J.B. Min

Glenn Research Center, Cleveland, Ohio

D. Xue and Y. Shi 

Analytical Services & Materials, Inc., Hampton, Virginia

Micromechanics Fatigue Damage Analysis 

Modeling for Fabric Reinforced Ceramic 

Matrix Composites

NASA/TM—2013-217870

May 2013
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J.B. Min

Glenn Research Center, Cleveland, Ohio

D. Xue and Y. Shi 

Analytical Services & Materials, Inc., Hampton, Virginia

Micromechanics Fatigue Damage Analysis 

Modeling for Fabric Reinforced Ceramic 

Matrix Composites

NASA/TM—2013-217870

May 2013

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

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

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

Boston, Massachusetts, April 8–13, 2013
Acknowledgments

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at the NASA Glenn Research Center.

Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2013-217870 1

Micromechanics Fatigue Damage Analysis Modeling for Fabric 

Reinforced Ceramic Matrix Composites

J.B. Min

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

D. Xue* and Y. Shi*

Analytical Services & Materials, Inc.

Hampton, Virginia 23666

Abstract

A micromechanics analysis modeling method was developed to analyze the damage progression and 

fatigue failure of fabric reinforced composite structures, especially for the brittle ceramic matrix material 

composites. A repeating unit cell concept of fabric reinforced composites was used to represent the global 

composite structure. The thermal and mechanical properties of the repeating unit cell were considered as 

the same as those of the global composite structure. The three-phase micromechanics, the shear-lag, and 

the continuum fracture mechanics models were integrated with a statistical model in the repeating unit 

cell to predict the progressive damages and fatigue life of the composite structures. The global structure 

failure was defined as the loss of loading capability of the repeating unit cell, which depends on the 

stiffness reduction due to material slice failures and nonlinear material properties in the repeating unit 

cell. The present methodology is demonstrated with the analysis results evaluated through the 

experimental test performed with carbon fiber reinforced silicon carbide matrix plain weave composite 

specimens.

1.0 Introduction

Historically, advances in aerospace engine performance and durability have been linked to

improvements in materials. Recent developments in high performance aerospace turbomachinery engines 

have led to increased interest in ceramic matrix composites (CMC) to achieve revolutionary gains in 

engine performance. The use of CMC promises many advantages in turbine development. The most 

beneficial aspects of CMC are the material’s ability to maintain its strength to over 2400 °F, the internal 

material damping, and the relatively low density. While CMC reinforced with woven and braided fabric 

preforms are being considered for potential candidates in the NASA’s next-generation aerospace 

turbomachinery engine applications (Refs. 1 and 2), one of challenge areas in the development of highly 

efficient and lighter aircraft engines is high performance rotating blades subject to high cycle fatigue 

(HCF) limitations as a result of high vibratory stresses (Ref. 3). 

While fabric reinforced composites have considerable attention as alternative to conventional 

laminate composites consisting of stacked unidirectional plies, the architecture of a fabric reinforced 

composite is very complex and, therefore, the parameters controlling its strength properties are numerous. 

Considerable advancement has been reported in the analytical methods for static properties of fabric 

reinforced composites (Refs. 4 to 13), reliable analytical capability to analyze the component’s fatigue 

strength and life prediction are still very limited and evolving. This necessitates the development of the 

effective approach with a feasible analysis procedure to design the fabric reinforced composite structures 

such as integrally bladed turbine disk (Figure 1). 

 * Formerly Analytical Services & Materials, Inc. 
NASA/TM—2013-217870 2

Figure 1.—Fabric reinforced ceramic matrix composite integrally bladed turbine 

disk specimens.

Figure 2.—Basic procedures of analysis techniques and micrographic image of failure of plain weave bundle yarn.

The present study has made an attempt to develop an analytical method especially with considering 

the applicable procedure that combines the analytical methods with limited testing for the fatigue life 

prediction analysis of fabric reinforced ceramic matrix composite structures.

The repeating unit cell (RUC) concept developed in the previous studies (Refs. 4 to 9) (as illustrated 

in Figure 4 and Figure 5) has been considered as a reasonable approach to perform stress analysis and

failure prediction of fabric reinforced composite structures without using up micro-level finite element 

modeling struggle and utilized in this study. By observing the periodicity of the repeating pattern in a 

woven (or braided) fabric, it was assumed that a small repeating unit cell can be isolated which is 

sufficient to describe the fabric architecture.

The geometry model of fabric composites assumed that the RUC (i.e., lamina) of a composite is a 

system consisting of matrix and yarn slices as depicted in Figure 2. This micro-mechanics analytical 

technique was used to predict the global mechanical and thermal properties of fabric reinforced 

composites. The calculation of the mechanical properties was based on a simple geometry analysis which 

transforms the RUC of a composite into a typical micro-structure under uniform global strains. The RUC 

of a composite was used to represent the global composite structure, and thermal and mechanical 

properties of the RUC were considered as the same as those of the global composite. The statistical failure 

criterion and maximum stress strength or maximum strain strength criteria were taken together to predict 

the failure of individual material slice in the RUC and overall composite structure.

For the progressive failure analysis, the composite failure was defined as the loss of loading 

capability of the RUC, which depends on the stiffness reduction due to material slice (matrix slice and 

yarn slice) failures and nonlinear material properties. Fiber failure and matrix cracking reduce the 

composite global stiffness and number of intact fibers to the point of failure.
NASA/TM—2013-217870 3

Figure 3.—(a) Idealized packing of fibers (or filaments) in a bundle yarn slice cut-section view (Ref. 14); (b) Damage 

mechanism in fiber-matrix yarn slice (side view); (c) Schematic representation of failure modes in fiber-matrix yarn

slice of the RUC. 

The failure mechanism of the yarn slice in the RUC is quite similar to that of a laminated composite, 

which involves fiber/matrix debonding, fiber-bridged matrix cracking, and statistical fiber failure. In the 

present study, a progressive failure analysis modeling technique based on the three-phase fiber-matrix 

micro-mechanics, fracture mechanics, and statistical models was developed at the yarn and matrix slice 

level for fiber-matrix yarn slice failures. While the main assumptions and procedures were used from the 

previous studies (Refs. 4 to 9), the present model included the fiber-bridged matrix crack model idealized 

by a continuum model in which the effect of the bridging fibers can be modeled by an equivalent closure 

pressure on the crack surface as illustrated in Figure 3.

The present paper describes the analysis procedure to compute the damage progression and fatigue 

failure of multilayer fabric reinforced composites. Experimental test results showed that inelastic strain 

can occur when the applied stress exceeds a “yielding stress”, accordingly a parameter related to the RUC 

(lamina) stress ratio was engaged to reduce the fiber and matrix stiffness after the “yielding”. Matrix 

stiffness reduction due to the cracking in the direction normal to the fibers was estimated according to the 

tensile stress in this direction. A stepwise loading procedure was required since the stress-strain curve is 

nonlinear due to the material degradation caused by fiber breakage, matrix cracking and inelastic yielding. 

The analysis steps are summarized in Figure 4.

2.0 Calculation of Effective Composite Material Properties Utilizing RUC 

Concept

Real fabric reinforced composites have very complicated geometry structures. As an example, 

Figure 5 shows a two-dimensional (2-D) carbon fiber reinforced silicon carbide matrix (C/SiC) plain 

weave composite laminate which is one of the simplest fiber architectures in fabric composites. It consists 

of stacked, pre-impregnated layers of woven fabric which are cured and consolidated by a process similar 

to tape laminates. Each yarn is a bundle of filaments (or fibers) and the yarn size is measured by the 

number of filaments in the yarn. To model this geometry is very difficult without certain simplifications.

A general geometry model was developed (Ref. 4) in details for fabric reinforced composites based 

on several assumptions to describe the fabric architectures and then calculate the mechanical and thermal 

properties of the fabric composites. Individual yarn architecture was discretely modeled using sinusoidal 

undulations at yarn crossovers and a straight portion. The iso-strain assumption was used to calculate the 

overall thermal and mechanical properties and average strains over the RUC. This section describes a 

brief derivation of geometry model for 2-D plain weave composites, and the calculation for overall 

thermal and mechanical properties of RUCs.
NASA/TM—2013-217870 4

Figure 4.—Incremental load step analysis procedures for fiber-matrix yarn slice failures in the RUC.

Figure 5.—Plain weave composite laminate (Ref. 15). (a) Top view of composite laminate with 

representaive RUC in dotted line (b) cross-section view of composite laminate.
NASA/TM—2013-217870 5

2.1 Geometric Model for 2-D Plain Weave Composites

The RUC for the 2-D plain weave composite is described as Figure 6. The sectional view (section AA) shows the undulation of a warp yarn as it crosses over and under the fill yarns. Only one layer of the 

plain weave is shown in the sectional view.

The plain weave composite is specified by known quantities of yarn spacing, Ly, in the fill and warp 

directions; yarn filament count, n, for the fill and warp yarns; yarn packing density, pd; filament diameter, 

df. These known quantities were used to determine the quantities of RUC thickness, H, yarn crosssectional area, A, yarn crimp angle, θc, yarn undulating length, Lu, projected yarn path length, Lp, vertical 

shift, Vs, and yarn thickness, t = H/2. These quantities are required to discretely model each yarn within 

the RUC.

The yarn spacing and yarn filament counts for the fill and warp directions were assumed to be equal. 

The RUC cross-sectional area (A) was assumed to be the same for both warp and fill directions and 

remain constant along the entire composite yarn path.

The projected length (Lp) of the yarn path of each yarn is Lp = 2Ly. The volume occupied by the four 

yarns within the RUC is given by 4 × A × Lp. The dimensions of the RUC are Lp × Lp × H. The volume of 

the RUC not occupied by the yarns was assumed to the interstitial matrix. The overall fiber volume 

fraction, Vf

R

, for the RUC can be derived as

y

R d

f HL

p A V 2 = (1)

Using the yarn filament count (n) and filament diameter (df), the yarn cross-sectional area (A) is 

calculated as

d

f

p

d n

A

4

2 π = (2)

For a given overall fiber volume fraction, the RUC thickness can be calculated using Equations (1) 

and (2). The yarn packing density (pd) may be determined either by image analysis of photomicrographs 

of sections through the thickness of the composite or may be assumed to be between 0.6 to 0.8.

Figure 6.—RUC of plain weave composite (Ref. 4).
NASA/TM—2013-217870 6

The yarn path in the RUC consists of two straight portions and three undulating portions. The yarns 

path was assumed to follow a sinusoidal path. The sine function used to describe the yarn centerline path 

has its origin at the cross-over points (COP) which was determined by the vertical shift (Vs) and the 

undulating length (Lu). For a warp yarn, the yarn centerline path, Zc, can be expressed as





 



 π = ± u

c c c L

H X Z X sin

4 ( ) (3)

where Xc is the quantity measured from the corresponding COP along the warp yarn direction. A negative 

sign in Equation (3) describes the undulation at the central COP and a positive sign describes the 

undulations at the edge COPs to describe three undulating portions of a yarn. Only a half sine wave was 

used for the centerline paths of both edges. The centerline path for other warp yarns in the RUC can be 

described in a similar manner with appropriate sign and sine wave portion at each COP. The undulations 

in the fill yarns can also be described along the fill yarn direction (i.e., Y-axis).

The undulating length, Lu, can be determined by the cross-sectional shape of the yarn consisting of a 

central straight portion of thickness, t, and two sinusoidal lenticular end portions assumed by the 

sinusoidal function of Equation (3). Also, the length of the sinusoidal end portions is equal to Lu/2. The 

cross-sectional area, A, can also be expressed as, 





 





π = − − 2 A Lyt Lut 1 (4)

The undulating length, Lu, can now be calculated using Equations (2) and (4). The total length of the 

straight portions, Lst, can be expressed as, 

Lst = 2Ly − 2Lu (5)

Equations (1) to (5) were used to determine all the required geometry parameters of the RUC, which 

includes yarn cross-sectional area and overall fiber volume fraction. Equation (4) is valid for Ly ≥ Lu.

The undulation in the yarn is described by its “crimp angle” defined as positive angle between the 

tangent to the sinusoidal yarn centerline path and the X-Y plane at a COP. The crimp angle, θc, can be 

expressed as,

0

tan( )

=















 θ =

Xc c

c c dX

dZ (6)

Generally, θmin ≤ θc ≤ (π/2) where θmin is minimum θc determined by the constraint Ly ≥ Lu for a given 

plain weave architecture. 

2.2 Effective Mechanical and Thermal Properties of RUC

All the analysis in this study is assumed to apply to the RUC. All the thermal and mechanical 

properties computed with the RUC can be considered as the same of the whole specimen. The overall 

composite properties are determined by discretizing all the yarns within the RUC. The straight portions of 

each yarn path are modeled as a single yarn slice length, Lst. Along an undulating portion, the yarn is 

divided into q, equal, piecewise straight slices made perpendicular to its in-plane direction and normal to 

the X-Y plane (Figure 5). Thus, the sinusoidal yarn undulating portion is approximated by q

interconnected straight yarn slices. The volume of each yarn slice of the undulating portion is computed 

as (A.Lu)/q. 
NASA/TM—2013-217870 7

The RUC of a textile composite was treated as a system consists of a set of spatially oriented yarn 

slices and an interstitial matrix material slice (Ref. 4). Thus, each yarn within the RUC was approximated 

by straight yarn slices known volumes and orientation angles. The total volume occupied by all the yarn 

slices was subtracted from the volume of the RUC to compute the volume occupied by the interstitial 

matrix pockets in the RUC. The matrix is assumed to be isotropic, while the yarn slices transversely 

isotropic (Ref. 16). The three-dimensional thermoelastic stress-strain relations in global XYZ coordinates 

for the RUC were defined using effective material properties to relate the average values of the stressstrain state variables. The average stress technique (Ref. 12) based on an iso-strain assumption within the 

RUC was used to compute overall composite properties as follows.

For the m

th material slice (yarn or matrix), the three-dimensional 6×6 stiffness matrix [ ] E m′ can be 

formed from given material constants (Young’s modulus and Poisson’s ratios). The five independent 

material constants (E11, E22, G12, ν12, and ν23; subscript 1 corresponds to the axial fiber direction) are 

required to define [ ] E m

' matrix for the yarn slice. Assuming the strains are uniform inside RUC, the 

stress-strain relation becomes

{σ}= [E]{ε} (7)

where the 6×6 matrix [E] is the overall stiffness matrix of RUC, which can be expressed as a summation 

over all the q yarn and matrix slices in terms of the materials slices stiffness matrix:

[ ] ∑( [ ] [ ] [ ] ) =

=

q

m

m m

T E Vm T m E T

1

' (8)

where Vm is the fiber volume fraction of the m

th material slice, matrix [ ] T m is the transformation between 

the global coordinates and the local coordinates of the m

th material slice, and superscript T indicates 

transpose. With a given loading, the average strains in a RUC which is also the strains of each material 

slice can be calculated from Equation (7). The stress components in global coordinates of each material 

slice can be obtained and transformed to the local material coordinates as

{σ} = ([ ] [ ] [ ] ){ε} T

m T m E m T m ' (9)

Thermal properties are also calculated based on the iso-strain assumption. For the m

th material slice, 

thermal residual stresses, σth, are given by

{ } [ ] {[ ] { } { }m} T

σth m = ∆T E′ m T m α − α′ (10)

where ∆T is the change in temperature from the stress free state, { } α m′ is the coefficients of thermal 

expansion (CTE) of the m

th material slice and {α}is assumed to be the overall CTE for the RUC, given as

{ } [ ] ∑( [ ] [ ] { } ) =

− α = ′ α′

q

m

m m

T E Vm T m E

1

1 (11)

This formulation enables the calculation of stresses and strains of each material slice when the thermal or 

mechanical loads are given. This forms the foundation for the damage analysis and failure analysis of 

fabric reinforced composites.
NASA/TM—2013-217870 8

2.3 Nonlinear Material Response

The material nonlinearity was approximated to represent the nonlinear material response of both 

interstitial matrix material slice and yarn material slice (Ref. 7).

( )

















β − + β

























































































ασ

σ = +

1 1

1

i ult

E i Eo (12)

where Eo represents the initial Young’s modulus or shear modulus of the matrix and yarn material, σ is 

the corresponding stress level, σult is the ultimate strength, and α and β are the empirical constants. 

Equation (12) was used to compute the instantaneous moduli for each material slice (matrix and yarn) in 

the RUC during incremental load step i. Thus, the overall stiffness for the RUC was computed at each 

incremental step. The Full Newton-Raphson Method (Ref. 17) was used for the incremental analysis. The 

incremental stresses in each yarn slice within the RUC were calculated as

{ } [ ] [ ]{ } i E i T i ∆σ = ′ ∆ε −1 (13)

where the 6×6 stiffness matrix [E′]i–1 defines the three-dimensional stress-strain relationship for the yarn 

slice and transformation matrix [T] defined in Reference 4. The [E′]i–1 matrix includes the effect of 

nonlinear response in Equation (12) and also the effects of damage accumulation from material slice 

failure. The cumulative stresses in each yarn slice were used together with appropriate failure criteria 

described in the following section to predict failure at each incremental step.

3.0 Micro-Mechanics Modeling For Composite Progressive Failure Criteria

The geometry model of textile composites assumes that the RUC of a textile composite is a system 

consisting of matrix and yarn slices. For the matrix slice, since the matrix material is assumed to be 

isotropic, only the volume of one matrix slice is necessary for the geometry modeling, while for a yarn 

slice other information like orientation and fiber volume fraction is also important. The overall stiffness 

and coefficients of thermal expansion of the RUC are calculated as a summation of all the material slices. 

The damage properties of the RUC, i.e., the stiffness reduction and strength of the RUC were also 

calculated based on each individual material slice. A failure criterion of yarn slices under tension based 

on micro-mechanics analysis and a statistical model is introduced. For the failure analysis, the failure 

criterion based on the fiber pullout concept of laminated composites was employed. The global composite 

structure failure was defined as the loss of loading capability of the RUC, which depends on the stiffness 

reduction due to material slice (matrix slice and yarn slice) failures and nonlinear material properties. The 

failure mechanism of the yarn slice in a fabric reinforced composite is quite similar to that of a laminated 

composite, which involves fiber/matrix debonding, fiber-bridged matrix cracking, and statistical fiber 

failure. In the present study, the fiber-bridged matrix crack was idealized by a continuum model in which 

the effect of the bridging fibers was modeled by an equivalent closure pressure on the crack surface. A 

shear-lag model was used to evaluate the closure pressure distribution along the crack surface and to 

compute the shear stress transfer along the debonded, frictional sliding fiber/matrix interface.

In this study, the three-phase micromechanics, the shear-lag (fiber pullout), and the continuum 

fracture mechanics models were integrated with a statistical model in the bundle yarn slices, as shown in 

Figure 3. At an increment of loading, σc , the matrix displacements, u(x), the fiber bridging tractions, T(x), 

and the interface shear stress, τi(z), using a shear-lag model (Ref. 18 to 19) were used in the continuum 

fracture mechanics analysis in which the bridging tractions were replaced by an effective crack closure 

pressure, p(x). The effective stress intensity factor, Ktip, at the fiber-bridged matrix crack tip was then used 
NASA/TM—2013-217870 9

along with a fracture criterion to predict matrix crack growth. Fiber failure along the slip regions, ls, in the 

wake of the crack was predicted using a Weibull statistical model (Ref. 20). The effect of failed fibers, 

predicted by the statistical analysis, was integrated into the shear-lag and fracture mechanics analyses by 

accounting for the local redistribution of the fiber loading and the reduction in material slice modulus at 

each incremental step to account for the accumulation of damage in each material slice within the RUC. 

Nonlinear material properties for both matrix and fiber were included in addition to the statistical failure 

progress of material slices to calculate the three-dimensional effective stiffness reduction of the RUC. The 

failure of fabric reinforced composites was predicted based on the overall stiffness calculation of RUC. 

The following sections briefly describe each of the analytical models used in the present method.

3.1 Three-Phase Micromechanics Model

A three-phase micromechanics model (Ref. 6) was utilized for computing thermal and mechanical 

stresses at the fiber-matrix interface. A classical elasticity approach in which the fiber (of radius a), the 

matrix (of radius b), and the surrounding composite were modeled as separate regions (Figure 7) was used. 

Upon carrying through a few mathematical manipulations, a generalized plane strain solution (which 

assumes εzz = constant in all the three regions) was obtained with the boundary conditions to solve the 

boundary value problem. The boundary conditions lead to a set of simultaneous equations involving a set of 

the unknown coefficients. The unknown coefficients in the fiber stresses, matrix stresses, composite 

stresses, and the unknown axial strain (εzz) were determined by using (i) the constitutive relationships in 

each phase, (ii) the strain displacement relationships, and (iii) the stress and displacement boundary 

conditions at the fiber-matrix interface, the matrix-composite interface, and the model external boundaries. 

These equations are solved numerically to arrive at the solution of the boundary value problem.

3.2 Shear-Lag and Fiber Pullout Models

A shear-lag model developed by Budiansky, Hutchinson, Evans (BHE) (Ref. 18) and McCartney 

(Ref. 19) was extended in the present analysis to establish the relationship between the fiber traction and 

the crack opening displacements. This relation was then provided to the fracture mechanics analysis to 

determine the fiber traction, which will be described in the following section. The effect of matrix shear 

deformation beyond the slip distance (Figure 7) was accounted for in the BHE model. For the shear-lag 

analysis, the matrix is hypothesized as being made of two regions: (i) the inner region of radius R is 

assumed to support only shear stresses (following BHE) and (ii) an outer region which supports only axial 

stresses.

Figure 7.—Three-phase shear-lag micro-mechanics model.
NASA/TM—2013-217870 10

Along the slip length, ls, the interfacial shear stress, τi, is given by

r a i rr = τ = μ ⋅σ (14)

where μ is the coefficient of friction and the radial stress σrr at the fiber-matrix interface. Since σrr

calculated from the generalized plane strain analysis in the previous Section 3.1 does not vary along the z

direction, the interfacial shear stress, τi, calculated from the Coulomb law will also be uniform along the 

length of the slip region. Note that the interfacial shear stress, τi, in the present development is a function 

of fiber radius, fiber volume fraction, thermal stresses, and applied load level in the yarn slice. The 

variation of the interfacial shear stress corresponding with applied load necessitates an incremental 

solution approach for the present analysis. For a given load level, however, the interfacial shear stress, τi, 

is constant and the solution to the shear-lag problem, at each load increment, can proceed in the same 

manner as was done in the BHE analysis.

The fiber pullout concept developed by Marshall, Cox, Evans (Ref. 21) was used (Figure 8) to 

determine the axial fiber stress, σf(z), as a function of the distance, z, from the crack plane, and the slip 

length, ls, at the interface for the fully propagated matrix crack.

As shown in Figure 8, assuming purely frictional fiber-matrix bond, the slip length can be determined 

by the length over which the interface shear stresses exceed the maximum shear stress which the fibermatrix interface can resist. The stress σm for matrix and σf for fiber under far-field stress σc can be related 

to each other as

( ) ( ) c m m f f f

f s

m

m s E E V E V E

l

E

l = +

σ = σ , (15)

where Ec, Em , and Ef are longitudinal Young’s modulus of composite, matrix , and fiber, respectively. 

Vm = (1 – Vf) is the volume fraction of matrix.

Figure 8.—Mechanics of fiber pullout concept.
NASA/TM—2013-217870 11

Assuming the extension due to stress σc is δ(x) for matrix, the extension for fiber will be u(x) + δ(x). 

By following the procedure in Reference 21 and considering the equilibrium of the matrix and fiber 

separately, a set of equations can be written as

( ) m Am az i σ z = 2π τ (16)

( ) f i f Af T(x)A = 2πRzτ + σ z (17)

( ) ( ) ( ) ∫ ∫

σ

δ + = σ δ =

s sl

f f

f

l

m m

m dz

A E

z

dz x u x

A E

z

x

0 0

( ) , ( ) ( ) (18)

where a is the radius of the fiber, Af is the fiber cross-section area, Am is the area of matrix per fiber, or 

Af/(Af + Am) = Vf is the fiber volume faction. Notice that when the shear stress on the fiber-matrix 

interface reaches its maximum value associated with the coefficient of friction μ and the radial stress σrr

at the fiber-matrix interface (which can also be a material constant for each bundle yarn by testing), 

Equation (18) becomes

( )

f

s i

f

s

m m

s i

aE

l

E

T x l

x u x

A E

al

x

τ δ + = − π τ δ =

2 2 ( ) ( ) , ( ) ( ) (19)

This set of equations yielded the relation between the crack opening displacement and fiber traction as 

( ) 2 ( ) 4

1

( ) T x

E E

aE V

u x

f c i

m f

τ

− = (20)

The slip length ls can also be expressed as 

c f

c m f s i

E V

E V

a

2l σ (1− ) = τ (21)

and the fiber stress within the slip region as

( )

a

z

V

z i

f

c f

τ − σ

σ = 2 (22)

3.3 3.2 Continuum Fracture Mechanics Model for Matrix Cracking

The fracture mechanics analysis for the continuum solution to the problem of a crack is required to 

provide the relationship between the unknown crack opening displacement u(x) and fiber bridging 

traction T(x), where x represents the position on the crack surface. Both the matrix and fibers of the 

composite (yarn) were assumed to deform as linear elastic solids having Young’s moduli Em and Ef

respectively. It was also assumed that both matrix and fibers have the same Poisson’s ratio so that the 

modulus of the yarn Ec in the direction of the fibers is given by Ec = Ef Vf + Em Vm .

As illustrated in Figure 3, considering a crack under far-field applied stress, σc, and a crack closure 

pressure, p(x), the crack opening displacement (Ref. 22) u(x),where X = x/c, can be written for a crack in 

an orthotropic material as
NASA/TM—2013-217870 12

( ) ( ) dt

t X

p t t X

E

c

u X

s

c c

c

2 2

2 2 1

0 1 1

1 1 1 ln 2

+ − +

− + − 



 





σ − π

σ = ∫ (23)

where Ec is the longitudinal Young’s modulus of the composite (i.e., yarn), c is the half crack length. The 

fiber bridging pressure (i.e., crack closure pressure), p(X), and the fiber bridging traction T(X) is related 

by the fiber volume fraction, Vf, as

p(X ) V T(X ) = f (24)

The nonlinear governing integral equation can be derived using Equations (20), (23), and (24) as

( ) ( ) ( ) 















+ − +

− + −

σ

π − − π

σ = τ

−

∫

s

dt

t X

t X T t

V

X

E

c T X

E E

aE V

c

f

c

c

f c i

m f

1

0

2 2

2 2

2 2

1 1

1 1 1 ln 2

4

1

(25)

Hence, the fiber bridging traction T(X) can be solved using iterative procedures developed by Marshall, 

Cox, and Evans (Ref. 21). By obtaining the crack closure pressure, the stress intensity factor for yarn 

material, K, at the fiber-bridged crack can be calculated as 

( ( ))

dX

X

c p X K

s

c

∫ −

σ −

π =

1

0

2 1

2 (26)

The matrix stress intensity factor, Km tip ( ) , can also be related to the yarn stress intensity factor K as

( f ) c

m m tip V E

E K K − = 1 ( ) (27)

where Em is the matrix Young’s modulus. With a specified critical stress intensity toughness of matrix, 

KIc, matrix crack propagation is predicted when 

Km tip ≥ KIc ( ) (28)

This fracture mechanics approach was integrated with the three-phase micromechanics and shear-lag 

models for the prediction of matrix crack propagation. An incremental scheme as described in Section 2.3

was used to calculate the matrix stress intensity factor at each load increment. When the criterion in 

Equations (28) was satisfied, the matrix crack was incremented by a small amount (for instance, 1/50 of 

the initial crack size), or a given crack growth rate (dc/dt) along with the numerical convergence where t

is time (i.e., loading increments), and the stress intensity was recalculated at the corresponding load level. 

3.4 Statistical Fiber Failure Model

The results from the combined micromechanics and fracture mechanics analysis indicate that under 

longitudinal tension loading, a unidirectional brittle fiber-matrix yarn system undergoes unstable matrix 

crack propagation soon after crack initiation. In sequence, predominantly a fiber-bridged matrix crack

propagates through the width of the yarn. The applied load is then carried by the bridging fibers. A shearlag model was used to model each fiber along the fully propagated matrix crack and the slipping region at 

the fiber-matrix interface.

The axial fiber stress, σf(z), as a function of the distance, z, from the crack plane was derived from the 

shear-lag and fiber pullout models in Section 3.1 and is given by Equations (22). This equation was used 
NASA/TM—2013-217870 13

to determine the probability of fiber failure in the slip region in the vicinity of the matrix crack plane 

where the fiber stress was a maximum.

The tensile strength of a brittle material under uniform loading was assumed to be governed by the 

Weibull distribution (Ref. 20) to describe the probability of fiber failure as

































σ

σ = − − π ∫

−

s

s

l

l

M

o

f Pf 1 exp 2 a dz (29)

where Pf denotes the fraction of the fibers that have already failed due to the stress distribution given in 

Equations (22), M is the Weibull modulus that is a measure of the scatter in the strength data. The 

practical ranges of M values are 2 < M < 60. Large scatter in the strength data accompanies low values of 

M. σo is the mean fiber strength. Substituting Equations (21) and (22) into Equations (29), the probability 

of fiber failure can be expressed as

( )

( ) ( )















































 − − + 













 − − + τ σ

π σ = −

+ +

+

+ 1 1

1

2 1 1

1

1

1

1

1 exp

M

c

m f

M

c

m f

M

f

M

i o

M

c f E

E V

E

E V

M V

a P (30)

The fiber volume fraction of the yarn at any given load level i, Vf

i

, was given by

( f ) f

i Vf =V 0 1− P (31)

where Vf

o was the initial fiber volume fraction of the undamaged yarn, i is the incremental load step as 

described in Section 2.3. The yarn was assumed to have failed for Vf

i

= 0.2 when the fiber failures would 

become unstable.

3.5 Failure Criteria for Repeating Unit Cell and Overall Composite Structures

The statistical failure criterion and maximum stress strength or maximum strain strength criteria were 

combined together to predict the failure of individual material slice in the RUC and overall composite 

structure. Yarn material slices failures are divided into matrix dominated failure and fiber dominated 

failure. The failure criterion of Equations (31) was used to predict the tensile failure of fibers in a yarn 

slice. A matrix crack failure criterion of Equations (28) was used for the transverse tensile failure (on the 

crack surface) for matrix crack propagation in a yarn slice. In other words, two additional criteria were 

combined with maximum stress strength or maximum strain strength criteria to test for yarn slice failure 

in the RUC: (1) Vf

i

= Vf

critical (e.g., 0.2) for fiber failure, or (2) Km ≥ KIc for matrix failure, where the 

critical stress intensity factor KIc is a material constant specified. If either condition was met, the yarn 

slice was assumed to have failed.

The stiffness reduction due to the material slice failure in the RUC was used for the damage 

accumulation in the RUC under tension loads. When a material slice is failed based on the failure criteria 

mentioned above, the Young’s modulus and shear modulus are reduced accordingly as described in 

Sections 2.2 and 2.3. 

Global composite structure failure is reached when the RUC stiffness corresponding to each load step 

is less than a critical value specified, for example, 10 percent of the original value E0 in Equations (12).

4.0 Fatigue Failure Prediction Model For Composite Structure

The progressive failure prediction method described in Sections 2.0 and 3.0 was extended to develop 

a fatigue damage prediction model based on the stiffness reduction due to the material slice failure in the 

RUC and the observation of experimental test results performed using 2-D plain weave carbon/silicon 
NASA/TM—2013-217870 14

Figure 9.—Photograph of plain weave C/SiC composite specimen produced with a 

chemical vapor infiltration process (CVIP) coating for protection from oxidation at 

high temperature.

carbide (C/SiC) specimens under tension loads (Figure 9). Based on the stiffness calculation as expressed 

in Equations (12) and the observation of experimental test data, the present model assumed that the 

stiffness reduction during the fatigue process contains all the constituents about the effect of fatigue 

damage to material strength, and also assumed that the fatigue damage to all the materials (matrix and 

yarn slice materials) is proportional.

Hence, the mean strength in the Weibull distribution and all the critical values in material failure 

criteria are related to the RUC stiffness reduction of the composite to reflect the effects of fatigue loading 

on the composite strength. Adopting a relationship between fatigue damage and changes in moduli of the 

composite laminate (i.e., RUC) (Ref. 23), Equations (12) can be related under fatigue loading as

1 ( )log( )

1

E c1 c2 max N

E

i o + + σ = 













 (32)

( ) ( ) M b b ult

N i mean i E

E + σ σ





 



 σ = σ 1 2 max

( )

0

(33)

where σmax is applied maximum stress (loading) during fatigue tests which is related to σ in 

Equations (12), σult is the static ultimate tensile strength of composite, Eo represents the initial Young’s 

modulus or shear modulus of the matrix and yarn slice materials, E is the corresponding moduli during 

the incremental load step, E0

E is the overall stiffness reduction at each load increment i, where b1 and b2,

c1 and c2 are material parameters determined with the experimental test data, M is the Weibull modulus, 

σmean is the mean stress of the fatigue loading cycle σmean = 0.5(σmax + σmin) = 0.5(1 + R) σmax, R is the 

stress ratio R = (σmean – σalt)/(σmean + σalt) = σmin/σmax, σalt is alternating stress (i.e., the amplitude of the 

cyclical stress) of the fatigue loading cycle, N is the number of cycle, and σN is the corresponding fatigue 

stress after N cycles. 

While fundamentally the fatigue behavior of fabric reinforced composites may be different from that 

of the metals, a method based on the well-known S-N approach for the fatigue life analysis was developed 

with considering the applicable procedure that combines the analytical methods with limited testing for 

the fatigue life prediction analysis for fabric reinforced ceramic matrix composite structures.

5.0 Results and Discussion With Specific Example Illustrating Method

Numerical analysis and experimental testing on C/SiC plain weave composite specimens were 

performed under uniaxial tension. The material properties for carbon fiber and silicon carbide matrix are 

listed in Table 1. Selected input parameters of the unit cell geometry input for the numerical calculation 

and calculated geometry parameters are shown in Table 2 and Table 3. It was assumed in the analysis that 

the warp and fill yarns are identical, hence the interfacial shear stress, τi, and other nonlinear parameters 

described in previous sections were also the same for those yarns. 
NASA/TM—2013-217870 15

TABLE 1.—FIBER AND MATRIX PROPERTIES USED 

FOR C/SiC PLAIN WEAVE UNIT CELL

Material Young’s Modulus,

GPa

Possion’s ratio

Fiber 231.0 0.20

Matrix 3.51 0.20

TABLE 2.—GEOMETRY PARAMETERS OF C/SiC PLAIN 

WEAVE UNIT CELL FOR NUMERICAL MODEL INPUT

Yarn size (K),

n

Weave angle Yarn spacing,

Ly

Yarn packing density,

pd

Unit cell thickness,

H

1 90.0 3.22 mm 0.6 0.60 mm

TABLE 3.—GEOMETRY PARAMETERS OF C/SIC PLAIN WEAVE

UNIT CELL CALCULATED BY NUMERICAL MODEL

Yarn thickness,

t

Crimp angle,

θc

Yarn length,

Lu+ Lst

Unit cell volume fraction,

Vf

R

0.30 mm 2.26 6.40 mm 0.40

In the analysis, the coefficient of friction μ= 0.4 for interfacial shear stress parameter, c0 = 0.01 mm 

for the initial crack length, critical stress intensity factor of KIc = 0.2 MPa√m, Weibull modulus M = 10, 

the mean fiber strength σo = 502 MPa, and 1.0 and 10.0 for material nonlinear parameter α and β were 

used, respectively.

An experimental investigation was performed to determine the fatigue and static behavior of the 

C/SiC specimens. The specimens were loaded in static tension and in tension-tension fatigue under 

isothermal fixed-frequency conditions. The room temperature level and a range of stress levels for fatigue 

were used to assess performance. The tests were conducted at 23 °C. Measurements were made of 

modulus and cycles to failure several times during the test.

The experimental investigation was divided into two parts to develop an understanding of the fatigue 

performance of the material. The first part consisted of an investigation to determine the static properties 

of the material and determine how the modulus of the material changes as a function of maximum applied 

static strain. The second portion of the testing consisted of an investigation of the high cycle fatigue 

behavior of the material. The specimens were tested to failure or 1,000,000 cycles.

Tests were performed on a C/SiC composite material system with a CVIP coating. The coating 

provided protection from oxidation at high temperature. This composite specimen system consisted of 

2-D plain weave carbon fibers in a silicon carbide matrix produced in 18 ply symmetric lay-ups. The test 

specimens were machined into 6 in. length specimens by approximately 0.50 in. wide gage section with a 

nominal thickness of 0.12 in. after manufacture.

The static test used on the C/SiC was performed in load control at a loading rate of 5,000 lb/min on a 

50 kip servo-hydraulic test stand equipped with an automated data acquisition system. Load and strain 

were recorded at half-second intervals during all of the tests. Property calculations consisted of ultimate 

strength and modulus. Ultimate stress was calculated using the highest load carried by the laminate. 

Modulus was calculated using a least squares fit to the unloading segments of data during static tests. 

To evaluate the accuracy of the analytical stiffness degradation predictions and stress-strain relations 

under static loading, comparisons were made with the experimental static test data. The present results 

and the experimental test results are compared as shown in Figure 10 and Figure 11.
NASA/TM—2013-217870 16

Figure 10.—Comparison of stiffness reductions of C/SiC plain 

weave laminate under static loading.

Figure 11.—Comparison of static stress-strain relations for 

C/SiC plain weave laminate. 

A comparison of the numerical analysis results and the experimental test data indicated a good 

agreement. Therefore, the numerical analysis method and the assumptions used in this analysis were 

considered reasonable to be applied for analyzing the static characteristics of the plain weave fabric 

reinforced composite structures.

With the acceptable results in the static loading analysis, the tension fatigue tests were performed on a 

50 kip capacity servo-hydraulic test stand. Load, strain, actuator displacement and time were recorded 

using an automated digital data acquisition system. All fatigue tests were performed under load control at 

a frequency of 5 Hz with a min/max stress ratio of R = 0.1 and a sinusoidal wave form. Tests were 

concluded after 1,000,000 cycles if failure did not occur. The strain and load were measured during 

specified cycles of the fatigue test as summarized in Table 4 and allowed for determination of changes in 

modulus.
NASA/TM—2013-217870 17

TABLE 4.—SPECIFIED CYCLES FOR FATIGUE DATA

MEASURED IN C/SiC FATIGUE TESTS

Cycle count range Reading interval

1 to 10 1 cycle

10 to 50 5 cycles

50 to 100 10 cycles

100 to 500 50 cycles

500 to 1,000 100 cycles

1,000 to 5,000 500 cycles

5,000 to 10,000 1,000 cycles

10,000 to 100,000 5,000 cycles

100,000 to 1,000,000 10,000 cycles

The 23 °C fatigue tests are summarized in Table 5 for the maximum stress and cycles to failure 

during fatigue.

TABLE 5.—MAXIMUM STRESS AND CYCLES TO 

FAILURE FOR C/SiC FATIGUE TESTS AT 23 °C 

[R = 0.1, frequency = 5 Hz]

Maximum stress,

σmax

ksi/MPa

Cycles to failure,

N

66.5/459 4,369

62.7/432 37,332

59.5/410 42,334

56.0/386 515,176

45.5/314 > 1,000,000

Applied load and measured strain were used to calculate an effective modulus for the gage section 

during fatigue tests. Stiffness calculations were performed by making a least squares linear fit to the stress 

versus strain data. Stress was calculated using the specimen gage section area and the load from the data 

acquisition system. Strain was taken from the extensometer attached to the gage section. The fit was 

performed on the loading and unloading data over the entire fatigue cycle. The parameters in 

Equations (32) were determined from experimental stiffness reduction data in Figure 12 for C/SiC plain 

weave composite. The two parameters in Equations (32) were determined as:

1.54 02, 7.32 05( MPa) c1 = − e − c2 = e −

Subsequently these parameters were substituted into Equations (32) to predict the stiffness reduction of 

C/SiC plain weave composite under fatigue loading. The comparison of stiffness reductions between 

analytical result and experimental test data for maximum stress at 410 MPa (80 percent of ultimate 

strength) is given in Figure 13 as an example. 

The parameters in Equations (33) were also determined from experimental test fatigue life diagrams 

S-N data from the C/SiC plain weave composite testing as shown in Figure 14.

The two coefficient were calculated as

b1 = 0.0415( MPa), b2 = −8.22e − 05

Subsequently these parameters were substituted into Equations (33) to predict the fatigue life calculations 

for plain weave composite. The predicted S-N curve was compared with experimental test data as shown in 

Figure 15. The numerical results appear to have reasonable agreement with the experimental test data.
NASA/TM—2013-217870 18

Figure 12.—Experimental test modulus change diagram as a 

function of cycle count (N) for C/SiC specimen at 23 °C.

Figure 13.—Stiffness reduction for max. stress at 410 MPa 

(80% ultimate strength) as a function of cycle count (N).
NASA/TM—2013-217870 19

Figure 14.—Experimental test fatigue life diagram for C/SiC 

composite specimen at 23 °C.

Figure 15.—Stress-life (S-N) curve for C/SiC plain weave 

composite at 23 °C.

6.0 Conclusion

When analytical methodologies are considered for the component fatigue life assessment in real flight 

composite hardware, it is dealt with as fully as possible to include all potential damage mechanisms and 

damage failure modes such as matrix cracking, delamination fracture, fiber failure, and time dependent 

effects such as viscoelasticity and polymer aging degradation under general service loading conditions. 

Analytical approach for the reliable fatigue life prediction for fabric reinforced composites is quite 

challenging.

A micromechanics analysis modeling method was developed to analyze the progressive fatigue 

failures of fabric reinforced composite structures, especially for the brittle ceramic matrix material 

composites. The repeating unit cell (RUC) concept of fabric reinforced composites was used to represent 

the composites. The thermal and mechanical properties of the repeating unit cell were considered as the 

same as those of the composite. The micromechanics, the shear-lag, and the continuum fracture 
NASA/TM—2013-217870 20

mechanics models were integrated with a statistical model in the repeating unit cell to predict the 

progressive fatigue damages and life prediction of the fabric reinforced composite structures.

While the present method would not be reachable to give an entirely theoretical solution satisfied for 

real engine operating conditions, the repeating unit cell geometry modeling concept was utilized by taking 

advantage of the periodically repeating pattern of textile composite materials to see whether the present 

repeating unit cell micromechanics model combined with an unique progressive damage analysis 

approach based on fracture mechanics, shear-lag theory, and fiber pullout concept with a statistical failure 

criterion of the yarn slice is practically applicable for predicting the fatigue life of global composite 

structures. The RUC was represented by a set of yarn and matrix slices based on the sinusoidal 

assumption of the undulation portion of the yarn paths. Thermal and mechanical properties were 

calculated based on an iso-strain assumption in the RUC.

Stiffness reduction of RUC comes from the material non-linearity and the failure of material slices. A 

set of failure criteria, including the statistical fiber failure criterion, was used for yarn and matrix slices 

within the RUC. When a material slice failed, the corresponding moduli were reduced. The present 

method was applied for the plain weave C/SiC ceramic matrix composite to predict composite laminate 

failure and fatigue life. It was assumed that the stiffness reduction due to fatigue loading when the 

composite laminate is under uniaxial tension-tension fatigue loading contains all the constituents about 

the effect of fatigue damage to material strength under complex stress states. The off-axis fatigue 

behavior was not considered for this study. Only the on-axis fatigue behavior subjected to the 

unidirectional tension-tension fatigue loading parallel to the fibers was considered in the methodologies 

developed in this study and experimental testing.

The present method demonstrated the capabilities including the geometry analysis for 2-D plain 

weave composite architectures, calculation of mechanical and thermal properties such as overall Young’s 

moduli and coefficients of thermal expansion for the RUC. It also calculated local strains and stresses, 

local material slice and overall composite progressive damage failure, and overall composite fatigue 

failure prediction through incremental loading steps by predicting damage progression. It predicted the 

fatigue life S-N curve based on a relatively few incremental loading steps.

The numerical predictions were compared with experimental test data for plain weave C/SiC 

composite specimens under uniaxial static tension loading and tension-tension fatigue loading to illustrate 

and demonstrate that the proposed analytical procedure is capable of predicting the fatigue life of fabric 

reinforced composite structures. The results correlated reasonably well for a specific 2-D plain weave 

C/SiC composite, and therefore the present analysis technique indicated well as a potential predictive 

capability for the fatigue cyclic damage progression and strength analysis for fabric reinforced composite 

structures.

References

1. T.P. Herbell, A.J. Eckel, and J.W. Brockmeyer, “Composites in High Speed Turbines for Rocket 

Engines,” in High Temperature High Performance Materials for Rocket Engines and Space 

Applications, K. Upadhaya, Ed., The Mineral, Metals, and Materials Society, pp. 13-20, 1994.

2. J.W. Brockmeyer: Fiber-Reinforced Ceramic Matrix Composites for Earth-to-Orbit Rocket Engine 

Turbines, Lewis Research Center, Cleveland, OH, NASA CR-198526, 1996.

3. J.B. Min, D.L Harris, J.M. Ting: “Advances in Ceramic Matrix Composite Blade Damping 

Characteristics for Aerospace Turbomachinery Applications,” AIAA 2011-1784, Proceedings, 52nd 

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, April 

2011.

4. R.A. Naik: “Analysis of Weave and Braided Fabric Reinforced Composite,” NASA CR-194930, 

1994.

5. R.A. Naik: “Multiaxial Stiffness and Strength Analysis of Woven and Braided Composites,” AIAA97-1323.
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6. R.A. Naik, J.H. Crew, Jr.: “Micromechanical Analysis of Fiber-Matrix Interfaces Stresses under 

Thermomechanical Loadings,” ASTM STP 1206, Composite Materials: Testing and Design, 1993, 

pp.205-219.

7. R.A. Naik: “Failure Analysis of Woven and Braided Fabric Reinforced Composite,” Journal of 

Composite Materials, Vol. 29, No. 17, pp. 2334-2363, 1995.

8. R.A. Naik: “Analysis of 2-D Triaxial and 3-D Multi-interlock Braided Textile Composites,” AIAA96-1530-CP, pp. 1804-1811, 1996.

9. C.C. Poe, Jr., H.B. Dexter, I.S. Raju: “A Review of the NASA Textile Composites Research,” 

AIAA-97-1321.

10. S. Kobayashi, T. Isikawa: “Elastic Properties of Unidirectional Fiber-Reinforced Composites,” 

Fukugo Zairyo Kenkyu (Composite Materials and Structures), Vol. 3, No. 3, 1974, pp. 12-20.

11. T. Ishikawa, S. Kobayashi: “Elastic Properties of Unidirectional Fiber-Reinforced Composites II,” 

Fukugo Zairyo Kenkyu (Composite Materials and Structures), Vol. 3, No. 4, 1974, pp. 23-31.

12. B.W. Rosen, S.N. Chatterjee, J.J. Kibler: “An Analysis Model for Spatially Oriented Fiber 

Composites,” Composite Materials: Testing and Design, ASTM STP 617, ASTM, 1977, pp. 243-

254.

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Plastics and Composites, Vol.6, July 1987, pp. 268-289.

14. Y. Gowayed, C. Pastore: Workshop on Mechanics of Textile Composite Materials, NASA GRC, 

2002.

15. D. Hull, T.W. Clyne: Introduction to Composite Materials, Cambridge University Press, 1996.

16. R.M Jones: Mechanics of Composite Materials, Hemisphere Publishing Corporation, New York, 

1975.

17. K.J. Bathe, A.P. Cimento: “Some Practical Procedures for th Solution of Nonlinear Finite Element 

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18. B. Budiansky, J.W. Hutchinson, A.G. Evans: “Matrix Fracture in Fiber-Reinforced Ceramics,” 

Journal of Mechanics and Physics of Solids, Vol. 34, No. 2, 1986. pp. 167-189.

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20. M. Sutcu, “Weibull Statistics Applied to Fiber Failure in Ceramic Composite and Work of Fracture,” 

Acta Metal., Vol. 37, No. 2, pp. 651-661, 1989.

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Composites,” Acta Metall., Vol. 33, No. 11, pp. 2013-2021, 1985.

22. L.N. McCartney: “Mechanics of Matrix Cracking in Brittle-Matrix Fiber-Reinforced Composites,” 

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Composite Design,” Int. J. Fatigue 14, No. 5, 1992, pp. 283-294.
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Micromechanics Fatigue Damage Analysis Modeling for Fabric Reinforced Ceramic Matrix 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

A micromechanics analysis modeling method was developed to analyze the damage progression and fatigue failure of fabric reinforced 

composite structures, especially for the brittle ceramic matrix material composites. A repeating unit cell concept of fabric reinforced 

composites was used to represent the global composite structure. The thermal and mechanical properties of the repeating unit cell were 

considered as the same as those of the global composite structure. The three-phase micromechanics, the shear-lag, and the continuum 

fracture mechanics models were integrated with a statistical model in the repeating unit cell to predict the progressive damages and fatigue 

life of the composite structures. The global structure failure was defined as the loss of loading capability of the repeating unit cell, which 

depends on the stiffness reduction due to material slice failures and nonlinear material properties in the repeating unit cell. The present 

methodology is demonstrated with the analysis results evaluated through the experimental test performed with carbon fiber reinforced 

silicon carbide matrix plain weave composite specimens.

15. SUBJECT TERMS

Micromechanics; Repeating unit cell; Fatigue damage prediction; Aerospace engine blade; Finite element; Vibration; Damping; 

Composites 

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF

 ABSTRACT

UU 

18. NUMBER

 OF 

 PAGES

28 

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