## Abstract

The initial mechanical response of ceramic matrix composites (CMCs) is linear until the proportional limit. This initial response is characterized by linear elastic properties, which are anisotropic due to the orientation and arrangement of fibers in the matrix. The linear elastic properties are needed during various phases of analysis and design of CMC components. CMCs are typically made with ceramic unidirectional (UD) or woven fiber preforms embedded in a ceramic matrix formed via various processing routes. The matrix processing of interest in this work is the polymer impregnation and pyrolysis (PIP) process. As this process involves pyrolysis to convert a preceramic polymer into ceramic, considerable volume shrinkage occurs in the material. This volume shrinkage can lead to significant defects in the final material in the form of porosity of various size, shape, and volume fraction. These defect structures can have a significant impact on the elastic and damage response of the material. In this paper, a multiscale micromechanics modeling framework is developed to study the effects of processing-induced defects on the linear elastic response of a PIP-derived CMC. A combination of analytical and computational micromechanics approaches is used to derive the overall elastic tensor of the CMC as a function of the underlying constituents and/or defect structures. It is shown that the volume fraction and aspect ratio of porosity at various length scales play an important role in accurate prediction of the elastic tensor. Specifically, it is shown that the through-thickness elastic tensor components cannot be predicted accurately using the micromechanics models unless the effects of defects are considered.

## 1 Introduction

Ceramic matrix composites (CMCs) are high-temperature materials suited for various applications such as hot sections of gas turbine engines, exhaust nozzles of aircraft engines, and hypersonic structures. These materials are engineered with multiple constituents at various length scales, and hence are heterogeneous in nature. It is of interest to determine the overall properties of CMCs as a function of the constituent properties using a mechanics-based approach. Such an approach can enable estimation of the material behavior prior to actually fabricating it, and, furthermore, it can enable model-based design and tailoring of the CMC for desired properties within the framework of an integrated computational materials science and engineering. In principle, once the properties and spatial distribution and arrangements of the constituents are known within a representative volume element (RVE) of the CMC, its overall properties can be determined using micromechanics theories (e.g., see Hashin [1], Mura [2], Nemat-Nasser and Hori [3]). Micromechanics theories can be applied to predict elastic as well as inelastic behavior of the material. However, it is easier to predict the elastic properties using analytical micromechanics theories. On the other hand, in order to predict inelastic behavior, typically, computational micromechanics approaches are needed.

Elastic properties of CMCs are needed in analysis and design of components. During the early stages of design, when the material is not fully characterized yet, estimates of elastic properties are made, typically, using existing data, extrapolation of properties from similar materials, simple rule-of-mixture approximation, etc. However, such approaches are limiting, and more rigorous micromechanics-based models can lead to better estimation of properties. In addition, micromechanics theories can allow for incorporation of expected defects in the material, thereby, establishing a link between processing, defects, and properties.

Most composite micromechanics codes when applied to CMCs can predict the in-plane elastic properties accurately, but significantly overpredict their through-thickness properties [4]. A computational study conducted by Goldberg et al. [5] showed that the reason for poor prediction of the through-thickness modulus is the presence of elongated defects in the material. Using a two-dimensional (2D) finite element analysis (FEA), wherein the defect structure derived from computed tomographic (CT) scan was explicitly mapped, they showed that the through-thickness modulus was sensitive to defect shape. While the computational model was useful in demonstrating the role of defects in knocking down the through-thickness properties, it is cumbersome to use, especially when a quick estimate of elastic properties is needed. In order to establish a faster analysis capability, a simplified approach was developed by Gowayed et al. [6] to model defects in the RVE of CMC. In their approach, the defect characteristics were established from optical micrograph images and mapped as area projection on three orthogonal planes in the RVE of the CMC. Using stiffness averaging approach, the effects of such projected defect areas on elastic properties were calculated. Their work showed that interlaminar separation defect leads to a significant knock-down in through-thickness elastic modulus of SiC/SiNC CMCs. Furthermore, the approach predicts both the in-plane elastic modulus and through-thickness modulus for this CMC in good agreement with the experimental data. However, they did not show whether the approach can predict the remaining elastic properties (shear moduli and Poisson's ratios) as well.

In the research presented in this paper, effort was focused on developing a micromechanics modeling framework for predicting the elastic response of woven CMCs with defect structures at various length scales. The motivation was to: (1) explain the reasons for low through-thickness properties, typically measured in woven CMCs such as polymer impregnation and pyrolysis (PIP)-derived S200 class of material and (2) establish a micromechanics framework that can predict all the elastic properties of any CMC material from the constituent properties and defect structure in a consistent manner, without requiring arbitrary calibration of constituent properties to fit the CMC properties. As discussed in the foregoing, while some progress has been made to address the lower through-thickness modulus of PIP CMCs, the author is not aware of a consistent modeling framework that has been shown to predict all the elastic properties of the CMC material as a function of constituent properties and defect structures.

## 2 Micromechanics Approach

In this work, a multiscale micromechanics framework is developed. This framework combines analytical and computational micromechanics models to determine the overall elastic properties of a woven CMC as a function of constituent volume fraction, their arrangement, and defect structures at various length scales. The RVE of the woven CMC consists of fiber tows in ceramic matrix. For the purposes of discussion in this paper, the term tow does not mean a bundle of dry fibers, but refer to an impregnated fiber bundle. Thus, a tow is essentially a unidirectional (UD) composite of ceramic fibers and matrix, and is sometimes referred to as a mini-composite. Similarly, matrix refers to pure matrix region of the RVE that is not occupied by the tows. While the framework can be applied to any 2D- or three-dimensional-woven architectures, it is demonstrated here for 2D woven 8-Harness Satin Weave (8HSW) architecture used in the S200 class CMCs. It should also be noted that the framework is also applicable for UD composites as each tow of the woven system is essentially a UD composite.

A schematic of the developed framework is shown in Fig. 1. This framework consists of computational and analytical micromechanics models at various length scales. At the highest length scale, the overall elastic properties of the CMC are determined via computational (FEA-based) micromechanics approach using multiscaledesigner code (Version 3.4.40; Altair Engineering). The RVE consists of tows and pure matrix regions, both of which are treated as homogeneous, but possibly anisotropic, continuum. While both the tows and the pure matrix region are modeled as homogeneous continuum at the RVE scale, it is clear that, in reality, both of these components are heterogeneous systems. The tow consists of coated fibers embedded in the matrix as well as defect structures such as porosity and cracks. Similarly, the pure matrix region contains defects in the forms of pores and cracks (Fig. 1). Now, to determine the effective homogeneous elastic properties of the tows and pure matrix, accounting of their respective constituent phases and defect structures, a second set of micromechanics models is used. Effective properties of the pure matrix region, accounting for defects, are determined using Eshelby's equivalent inclusion method (see Mura [2]). In this method, the defects are idealized as ellipsoidal voids distributed in the pristine (i.e., defect free) matrix. On the other hand, the effective elastic properties of tows are determined using two analytical micromechanics models. The first micromechanics model determines the effective elastic properties of the pristine tow by ignoring any defect structures. This is followed by a second micromechanics model that determines the effect of defects on the elastic properties of the pristine tow obtained in the first step. The elastic properties of the pristine tow are determined from fiber, interface coating, and matrix properties using the composite cylinder assemblage (CCA) model developed by Hashin and Rosen [7], generalized to include multiple interface coating layers using the procedure developed by Qiu and Weng [8] and Chu and Rokhlin [9]. The effect of defects on the pristine elastic properties of the tow is determined using Eshelby's equivalent inclusion method (Mura [2]) similar to that used for the pure matrix region, but accounting for the transverse isotropy of the pristine tows. As shown in Fig. 1, the effective properties of the tows and pure matrix regions, as determined via analytical micromechanics models, feed into the RVE-scale model, which is solved using FEA-based homogenization approach [10]. The elastic properties thus obtained are the effective elastic properties of the CMC material accounting for constituents and defect structures. The details of the various analytical and computational micromechanics models are discussed next.

### 2.1 Model for Pristine Tows (Tows Without Defects).

An individual tow in the RVE of a woven composite is essentially a UD composite in an undulated form. Thus, the tow's effective elastic properties, in its material coordinate system, can be determined by classical analytical micromechanics approaches such as the CCA model developed by Hashin and Rosen [7] for polymer matrix composites. However, in contrast with the polymer matrix composites, fibers in SiC-based CMCs are coated with one or more layers of interface coating. Thus, a generalized version of the CCA model, as developed by Qiu and Weng [8] and Chu and Rokhlin [9], is adopted in this work. In this section, the CCA model and its generalized form are summarized with the key equations listed for completeness. The reader is referred to the original papers cited earlier for further details.

The CCA model and its generalized version treat the UD composite as a collection of composite cylinder of various sizes while maintaining the volume fractions of individual constituents (fiber, interface coating, matrix) in each cylinder. A number of elastic boundary value problems are solved on a single composite cylinder with appropriate boundary conditions and used in conjunction with the principle of minimum potential energy or minimum complementary energy to determine the bounds on the elastic constants. The original CCA model assumed the individual constituents of the composite cylinder to be isotropic; however, generalization to transversely isotropic constituents has been made in Ref. [8]. In this work, it is assumed that fibers, matrix, and the interface coating are all isotropic in their elastic behavior. The fiber and the matrix material of the CMC considered in this work are isotropic. While the interface coating layers are likely to be transversely isotropic, they are modeled here as isotropic for simplicity, and also because, it is difficult to determine the anisotropic elastic properties of the interface coating experimentally.

where $c(j)$, $E(j)$, $\nu (j)$, $\kappa (j)$, and $\mu (j)$ are the volume fraction, Young's modulus, Poisson's ratio, plane strain (or transverse) bulk modulus, and shear modulus, respectively, of the phase $(j)$. In the above equations, the first phase, represented by superscript $(1)$, is the core of the composite cylinder and the phase represented by superscript $(2)$ is the outside jacket. Note that for isotropic constituent phases, the plane strain (or transverse) bulk modulus can be expressed in terms of elastic modulus and Poisson's ratio as $\kappa (j)=\mu (j)/(1-2\nu (j))$ .

For a two-phase composite consisting of fiber and matrix, the above set of equations are used with fiber represented by superscript $(1)$ and matrix by superscript $(2)$ to derive the effective composite properties. For a composite with three phases, namely, fiber, interface coating, and matrix, the above set of equations are used recursively two times. First, the effective properties of the coated fiber are determined using the above equations with superscript (1) representing fiber and superscript (2) representing the interface coating. Second, the same set of equations, with modifications noted below, are used with superscript (1) representing the coated fiber and its effective properties (determined from step 1) and superscript (2) representing the matrix and its properties. As the effective elastic properties of the coated fiber derived from the first step will be transversely isotropic, the following substitutions must be made when using Eqs. (1)–(4) in the second step: $E(1)\u2192E11*\u2192\nu (1)\u2192\nu 12*,k(1)\u2192k23*,\mu (1)\u2192\mu 12*$, where superscript (*) denotes the effective properties of the coated fiber determined in step 1. This process can be repeated for arbitrary number of interface coating layers, if present in the composite.

### 2.2 Model for Tows With Defects.

As discussed in Sec. 2.1, the tows can be represented as transversely isotropic material when it contains no defect structures. However, tows in PIP-derived as well as other CMCs contain various processing-induced defects. These defects are typically in the form on distributed porosity of various shapes and shrinkage cracks. In the framework developed in this work, the effects of defects in the tows are accounted for after the pristine tow elastic properties have been obtained using the method discussed in Sec. 2.1 (see also introductory remarks at the beginning of Sec. 2 and Fig. 1).

The effective properties of the transversely isotropic tow with defects are obtained using Eshelby's equivalent inclusion method, which is summarized in this section. Eshelby's equivalent inclusion method treats the defects in the pristine tow as ellipsoidal inhomogeneity, which is defined as a region with different elastic constants compared to the surrounding parent material. The inhomogeneity is then converted into an equivalent inclusion (defined as material with same elastic constants as the parent material) of the same ellipsoidal shape as the inhomogeneity, but with an eigenstrain specified. This eigenstrain is fully specified through the so-called Eshelby tensors that have been derived for inclusions of various shapes (see Mura [2]). In the present case, the parent material is the homogenized pristine tow with properties obtained in Sec. 2.1. The defects are treated as inhomogeneity with zero elastic properties. It should be noted that the ellipsoidal representation is very general and can be used to represent defects such as spherical porosity, long ellipsoid, or a crack by changing the aspect ratio.

In Eq. (8), $c(r)$ and $S(r)$ represent, respectively, the volume fraction and Eshelby tensor of the defect type $(r)$ in the tow. Note that all defects of type $(r)$ will have same characteristics (shape and orientation, for example) so that they are defined by the same Eshelby tensor. There can be any number of defect types in the domain. It should also be noted that all the tensors described in Eq. (8) are fourth-rank tensors. The Eshelby tensor depends on the defect shape as well as the elastic properties of the parent medium. The expression for Eshelby tensor for defects in isotropic parent medium is available in closed form for many defect shapes (see Mura [2]). On the other hand, the Eshelby tensor for defects in anisotropic medium is not available in closed form, and hence requires numerical evaluation of certain integrals (see Mura [2]).

### 2.3 Model for Pure Matrix Region With Defects.

and, the stiffness tensor $L(0)$ obtained via tensor inverse (see Eq. (10)). Furthermore, the Eshelby tensor for defects of various shapes in the pure matrix can be expressed in closed form (see Mura [2], Chapter 2). As before, multiple defect types can be considered such as spherical pores and elliptical cylindrical pores in various orientations. Finally, once the effective compliance tensor of the matrix with defect is obtained, the usual engineering elastic constants can be derived.

### 2.4 Model for Composite Response: Analysis of the Representative Volume Element.

The overall elastic properties of the composite RVE with defects are determined by FEA-based forward homogenization method of Yuan and Fish [10], as implemented in the multiscaledesigner software (Version 3.4.40; Altair Engineering). The tows in the RVE along with their defect structures are modeled as homogeneous continuum with effective properties determined using approach outlined in Secs. 2.1 and 2.2. Similarly, the pure matrix region is modeled as homogeneous continuum with effective properties derived using procedure discussed in Sec. 2.3. The forward homogenization procedure was conducted using multiscaledesigner code (Version 3.4.40; Altair Engineering). In this approach, the 8HSW unit cell is defined directly by specifying the relevant geometrical parameters of the weave: semimajor and semiminor axis of the tow's elliptical cross section, spacing between the tows in two orthogonal directions, and volume faction of tows in the unit cell. The elastic properties of the tow and matrix are specified in terms of engineering constant. A finite element (FE) mesh of the RVE is generated by specifying suitable meshing parameter. The code automatically applies the periodic boundary conditions onto the unit cell and subjects it to appropriate boundary conditions to derive the effective elastic tensor for the RVE via homogenization method.

## 3 Results and Discussion

### 3.1 Verification of Analytical Micromechanics Models.

In order to verify various components of the micromechanics modeling framework, as outlined in Sec. 2, simple verification problems are considered for analysis and their results are compared against the corresponding computational micromechanics predictions. The computational micromechanics code used for this purpose is the multiscaledesigner code (Version 3.4.40, Altair Engineering), which can predict overall elastic properties of various heterogeneous materials where its unit cell is modeled via FE with appropriate periodic boundary conditions.

#### 3.1.1 Pristine Tow Model.

Verification of the generalized CCA model is conducted on two problems. The first problem considers a fiber/matrix tow composite without any interface coating layer. The second problem is a fiber/matrix tow composite with a single layer of interface coating. The thickness of the interface coating is assumed and is in the range of thicknesses explored in Ref. [12]. No defects are considered as the generalized CCA model does not account for defects. The input properties used for the analyses are shown in Table 1. The prediction from the generalized CCA model is compared against the computational micromechanics (FE-based analysis in multiscaledesigner) in Tables 2 and 3 for the two cases, respectively. It is seen that the prediction using the generalized CCA model is in good agreement with the FEA solution. Thus, the generalized CCA code is considered to be verified.

Property | Value |
---|---|

Fiber volume fraction | 68% |

Fiber | |

Radius | 7 μm (2.756 × 10^{−4} in.) |

Elastic modulus | 204.71 GPa (29.69 Msi) |

Poisson's ratio | 0.16 |

Matrix | |

Elastic modulus | 139.69 GPa (20.26 Msi) |

Poisson's ratio | 0.16 |

Interface | |

Thickness | 0.35 μm (1.38 × 10^{−5} in.) |

Elastic modulus | 13.79 GPa (2.0 Msi) |

Poisson's ratio | 0.16 |

Property | Value |
---|---|

Fiber volume fraction | 68% |

Fiber | |

Radius | 7 μm (2.756 × 10^{−4} in.) |

Elastic modulus | 204.71 GPa (29.69 Msi) |

Poisson's ratio | 0.16 |

Matrix | |

Elastic modulus | 139.69 GPa (20.26 Msi) |

Poisson's ratio | 0.16 |

Interface | |

Thickness | 0.35 μm (1.38 × 10^{−5} in.) |

Elastic modulus | 13.79 GPa (2.0 Msi) |

Poisson's ratio | 0.16 |

Tow composite property | Generalized CCA | FEA (multiscaledesigner) |
---|---|---|

$E11$ | 183.88 GPa (26.67 Msi) | 183.47 GPa (26.61 Msi) |

$E22=E33$ | 179.95 GPa (26.10 Msi) | 179.88 GPa (26.09 Msi) |

$\nu 12=\nu 13$ | 0.16 | 0.16 |

$\nu 23$ | 0.161 | 0.162 |

$G12=G13$ | 77.98 GPa (11.31 Msi) | 77.84 GPa (11.29 Msi) |

$G23$ | 77.50 GPa (11.24 Msi) | 77.43 GPa (11.23 Msi) |

Tow composite property | Generalized CCA | FEA (multiscaledesigner) |
---|---|---|

$E11$ | 183.88 GPa (26.67 Msi) | 183.47 GPa (26.61 Msi) |

$E22=E33$ | 179.95 GPa (26.10 Msi) | 179.88 GPa (26.09 Msi) |

$\nu 12=\nu 13$ | 0.16 | 0.16 |

$\nu 23$ | 0.161 | 0.162 |

$G12=G13$ | 77.98 GPa (11.31 Msi) | 77.84 GPa (11.29 Msi) |

$G23$ | 77.50 GPa (11.24 Msi) | 77.43 GPa (11.23 Msi) |

Tow composite property | Generalized CCA | FEA (multiscaledesigner) |
---|---|---|

$E11$ | 175.13 GPa (25.40 Msi) | 174.64 GPa (25.33 Msi) |

$E22=E33$ | 116.11 GPa (16.84 Msi) | 117.62 GPa (17.06 Msi) |

$\nu 12=\nu 13$ | 0.16 | 0.16 |

$\nu 23$ | 0.17 | 0.18 |

$G12=G13$ | 53.43 GPa (7.75 Msi) | 53.43 GPa (7.75 Msi) |

$G23$ | 49.64 GPa (7.20 Msi) | 49.85 GPa (7.23 Msi) |

Tow composite property | Generalized CCA | FEA (multiscaledesigner) |
---|---|---|

$E11$ | 175.13 GPa (25.40 Msi) | 174.64 GPa (25.33 Msi) |

$E22=E33$ | 116.11 GPa (16.84 Msi) | 117.62 GPa (17.06 Msi) |

$\nu 12=\nu 13$ | 0.16 | 0.16 |

$\nu 23$ | 0.17 | 0.18 |

$G12=G13$ | 53.43 GPa (7.75 Msi) | 53.43 GPa (7.75 Msi) |

$G23$ | 49.64 GPa (7.20 Msi) | 49.85 GPa (7.23 Msi) |

#### 3.1.2 Model for Defects in Tows.

Defects in homogenized tows are modeled as ellipsoidal voids in transversely isotropic material using the Eshelby's equivalent inclusion approach, as discussed in Sec. 2.2. The implementation of the model is verified by solving the problem shown in Fig. 2. Elliptical cylindrical voids in transversely isotropic medium are considered. The orientation of the voids is such that they are aligned with the fiber in the tows. Of course, the fiber and matrix are not modeled explicitly, but they are homogenized into an equivalent transversely isotropic material. The elastic properties of the transversely isotropic medium used are listed in Table 4. Void's aspect ratio $AR=a/b$ ($a$ is semimajor axis and $b$ is semiminor axis of the ellipse) is taken as 10, and void's volume fraction of 10% is considered. The prediction from the analytical micromechanics model is compared against corresponding computational micromechanics (FE) solution in Table 5. The FE analysis was carried using multiscaledesigner code by importing a meshed unit cell of the material with elliptical cylinder void generated using abaqus/cae (Version 2017, Dassault Systemes Simulia Corporation). Again, a good agreement between the analytical model and the benchmark FE solution verifies the implementation of the analytical model.

Property | Value |
---|---|

$E11$ | 182.71 GPa (26.5 Msi) |

$E22=E33$ | 155.82 GPa (22.6 Msi) |

$\nu 12=\nu 13$ | 0.17 |

$\nu 23$ | 0.18 |

$G12=G13$ | 68.60 GPa (9.95 Msi) |

$G23$ | 66.05 GPa (9.58 Msi) |

Property | Value |
---|---|

$E11$ | 182.71 GPa (26.5 Msi) |

$E22=E33$ | 155.82 GPa (22.6 Msi) |

$\nu 12=\nu 13$ | 0.17 |

$\nu 23$ | 0.18 |

$G12=G13$ | 68.60 GPa (9.95 Msi) |

$G23$ | 66.05 GPa (9.58 Msi) |

Property | Analytical micromechanics | FEA |
---|---|---|

$E11$ | 166.09 GPa (24.09 Msi) | 164.44 GPa (23.85 Msi) |

$E22$ | 139.21 GPa (20.19 Msi) | 138.24 GPa (20.05 Msi) |

$E33$ | 51.09 GPa (7.41 Msi) | 66.05 GPa (9.58 Msi) |

$\nu 12$ | 0.17 | 0.17 |

$\nu 13$ | 0.17 | 0.17 |

$\nu 23$ | 0.25 | 0.186 |

$G12$ | 61.78 GPa (8.96 Msi) | 61.02 GPa (8.85 Msi) |

$G13$ | 32.68 GPa (4.74 Msi) | 33.03 GPa (4.79 Msi) |

$G23$ | 33.03 GPa (4.79 Msi) | 19.79 GPa (2.87 Msi) |

Property | Analytical micromechanics | FEA |
---|---|---|

$E11$ | 166.09 GPa (24.09 Msi) | 164.44 GPa (23.85 Msi) |

$E22$ | 139.21 GPa (20.19 Msi) | 138.24 GPa (20.05 Msi) |

$E33$ | 51.09 GPa (7.41 Msi) | 66.05 GPa (9.58 Msi) |

$\nu 12$ | 0.17 | 0.17 |

$\nu 13$ | 0.17 | 0.17 |

$\nu 23$ | 0.25 | 0.186 |

$G12$ | 61.78 GPa (8.96 Msi) | 61.02 GPa (8.85 Msi) |

$G13$ | 32.68 GPa (4.74 Msi) | 33.03 GPa (4.79 Msi) |

$G23$ | 33.03 GPa (4.79 Msi) | 19.79 GPa (2.87 Msi) |

#### 3.1.3 Model for Defects in Matrix.

As discussed in Sec. 2.3, voids in the form of ellipsoids are modeled in isotropic medium using the Eshelby's equivalent inclusion method in order to represent defects in pure matrix region of the CMC RVE. The problems considered for model verification are shown in Fig. 3. Defects in the form of spherical and circular cylindrical voids in isotropic matrix are considered. The matrix properties used are $Em=139.96\u2009GPa\u2009(20.3\u2009Msi)$ and $\nu m=0.2$, and two void volume fractions of 10% and 20% are considered. The prediction from the analytical model is compared against the FE multiscaledesigner code in Table 6 for spherical voids and Table 7 for cylindrical voids. It should be noted that in the multiscaledesigner simulation, the voids are modeled as inclusions with very small elastic modulus of $6.89\u2009\xd710\u22126\u2009GPa\u2009(1.0\u2009\xd710\u22126\u2009Msi)$. It is seen that the analytical model is in good agreement with the benchmark FE solution. Hence, the analytical model and its implementation are verified.

Void V_{f} = 10% | Void V_{f} = 20% | |||
---|---|---|---|---|

Property | Analytical | FEA | Analytical | FEA |

$E$ | 116.66 GPa (16.92 Msi) | 118.59 GPa (17.2 Msi) | 99.97 GPa (14.5 Msi) | 99.97 GPa (14.5 Msi) |

$\nu $ | 0.2 | 0.195 | 0.2 | 0.185 |

$G$ | 48.61 GPa (7.05 Msi) | 48.40 GPa (7.02 Msi) | 41.64 GPa (6.04 Msi) | 38.54 GPa (5.59 Msi) |

Void V_{f} = 10% | Void V_{f} = 20% | |||
---|---|---|---|---|

Property | Analytical | FEA | Analytical | FEA |

$E$ | 116.66 GPa (16.92 Msi) | 118.59 GPa (17.2 Msi) | 99.97 GPa (14.5 Msi) | 99.97 GPa (14.5 Msi) |

$\nu $ | 0.2 | 0.195 | 0.2 | 0.185 |

$G$ | 48.61 GPa (7.05 Msi) | 48.40 GPa (7.02 Msi) | 41.64 GPa (6.04 Msi) | 38.54 GPa (5.59 Msi) |

Void V_{f} = 10% | Void V_{f} = 20% | |||
---|---|---|---|---|

Property | Analytical | FEA | Analytical | FEA |

$E11$ | 127.21 GPa (18.45 Msi) | 126.52 GPa (18.35 Msi) | 116.66 GPa (16.92 Msi) | 112.59 GPa (16.33 Msi) |

$E22=E33$ | 108.32 GPa (15.71 Msi) | 112.04 GPa (16.25 Msi) | 88.39 GPa (12.82 Msi) | 89.84 GPa (13.03 Msi) |

$\nu 12=\nu 13$ | 0.2 | 0.2 | 0.2 | 0.199 |

$\nu 23$ | 0.23 | 0.202 | 0.24 | 0.189 |

$G12=G13$ | 48.61 GPa (7.05 Msi) | 48.75 GPa (7.07 Msi) | 41.64 GPa (6.04 Msi) | 39.85 GPa (5.78 Msi) |

$G23$ | 44.20 GPa (6.41 Msi) | 43.99 GPa (6.38 Msi) | 35.58 GPa (5.16 Msi) | 30.13 GPa (4.37 Msi) |

Void V_{f} = 10% | Void V_{f} = 20% | |||
---|---|---|---|---|

Property | Analytical | FEA | Analytical | FEA |

$E11$ | 127.21 GPa (18.45 Msi) | 126.52 GPa (18.35 Msi) | 116.66 GPa (16.92 Msi) | 112.59 GPa (16.33 Msi) |

$E22=E33$ | 108.32 GPa (15.71 Msi) | 112.04 GPa (16.25 Msi) | 88.39 GPa (12.82 Msi) | 89.84 GPa (13.03 Msi) |

$\nu 12=\nu 13$ | 0.2 | 0.2 | 0.2 | 0.199 |

$\nu 23$ | 0.23 | 0.202 | 0.24 | 0.189 |

$G12=G13$ | 48.61 GPa (7.05 Msi) | 48.75 GPa (7.07 Msi) | 41.64 GPa (6.04 Msi) | 39.85 GPa (5.78 Msi) |

$G23$ | 44.20 GPa (6.41 Msi) | 43.99 GPa (6.38 Msi) | 35.58 GPa (5.16 Msi) | 30.13 GPa (4.37 Msi) |

### 3.2 Prediction of Ceramic Matrix Composite Elastic Properties.

The analytical micromechanics models developed and verified in the foregoing are used in the overall micromechanics framework, as outlined in Sec. 2 (see Fig. 1), to predict the overall elastic properties of the S200 CMC material. This material is chosen for the present analysis due to availability of experimental data for validation purposes. The approach can be easily extended to other CMC material system. The fiber in the S200 material is CG Nicalon^{TM} woven in 8HSW architecture. The fibers are coated with an interface coating, and the matrix is a SI–N–C matrix derived using the PIP process. The input constituent properties are listed in Table 8. Here, the fiber properties are taken from manufacturer's datasheet. The matrix properties are taken from the nano-indentation work by Ojard et al. [4]. The interface coating thickness is assumed to be in the range of thicknesses explored in Ref. [12], and its elastic properties are assumed based on various sources in the open literature. Typical void volume fraction in this material ranges from 10% to 20%. Here, it is assumed that the total void volume fraction is 20% with 10% intralaminar voids and 10% interlaminar voids, both in the pure matrix region of the RVE. Both types of voids are considered to be elongated elliptical cylinder with aspect ratio of the ellipse as 10. Intralaminar voids are oriented such that their normal vector is along the in-plane directions (i.e., along the two directions), whereas interlaminar void's normal directions are along the through-thickness direction of the unit cell. The results from the micromechanics modeling framework developed in this work are compared against experimental data (taken from Kumar and Welsh [13]) in Table 9. For comparison, results from analysis without any defects are also shown.

Property | Value |
---|---|

Overall fiber volume fraction in CMC | 42% |

Fiber | |

Radius | 7.0 μm (2.756 × 10^{−4} in.) |

Elastic modulus | 199.95 GPa (29.0 Msi) |

Poisson's ratio | 0.16 |

Matrix | |

Elastic modulus | 139.96 GPa (20.3 Msi) |

Poisson's ratio | 0.2 |

Interface coating | |

Thickness | 0.4 μm (1.57 × 10^{−5} in.) |

Elastic modulus | 275.79 GPa (40.0 Msi) |

Poisson's ratio | 0.2 |

Defect (elliptical cylinder voids) | |

Intralaminar | V_{f} = 10%; AR = 10 |

Interlaminar | V_{f} = 10%; AR = 10 |

Property | Value |
---|---|

Overall fiber volume fraction in CMC | 42% |

Fiber | |

Radius | 7.0 μm (2.756 × 10^{−4} in.) |

Elastic modulus | 199.95 GPa (29.0 Msi) |

Poisson's ratio | 0.16 |

Matrix | |

Elastic modulus | 139.96 GPa (20.3 Msi) |

Poisson's ratio | 0.2 |

Interface coating | |

Thickness | 0.4 μm (1.57 × 10^{−5} in.) |

Elastic modulus | 275.79 GPa (40.0 Msi) |

Poisson's ratio | 0.2 |

Defect (elliptical cylinder voids) | |

Intralaminar | V_{f} = 10%; AR = 10 |

Interlaminar | V_{f} = 10%; AR = 10 |

Micromechanics model | |||
---|---|---|---|

Property | No defects | With defects | Experiments |

$E11=E22$ | 155.55 GPa (22.56 Msi) | 104.80 GPa (15.2 Msi) | 104.11 GPa (15.1 Msi) |

$E33$ | 148.51 GPa (21.54 Msi) | 37.23 GPa (5.4 Msi) | 34.47 GPa (5.0 Msi) |

$\nu 12$ | 0.175 | 0.13 | 0.14 |

$\nu 13=\nu 23$ | 0.188 | 0.18 | 0.19 |

$G12$ | 63.71 GPa (9.24 Msi) | 42.06 GPa (6.1 Msi) | 44.13 GPa (6.4 Msi) |

$G13=G23$ | 62.95 GPa (9.13 Msi) | 22.75 GPa (3.3 Msi) | 7.58 GPa (1.1 Msi) |

Micromechanics model | |||
---|---|---|---|

Property | No defects | With defects | Experiments |

$E11=E22$ | 155.55 GPa (22.56 Msi) | 104.80 GPa (15.2 Msi) | 104.11 GPa (15.1 Msi) |

$E33$ | 148.51 GPa (21.54 Msi) | 37.23 GPa (5.4 Msi) | 34.47 GPa (5.0 Msi) |

$\nu 12$ | 0.175 | 0.13 | 0.14 |

$\nu 13=\nu 23$ | 0.188 | 0.18 | 0.19 |

$G12$ | 63.71 GPa (9.24 Msi) | 42.06 GPa (6.1 Msi) | 44.13 GPa (6.4 Msi) |

$G13=G23$ | 62.95 GPa (9.13 Msi) | 22.75 GPa (3.3 Msi) | 7.58 GPa (1.1 Msi) |

It is clear that the micromechanics model without incorporating the effects of defects leads to a poor prediction of elastic constants, especially the through-thickness properties such as $E33$ and $G13$. These properties are overpredicted by approximately 4 and 9 times, respectively. In fact, recall that the overprediction of $E33$ was the main motivation for pursuing this research. It is seen that incorporation of the defect structures leads to a better prediction of the overall elastic properties of the CMC. It is also seen that most of the elastic properties are in good agreement with the experimental data; however, the transverse shear modulus is still overpredicted, though by a smaller factor than the model without defects. The reason for the discrepancy in transverse shear modulus is not clear. Possible reason could be that the real void structures are more complex than the distributed ellipsoids assumed in the models and there might be stronger interaction between the defects, and both these factors have a stronger effect on the transverse shear moduli. It is also possible that the experimental method for estimating transverse shear modulus is not very accurate as this property is estimated indirectly via bending tests. Comparison of the model prediction against experimental scatter in this property may also provide insights. However, unfortunately, the experimental scatter in the properties of the material considered in this work is not available. In any case, further work is necessary to establish the reasons for this discrepancy. However, it is important to note that the framework developed in this work is able to predict all the elastic constants of CMCs with defects in a consistent manner.

## 4 Conclusions

In this paper, we presented a new micromechanics modeling framework to predict the overall elastic properties of woven CMCs incorporating processing-induced defect structures. The framework uses FE-based approach to model the RVE of the CMC. However, the lower length-scale homogenization is achieved by a suite of analytical micromechanics models. The tow is homogenized in two steps: the first step homogenizes the fiber, matrix, and interface coatings, and the second step considers the effects of processing defects. The pure matrix region of the RVE is homogenized in one step incorporating the effects of defects. For determining the elastic properties of pristine tows, the CCA model is generalized to incorporate arbitrary number of interface coating layers as multiple layers are often present in SiC/SiC CMCs of interest. The effects of defects in both the tow and the pure matrix region are derived using Eshelby's equivalent inclusion method which allows for multiple defects of ellipsoidal shapes and orientations within the domain.

As the modeling framework consists of various individual micromechanics models, it was necessary to verify them for ensuring that they have been implemented correctly. The verification was conducted by solving a number of simple problems and comparing the results against commercial FEA-based computational micromechanics results. Once verified, the micromechanics framework was applied to predict the overall elastic properties of the S200 CMC with experimentally estimated input parameters consisting of constituent properties and defect characteristics. The framework was exercised with and without considering defects. It is shown that without incorporating the effects of defects, the in-plane fiber-dominated properties of the CMC are predicted reasonably well; however, the through-thickness properties are overpredicted by several times. Furthermore, it is shown that incorporating the defect structures, especially the elliptical cylindrical shaped porosity in the pure-matrix region, leads to a better prediction of all the elastic constants. With this new framework, both the in-plane and through-thickness properties are predicted and they are in good agreement with the experimental data. However, some discrepancy between the model and the test data remains for the transverse shear modulus. Further work is needed to understand and source of this discrepancy and improve the modeling framework.

It should be noted that the defect characteristics (volume fraction, aspect ratios, etc.), used in the prediction of the S200 CMC properties, are not exact, but estimated values based on experimental measurements. Future application of the framework could include exact representation of the defect characteristics as derived from statistical analysis of the experimental CT-scan data of defect structures.

## Acknowledgment

The work reported in this paper was funded by the U.S. Air Force Research Laboratory (AFRL) under Contract No. FA8650-13-C-5213. The author would like to thank Dr. George Jefferson, program monitor, for his support throughout the duration of this program. The paper has been approved for public release by the AFRL (Case No. 88ABW-2019-5601) and Raytheon Technologies Research Center.

## Funding Data

Air Force Research Laboratory (AFRL) (Contract No. FA8650-13-C-5213; Funder ID: 10.13039/100006602).