaluating damage initiation, caution 

should be used as the magnitude of the local stress value can vary significantly based on mesh density and 

thus the damage initiation. Future work will incorporate more realistic composite architectures, including 

non-uniform matrix thickness, and damage models more suitable for ceramic materials. Efforts are 

currently underway to compare the predicted damage patterns to the results from experimental 

observations. Furthermore, the effects of microstructural features on damage will be investigated in more 

detail and work will be conducted to quantitatively link the variability in the architecture to the material 

response.
NASA/TM—2012-217297 23

References

1. Nemeth, N.N.; Mital, S.K; and Lang, J.: “Evaluation of Solid Modeling Software for Finite Element 

Analysis of Woven Ceramic Matrix Composites”, NASA/TM-2010-216250, 2010.

2. Mital, S.K.; Bednarcyk, B.A; Arnold, S.M.; and Lang, J.: “Modeling of Melt-Infiltrated SiC/SiC 

Composite Properties”, NASA/TM-2009-215806, October 2009.

3. Goldberg, R. K.; Bonacuse, P. J.; and Mital, S. K.: “Investigation of Effects of Material Architecture 

on the Elastic Response of a Woven Ceramic Matrix Composite,” NASA Technical Memorandum

NASA/TM—2012-217269, January 2012.

4. Bonacuse, P.J.; Mital, S.K.; and Goldberg, R.K.: “Characterization of the As Manufactured 

Variability in a CVI SiC/SiC Woven Composite”, Proc. of ASME Turbo Expo 2011, June 6-10, 

2011, Vancouver, British Columbia, Canada.

5. Nagpal, V.K., Tong, M.T., Murthy, P.L.N. and Mital, S.K.: Probabilistic Modeling of High 

Temperature Material Properties of a 5-Harness 0/90 Sylramic Fiber/CVI-SiC/MI-SiC Woven 

Composite, NASA TM-208497, 1998.

6. Mital, S.K. and Murthy, P.L.N.: Probabilistic Micromechanical Analysis of a C/SiC Composite,

NASA TM-211339, 2002.

7. Liu, K. 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 in print, 2011.

8. Reid, A. et al.: “Modeling Microstructures with OOF2”, Int. J. Materials and Product Technology, 

Vol. 35, pp. 361-373, 2009. 

9. ABAQUS General Purpose Finite Element Program. Version 6.8, SIMULIA, Providence, RI

10. Morscher, G.N., Singh M., Kiser, J.D., Freedman, M., Bharr, R. “Modelling 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. 

11. Kiser, J; Calomino, A.; Brewer, D.; DiCarlo, J.; and Morscher, G.: “SiC/SiC Composites for High 

Temperature Applications.” Proc. of HTCMC-7 (7th International Conference on High Temperature 

Ceramic Matrix Composites), Sept. 19-22, 2010, Bayreuth, Bavaria, Germany. 

12. Morscher, G. “Stress, Matrix Cracking, Temperature, Environment and Life of SiC/SiC Woven 

Composites”, Proc. of HTCMC-7 (7th International Conference on High Temperature Ceramic 

Matrix Composites), Sept. 19-22, 2010, Bayreuth, Bavaria, Germany. 
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4. TITLE AND SUBTITLE

Two-Dimensional Nonlinear Finite Element Analysis of CMC Microstructures 

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Mital, Subodh, K.; Goldberg, Robert, K.; Bonacuse, Peter, J. 

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WBS 599489.02.07.03.02.11.01 

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14. ABSTRACT

A research program has been developed to quantify the effects of the microstructure of a woven ceramic matrix composite and its variability 

on the effective properties and response of the material. In order to characterize and quantify the variations in the microstructure of a five 

harness satin weave, chemical vapor infiltrated (CVI) SiC/SiC composite material, specimens were serially sectioned and polished to 

capture images that detailed the fiber tows, matrix, and porosity. Open source quantitative image analysis tools were then used to isolate the 

constituents, from which two dimensional finite element models were generated which approximated the actual specimen section geometry. 

A simplified elastic-plastic model, wherein all stress above yield is redistributed to lower stress regions, is used to approximate the 

progressive damage behavior for each of the composite constituents. Finite element analyses under in-plane tensile loading were performed 

to examine how the variability in the local microstructure affected the macroscopic stress-strain response of the material as well as the local 

initiation and progression of damage. The macroscopic stress-strain response appeared to be minimally affected by the variation in local 

microstructure, but the locations where damage initiated and propagated appeared to be linked to specific aspects of the local microstructure.

15. SUBJECT TERMS

Ceramic matrix composites; Woven composites; Finite element analysis; Microstructure; Porosity 

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