Evan J. Pineda

Glenn Research Center, Cleveland, Ohio

Subodh K. Mital

University of Toledo, Toledo, Ohio

Brett A. Bednarcyk and Steven M. Arnold

Glenn Research Center, Cleveland, Ohio

The Effect of Stochastically Varying Creep

Parameters on Residual Stresses in Ceramic

Matrix Composites

NASA/TM—2015-218759

May 2015

AIAA–2015–0389
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Evan J. Pineda

Glenn Research Center, Cleveland, Ohio

Subodh K. Mital

University of Toledo, Toledo, Ohio

Brett A. Bednarcyk and Steven M. Arnold

Glenn Research Center, Cleveland, Ohio

The Effect of Stochastically Varying Creep

Parameters on Residual Stresses in Ceramic

Matrix Composites

NASA/TM—2015-218759

May 2015

AIAA–2015–0389

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Constituent properties, along with volume fraction, have a first order effect on the microscale fields within a composite material and influence the macroscopic response. Therefore, there is a need to assess the significance of stochastic variation in the constituent

properties of composites at the higher scales. The effect of variability in the parameters

controlling the time-dependent behavior, in a unidirectional SCS-6 SiC fiber-reinforced

RBSN matrix composite lamina, on the residual stresses induced during processing is investigated numerically. The generalized method of cells micromechanics theory is utilized

to model the ceramic matrix composite lamina using a repeating unit cell. The primary

creep phases of the constituents are approximated using a Norton-Bailey, steady state,

power law creep model. The effect of residual stresses on the proportional limit stress and

strain to failure of the composite is demonstrated. Monte Carlo simulations were conducted

using a normal distribution for the power law parameters and the resulting residual stress

distributions were predicted.

I. Introduction

C

eramic matrix composites (CMCs) provide excellent high-temperature capabilities with much lower mass

compared to high-temperature superalloys. They are thus under development for aerospace applications

such as entry, ascent, and cruise vehicles, leading edges, control surfaces, combustor liners, and gas turbine

engine vanes and blades. Examples of three CMCs are shown in Figure 1. Woven melt infiltrated (MI)

CMCs tend to have voids within the tows, but good matrix density between the tows. Woven chemical

vapor infiltrated (CVI) CMCs have voids within the tows as well as large voids between the tows. With

unidirectional and laminated MI CMCs, excellent matrix density can be achieved both within and between

the tows with a simpler manufacturing. In all three types of CMCs, the fiber filaments are coated, which

provides a very compliant interface to inhibit matrix cracks from entering the fibers.

Variability in the stiffness, strength and proportional limit stress (PLS) of CMCs has been demonstrated

to be significant, compared to polymer matrix composites (PMCs).1 The mismatch in stiffness between the

fibers and matrix in CMCs tend to be far lower than that of PMCs. As such, the response of CMCs is not

“fiber-dominated,” and the performance of CMCs is highly sensitive to defects and imperfections arising

from the manufacturing process. These imperfections can be architectural, including fiber volume fraction,

variability in fiber diameter and coating thickness, fiber clustering, and voids. Moreover, the constituent

properties may be non-uniform due to thermal gradients and microstructral variability (grain size, grain

boundary phase, defects, etc) induced during processing . For instance, grain growth in the ceramic matrix

is a key aspect resulting directly from thermal processing. Fine grains tend to creep more, but give higher

strengths than coarse grains. So, there is a tradeoff between strength and creep resistance. The effects

of all these imperfections can be taken into consideration by assuming variability in the effective material

properties of the constituents. Finally, the presence of residual stresses (introduced during the manufacturing

process) is highly influenced by the in-situ constituent properties, and also has a significant impact on the

performance of CMC components.2–4

The Effect of Stochastically Varying Creep Parameters on 

Residual Stresses in Ceramic Matrix Composites 

Evan J. Pineda 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Subodh K. Mital 

University of Toledo 

Toledo, Ohio 43606 

Brett A. Bednarcyk and Steven M. Arnold 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

NASA/TM—2015-218759 1
Multiscale modeling is a useful technique for capturing the effect of variability at one scale on the

response at higher scales. Variability in the CMC microstructure and material parameters can be modeled

directly using micromechancs and linked to the higher scales using hierarchical, concurrent, or synergistic

“handshaking techniques”.5 Since the constituents are modeled explicitly, residual stresses can be predicted

by applying the appropriate processing history. Through multiscale modeling, combined with the flexibility

available in the design of CMC materials afforded by the presence of three constituents, and the significance

of residual stresses on performance, integrated computational materials engineering (ICME), which aims to

tailor the material for a given purpose, is achievable. With concurrent and synergistic multiscale modeling,

the fidelity of the micromechanics model can have adverse effects on the computational cost of the multiscale

analysis. However, multiscale modeling has been achieved successfully using a variety of suitable analytical,

semi-analytical, and numerical micromechanics models.6–9

The generalized method of cells (GMC) provides a semi-analytical solution for the local fields (stresses,

elastic strains, thermal strains and inelastic strains) within a repeating unit cell (RUC) that is discretized

into a number of subcells.9, 10 Traction and displacement continuity conditions, along with periodic boundary

conditions, are utilized with local constitutive laws to formulate a strain concentration matrix relating the

globally applied strains to the local subcell strains. The generality of the method admits any linear or

nonlinear, elastic or inelastic constitutive model. GMC has been utilized to capture the stochastic response

of woven CMCs exhibiting damage, and the effect of residual stresses on performance.3, 4, 11 Furthermore,

the semi-analytical formulation of GMC results in ultra-efficient computation making the method ideal for

inclusion within a multiscale ICME framework.12–14 In addition, the use of GMC for stochastic multiscale

analysis has been demonstrated.11, 15–17

As stated previously, residual stresses (occurring during processing) can have a substantial influence on

the performance of CMCs. The main driver for these residual stresses is the mismatch in thermo-elasticinelastic constituent properties. Thus, understanding the evolution of residual stresses is important for

predicting performance. Although it is understood that variation in the elastic properties and coefficients of

thermal expansion (CTEs) of the constituents will affect the residual stress state, the primary focus of this

work is to investigate the effects of variability, and corresponding disparity, in the inelastic (creep) properties

of the ceramic constituents on the post-manufactured residual stresses in CMCs. While variability in the

fiber architecture is evident (see Figure 1) and it affects the residual stresses in the CMC, herein an ordered,

square-packed RUC is utilized throughout; 1) for computational efficiency and 2) to isolate the effects

of constituent property variation from variability in the microstructure. The influence of microstructural

variability on the deformation and life of directional composites has been studied by others.18

For this study a demonstrative material system composed of continuous SCS-6 silicon carbide (SiC) fiberreinforced reaction bonded silicon nitride (RBSN) matrix with a boron nitride (BN) coating around the SiC

fibers was chosen.19, 20 Residual stresses arise in this material as a result of differences in the coefficients

of thermal expansion (CTE) in the fibers and matrix.2 During processing the residual stresses redistribute

(relax), due to the time-dependent behavior of the ceramic constituents (SiC fiber and RBSN matrix).20, 21

For simplicity, the time-dependent behavior is modeled using power law creep in GMC.9, 22, 23 Thermal

loads are applied to a single fiber RUC to simulate processing using the MAC/GMC micromechanics code,

developed at the NASA Glenn Research Center (GRC).24, 25 A normal distribution for each creep parameter is

assumed and Monte Carlo simulations are performed to probe the effect of variability in the creep parameters

(A and n) for each constituent (individually) on the residual stresses after manufacturing.

The effect of residual stresses on the performance (measured by changes in PLS and strain to failure εf )

of a CMC is demonstrated in Section II. The stochastic modeling approach used is described in detail in

Section III. Finally, the predicted residual stress distributions are examined in Section IV and compared to

a deterministic approach.

II. Effect of Residual Stresses in Ceramic Matrix Composites

Previous work has shown the effect of residual stresses on CMCs; that work is reviewed briefly in this

section.4, 26 A CMC RUC, shown in Figure 2 was modeled using GMC. Although, the previous study

(summarized here) involved a SiC-SiC CMC, the novel work in this manuscript focuses on a SiC-RBSN CMC

material due to the availability of material properties, and other information, in the open literature. The

same observed relationship between residual stresses and the macroscopic composite response is applicable,

regardless of the specific CMC system. The matrix subcells are represented in green, the fiber subcell in

NASA/TM—2015-218759 2
blue and the coating subcells in pink. The RUC was subjected to an idealized processing history, during

which the time-dependent behavior of the fiber and matrix is approximated using a power law creep model,

and the ensuing residual stresses are shown in Figure 3. The fundamental cause of the residual stresses is

the discrepancy between the constituent properties. The magnitude of residual stresses developed is a result

of stress redistribution induced by a disparity in the thermo-elastic properties (e.g. stiffness and CTEs).

However, the rate and magnitude of redistribution between constituents is driven by differences in the creep

properties of the constituents. Thus, the assumed creep properties and process history will significantly

affect the final residual stress state.

After processing, the matrix is under a longitudinal compressive stress, and the fiber is subjected to a

longitudinal tensile stress since the CTE of the fiber is greater than the CTE of the matrix in the longitudinal direction.A bilinear progressive damage model was employed for the matrix, a maximum stress failure

criterion was used for the fiber, and the RUC was subjected to longitudinal tension after application of processing. The impact of the residual stresses on the tensile response of a CMC is shown in Figure 4. Residual

compression in the matrix delays matrix cracking which leads to significant increase in the predicted “knee”

in the stress strain curve (PLS increases). Conversely, residual longitudinal tension in the fiber yields a

reduction in predicted strain to failure (εf ) because of the pre-strain existing in the fiber after processing.

Thus, there will always be a trade off between PLS and strain to failure. Matrix cracking can result in

oxidation and other adverse environmental effects. Therefore, it may be desirable to process the CMC such

that the εf is reduced in exchange for an increase in the PLS. So, if the PLS is used as a design criterion for

the composite material, then clearly the compressive matrix residual stresses are very beneficial.

Figure 5 shows the predicted effect of heat treatment on the room temperature response the composite.

Heat treatment at a moderate temperature T3, even for a long duration t1, only allows for a small reduction

in the residual stress, and thus only changes the tensile response slightly. However, heat treatment at

T1 (T1 > T3) for a shorter time t2 (t2 = 0.1t1) yields a substantial reduction in the residual stress, and

consequently the composite tensile response is much closer to the predicted response with no residual stresses.

Figure 5 demonstrates the competing effect of processing time and temperature on the post-manufactured

residual stress state, and hence, performance of a CMC. Clearly, application at elevated temperature can

eventually lead to a reduction of PLS. Then, given enough time and thermal driving force, matrix cracking

will occur when the CMC is loaded beyond the PLS corresponding to zero residual stresses. This is similar

to the concept of creep buckling.

Figure 6 shows the impact of heat treatment at T1 for t2 on the predicted longitudinal creep response of

the composite at T1. The applied stress level is 200 MPa, which is below the predicted PLS of the composite.

The effect is noticeable, but not large (due to the higher creep resistance of the fiber compared to the matrix)

because the composite does not creep very much in the longitudinal direction. Further heating up to T2,

prior to applying the simulated creep loading, eliminates some of the residual stresses.

Figures 4-6 show the importance of stress relaxation and processing on the residual stresses in, and

ultimately the performance of, the composite. However, these results do not account for any statistical

variation in the material properties or fiber-matrix architecture. These unaccounted for distributions may

significantly affect the performance of the material. Understanding the influence of stochastic distributions

at the microscale is critical for ICME.

III. Stochastic Modeling Approach

The previous section demonstrated the impact residual stresses can have on a CMC, supporting other

work in the literature.2, 3 However, the focus of this work is to examine the effect of variability in the creep

parameters of the CMC constituents on the residual stresses. Monte Carlo simulations utilizing GMC as the

micromechanics theory were preformed on a unidirectional SiC-RBSN system using the MAC/GMC software

package. The same processing history was used for each simulation, but the creep parameters for the SiC

fiber and RBSN matrix were chosen randomly using a normal distributions.

III.A. The Generalized Method of Cells

With GMC, an RUC must be identified that represents the composite microstructure, and the RUC is

discretized into a number of subcells. The RUC chosen to represent the unidirectional SiC-RBSN system

is shown in Figure 7. The unidirectional lamina is composed of continuous reinforcement in the axial, x1-

NASA/TM—2015-218759 3
direction. Thus, it is assumed that the RUC extends infinitely in that direction, and the RUC is subjected

to a state of generalized plane strain; therefore, the doubly-periodic formulation of GMC is presented herein.

Each subcell can be occupied by a single constituent (obeying the constitutive law of that constituent), but

there is no limit on the number of different constituents that can be present in the RUC. Displacement

and traction continuity is enforced in an average, or integral, sense at each of the subcell interfaces and

the periodic boundaries of the RUC. These continuity conditions are used to formulate elastic and inelastic

strain concentration matrices A and D, respectively.

εS = A ̄ε + D

(

ε

I

S + ε

T

S

)

(1)

which gives all the local generalized Hookean subcell strains εS in terms of the six, global, average, applied

strains ε ̄ and the local inelastic and thermal strains ε

I

S

and ε

T

S

. Note that, even though the formulation is

doubly-periodic the stress and strain fields are fully-3D both globally and locally.

Once the strain concentration matrix is formulated and the local subcell strains have been calculated, the

local subcell stresses σ

(βγ)

can be calculated using the local constitutive law and the local subcell strains.

σ

(βγ) = C(βγ)

(

ε

(βγ) − εI

(βγ)

− ε

T (βγ)

)

(2)

where β, and γ are the subcell indices in the x2-, and x3-directions, respectively, C(βγ)is the local subcell

stiffness, ε

(βγ)

contains the six local subcell strains, ε

I

(βγ)

are the inelastic strains, and ε

T (βγ)

are the thermal

strains. Finally, the global RUC (composite) stiffness can be calculated.

B =

1

HL

∑

Nβ

β=1

∑

Nγ

γ=1

hβlγC(βγ)A(βγ)(3)

where Nβ, and Nγ are the total number of subcells in the x2-, and x3-directions, respectively, H, and L are

the RUC dimensions in the x2-, and x3-directions, hβ, and lγ are the local subcell dimensions in the x2-, and

x3-directions for subcell βγ, and A(βγ)is a submatrix of the strain concentration matrix A corresponding

to subcell βγ.

Table 1 contains the volume fractions (V ) for each constituent used in all analyses. The SiC fiber volume

fraction V

f was 30%, the RBSN matrix volume fraction Vm was 62% and the BN coating volume fraction

V

c was 8%. The elastic properties and CTEs, at room temperature, for the SiC fiber, RBSN matrix, and

BN coating used in all simulations are also presented in Table 1, where E indicates Young’s modulus, ν

indicates Poisson’s ratio, and α indicates CTE. The elastic properties of all three materials are isotropic but

the longitudinal and transverse CTEs for the SiC fiber are different and are designated with a subscript L

and T, respectively. The Young’s modulus and CTEs of the fiber and matrix are also temperature dependent.

The values of these parameters at specific temperatures are given in Table 2 with linear interpolation being

applied to arrive at intermediate temperatures. In Table 2, SiC fiber properties are marked with a superscript

f, RBSN matrix properties are marked with a superscript m.

III.B. Stochastic Modeling of Time- and Temperature-Dependent Response

The aim of this work is to investigate the effect of variability in the time-dependent deformation, during

processing, on the resulting residual stresses in the SiC-RBSN CMC. To achieve this, the inelastic strain rate

components  ̇ε

I

(βγ)

ij are related to the second stress invariant J

(βγ)

2

and components of the deviatoric stress

tensor s

(βγ)

ij = σ

(βγ)

ij − δijσ

(βγ)

kk /3 (where δij is the Kronecker delta) in each subcell through a Norton-Bailey

power law creep relation.9

ε ̇

I

(βγ)

ij =

3

2

A ̃ (T)

(

3J

(βγ)

2

) n−1

2

s

(βγ)

ij (4)

where n is the power law exponent, and A ̃ (T) is a temperature dependent material constant (scaling factor)

related to the activation energy for the material, Q, and the ideal gas constant, R.

A ̃ (T) = A exp (−

Q

RT )

(5)

NASA/TM—2015-218759 4
Under uniaxial loading, Eq. (4) simplifies to

ε ̇

I

(βγ)

= A ̃ (T) σ

(βγ)

n

(6)

Although the constituent materials often times exhibit a primary creep regime (i.e., a zone in which the

inelastic strain rate decreases with increasing time), in this study a simple steady state creep model will

be used. The influence of higher creep rates associated with the primary creep will be accounted for by

characterizing the material parameters A and n using the concept of secant inelastic strain rate.

The strain versus time profile for primary stage, tensile creep of SCS-6 SiC fibers has been characterized

previously.27

ε

I = A0σmtp

exp (−

B

T

)

(7)

where t is time and A0, m, p and B are material parameters, given in Table 3 at an applied uniaxial stress

of 500 MPa.27 Taking the time derivative of Eq. (7) yields an expression for the inelastic strain rate of an

SCS-6 SiC fiber as a function of time under a uniaxial stress.

ε ̇

I = pA0σmtp−1

exp (−

B

T

)

(8)

In order to obtain the parameters used in the steady state power law approximation of the primary creep

regime of the SCS-6 SiC fibers, Eq. (6) is equated to the average creep strain rate ε ̄ ̇

I during processing. This

is obtained by time averaging Eq. (8) over a given processing time tf (herein taken to be the total time).

ε ̄ ̇

I =

∫ tf

0

ε ̇

I

dt (9)

ε ̄ ̇

I = A0σmt

p−1

f

exp (−

B

T

)

(10)

Assuming a uniaxial stress state (σ

(βγ)

11 = σ = 500MP a) and setting Eq. (6) equal to Eq. 10) yields

expressions for the steady state creep approximation constants in terms of the creep strain profile constants.

A ̃ (T) = A0t

p−1

f

exp (−

B

T

)

(11)

n = m (12)

Table 4 contains tabulated values of A ̃ at critical processing temperatures. Figure 8 shows a comparison

between the primary creep strain, given by Eq. (7), and the steady state secant approximation, calculated

using a constant inelastic strain rate given by Eq. (6) with the values in Table 4, under a uniaxial stress

of 500 MPa over the duration of the processing at different critical processing temperatures. Clearly, this

averaging procedure resulted in the steady state creep approximation achieving the same final inelastic strain

as the primary creep equation at the end of processing, tf .

The creep behavior of RBSN is not as well characterized as the SCS SiC fiber. Furthermore, it is difficult to

measure the in situ matrix properties from experiments on bulk RBSN because the ceramic microstructures

(grain size, grain boundary phases, defects, void content, etc.) are not the same. Thus, the creep properties

of the RBSN matrix were adjusted such that the reduction in residual stresses (when compared to thermoelastic analysis) in the fiber, and matrix, upon completion of the thermal processing is discernible. Since

the interest of this work is to investigate the effect of variability in creep properties on the residual stresses

in the material, not to predict the performance of the CMC, the method used to approximate the creep

parameters of the RBSN matrix is reasonable. The final values employed are presented in Table 4. Note

that, it is assumed that the BN fiber coating does not exhibit any creep.

The RUC is subjected to the thermal process history given in Figure 9. The exact details of the thermal

process history of this material are export controlled; thus, an idealized processing is given in Figure 9. The

processing starts at T1 and is held until time t1, after which the temperature is ramped down to T3 at t2. An

isothermal hold is performed at T3 until t3 is reached; whereupon the temperature is again ramped down to

room temperature (20◦C). The total processing time is tf . The resulting residual stresses, and subsequent

material performance, obviously heavily rely on the process history. However, to isolate the effect of material

NASA/TM—2015-218759 5
property variation, the specific process history is held fixed (dictated by Figure 9). Although not done here,

additional simulations could be performed to examine fluctuations or uncertainty in the process history on

residual stresses.

Variability is introduced into each creep parameter by assuming a normal distribution with a standard

deviation (σ) equal to 10% of the mean (μ) values (given in Table 4). The stochastic value of the parameter

X is determined using

X = σ

√

−2 ln r1 sin (2πr2) + μ (13)

where r1 and r2 are random numbers. The probability density functions (PDF) for each creep parameter of

each constituent (A ̃f, A ̃m, n

f and nm) are displayed in Figures 10-11. The PDFs for A ̃ are given for critical

processing temperatures (shown in Figure 9). Although the mean values of A ̃ are a function of temperature,

the random numbers r1 and r2 used to determine the stochastic value of A ̃ are fixed for each temperature

during a single analysis. It is assumed that n is independent of temperature.

To investigate the effect of variability in the creep parameters on the post-manufactured residual stress

state, the Monte Carlo method is utilized. Multiple simulations are performed, and the creep parameter

of interest is varied using Eq. (13) with different random values for r1 and r2 during each simulation.

Consequently, each simulation performed has a different disparity in the inelastic (creep) properties among

the constituents. However, for each simulation it is assumed that the creep properties of each material are

spatially uniform; i.e. there is no variation in the creep parameters among the different subcells within the

RUC that are occupied by the same constituent material. The RUC represents a material point in a larger

continuum domain; thus, it is assumed that spatial variability occurs at the higher length scales than that

represented with the RUC. To model spatial variability of creep parameters at the microscale would require

the utilization of a statistical representative volume element (RVE), which inherently includes numerous

fibers.9, 17, 28

IV. Results

IV.A. Residual Stresses in CMC Using a Deterministic Approach

To observe the effect stress redistribution has on the residual stresses (which affect performance) in a SiCRBSN lamina, the SiC-RBSN RUC (see Figure 7) is subjected to the thermal profile given in Figure 9. First,

it was assumed there was no time-dependent behavior in the constituents (wherein dwell periods within

process histories would make no difference), and then the time-dependent response was modeled using the

mean power law creep parameters from Table 4. The predicted, full-field, axial residual stress contours for

these two cases are shown in Figure 12 at the end of processing, and the maximum and minimum residual

stresses in the composite are given as a function of time in Figure 13.

Figure 12 shows that, subsequent to thermal processing, the axial residual stresses in the fiber are tensile

and compressive in the matrix. This is consistent with experimental observations and it is deemed to improve

the PLS of the composite provided the stresses in the fiber and matrix do not exceed the relative strengths

in tension and compression, respectively.2 Comparing the longitudinal residual stresses in the fiber indicates

that there is a 15.5% reduction in tensile stress from 437.4 MPa to 369.5 MPa if the stresses are allowed

to relax using the parameters in Table 4. Meanwhile, the compressive axial residual stresses in the matrix

are similarly reduced by 15.5% from -212.0 MPa to -179.0 MPa when reduction in the residual stresses is

enabled due to the accumulation of inelastic strain.

The predicted longitudinal residual stress history in each constituent is displayed in Figure 13. It can

be seen that from the outset t1 to t2, stress relaxation enables some moderate reduction in the residual

stresses, when compared to the residual stress accumulation in the purely thermo-elastic example. Although

the majority of the residual stress and residual stress relaxation develops during the final cooldown from t3

to tf , as expected. To validate these residual stress predictions, a comparison with sophisticated multiscale

experimental techniques would be required.29

As stated previously, the driving forces for the accumulation of residual stresses is the discrepancy of the

thermo-elastic properties (E and α) between the constituents and the total change in temperature during

processing. Conversely, the major driving forces enabling the development of inelastic strain, which effectively

reduces the residual stresses, is the difference in the creep properties among the constituents, temperature and

time. Therefore, the final residual stress state is predicated upon a competition between the advancement of

residual stresses due to constraints enacted by the difference in thermal expansion between the constituents,

NASA/TM—2015-218759 6
and the “relaxation” of those stress through time and temperature-dependent, inelastic dissipation. As

such, the residual stresses in the post-manufactured material can be tailored through changes in the thermomechanical properties (elastic and inelastic) and applied process history.

It should be noted that, the term “stress relaxation” is used loosely to describe the reduction in residual

stresses due to time-dependent development of inelastic strain. In fact, the mechanics of the thermal processing of a CMC are far more complex than isothermal stress relaxation or creep. Inelastic strain is developing

within the constituents while the stresses are evolving, and yet the development of inelastic strain affects the

stress evolution. Furthermore, the response is transversely isotropic. So, longitudinally the behavior might

be closer to stress relaxation at a constant strain, but transversely the response may be more similar to creep

at a fixed stress.

IV.B. Effect of Variability in Creep Parameters on Residual Stresses

The sensitivity of the inelastic strain rate  ̇ε

I

(βγ)

to changes in the material parameters A ̃ and n can be

obtained by taking the partial derivative of Eq. (6) with respect to each parameter. As such, the sensitivity

of the inelastic strain rate to changes in A ̃ is

∂ε ̇

I

(βγ)

∂A ̃

= σ

(βγ)

n

(14)

which is constant for a given stress, and n, and is independent of A ̃. However, the sensitivity of  ̇ε

I

(βγ)

to

changes in n is a nonlinear function of n

∂ε ̇

I

(βγ)

∂n = nAσ ̃ (βγ)

n−1

(15)

for a given A ̃ and σ

(βγ)

To determine the effect of variability in the creep properties of the constituents on the post-processed

residual stresses, the Monte Carlo methodology, described in Section III.B was employed. The effect of

variability in each of the four power law properties (A ̃ for the fiber and matrix, and n for the fiber and matrix)

was studied independently. Clearly in reality, some interdependence of these parameters would be expected

– the investigation of correlation effects will be reserved for a future study. One thousand MAC/GMC

simulations were conducted studying the effect of variability in each of the four creep parameters separately.

The analyses were performed on a desktop workstation using a single Intel Xeon E5649 processor at 2.53 GHz

and with 24 GB of RAM. The average runtime for each of the four Monte Carlo simulation (which included

1000 serial analyses) was 436.4 seconds. Figures 14-17 show the resulting variability in the post-processed,

longitudinal residual stresses in the fiber and matrix of the unidirectional SiC-RBSN RUC.

Figure 14 shows histograms for the variability in the residual longitudinal tensile stress in the fiber, and

compressive stress in the matrix, assuming a normal distribution in A ̃m (see Eq. (4)) in the SCS-6 fiber with

a standard deviation of 10% of the mean. The histograms were fit with PDFs assuming normal distributions

for the residual stresses in the fiber and matrix. The mean tensile fiber and compressive matrix residual

stresses were predicted to be 369.5 MPa and 179.0 MPa, respectively. The corresponding standard deviations

were predicted to be 0.43 MPa (0.1% of the mean) and 0.21 MPa (0.1% of the mean) in the fiber and matrix,

respectively.

The residual stress histograms obtained by varying A ̃f of the RBSN matrix are exhibited in Figure

15. The normal PDFs for these distributions predicted similar mean fiber tensile (369.8 MPa) and matrix

compressive (179.2 MPa) as the case when A ̃m of the fiber was varied. However, the predicted standard

deviations are 4.34 MPa (1% of the mean) and 2.13 MPa (1% of the mean) for the fiber and matrix.

It was expected that imposing normal distribution on both A ̃f and A ̃m would result in a normal distribution of the resulting residual stresses because the sensitivity of the inelastic strain rate to changes in A ̃

(in the fiber or matrix) is constant as given in Eq. (14). Moreover, the magnitude of A ̃ does not affect the

sensitivity of  ̇ε

I

(βγ)

to changes in A ̃. The small standard deviations in the predicted residual stresses, shown

in Figures 14 and 15, further support that variability in A ̃ (for the fiber or matrix) is inconsequential.

Utilizing a normal distribution for the parameter n

f

in the SCS-6 SiC fiber resulted in a distribution of

residual stresses for both fiber and matrix represented by the histograms in Figure 16. Note that, although a

symmetric, normal distribution was used for n

f

, the resulting residual stress distributions were not normal.

NASA/TM—2015-218759 7
A Weibull PDF best fit the residual stress data. This discrepancy between the distribution imposed on

the input and the resulting distribution of the output was expected due to the non-linear sensitivity of the

inelastic strain rate to changes in n, given by Eq. (15). Similar discrepancies between the shape of the

distribution used to control the input parameters and the shape of the resulting output distribution have

also been observed with respect to void distributions and damage.11 The mean residual tensile stress in

the fiber was 368.9 MPa with a standard deviation of 2.34 MPa (0.6% of the mean), and the mean residual

compressive stresses in the matrix 178.7 MPa with a standard deviation of 1.16 MPa (0.6% o f the mean).

Finally, Figure 17 shows the histogram for the residual stress distributions when n

m for the RBSN matrix

was varied. Again a normal distribution of n

m did not result in a normal distribution of the predicted residual

stresses. A generalized extreme value (GEV) PDF was used to fit the data. Variability in this paramter

yielded a slight reduction in the mean residual stresses: 354.4 MPa (tensile stress in fiber) and 171.8 MPa

(compressive stress in the fiber). Furthermore, the range of the variability in residual stresses was much

larger as indicated by the standard deviations: 58.1 MPa (16% of the mean) for the tensile residual stresses

in the fiber and 28.0 MPa (16% of the mean) for the compressive residual stresses in the matrix. The effect

of variability of n

m was markedly more significant than nf because the mean value of nm was 3 as opposed

to 1.1 for n

f

. If Eq. (15) is evaluated with n = n

f = 1.1, then the sensitivity of the inelastic strain rate with

respect to changes in n

f

is on the order of ∝ σ

(βγ)

0.1 Whereas, if Eq. (15) is evaluated at n = n

m = 3, then

the sensitivity of  ̇ε

I

(βγ)

with respect to changes in n

m is on the order of ∝ σ(βγ)

2

. Thus, it is clear that the

standard deviation of residual stresses is a product of the sensitivity of the inelastic strain rate to changes

in n with respect to a particular mean.

Figures 14 - 17 show the influence in the variability of the creep properties of the constituents on the

predicted residual stresses in the CMC. Variation in the power law properties for the SCS-6 SiC fiber did

not yield any major variability in the residual stresses. However, variation of the exponent n

f produced

a wider distribution in the residual stresses. The impact of variation in the RBSN matrix parameters was

more significant. Using a normal distribution on A ̃m for the matrix results in standard deviations an order

of magnitude larger than those obtained by varying A ̃ffor the fiber. However, these standard deviations

still remained within 1% of the mean. By far, the most influential parameter was n

m of the RBSN matrix

which exhibited standard deviations for the residual stress distributions in the fiber and matrix greater than

16% of he mean. Moreover, the resulting PDFs were not symmetric. The bias shown in the PDF indicates

that there is a higher probability that variation in the power exponent n

m of the RBSN matrix would yield

a reduction in the residual stresses, rather than an increase. This study did not take into account variability

in the distribution of fibers within the RUC because the focus was on material property variation, but it is

expected that geometrical variability would also have marked effect on the resulting residual stresses.

V. Conclusions

The influence of variability in the material parameters controlling the time dependent response of the

constituents in a SCS-6 SiC fiber-reinforced RBSN matrix unidirectional composite on the residual stresses

induced during processing was demonstrated. The residual stress state is of particular importance for CMCs

because it influences the performance (PLS and strain to failure) of the composite.2–4

In order to predict the evolution of residual stresses reasonably, micromechanics or multiscale modeling

must be utilized. GMC, implemented within the MAC/GMC suite of micromechanics codes, was chosen

as the modeling platform for all analyses. The time-dependent behavior of the SCS-6 SiC fiber and RBSN

matrix, during processing, was approximated using power law creep. It was shown that time-dependent

behavior in the fiber and matrix yields a redistribution of stresses in the constituents during processing.

Four sets of Monte Carlo simulations were conducted, varying each of the power law creep properties

individually according to normal distributions. One thousand analyses were conducted for each Monte Carlo

simulation and the average computational time for each Monte Carlo simulation was 436.4 seconds. The

results of this study indicated that the creep exponent of the RBSN matrix had the largest effect on variability

in the residual stresses within the processed composite. This was to be expected because of the relatively

large mean value that was used and from a preliminary sensitivity analysis. Moreover, the sensitivity analysis

revealed a constant sensitivity of the creep strain rate on changes in the power law scaling factor A, but

a nonlinear dependence on the power law exponent. This was supported by the resulting shapes of the

predicted residual distributions. The residual stress distribution takes the same form as the distribution

used to introduce variability into the scaling factor A. However, if there is variability in the power law

NASA/TM—2015-218759 8
exponent, the resulting shape of the residual stress distributions cannot be inferred.

To achieve ICME, processing and performance must be linked in a fully coupled manner. This exercise

has shown that processing effects, including variability, can be predicted for CMCs. If the impact of manufacturing on the creep parameters could be quantified then the resulting effects on residual stresses, and

subsequently performance, could be estimated. Furthermore, if manufacturing affects only the power law

constant A (intercept of creep rate versus stress), then deterministic approaches could be utilized with more

confidence because any statistical distribution of this parameter coincide with a similarly shaped distribution

of the residual stresses. Yet, if manufacturing significantly affects the uncertainty of the creep exponent n

(slope of creep rate versus stress), a stochastic study is warranted. Although the present investigation did

not account for microstructure variability, MAC/GMC is an ideal numerical tool for studying these effects in

conjunction with material property variation.18, 30 Furthermore, it has been demonstrated that, due to the

computational efficiency of MAC/GMC, it is a viable software package for conducting statistical analyses

and integration into an ICME framework. Future work will look at the effects of processing of laminates and

will also include statistical modeling of thermo-elastic properties in conjunction with the creep properties.

NASA/TM—2015-218759 9
Material V (%) E (GPa) ν αL/αT (

◦C−1

)

SiC Fiber 30 390 0.17 4.1E-6/1.84E-6

RBSN Matrix 62 110 0.22 2.2E-6

BN Coating 8 1.8 0.22 2.0E-6

Table 1: Elastic properties and CTE for SiC-RBSN constituents at room temperature.

Temperature Ef(GPa) α

f

L

/α

f

T

(

◦C−1

) Em (GPa) α

m (◦C−1

)

T3 357.2 5.83E-6/2.62E-6 100.7 3.13E-6

T2 354.1 5.99E-6/2.69E-6 99.9 3.22E-6

T1 351.0 6.15E-6/2.76E-6 99.0 3.30E-6

Table 2: Elastic properties and CTE for SiC fiber (f) and RBSN matrix (m) constituents as a function of

temperature.

A0 (MPa−m·s

−p

) m p B (K)

1,995 1.1 0.37 25,800

Table 3: Parameters used to characterize primary creep strain of SCS-6 fiber.27

NASA/TM—2015-218759 10
Temperature A ̃f

(MPa−n

f

) n

f A ̃m (MPa−nm

) n

m

Room Temp 0. 1.1 0. 3.0

T3 9.37E-8 1.1 3.41E-10 3.0

T2 3.34E-7 1.1 2.35E-8 3.0

T1 1.12E-6 1.1 9.48E-7 3.0

Table 4: Temperature dependent baseline (mean) Norton-Bailey power law creep parameters for SCS-6 SiC

fiber (f) and RBSN matrix (m).

NASA/TM—2015-218759 11
(a) Woven melt infiltrated (MI).

(b) Woven chemical vapor infiltration (CVI).

(c) Unidirectional or laminated MI.

Figure 1: Three types of ceramic matrix composites.

NASA/TM—2015-218759 12
Figure 2: RUC used to model unidirectional CMC lamina. Matrix subcells are green, fiber subcell is blue,

and coating subcell is pink.

Figure 3: Predicted residual stresses in CMC RUC after being subjected to a thermal processing history.

Figure 4: Predicted room temperature longitudinal tensile response of the unidirectional CMC with and

without residual stresses resulting from processing.

NASA/TM—2015-218759 13
Figure 5: Predicted room temperature longitudinal tensile response of the unidirectional CMC after heat

treatment at temperatures T1 > T3 for times t2 = t1/10.

Figure 6: Predicted longitudinal creep response of the unidirectional CMC at T1 with and without heat

treatment, in response to an applied stress of 200 MPa.

NASA/TM—2015-218759 14
Figure 7: RUC used in stochastic analyses.

Time (hrs.)

0 2 4 6 8

Inelastic (Creep) Strain 

ǫ

I

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

RT Primary (DiCarlo and Morscher, 1991)

T

3

 Primary (DiCarlo and Morscher, 1991)

T

2

 Primary (DiCarlo and Morscher, 1991)

T

1

 Primary (DiCarlo and Morscher, 1991)

RT Steady State Approximation

T

3

 Steady State Approximation

T

2

 Steady State Approximation

T

1

 Steady State Approximation

Figure 8: Comparison of primary creep and steady state, power law approximation over processing duration

at critical processing temperatures.

Figure 9: Thermal loading applied to RUC to simulate processing.

NASA/TM—2015-218759 15
(a) SCS-6 SiC fiber.

(b) RSBN matrix.

Figure 10: Probability density functions of power law creep parameter A ̃ at critical processing temperatures

(T1 > T2 > T3) for SCS-6 fiber and RSBN matrix.

(a) SCS-6 SiC fiber. (b) RSBN matrix.

Figure 11: Probability density functions of power law creep parameter n for SCS-6 fiber and RSBN matrix.

NASA/TM—2015-218759 16
(a) No time-dependent deformation. (b) Deterministic time-dependent deformation.

Figure 12: Predicted residual stresses in SiC-RBSN unidirectional RUC after processing with and without

time-depndent behavior in the constituents.

Figure 13: Predicted residual stress history in SCS-6 SiC fiber and RBSN matrix during processing of

unidirectional CMC RUC with and without time-dependent behavior in constituents.

NASA/TM—2015-218759 17
Figure 14: Histogram and PDF of predicted residual stresses after processing in a SiC-RBSN unidirectional

RUC assuming a normal distribution in the SCS-6 SiC fiber power law creep parameter A ̃f. Left: Maximum

Axial Tensile Stress in Fiber, Mean Stress = 369.5 MPa, Standard Deviation = 0.43 MPa. Right: Maximum

Axial Compressive Stress in Matrix, Mean Stress = 179.0 MPa, Standard Deviation = 0.21 MPa.

Figure 15: Histogram and PDF of predicted residual stresses after processing in a SiC-RBSN unidirectional

RUC assuming a normal distribution in the RBSN matrix power law creep parameter A ̃m. Left: Maximum

Axial Tensile Stress in Fiber, Mean Stress = 369.8 MPa, Standard Deviation = 4.34 MPa. Right: Maximum

Axial Compressive Stress in Matrix, Mean Stress 179.2 = MPa, Standard Deviation = 2.13 MPa.

NASA/TM—2015-218759 18
Figure 16: Histogram and PDF of predicted residual stresses after processing in a SiC-RBSN unidirectional

RUC assuming a normal distribution in the SCS-6 SiC fiber power law creep parameter n

f

. Left: Maximum

Axial Tensile Stress in Fiber, Mean Stress = 369.0 MPa, Standard Deviation = 2.34 MPa. Right: Maximum

Axial Compressive Stress in Matrix, Mean Stress = 178.8 MPa, Standard Deviation = 1.16 MPa.

Figure 17: Histogram and PDF of predicted residual stresses after processing in a SiC-RBSN unidirectional

RUC assuming a normal distribution in the RBSN matrix power law creep parameter n

m. Left: Maximum

Axial Tensile Stress in Fiber, Mean Stress = 354.4 MPa, Standard Deviation = 58.1 MPa. Right: Maximum

Axial Compressive Stress in Matrix, Mean Stress = 171.8 MPa, Standard Deviation = 28.1 MPa.

NASA/TM—2015-218759 19
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