Marlana B. Goldsmith, Bhavani V. Sankar, and Raphael T. Haftka

University of Florida, Gainesville, Florida

Robert K. Goldberg

Glenn Research Center, Cleveland, Ohio

Quantifying Effects of Voids in Woven Ceramic

Matrix Composites

NASA/TM—2013-217874

June 2013

AIAA–2013–1619
NASA STI Program . . . in Profi le

Since its founding, NASA has been dedicated to the 

advancement of aeronautics and space science. The 

NASA Scientifi c 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 Offi cer. 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 signifi cant phase 

of research that present the results of NASA 

programs and include extensive data or theoretical 

analysis. Includes compilations of signifi cant 

scientifi c 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. Scientifi c 

and technical fi ndings 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. Scientifi c and 

technical fi ndings by NASA-sponsored 

contractors and grantees.

• CONFERENCE PUBLICATION. Collected 

papers from scientifi c and technical 

conferences, symposia, seminars, or other 

meetings sponsored or cosponsored by NASA.

• SPECIAL PUBLICATION. Scientifi c, 

technical, or historical information from 

NASA programs, projects, and missions, often 

concerned with subjects having substantial 

public interest.

• TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientifi c 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
Marlana B. Goldsmith, Bhavani V. Sankar, and Raphael T. Haftka

University of Florida, Gainesville, Florida

Robert K. Goldberg

Glenn Research Center, Cleveland, Ohio

Quantifying Effects of Voids in Woven Ceramic

Matrix Composites

NASA/TM—2013-217874

June 2013

AIAA–2013–1619

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

54th Structures, Structural Dynamics, and Materials Conference (SDM)

cosponsored by AIAA, ASME, ASCE, AHS, and ASC

Boston, Massachusetts, April 8–11, 2013
Acknowledgments

The funding for this work was provided by the NASA Graduate Student Research Program, grant number NNX10AM49H. The 

authors are thankful to Kim Bey of NASA Langley Research Center (LaRC), Peter Bonacuse of NASA Glenn Research Center 

(GRC), and Subodh Mital of The University of Toledo and GRC for many helpful discussions.

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

Trade names and trademarks are used in this report for identifi cation 

only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2013-217874 1 

Quantifying Effects of Voids in Woven Ceramic 

Matrix Composites 

Marlana B. Goldsmith, Bhavani V. Sankar, and Raphael T. Haftka 

University of Florida 

Gainesville, Florida 32611 

Robert K. Goldberg 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

Randomness in woven ceramic matrix composite architecture has been found to cause large 

variability in stiffness and strength. The inherent voids are an aspect of the architecture that may cause a 

significant portion of the variability. A study is undertaken to investigate the effects of many voids of 

random sizes and distributions. Response surface approximations were formulated based on void 

parameters such as area and length fractions to provide an estimate of the effective stiffness. Obtaining 

quantitative relationships between the properties of the voids and their effects on stiffness of ceramic 

matrix composites are of ultimate interest, but the exploratory study presented here starts by first 

modeling the effects of voids on an isotropic material. Several cases with varying void parameters were 

modeled which resulted in a large amount of variability of the transverse stiffness and out-of-plane shear 

stiffness. An investigation into a physical explanation for the stiffness degradation led to the observation 

that the voids need to be treated as an entity that reduces load bearing capabilities in a space larger than 

what the void directly occupies through a corrected length fraction or area fraction. This provides 

explanation as to why void volume fraction is not the only important factor to consider when computing 

loss of stiffness. 

Nomenclature 

a Void area 

A FEA model area 

C Material stiffness matrix 

E Young’s modulus 

G Shear modulus 

l Height of void 

L FEA model height 

 Height correction factor 

Vv Void volume fraction 

ν Poisson’s ratio 

1.0 Introduction 

Aircraft and spacecraft components that undergo extreme thermo-structural loads have reached some 

material limitations in terms of strength and weight. Woven ceramic matrix composites (CMCs), in 

particular, are candidate materials for future hypersonic vehicle components such as thermal protection 

and aero-propulsion systems due to their high strength and fracture toughness at elevated temperatures 
NASA/TM—2013-217874 2 

(Ref. 1). However, variability in the stiffness and strength may limit widespread implementation. Some of 

the variability is believed to be due to randomness in the architecture (tow spacing, tow size, tow nesting), 

as well as the unevenly shaped and spaced voids created as a result of the randomness in architecture and 

the nature of the manufacturing process. Variability also exists in the material properties of the constituents, but its effects on stiffness and strength are thought to be smaller than the variability due to 

architectural randomness. 

Recent work completed by the authors found that variation in tow size and tow spacing alone does not 

explain all of the variability found in the stiffness of the ceramic matrix composite system under 

investigation (Ref. 2). Instead, we hypothesized that the variable size and spacing of the voids plays a 

larger role in the variability of the composite properties. The effects of voids have been studied in the past 

with a variety of methods and goals. There is a large body of work in which the results are related mostly 

to void volume fraction (Refs. 3 to 6). The relationships found may be relevant for some applications, but 

as will be shown later, the volume fraction of the voids is not the only thing that must be considered in the 

woven ceramic matrix composite under consideration in this work. The voids considered here are 

unevenly distributed and many have large aspect ratios. Others have shown the importance of the 

microstructure including Tsukrov and Kachanov (Ref. 7) who accounted for elliptical voids with arbitrary 

orientations and eccentricities. This work is limited to a 2D anisotropic solid and the holes are noninteracting. Huang and Talreja (Ref. 8) demonstrated that the void shape and size was an important factor 

to consider, especially for transverse and shear stiffness of a unidirectional composite. They also observed 

that long, flat voids are most detrimental to the transverse stiffness. This agrees well with preliminary 

analysis on the woven Chemical Vapor Infiltration (CVI) SiC/SiC composite where void volume fraction 

is not the sole mechanism of stiffness degradation (Ref. 9). Uniform distribution of voids was assumed 

for the analysis which was shown to compare well for experimental data in unidirectional composites. 

However, woven composites appear to have significantly greater variability in void size and distribution 

as compared to unidirectional composites. 

The previous work on the effects of voids as discussed above provides insight into some of the 

mechanisms that contribute to stiffness degradation, and provide a benchmark to which observations in 

this paper can be qualitatively compared. However, the intent of the current work is to be able to use 

quantitative information about randomly spaced and shaped voids to make quantitative predictions of the 

resulting stiffness. The motivation and background for this work is provided in Section 2.0. The 

investigation begins with using finite element analysis to study the effects of non-overlapping voids 

(aligned in one plane) in an isotropic material, rather than a composite, so that the results are not 

convoluted by the other architectural variants. As will be demonstrated below, this is a good assumption 

to begin with since the variability in stiffness due to other architectural variations such as tow size and 

tow spacing contribute a relatively small amount to the variability in stiffness. Once the physical 

mechanisms of the effects of voids are understood at this level, the procedures can be extended to 

understand more complex material systems such as a composite. 

2.0 Characterization of Composite Variability 

The composite system under investigation is a CVI SiC/SiC eight ply 5HS (harness satin) weave 

material. The composite has continuous Sylramic-iBN fiber tows (20 ends per inch) woven into a fiveharness woven fabric preform in a [0/90] pattern. A silicon-doped boron nitride coating is deposited on 

the surface of the individual fibers in the tows. The fiber preform is then infiltrated with a CVI-SiC matrix 

which fills the tows and forms a thin matrix coating around the tows. The constituent material properties 

can be found in Reference 9. A 2D image of one cross section of the SiC/SiC composite, obtained by 

Goldberg, Bonacuse, and Mital is shown in Figure 1 (Ref. 9). The black areas in the interior of the cross 

section represent voids (the black area comprising the border of the image is not voids), which vary in 

location, size, and shape. Other 2D cross sections exhibit different random distributions of the voids and 

the architectural characteristics such as tow size, shape, and spacing. 
NASA/TM—2013-217874 3 

Finite element analysis of three cross sections similar to that of Figure 1 revealed significant 

variability in the transverse modulus (E3) that did not correlate to constituent volume fractions (VF), as 

shown in Table 1 (Ref. 9). Note that E1 is the modulus in the longitudinal direction as labeled by the 

coordinate system in Figure 1. This will be referred to as the in-plane modulus. Since the composite is 

stacked in a [0/90] pattern, E1 is assume to be equal to E2 E3 will be referred to as the transverse, or 

out-of-plane, modulus. 

In an attempt to capture the variability, finite element analysis based micromechanics and monte carlo 

simulations were used to analyze a large number of hypothetical cross sections (modeled as representative 

volume elements (RVEs)) that were generated based on quantified variability found in the tow width, tow 

height, and tow spacing. A typical RVE is represented in Figure 2. It consists of two stacked unit cells 

that are offset from one another by the width of one tow. This ply shifting of one tow offset was held 

constant for all RVEs with varying tow parameters. The results generated are summarized in Table 2, and 

they indicate that the architectural parameters that were varied were not enough to capture the variability 

exhibited by the full cross sections. In addition, the stiffnesses were correlated mostly to the volume 

fractions of the constituents which we know is not entirely the case. Details of the analysis can be found 

in Reference 2. 

TABLE 1.—FEA RESULTS OF MODULI FOR FULL CROSS SECTIONS 

 Void VF Tow VF Matrix VF E1 E3 

Cross section 1 3.2 63.0 33.8 237 103 

Cross section 2 4.8 62.8 32.4 227 77 

Cross section 3 3.5 63.9 32.6 234 51 

TABLE 2.—STATISTICAL PROPERTIES OF MODULI 

FROM MONTE CARLO SIMULATIONS 

OF RVE ANALYSIS 

 Mean Standard deviation 

E1 (GPa) 231.0 5.0 

E3 (GPa) 105.8 6.2 

 
NASA/TM—2013-217874 4 

As previously mentioned, one major component of the architectural variation not considered in the 

RVE analysis was ply shifting since it could not be quantified in the same manner as the tow size and 

spacing. An investigation into the effects of ply shifting indicated that it significantly affected the 

variability of the stiffness, as shown in Table 3. The results shown are for one RVE (tow size and spacing 

remains constant), with varying tow offsets. The magnitude of the tow offset is defined by assuming 

perfectly aligned tows or unit cells initially, then prescribing one unit cell to be offset by a certain fraction 

of a tow width. The variation in the shifting affects the out-of-plane modulus significantly. This effect is 

mostly due to rearrangement of macroscopic voids. The variability in ply shifting decreases the average 

value of the modulus and increases the amount of variability as compared to the results in Table 2. 

A visual assessment of the voids in Figures 3 to 5 provides clarity into the increased variability in the 

moduli due to shifting. It is known that voids have a significantly more detrimental effect on the out of 

plane moduli than the in-plane moduli for varying void content as well as for flat shapes (Ref. 8). The 

RVE with the tow offset of one tow has one void with a large aspect ratio, and several that are square in 

shape. The cross section in Figure 5 has several voids with large aspect ratios distributed throughout the 

composite, which is better represented by the RVE in Figure 4. As previously mentioned, the work of 

Huang and Talreja (Ref. 8) emphasize the importance of considering the size and aspect ratio of the voids. 

Clearly, when there are more voids with large aspect ratios the stiffness is significantly reduced. 

TABLE 3.—MODULI FOR ONE RVE 

DUE TO SHIFTING VARIATION 

(FIGS. 3 AND 4) 

Shifting (tow offset) E1 E3

1.00 (current RVE) 224 106 

0.75 221 92 

2.50 231 82 

3.25 234 89 

4.50 218 70 
NASA/TM—2013-217874 5 

3.0 Analysis Methods 

With each new ply shifting applied, the key architectural change is the size, shape, and location of the 

voids. Some qualitative effects have been studied, but it would be useful to have predictive models to 

estimate the mechanical properties, as well as to provide a physical understanding. The microstructure of the 

5HS SiC/SiC composite is very complicated. In order to develop an understanding of the effects of voids, 

without the results being convoluted by other aspects of the material geometry, the preliminary analysis will 

be completed on an isotropic material with a Young’s modulus of 100 GPa and a Poisson’s ratio of 0.3. 

3.1 Finite Element Model 

Previous analysis of the effects of tow variability was completed in 2D for simplicity. This is not the 

best way to model the woven composite, but it gives reasonable estimates in regard to modeling the 

variability. Since modeling the variability in the tow size and tow spacing led to relatively small 

variability in the stiffness, it may be possible to capture the variability in the composite specimens by 

modeling only the voids, and not the other details of the microstructure. At the least, studying the effects of 

voids in an isotropic material will provide valuable preliminary insight into the physical effects of voids on 

the composite. This also simplifies the problem so that the effects of voids can be studied in 3D, eliminating 

some of the shortcomings of a 2D approximation, especially when considering the shear stiffness. 

Finite element analysis was completed using Abaqus (Ref. 10), with 4-node tetrahedron elements. 

Periodic boundary conditions were applied such that one of the macro strains was non-zero and all other 

strains and the temperature differential T were zero. The macro-stresses were calculated by averaging 

the micro-stresses in the RVE. Using the six macro-stresses one can determine the first column of the 

stiffness matrix, C. The procedure was repeated for the other five macro stains to calculate the entire C

matrix. From C one can calculate the elastic constants using the relations of the type shown in 

Equation (1) where C is the 

 

 

11 16

1

26

61 66

6 6

21

11 12 66

1 2 12

...

[ ] ... ...

...

1 1

, , ,etc.

S S

CS S

S S

SS S

EEG





 

     

   

  

 (1) 

material stiffness matrix, S is the material compliance matrix, is Poisson’s ratio, and E and G are 

Young’s moduli and shear moduli, respectively. 

The finite element model depicted in Figure 6 is a cuboid of equivalent length and width. The aspect 

ratio (ratio of the length to the height) is 4 to 1, similar to that of the composite cross sections that were 
NASA/TM—2013-217874 6 

available. The voids were modeled as cuboids. The sharp corners do not affect the stiffness results; 

however they will be avoided if a similar model is used for examining the effects of voids on strength. 

The length and width of a given void are also equivalent. Since the composite’s weave is balanced, it is 

safe to assume that if the voids are of a given length when scanning from the 1-3 plane, they will be 

similar when scanning from the 2-3 plane. The voids can occur anywhere in the 1-2 plane. However, the 

location on the 3-axis is constrained to 7 different layers. This reflects the fact that the voids generally 

occur in the interlaminar positions, or between the plies. The descriptions of the specific void sizes and 

locations studied are given in the following section. 

3.2 Description of Void Cases 

The variables for this study are the number of voids, the aspect ratio, and the position of the voids. 

The aspect ratio (AR) is defined by the width (1-direction) divided by the height (3-direction). The size 

(8 by 8 by 2 mm) and isotropic material properties of the specimen will remain constant, in addition to the 

total void volume fraction of 4 percent. The specimen itself has an aspect ratio of 4. The initial 

exploratory results encompass 20 cases summarized pictorially in Figures 7 to 12. Additional 1-void cases 

in which the length did not equal the width were analyzed for the shear stiffness analysis. 

 

 

 
NASA/TM—2013-217874 7 

A more thorough description of each case and the FEA results relating to them can be found in Tables A.1 

and A.2. The variables include the number of voids, the aspect ratio of the voids (aspect ratio of length to 

height, since length and width are assume to equivalent), and the position of the voids. Some positions are 

not surveyed due to symmetry. 

4.0 Results and Analysis 

After analyzing 20 exploratory void cases, a few trends became obvious. The longitudinal stiffness 

and in-plane shear stiffness were not significantly affected by changes in the void aspect ratio and 

location. However, the transverse stiffness and out-of-plane shear stiffness were significantly affected. 

For a constant void volume fraction of 4.0 percent, the loss in stiffness averaged up to 15 percent. This is 

significant in that many models accounting for voids rely on void volume fraction alone. For the 

particular composite studied here, there are other factors besides volume fraction that must be significant 

and important to consider. The average loss of stiffness and the respective standard deviation is displayed 

in Table 4. When considering the effects of placing the voids in various locations with respect to one 

another, the stiffness is minimally impacted. This may not be true for cases in which voids are 

overlapping which will be explored in future work. 

TABLE 4.—LOSS OF STIFFNESS BASED ON 20 VOID CASES WITH A CONSTANT 

VOLUME FRACTION OF 4.0 PERCENT, DESCRIBED IN 

SECTION 3.0 AND THE APPENDIX 

E1 E2 E3 G12 G13 G23

Average percent loss of stiffness 0.7 1.0 12.5 1.2 12.4 14.9 

Standard deviation 0.5 0.5 4.1 0.7 3.1 4.4 

4.1 Transverse Stiffness Analysis and Results 

The remaining results and analysis shown in this paper will consider the transverse stiffness and shear 

stiffness. When examining the results of the transverse stiffness, there is an obvious pattern that shows 

that as the aspect ratio of the void increases, the stiffness decreases. The key feature of the void that is 

changing as the aspect ratio increases, is the area of the void on the 1-2 plane. It became clear that the 

projected area of the voids in the transverse direction (or the area of the void in the 1-2 plane) was 

important to determining the stiffness. The plot in Figure 13 plots the transverse stiffness as a function of 

the area fraction, a3 , which can be define as the ratio of the projected void area, a3, to the total area of the 

12 plane, A3, and written as 

3

3

3 A

a

a  (2) 

A strong linear relationship is found between the projected area of the voids and the resulting 

stiffness. The equation of the line is 

E3  70.96a3 102.79 (3) 

and the coefficient of determination, R2, is 0.98. There is obviously a small error in the fit because we 

know that the intercept should be 100 (at a void ratio of 0, the stiffness should be 100 GPa). Using the 

linear equation, the void cases not used for the fit (cases with various void spacing) can be used to check 

how well the fit predicts the stiffness for other void cases. When using the equation, the RMS (root mean 

square) error was 0.45 GPa which is very small. Using the equation does not account for any variability 

due to spacing, since the area ratio is the only input. It is important to note again that the voids’ placement 

did not significantly affect the stiffness, so the relative variability in that data set is small. 
NASA/TM—2013-217874 8 

The relationship of the projected area to the transverse stiffness may be useful, but it does not provide 

any physical understanding about the effects of voids. Intuitively one would assume the slope of the line 

in Figure 13 would be –100, which is not the case. In order to arrive at a physical explanation we can 

examine a problem in which we have two materials stacked on top of one another as shown in Figure 14. 

The displacement of the material can be written as 

 

  3 3

3

3 3 3

3

3 3

3 3

3 A E'

FL

A a E

Fl

A E

F L l       (4) 

where F is a force applied to the material, A3 is the area of the material, a3 is the area of the void in 

Material 3, L3 is the total length of the isotropic material in the 3-direction, l3 is the length of the void, 

E3 is the material’s nominal stiffness, and 3 E' is the resultant stiffness due to the void. A 3D image is 

provided in Figure 15. 

Equation (4) simplifies to 

  3

3

3 1

1 1

1

1

a

V

E

E v     

 (5) 

where Vv is the void volume fraction, E3 is the change in trasverse stiffness due to the voids, and 3 a was 

defined in Equation (2). If 3 a and 

3

3

E

E are small, the following approximation can be used: 

x

x    1

1

1
NASA/TM—2013-217874 9 

Equation (5) then simplifies to 

  3

3

3 V 1 a

E

E  v  (6) 

This relationship poorly predicts the FEA result, with an RMS error of 8.2 GPa. It was hypothesized that 

the height of the void, l3, has an effect on the stiffness that extends beyond the nominal height. 

Equation (6) can then be rewritten as 

       3 3 3 3 3 3

3

3 a l 1 a V 1 a a 1 a

E

E      v     (7) 

where ξ is a correction factor to the height fraction of the void and 3l is the height fraction of the void 

3

3

3 L

l l 

The height fraction may be referred to as the length fraction, also. In order to determine what the 

correction factor should be, the results from the basic approximation given by Equation (6) and the FEA 

results can be used to solve for ξ with the equation as derived below. 

      3 3 3 3

3

analytical

3

3

FEA

3 V 1 a a 1 a V 1 a

E

E

E

E  v      v   

  3 3

3

analytical

3

3

FEA

3

a 1 a

E

E

E

E



  

  (8) 

 
NASA/TM—2013-217874 10 

When the values for ξ are solved and plotted against the original length fraction, 3l , a strong linear 

relationship exists, shown in Figure 16. The effective length fraction, 3 l ' , can also be plotted against the 

original length as shown in Figure 17 so that there is a relationship between the original length fraction 

and the effective length fraction. 

   3 3 l ' l (9) 

The relationship between the original length fraction and the corrected length fraction is 

l '

3  0.33l3  0.51 (10) 

The need for a corrected length implies that there is more material that is not load-bearing than simply the 

space of the void in the total RVE volume. Instead, it extends farther through the thickness. The 45 line 

on the graph of Figure 17 represents the result if no correction was applied (the length fraction is equal to 

the corrected length fraction). The line demonstrates a clear difference between the original length 

fraction and that of the corrected length fraction. As the original length of the void increases, the 

correction factor and thus the amount the length needs to be corrected decreases. This reflects the fact that 

as the void’s size increases, its influence on the material around it is reaching the boundaries of the 

material and thus cannot be further corrected. 
NASA/TM—2013-217874 11 

TABLE 5.—COMPARISON OF ΔE3 FOR SAMPLE CASES NOT 

USED IN LENGTH FRACTION LINEAR REGRESSION 

ΔE3 linear regression,

GPa 

ΔE3 corrected equation, 

GPa 

ΔE3 FEA, 

GPa 

1 void, AR = 3 4.1 4.8 6.3 

1 void, AR = 5 6.8 6.4 7.3 

1 void, AR = 8 10.4 9.7 11.3 

1 void, AR = 10 12.5 11.8 13.7 

2 voids, s3 = max 13.8 13.2 14.0 

4 voids, 4 different AR’s, aligned 7.7 7.3 7.7 

Table 5 provides a comparison of the predicted decrease in the transverse stiffness for a few sample 

cases using Equation (3), for linear regression to the projected area, and Equation (7), using the corrected 

length fraction, versus the results from FEA. Both methods appear to be reasonable at predicting the 

stiffness loss due to voids for the given cases. 

5.0 Shear Stiffness Analysis and Results 

The shear stiffness discussion presented here is based on analysis of only one void of varying size in 

order to more easily see the effects of the geometry of the void on the stiffness reduction. The shear 

stiffnesses as the area fraction of the void, a3 , changes are plotted in Figure 18. For all data the height of 

the void, l3, are held constant. Varying heights affected the stiffness by less than 5 percent. A quadratic 

relationship is found between the projected area of the voids and the resulting stiffness. The equation of 

the line is 

10.9 26.77 3 38.46 2 G13  G23   a3  a  (11) 

In the same way that the transverse stiffness calculation must account for a corrected length, the shear 

stiffness calculation must account for a corrected area. The shear stress at the wall of the void is zero, and 

small near the boundary of the void. In this way, the effect of the void on the stresses extends further than 

just the space the void occupies. This is illustrated in Figure 19. The equations below are derived with 

respect to G13, but are written similarly for G23 also. 

A similar relationship to that of Equation (5) can be derived for the shear stiffness reduction 

  3

3 3

3

13

13 1

1

1

1

1

1

a

l a

a

V

G

G

v

       

 (12) 

Assuming a corrected area is necessary, a relationship between the corrected area 3 a' and the 

calculated shear stiffness reduction according to finite element analysis can be written as 

1

'

3

13

13

13

13

13

13

3



  





l

G

G

G

G

G

G

a (13) 

Knowing that 1

l will affect G13 differently than, 2l 3 a' can be thought of as 1

'

l 2

'

l . Therefore, the 

analysis was completed by using only one void while holding 1

l constant and varying 2l . This was done 
NASA/TM—2013-217874 12 

for several values of 1

'

l . The data can be found in Table A3 of the Appendix. Similarly to the length 

correction in the previous section, a relationship between the actual area, 3 a and the corrected area, 3 a'

can be found. The data are plotted in Figure 20. Through curve fitting, we found that the areas are related 

by the equation 

3

2

3 3 a'  1.84a  2.60a (14) 

with an R2

 value of 0.97. The length fractions are plotted against the corrected area in Figure 21. For 

small l1’s, the effect of l2 on the area is linear. As l1 increases the relationship between l2 and the corrected 

area becomes quadratic. This explains the deviation from linearity in the curve of Figure 20. The 

deviation from linearity between the uncorrected and corrected areas is likely caused by effects of the 

void approaching the boundary, as was observed in the transverse stiffness results. 

 
NASA/TM—2013-217874 13 

 
NASA/TM—2013-217874 14 

TABLE 6.—COMPARISON OF G13 FOR SAMPLE CASES 

A3 ΔG13 FEA,

GPa 

ΔG13 quadratic fit,

GPa 

ΔG13 corrected area, 

GPa 

1 void 0.22 6.43 6.27 5.53 

2 voids 0.27 5.78 8.07 5.48 

3 voids 0.27 5.30 7.95 6.23 

2 voids misaligned 0.23 4.74 6.81 5.76 

4 voids, various AR 0.15 3.64 4.18 4.30 

RMS error (GPa) ----- ----- 1.85 0.81 

TABLE 7.—COMPARISON OF G23 FOR SAMPLE CASES 

A3 ΔG23 FEA,

GPa 

ΔG23 quadratic fit,

GPa 

ΔG23 corrected area, 

GPa 

1 void 0.22 6.42 6.27 5.53 

2 voids 0.27 7.36 8.07 5.48 

3 voids 0.27 7.29 7.95 6.23 

2 voids misaligned 0.23 6.14 6.81 5.76 

4 voids, various AR 0.15 3.79 4.18 4.30 

RMS error (GPa) ----- ----- 0.57 1.10 

Tables 6 and 7 provide a comparison of the predicted decrease in shear stiffnesses for a few sample 

cases using Equation (11), for the quadratic fit to the projected area, and Equation (13), using the 

corrected area fraction calculated from the empirical relationship in Equation (14), versus the results from 

FEA. When comparing the results of the two tables, it is clear that G13 and G23 are not equivalent and 

likely dependent on some directional component that a relationship to area fraction cannot capture. It is 

not clear which method is more accurate. Further investigation regarding how the lengths, instead of only 

the area will be considered for future work. 

6.0 Concluding Remarks 

After completing finite element analysis on several cases of void size and spacing while the void 

volume fraction remained constant, it was clear that the voids affect the transverse stiffness and the outof-plane shear stiffness the most (up to 15 percent stiffness degradation on average for a void volume 

fraction of 4 percent). In an attempt to determine the driving factors of the reduction in stiffness, we found 

that for the transverse stiffness, the projected area of the voids onto the 1-2 plane had a significant impact. 

A strong linear relationship was found between the area and the resulting stiffness, but the relationship 

involving a corrected length fraction provided a better physical understanding. The voids effect on the 

load bearing volume extends further than the space that the void occupies, which offers an explanation of 

why relationships based solely on volume fraction are not sufficient when the voids of the composite have 

large aspect ratios and are unevenly distributed. Similarly, we found that the shear stiffness is also related 

to the projected area of the voids onto the 1-2 plane. However, an area correction, rather than a length 

correction is most appropriate due to the small shear stress around the walls of the voids. Future work will 

consider the effects of non-symmetry in the shear stiffness. The problems that arise when there are more 

voids that are potentially overlapping will also be addressed. 
NASA/TM—2013-217874 15 

Appendix—Effect of Void Geometry on Stiffness Properties 

The void cases studied are described below and their respective data are found in Tables A1 and A2. 

The cases are shown pictorially in Section 3.0. 

 1 void; Centered; Aspect ratio (AR) = 3, AR = 5, AR = 8, AR = 10 

 2 voids; Centered and aligned in “1” direction; d1 constant; AR = 3, AR = 5, AR = 8, AR = 10 

 3 voids; Centered and aligned in “1” direction; d1 constant; AR = 3, AR = 5, AR = 8, AR = 10 

 2 voids; AR = 8; Aligned in “1” direction; s1 = 0.15 mm, s1 = 0.57 mm, s1 = 1 mm, s1 = 1.6 mm 

 2 voids; AR = 8; Aligned in “1” direction; s1 constant; s3—maximum possible distance apart 

(one void on bottom, one void on top) 

 2 voids; AR = 8; NOT aligned in “1” direction (located in opposite corners); s1 constant; 

s3—maximum possible distance apart (one void on bottom, one void on top); 

 3 voids; AR = 10; Voids diagonal, but in same plane 

 3 voids; AR = 10; Voids diagonal, but in different planes; 

TABLE A1.—DATA FROM VOID CASES WITH VARYING ASPECT RATIOS 

No. voids AR E1, 

GPa 

E2, 

GPa 

E3, 

GPa 

G12, 

GPa 

G13, 

GPa 

G23, 

GPa 

1 3 98.0 98.0 95.7 37.4 35.4 35.4 

1 5 98.7 98.7 92.7 37.8 34.4 34.4 

1 8 99.1 99.1 88.7 38.0 32.9 32.9 

1 10 99.4 99.4 86.3 38.1 32.0 32.0 

2 3 98.5 98.1 94.3 37.5 35.8 35.3 

2 5 99.1 98.8 90.9 37.8 34.9 34.1 

2 8 99.5 99.3 86.1 38.1 33.6 32.3 

2 10 99.6 99.4 83.2 38.2 32.6 31.0 

3 3 99.0 98.2 93.8 37.5 36.0 35.1 

3 5 99.4 98.8 89.9 37.9 34.8 33.6 

3 8 99.8 99.3 84.7 38.1 33.2 31.2 

3 10 99.9 99.4 80.2 38.2 31.7 28.7 

TABLE A2.—DATA FROM VOID CASES WITH ASPECT RATIO OF 8 AND VARYING SPATIAL DISTRIBUTIONS 

No. voids Spacing description E1, 

GPa 

E2, 

GPa 

E3, 

GPa 

G12, 

GPa 

G13, 

GPa 

G23, 

GPa 

2 s1 = 0.16 mm 99.6 99.3 86.0 38.1 33.0 31.7 

2 s1 = 0.57 mm 99.6 99.3 86.2 38.1 33.5 32.2 

2 s1 = 1 mm 99.5 99.3 86.2 38.1 33.6 32.3 

2 s1 = 1.6 mm 99.5 99.3 86.2 38.1 33.6 32.3 

2 s3 = max 99.5 99.3 86.0 38.1 33.7 32.3 

2 Opposite corners 99.3 99.3 86.1 38.2 33.0 33.0 

3 Diagonal; same plane 99.4 99.3 84.7 38.2 33.1 33.0 

3 Diagonal; tiered planes 99.5 99.3 84.5 38.2 33.3 33.0 
NASA/TM—2013-217874 16 

TABLE A3.—DATA FROM VOID CASES WITH 1 VOIDS AND CONSTANT VOID HEIGHT 

l1 l2 E1, 

GPa 

E2, 

GPa 

E3, 

GPa 

G12, 

GPa 

G13, 

GPa 

E1, 

GPa 

0.001 0.001 99.9 99.9 99.5 38.4 38.3 38.3 

0.001 0.002 99.8 99.9 98.8 38.4 38.1 38.1 

0.001 0.003 99.7 99.9 98.1 38.4 37.8 37.9 

0.001 0.004 99.5 99.9 97.4 38.3 37.5 37.7 

0.001 0.005 99.4 99.9 96.6 38.3 37.1 37.5 

0.001 0.006 99.3 99.9 95.9 38.2 36.8 37.2 

0.001 0.007 99.1 99.9 95.2 38.2 36.5 37.1 

0.002 0.001 99.9 99.8 98.8 38.4 38.1 38.1 

0.002 0.002 99.7 99.7 96.9 38.3 37.3 37.3 

0.002 0.003 99.6 99.7 94.9 38.3 36.3 36.5 

0.002 0.004 99.4 99.7 93.0 38.2 35.2 35.8 

0.002 0.005 99.3 99.7 90.9 38.2 34.1 35.1 

0.002 0.006 99.1 99.8 88.9 38.1 32.9 34.5 

0.002 0.007 98.9 99.8 86.8 38.1 31.5 34.0 

0.003 0.001 99.9 99.7 98.1 38.4 37.9 37.8 

0.003 0.002 99.7 99.6 95.0 38.3 34.8 36.3 

0.003 0.003 99.5 99.5 91.7 38.2 34.8 34.8 

0.003 0.004 99.4 99.5 88.4 38.2 32.8 33.4 

0.003 0.005 99.2 99.6 85.1 38.1 30.7 32.2 

0.003 0.006 99.0 99.6 81.6 38.0 28.4 31.2 

0.003 0.007 98.8 99.7 78.1 38.0 25.8 30.3 

0.005 0.001 99.9 99.4 96.7 38.3 37.5 37.1 

0.005 0.002 99.7 99.3 90.9 38.2 35.1 34.1 

0.005 0.003 99.6 99.2 85.1 38.1 32.2 30.7 

0.005 0.004 99.4 99.2 78.9 38.0 29.0 27.7 

0.005 0.005 99.2 99.2 72.5 38.0 25.4 25.4 

0.005 0.006 99.0 99.3 65.8 37.9 21.6 23.6 

0.005 0.0075 98.7 99.7 55.3 37.9 14.9 21.4 

0.007 0.001 99.9 99.1 95.2 38.2 37.1 36.5 

0.007 0.002 99.8 98.9 86.8 38.1 34.0 31.5 

0.007 0.003 99.7 98.8 78.1 38.0 30.3 25.8 

0.007 0.004 99.6 98.8 68.8 37.9 26.4 21.0 

0.007 0.005 99.5 98.8 58.9 37.9 22.2 17.4 

0.007 0.006 99.4 99.0 48.4 37.9 17.7 14.8 

0.007 0.0075 99.2 99.5 31.5 38.0 10.1 11.8 
NASA/TM—2013-217874 17 

References 

1. Ohnabe, H., Masaki, S., Onozuka, M., Miyahara, K., and Sasa, T., “Potential application of ceramic 

matrix composites to aero-engine components,” Composites Part A: Applied Science and 

Manufacturing, vol. 30, Apr. 1999, pp. 489–496. 

2. Goldsmith, M.B., Sankar, B.V, Haftka, R.T., and Goldberg, R.K., “Effects of Microstructural 

Variability on Thermo-Mechanical Properties of a Woven Ceramic Matrix Composite,” 

NASA/TM—2013-217817, 2013. 

3. Madsen, B., and Lilholt, H., “Physical and mechanical properties of unidirectional plant fibre 

composites—an evaluation of the influence of porosity,” Composites Science and Technology, 

vol. 63, Jul. 2003, pp. 1265–1272. 

4. Pal, R., “Porosity-dependence of Effective Mechanical Properties of Pore-solid Composite 

Materials,” Journal of Composite Materials, vol. 39, Jul. 2005, pp. 1147–1158. 

5. Peters, P., Martin, E., and Pluvinage, P., “Influence of porosity and fibre coating on engineering 

elastic moduli of fibre-reinforced ceramics (SiC/SiC),” Composites, vol. 26, 1995, pp. 108–114. 

6. Wu, Y., Shivpuri, R., and Lee, L.J., “Effect of Macro and micro Voids on Elastic Properties of 

Polymer Composites,” Journal of Reinforced Plastics and Composites, vol. 17, 1998. 

7. Tsukrov, I., and Kachanov, M., “Effective moduli of an anisotropic material with elliptical holes of 

arbitrary orientational distribution,” International Journal of Solids and Structures, vol. 37, 2000, 

pp. 5919–5941. 

8. Huang, H., and Talreja, R., “Effects of void geometry on elastic properties of unidirectional fiber 

reinforced composites,” Composites Science and Technology, vol. 65, Oct. 2005, pp. 1964–1981. 

9. 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/TM—2012-217269, 2012. 

10. “ABAQUS Software Package, Ver. 6.8, SIMULIA, Providence, RI.” 
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

Quantifying Effects of Voids in Woven Ceramic Matrix Composites 

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Goldsmith, Marlana, B.; Sankar, Bhavani, V.; Haftka, Raphael, T.; Goldberg, Robert, K. 

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

WBS 794072.02.03.03.03 

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-18673 

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-2013-217874 

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited 

Subject Categories: 24 and 39 

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

Randomness in woven ceramic matrix composite architecture has been found to cause large variability in stiffness and strength. The 

inherent voids are an aspect of the architecture that may cause a significant portion of the variability. A study is undertaken to investigate the 

effects of many voids of random sizes and distributions. Response surface approximations were formulated based on void parameters such 

as area and length fractions to provide an estimate of the effective stiffness. Obtaining quantitative relationships between the properties of 

the voids and their effects on stiffness of ceramic matrix composites are of ultimate interest, but the exploratory study presented here starts 

by first modeling the effects of voids on an isotropic material. Several cases with varying void parameters were modeled which resulted in a 

large amount of variability of the transverse stiffness and out-of-plane shear stiffness. An investigation into a physical explanation for the 

stiffness degradation led to the observation that the voids need to be treated as an entity that reduces load bearing capabilities in a space 

larger than what the void directly occupies through a corrected length fraction or area fraction. This provides explanation as to why void 

volume fraction is not the only important factor to consider when computing loss of stiffness.

15. SUBJECT TERMS

Ceramic matrix composites; Micromechanics 

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF

 ABSTRACT

UU 

18. NUMBER

 OF 

 PAGES

24 

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