Micromechanical Modeling Tensile and Fatigue Behavior of Fiber-Reinforced Ceramic-Matrix Composites Considering Matrix Fragmentation and Closure

Abstract

In this paper, micromechanical constitutive models are developed to predict the tensile and fatigue behavior of fiber-reinforced ceramic-matrix composites (CMCs) considering matrix fragmentation and closure. Damage models of matrix fragmentation, interface debonding, and fiber’s failure are considered in the micromechanical analysis of tensile response, and the matrix fragmentation closure, interface debonding and repeated sliding are considered in the hysteresis response. Relationships between the matrix fragmentation and closure, tensile and fatigue response, and interface debonding and fiber’s failure are established. Experimental matrix fragmentation density, tensile curves, and fatigue hysteresis loops of mini, unidirectional, cross-ply, and 2D plain-woven SiC/SiC composites are predicted using the developed constitutive models. Matrix fragmentation density changes with increasing or decreasing applied stress, which affects the nonlinear strain of SiC/SiC composite under tensile loading, and the interface debonding and sliding range of SiC/SiC composite under fatigue loading.

1. Introduction

SiC/SiC ceramic matrix composite (CMC) is a type of composite with the SiC fiber as reinforcement and SiC ceramic as matrix. It not only maintains the advantages of SiC ceramic, such as, high temperature resistance, high strength, low density, and oxidation resistance, but also possesses the toughening effect of the SiC fiber, and effectively overcomes the dangerous shortcoming of monolithic ceramics, such as, brittleness, sensitivity to defects or cracks and poor reliability. Compared with superalloy material, SiC/SiC composites possess lower density (i.e., usually 2.0~3.0 g/cm³, which is only 1/3~1/4 of superalloy), and higher temperature resistance (i.e., the operating temperature with cooling is higher than 1200 °C). For the application in aeroengine, SiC/SiC composites can reduce the structural weight, simplify cooling structure, reduce cooling air consumption, improve combustion efficiency, and increase thrust-weight-ratio. It is one of the most potential thermal structural materials for aeroengine hot-section components [1,2,3,4].

Under tensile and cyclic fatigue loading, multiple micro damage mechanisms occur and contribute to the nonlinear behavior of the composite [5,6,7,8,9,10]. With increasing applied tensile stress, matrix fragmentation occurs first, and the matrix fragmentation density increases to saturation. Sevener et al. [11] performed experimental investigation on crack opening behavior in 2D plain-woven melt infiltrated SiC/SiC composite under uniaxial tension using scanning electron microscopy (SEM) combined with digital image correlation (DIC) and manual crack opening displacement (COD) measurements. Crack openings were found to increase linearly with increasing applied stress. Li [12] developed a micromechanical model to predict the COD behavior of mini-SiC/SiC composite. Relationships between COD, crack opening stress, interface debonding stress and interface debonding ratio were established. Chen et al. [13] investigated the crack initiation and propagation in braided SiC/SiC tubes with different braiding angle using in situ tensile tests with synchrotron micro-computed tomography. The results show that braiding angle has no obvious effect on the location of crack onsets, whereas it significantly affects the paths of crack propagation. Li [14] predicted the first matrix fragmentation stress in fiber-reinforced CMCs using the energy balance approach considering fiber’s debonding and fracture. Morscher et al. [15], Goulmy et al. [16] performed experimental investigations on matrix multiple cracking using the electrical resistance. Li [17] investigated multiple matrix fragmentation behavior of mini, unidirectional, and 2D plain-woven SiC/SiC composites using critical matrix strain energy (CMSE) criterion. Matrix fragmentation density increased with the increasing of fiber’s volume, fiber’s elastic modulus, interface shear stress, and interface debonding energy, and the decreasing of the fiber’s radius and matrix elastic modulus. Matrix fragmentation affects the tensile and fatigue behavior of fiber-reinforced CMCs [18,19,20,21,22,23,24,25]. Ahn and Curtin [26] investigated matrix stochastic fragmentation on hysteresis loops in unidirectional CMCs. Liu et al. [27] analyzed the hysteresis loops of 3D needle-punched C/SiC composite and predicted the mechanical hysteresis loops for different tensile peak stress. The hysteresis-based damage parameters of inverse tangent modulus and interface debonding ratio were adopted to describe the composite’s internal damage evolution. Li [28] predicted the non-closure hysteresis loops in fiber-reinforced CMCs at high tensile stress, due to the change of matrix fragmentation density with loading/unloading tensile stress. However, in the research mentioned above, the synergistic effect of matrix fragmentation and closure with increasing or decreasing tensile stress on tensile and fatigue behavior of fiber-reinforced CMCs have not been analyzed.

The objective of this paper is to develop a micromechanical constitutive model to analyze the effects of matrix fragmentation and closure on tensile and fatigue behavior of fiber-reinforced CMCs. Damage models of matrix fragmentation, interface debonding, and fiber’s failure are considered in the micromechanical analysis for tensile response. Matrix fragmentation and closure, interface debonding and repeated sliding are considered in the micromechanical analysis for hysteresis response. Relationships between composite’s tensile and fatigue response and matrix fragmentation and closure are established. Experimental tensile curves and fatigue hysteresis loops of mini, unidirectional, cross-ply, and 2D plain-woven SiC/SiC composites are predicted.

2. Micromechanical Constitutive Models

Under tensile and cyclic loading/unloading tensile testing, matrix fragmentation with interface debonding occurs, and the fragmentation density increases with tensile peak stress. However, with decreasing applied stress, the matrix fragmentation density changes with applied stress, which would affect the mechanical hysteresis loops of fiber-reinforced CMCs. In this section, the micromechanical constitutive models for monotonic tensile and cyclic loading/unloading hysteresis loops were developed considering matrix fragmentation and closure.

2.1. Micromechanical Tensile Constitutive Model Considering Matrix Fragmentation

For fiber-reinforced CMCs under tensile loading, damage mechanisms of matrix fragmentation, interface debonding and fiber’s failure occur, and contribute to the nonlinear damage and fracture of composite’s tensile response.

For the initial tensile loading, the composite’s strain response is, [5]

$${\epsilon }_{\mathrm{c}}=\frac{\sigma }{E_{\mathrm{c}}}$$

(1)

where ε_c is the strain of composite, σ is applied stress, and E_c is elastic modulus of composite.

Considering damage mechanisms of matrix fragmentation, interface debonding and fiber’s failure, the nonlinear composite’s strain response is,

$${\epsilon }_{\mathrm{c}}\left(\sigma \right)=\{\begin{array}{l}\frac{\mathsf{\Phi }}{E_{\mathrm{f}}}\eta -\frac{{\tau }_{\mathrm{i}}}{E_{\mathrm{f}}}\frac{l_{\mathrm{d}}}{r_{\mathrm{f}}}\eta +\frac{{\sigma }_{\mathrm{fo}}}{E_{\mathrm{f}}}\left(1-\eta \right)-\frac{1}{\rho E_{\mathrm{f}}}\frac{2r_{\mathrm{f}}}{l_{\mathrm{c}}}\left(\mathsf{\Phi }-{\sigma }_{\mathrm{fo}}-2\frac{l_{\mathrm{d}}}{r_{\mathrm{f}}}{\tau }_{\mathrm{i}}\right)\\ \times \left[\exp \left(-\frac{\rho }{2}\frac{l_{\mathrm{c}}}{r_{\mathrm{f}}}\left(1-\eta \right)\right)-1\right]-\left({\alpha }_{\mathrm{c}}-{\alpha }_{\mathrm{f}}\right)\Delta \mathsf{{\rm T}},\eta <1\\ \frac{\mathsf{\Phi }}{E_{\mathrm{f}}}-\frac{{\tau }_{\mathrm{i}}}{E_{\mathrm{f}}}\frac{l_{\mathrm{c}}}{2r_{\mathrm{f}}}-\left({\alpha }_{\mathrm{c}}-{\alpha }_{\mathrm{f}}\right)\Delta \mathsf{{\rm T}},\eta =1\end{array}$$

(2)

where E_f is the fiber’s elastic modulus, Φ represents the intact fiber stress, τ_i represents the shear stress at the interface, and r_f represents the fiber’s radius; l_d and l_c are the lengths of the debonding and the space between the cracks in the matrix, respectively; η is the ratio between l_d and half of l_c; σ_fo represents the fiber’s axial stress levels in the interface bonding region; ρ represents a shear-lag model parameter; α_f and α_c represent the coefficients of axial thermal expansion of the fiber and composite, respectively; ΔT is the temperature difference between the test and fabrication temperatures. Curtin [29] developed a stochastic model to analysis matrix stochastic cracking inside CMCs, and the relationship between matrix crack spacing and applied stress can be determined by Equation (3). Gao et al. [30] developed a fracture mechanical approach to determine the interface debonding length when matrix crack propagates to the interface, and the relationship between the interface debonding length and the applied stress can be determined by Equation (4). The Global Load Sharing (GLS) criterion is adopted to determine the stress distribution between the intact and breakage fibers, which is given by Equation (5) [31].

$$l_{\mathrm{c}}=l_{\mathrm{sat}}{\left\{1-\exp \left[-{\left(\frac{{\sigma }_{\mathrm{m}}}{{\sigma }_{\mathrm{R}}}\right)}^m\right]\right\}}^{-1}$$

(3)

$$l_{\mathrm{d}}=\frac{r_{\mathrm{f}}}{2}\left(\frac{V_{\mathrm{m}}{E}_{\mathrm{m}}}{E_{\mathrm{c}}{\tau }_i}\mathsf{\Phi }-\frac{1}{\rho }\right)-\sqrt{{\left(\frac{r_{\mathrm{f}}}{2\rho }\right)}^2-\frac{r_{\mathrm{f}}^2{V}_{\mathrm{f}}{V}_{\mathrm{m}}{E}_{\mathrm{f}}{E}_{\mathrm{m}}\mathsf{\Phi }}{4E_{\mathrm{c}}^2{\tau }_{\mathrm{i}}{}^2}\left(\mathsf{\Phi }-\frac{\sigma }{V_{\mathrm{f}}}\right)+\frac{r_{\mathrm{f}}{V}_{\mathrm{m}}{E}_{\mathrm{m}}{E}_{\mathrm{f}}}{E_{\mathrm{c}}{\tau }_{\mathrm{i}}{}^2}{\mathsf{\Gamma }}_{\mathrm{i}}}$$

(4)

$$\frac{\sigma }{V_{\mathrm{f}}}=\mathsf{\Phi }\left(1-P\right)+\frac{2{\tau }_{\mathrm{i}}}{r_{\mathrm{f}}}\langle L\rangle P$$

(5)

where V_f and V_m are the volume fraction of the fiber and the matrix, respectively; E_m is the elastic modulus of the matrix; l_sat is the saturation length of matrix cracking; σ_m represents the stress loaded on the matrix; σ_R represents the characteristic stress for cracks in the matrix; Γ_i represents the debonding energy at the interface; <L> corresponds to mean fiber pullout length, and P represents the probability of fiber fragmentation.

2.2. Micromechanical Cyclic Hysteresis Loops Constitutive Model Considering Matrix Fragmentation and Closure

Upon unloading, the composite’s strain is divided into two stages, as following:

▪

Stage I, when σ_unloading > σ_{tr_unloading}, the unloading interface reverse slip length is less than the interface debonding length.

▪

Stage II, when σ_min < σ_unloading < σ_{tr_unloading}, the unloading interface reverse slip length is equal to the interface debonding length.

Upon unloading, when σ_unloading > σ_{tr_unloading}, the composite’s strain can be determined by Equation (6), [28]

$$\begin{array}{l}{\epsilon }_{\mathrm{unloading}}=\frac{{\sigma }_{\mathrm{unloading}}}{V_{\mathrm{f}}{E}_{\mathrm{f}}}\eta +2\frac{{\tau }_{\mathrm{i}}}{E_{\mathrm{f}}}\frac{l_{\mathrm{d}}}{r_{\mathrm{f}}}\eta {\gamma }^2-\frac{{\tau }_{\mathrm{i}}}{E_{\mathrm{f}}}\frac{l_{\mathrm{d}}}{r_{\mathrm{f}}}\eta {\left(2\gamma -1\right)}^2+\frac{{\sigma }_{\mathrm{fo}}}{E_{\mathrm{f}}}\left(1-\eta \right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{\rho E_{\mathrm{f}}}\left[2{\sigma }_{\mathrm{mo}}\frac{V_{\mathrm{m}}}{V_{\mathrm{f}}}\frac{r_{\mathrm{f}}}{l_{\mathrm{c}}}-2{\tau }_{\mathrm{i}}\eta \left(1-2\gamma \right)\right]\left[1-\exp \left(-\frac{\rho l_{\mathrm{c}}}{2r_{\mathrm{f}}}\left(1-\eta \right)\right)\right]-\left({\alpha }_{\mathrm{c}}-{\alpha }_{\mathrm{f}}\right)\Delta \mathsf{{\rm T}}\end{array}$$

(6)

where

$$\eta =2\frac{l_{\mathrm{d}}}{l_{\mathrm{c}}},\gamma =\frac{l_{\mathrm{counter}\text{\_}\mathrm{slip}}}{l_{\mathrm{d}}}$$

(7)

When σ_min < σ_unloading < σ_{tr_unloading}, the composite’s strain can be determined by Equation (8), [28]

$$\begin{array}{l}{\epsilon }_{\mathrm{unloading}}=\frac{{\sigma }_{\mathrm{unloading}}}{V_{\mathrm{f}}{E}_{\mathrm{f}}}\eta +2\frac{{\tau }_{\mathrm{i}}}{E_{\mathrm{f}}}\frac{l_{\mathrm{d}}}{r_{\mathrm{f}}}\eta -\frac{{\tau }_{\mathrm{i}}}{E_{\mathrm{f}}}\frac{l_{\mathrm{d}}}{r_{\mathrm{f}}}\eta +\frac{{\sigma }_{\mathrm{fo}}}{E_{\mathrm{f}}}\left(1-\eta \right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{\rho E_{\mathrm{f}}}\left[2{\sigma }_{\mathrm{mo}}\frac{V_{\mathrm{m}}}{V_{\mathrm{f}}}\frac{r_{\mathrm{f}}}{l_{\mathrm{c}}}+2{\tau }_{\mathrm{i}}\eta \right]\left[1-\exp \left(-\frac{\rho l_{\mathrm{c}}}{2r_{\mathrm{f}}}\left(1-\eta \right)\right)\right]-\left({\alpha }_{\mathrm{c}}-{\alpha }_{\mathrm{f}}\right)\Delta \mathsf{{\rm T}}\end{array}$$

(8)

Upon reloading, the composite’s strain is divided into two stages, as following:

▪

Stage I, when σ_reloading > σ > σ_min, the reloading interface new slip length is less than the interface debonding length.

▪

Stage II, when σ > σ_reloading, the reloading interface new slip length is equal to the interface debonding length.

Upon reloading, when σ_reloading > σ > σ_min, the composite’s strain can be determined by Equation (9), [28]

$$\begin{array}{l}{\epsilon }_{\mathrm{reloading}}=\frac{\sigma }{V_{\mathrm{f}}{E}_{\mathrm{f}}}\eta -2\frac{{\tau }_{\mathrm{i}}}{E_{\mathrm{f}}}\frac{l_{\mathrm{d}}}{r_{\mathrm{f}}}\eta {\phi }^2+2\frac{{\tau }_{\mathrm{i}}}{E_{\mathrm{f}}}\frac{l_{\mathrm{d}}}{r_{\mathrm{f}}}\eta {\left(1-2\phi \right)}^2\\ \hspace{1em}\hspace{1em}\hspace{1em} -\frac{{\tau }_{\mathrm{i}}}{E_{\mathrm{f}}}\frac{l_{\mathrm{d}}}{r_{\mathrm{f}}}\eta {\left(1-2\phi \right)}^2+\frac{{\sigma }_{\mathrm{fo}}}{E_{\mathrm{f}}}\left(1-\eta \right)\\ \hspace{1em}\hspace{1em}\hspace{1em} +\frac{2}{\rho E_{\mathrm{f}}}\left[{\sigma }_{\mathrm{mo}}\frac{V_{\mathrm{m}}}{V_{\mathrm{f}}}\frac{r_{\mathrm{f}}}{l_{\mathrm{c}}}-{\tau }_{\mathrm{i}}\eta \left(1-2\eta +2\phi \right)\right]\left[1-\exp \left(-\frac{\rho }{2}\frac{l_{\mathrm{c}}}{r_{\mathrm{f}}}\left(1-\eta \right)\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em} -\left({\alpha }_{\mathrm{c}}-{\alpha }_{\mathrm{f}}\right)\Delta \mathsf{{\rm T}}\end{array}$$

(9)

where

$$\phi =\frac{l_{\mathrm{new}\text{\_}\mathrm{slip}}}{l_{\mathrm{d}}}$$

(10)

When σ > σ_reloading, the reloading composite’s strain can be determined by Equation (11), [28]

$$\begin{array}{l}{\epsilon }_{\mathrm{reloading}}=\frac{\sigma }{V_{\mathrm{f}}{E}_{\mathrm{f}}}\eta -2\frac{{\tau }_{\mathrm{i}}}{E_{\mathrm{f}}}\frac{l_{\mathrm{d}}}{r_{\mathrm{f}}}\eta +2\frac{{\tau }_{\mathrm{i}}}{E_{\mathrm{f}}}\frac{l_{\mathrm{d}}}{r_{\mathrm{f}}}\eta -\frac{{\tau }_{\mathrm{i}}}{E_{\mathrm{f}}}\frac{l_{\mathrm{d}}}{r_{\mathrm{f}}}\eta +\frac{{\sigma }_{\mathrm{fo}}}{E_{\mathrm{f}}}\left(1-\eta \right)\\ \hspace{1em}\hspace{1em}\hspace{1em} +\frac{2}{\rho E_{\mathrm{f}}}\left[{\sigma }_{\mathrm{mo}}\frac{V_{\mathrm{m}}}{V_{\mathrm{f}}}\frac{r_{\mathrm{f}}}{l_{\mathrm{c}}}-{\tau }_{\mathrm{i}}\eta \left(3-2\eta \right)\right]\left[1-\exp \left(-\frac{\rho }{2}\frac{l_{\mathrm{c}}}{r_{\mathrm{f}}}\left(1-\eta \right)\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em} -\left({\alpha }_{\mathrm{c}}-{\alpha }_{\mathrm{f}}\right)\Delta \mathsf{{\rm T}}\end{array}$$

(11)

3. Experimental Comparisons

Experimental monotonic tensile, cyclic loading/unloading hysteresis loops, inverse tangent modulus (ITM), and interface counter slip ratio (ICSR)/interface new slip ratio (INSR) of mini, unidirectional, cross-ply, and 2D plain-woven SiC/SiC composites are predicted using the developed models. Material properties of SiC/SiC composites are listed in

Table 1. Material properties of mini, unidirectional, cross-ply and 2D plain-woven SiC/SiC composites.

Items	Mini	Unidirectional	Cross-Ply	2D Plain-Woven
Vf	0.43 [32]	0.25 [33]	0.25 [34]	0.36 [35]
rf/(μm)	3.5 [32]	7.5 [34]	7.5 [34]	7.5 [35]
Ef/(GPa)	387 [35]	420 [34]	420 [34]	350 [35]
Em/(GPa)	400 [35]	339 [34]	339 [34]	400 [35]
αf/(10−6/K)	2.3 [34]	5.1 [34]	5.1 [34]	4 [29]
αm/(10−6/K)	1.48 [34]	4.38 [34]	4.38 [34]	4.6 [29]
ΔT/(°C)	−1000 [32]	−1400 [34]	−1400 [34]	−1400 [35]
τi/(MPa)	100 [33]	60 [34]	30 [34]	100 [33]
ΓI/(J/m2)	4.0 [33]	5 [29]	2.0 [33]	6.0 [33]

.

3.1. Prediction of Tensile and Fatigue Hysteresis Loops of Mini-SiC/SiC Composite

Sauder et al. [33] performed experimental study on the tensile and cyclic loading/unloading fatigue hysteresis behavior of mini-SiC/SiC composite. The mini-SiC/SiC composite was fabricated using the chemical vapor infiltration (CVI) method. Tyranno^TM SA3 (UBE Industries, Tokyo, Japan) tows coated by a single layer of pyrocarbon (PyC) were used as reinforcement.

Figure 1 shows the experimental and predicted tensile stress-strain curves and matrix fragmentation density versus applied stress curves. Matrix fragmentation starts from approximately σ_mc = 600 MPa, and approaches saturation at approximately σ_sat =1000 MPa, and the saturation fragmentation density is approximately λ_sat = 25/mm. During tensile experiment, acoustic emission was used to monitor the matrix crack evolution, and after the fracture of tensile specimen, the saturation matrix cracking density was obtained [33]. The occurrence of matrix fragmentation causes nonlinear behavior of tensile curve, and the composite tensile strength is approximately σ_UTS = 1116 MPa with the fracture strain ε_f = 0.58%.

Figure 2 shows the experimental and predicted cyclic loading/unloading hysteresis loops, ITM, and ICSR/INSR of mini-SiC/SiC composite without/with considering matrix fragmentation closure under σ_max = 935 MPa. Theoretical predicted hysteresis loops considering closure of matrix fragmentation agreed with experimental data, as shown in Figure 2a.

(a)

Under σ_max = 935 MPa, when the closure of matrix fragmentation is not considered, the unloading ITM increases with unloading stress due to internal damage of interface debonding and counter sliding, i.e., from ITM = 3.1 TPa⁻¹ at the tensile peak stress σ_max = 935 MPa to ITM = 3.48 TPa⁻¹ at the unloading transition stress σ_{tr_unloading} = 427 MPa (i.e., A–B₁ in Figure 2b), and then increases to ITM = 3.73 TPa⁻¹ at the valley stress σ_min = zero MPa (i.e., B₁–C₁ in Figure 2b); the reloading ITM increases with reloading stress due to internal damage of interface debonding and new sliding, i.e., from ITM = 3.1 TPa⁻¹ at the valley stress σ_min = zero MPa to ITM = 3.48 TPa⁻¹ at the reloading transition stress σ_{tr_reloading} = 508 MPa (i.e., A–B₁ in Figure 2c), and increases to ITM = 3.73 TPa⁻¹ at the tensile peak stress σ_max = 935 MPa (i.e., B₁–C₁ in Figure 2c); the unloading ICSR increases with unloading stress due to the increase of interface counter slip length, i.e., from ICSR = zero at the tensile peak stress σ_max = 935 MPa to ICSR = 0.194 at the unloading transition stress σ_{tr_unloading} = 427 MPa (i.e., A–B₁ in Figure 2d), and remains constant till the unloading valley stress σ_min = zero MPa (i.e., B₁–C₁ in Figure 2d); and the reloading INSR increases with reloading stress due to the increase of the interface new slip length, i.e., from INSR = zero at the valley stress σ_min = zero MPa to INSR = 0.194 at the reloading transition stress σ_{tr_reloading} = 508 MPa (i.e., A–B₂ in Figure 2e), and then remains constant till the tensile peak stress σ_max = 935 MPa (i.e., B₂–C in Figure 2e).

(b)

Considering the closure of matrix fragmentation, the unloading ITM increases with unloading stress due to internal damage of interface debonding and counter sliding, i.e., from ITM = 3.1 TPa⁻¹ at the tensile peak stress σ_max = 935 MPa to ITM = 4.15 TPa⁻¹ at the unloading stress σ_unloading = 229 MPa (i.e., A–B₂ in Figure 2b), and then decreases to ITM = 3.97 TPa⁻¹ at the valley stress σ_min = zero MPa (i.e., B₂–C₂ in Figure 2b); the reloading ITM increases with reloading stress due to internal damage of interface debonding and new sliding, i.e., from ITM = 3.1 TPa⁻¹ at the valley stress σ_min = zero MPa to ITM = 3.9 TPa⁻¹ at the reloading stress σ_reloading = 508 MPa (i.e., A–B₂ in Figure 2c), and increases slowly to ITM = 3.97 TPa⁻¹ at the tensile peak stress σ_max = 935 MPa (i.e., B₂–C in Figure 2c); the unloading ICSR increases with unloading stress due to the increase of interface counter slip length, i.e., from ICSR = zero at the tensile peak stress σ_max = 935 MPa to ICSR = 0.143 at the unloading stress σ_unloading = 486 MPa (i.e., A–B₂ in Figure 2d), and decreases to ICSR = zero the valley stress σ_min = zero MPa (i.e., B₂–C₂ in Figure 2d); and the reloading INSR increases with reloading stress due to the increase of the interface new slip length, i.e., from INSR = zero at the valley stress σ_min = zero MPa to INSR = 0.194 at the reloading stress σ_reloading = 589 MPa (i.e., A–B₂ in Figure 2e), and then remains constant till the peak stress σ_max = 935 MPa (i.e., B₂–C in Figure 2e).

Table 2. Hysteresis loops parameters of mini-SiC/SiC composite considering closure of matrix fragmentation.

Items	σmax = 890 MPa	σmax = 935 MPa	σmax = 1078 MPa
η	0.075	0.194	0.38
σtr_unloading/(MPa)	563	427	195
σtr_reloading/(MPa)	327	508	883
Unloading Peak ITM/(TPa−1)	3.58	4.15	5.18
Reloading Peak ITM/(TPa−1)	3.486	3.97	4.87
Unloading Peak ICSR	0.07	0.143	0.17
Reloading Peak INSR	0.075	0.194	0.38

shows the hysteresis loops related parameters of mini-SiC/SiC composite under σ_max = 890, 935, and 1078 MPa. When the tensile peak stress increases, the interface debonding ratio increases from η = 0.075 at σ_max = 890 MPa to η = 0.38 at σ_max = 1078 MPa; the unloading transition stress decreases from σ_{tr_unloading} = 563 MPa at σ_max = 890 MPa to σ_{tr_unloading} = 195 MPa at σ_max = 1078 MPa; the unloading peak ITM increases from ITM = 3.58 to 5.18 TPa⁻¹; the unloading peak ICSR increases from 0.07 at σ_max = 890 MPa to 0.17 at σ_max = 1078 MPa; the reloading transition stress increases from σ_{tr_reloading} = 327 MPa at σ_max = 890 MPa to σ_{tr_unloading} = 883 MPa at σ_max = 1078 MPa; the reloading ITM increases from ITM = 3.486 to 4.87 TPa⁻¹; and the reloading peak INSR increases from 0.075 at σ_max = 890 MPa to 0.38 at σ_max = 1078 MPa.

3.2. Prediction of Fatigue Hysteresis Loops of Unidirectional SiC/SiC Composite

Gordon [34] performed experimental study on cyclic loading/unloading hysteresis behavior of unidirectional SiC/SiC composite. The composite was fabricated using the pre-impregnated (pre-preg) melt-infiltrated (MI) method. The Hi-Nicalon Type S^TM fibers were coated with a boron-nitride (BN) interphase. Experimental and predicted tensile loading/unloading curves, ITM, and ICSR/INSR curves are shown in Figure 3. Theoretical predicted hysteresis loops considering closure of matrix fragmentation agreed with experimental data, as shown in Figure 3a.

(a)

Under σ_max = 415 MPa, when the closure of matrix fragmentation is not considered, the unloading ITM increases with unloading stress due to internal damage of interface debonding and counter sliding, i.e., from ITM = 2.76 TPa⁻¹ at the tensile peak stress σ_max = 415 MPa to ITM = 3.68 TPa⁻¹ at the unloading transition stress σ_{tr_unloading} = 209 MPa (i.e., A–B₁ in Figure 3b), and then increases to ITM = 4.35 TPa⁻¹ at the valley stress σ_min = zero MPa (i.e., B₁–C₁ in Figure 3b); the reloading ITM increases with reloading stress due to internal damage of interface debonding and new sliding, i.e., from ITM = 2.76 TPa⁻¹ at the valley stress σ_min = zero MPa to ITM = 3.68 TPa⁻¹ at the reloading transition stress σ_{tr_reloading} = 206 MPa (i.e., A–B₁ in Figure 3c), and increases to ITM = 4.35 TPa⁻¹ at the tensile peak stress σ_max = 415 MPa (i.e., B₁–C₁ in Figure 3c); the unloading ICSR increases with unloading stress due to the increase of interface counter slip length, i.e., from ICSR = zero at the tensile peak stress σ_max = 415 MPa to ICSR = 0.14 at the unloading transition stress σ_{tr_unloading} = 209 MPa (i.e., A–B₁ in Figure 3d), and remains constant till the valley stress σ_min = zero MPa (i.e., B₁–C₁ in Figure 3d); and the reloading INSR increases with reloading stress due to the increase of the interface new slip length, i.e., from INSR = zero at the valley stress σ_min = zero MPa to INSR = 0.14 at the reloading transition stress σ_{tr_reloading} = 206 MPa (i.e., A–B₁ in Figure 3e), and then remains constant till the tensile peak stress σ_max = 415 MPa (i.e., B₁–C in Figure 3e).

(b)

When the closure of matrix fragmentation is considered, the unloading ITM increases with unloading stress due to internal damage of interface debonding and counter sliding, i.e., from ITM = 2.76 TPa⁻¹ at the peak stress σ_max = 415 MPa to ITM = 5.34 TPa⁻¹ at the unloading stress σ_unloading = 105.8 MPa (i.e., A–B₂ in Figure 3b), and then decreases to ITM = 4.88 TPa⁻¹ at the valley stress σ_min = zero MPa (i.e., B₂–C₂ in Figure 3b); the reloading ITM increases with reloading stress due to internal damage of interface debonding and new sliding, i.e., from ITM = 2.75 TPa⁻¹ at the valley stress σ_min = zero MPa to ITM = 4.53 TPa⁻¹ at the reloading stress σ_reloading = 206 MPa (i.e., A–B₂ in Figure 3c), and increases slowly to ITM = 4.88 TPa⁻¹ at the tensile peak stress σ_max = 415 MPa (i.e., B₂–C₂ in Figure 3c); the unloading ICSR increases with unloading stress due to the increase of the interface counter slip length, i.e., from ICSR = zero at the tensile peak stress σ_max = 415 MPa to ICSR = 0.11 at the unloading stress σ_unloading = 217.8 MPa (i.e., A–B₂ in Figure 3d), and decreases to ICSR = zero the valley stress σ_min = zero MPa (i.e., B₂–C₂ in Figure 3d); and the reloading INSR increases with reloading stress due to the increase of the interface new slip length, i.e., from INSR = zero at the valley stress σ_min = zero MPa to INSR = 0.14 at the reloading stress σ_reloading = 236 MPa (i.e., A–B₂ in Figure 3e), and then remains constant till the tensile peak stress σ_max = 415 MPa (i.e., B₂–C in Figure 3e).

Table 3. Hysteresis loops parameters of unidirectional SiC/SiC composite considering closure of matrix fragmentation.

Items	σmax = 415 MPa	σmax = 449 MPa
η	0.14	0.295
σtr_unloading/(MPa)	209	170
σtr_reloading/(MPa)	206	279
Unloading Peak ITM/(TPa−1)	5.34	7.21
Reloading Peak ITM/(TPa−1)	4.88	6.76
Unloading Peak ICSR	0.11	0.188
Reloading Peak INSR	0.14	0.295

shows the hysteresis loops related parameters of unidirectional SiC/SiC composite under σ_max = 415 and 449 MPa. When the peak stress increases from σ_max = 415 to 449 MPa, the interface debonding ratio increases from η = 0.14 to 0.295; the unloading transition stress decreases from σ_{tr_unloading} = 209 to 170 MPa; the unloading ITM increases from ITM = 5.34 to 7.21 TPa⁻¹; the unloading peak ICSR increases from 0.11 to 0.1888; the reloading transition stress increases from σ_{tr_reloading} = 206 to 279 MPa; the reloading peak INSR increases from 0.14 to 0.295; and the reloading ITM increases from ITM = 4.88 to 6.76 TPa⁻¹.

3.3. Prediction of Fatigue Hysteresis Loops of Cross-Ply SiC/SiC Composite

Gordon [34] performed experimental study on the cyclic loading/unloading hysteresis behavior of cross-ply [0/90]_2s SiC/SiC composite. The composite was fabricated using the pre-impregnated (pre-preg) melt-infiltrated (MI) method. The Hi-Nicalon Type S^TM fibers were coated with a boron-nitride (BN) interphase. Experimental and predicted tensile loading/unloading hysteresis loops, ITM, and ICSR/INSR are shown in Figure 4. Theoretical predicted hysteresis loops considering closure of matrix fragmentation agreed with experimental data, as shown in Figure 4a.

(a)

Under σ_max = 189 MPa, when the closure of matrix fragmentation is not considered, the unloading ITM increases with unloading stress due to internal damage of interface debonding and counter sliding, i.e., from ITM = 4.01 TPa⁻¹ at the tensile peak stress σ_max = 189 MPa to ITM = 5.7 TPa⁻¹ at the transition stress σ_{tr_unloading} = 52 MPa (i.e., A–B₁ in Figure 4b), and then increases to ITM = 6.29 TPa⁻¹ at the valley stress σ_min = zero MPa (i.e., B₁–C₁ in Figure 4b); the reloading ITM increases with reloading stress due to internal damage of interface debonding and new sliding, i.e., from ITM = 4.01 TPa⁻¹ at the valley stress σ_min = zero MPa to ITM = 5.7 TPa⁻¹ at the reloading transition stress σ_{tr_reloading} = 137 MPa (i.e., A–B₁ in Figure 4c), and increases to ITM = 6.29 TPa⁻¹ at the tensile peak stress σ_max = 189 MPa (i.e., B₁–C₁ in Figure 4c); the unloading ICSR increases with unloading stress due to the increase of the interface counter slip length, i.e., from ICSR = zero at the tensile peak stress σ_max = 189 MPa to ICSR = 0.214 at the unloading transition stress σ_{tr_unloading} = 52 MPa (i.e., A–B₁ in Figure 4d), and remains constant till the valley stress σ_min = zero MPa (i.e., B₁–C₁ in Figure 4d); and the reloading INSR increases with reloading stress due to the increase of the interface new slip length, i.e., from INSR = zero at the valley stress σ_min = zero MPa to INSR = 0.214 at the transition stress σ_{tr_reloading} = 137 MPa (i.e., A–B₁ in Figure 4e), and then remains constant till the peak stress σ_max = 189 MPa (i.e., B₁–C in Figure 4e).

(b)

When the closure of matrix fragmentation is considered, the unloading ITM increases with unloading stress due to internal damage of interface debonding and counter sliding, i.e., from ITM = 4.01 TPa⁻¹ at the tensile peak stress σ_max = 189 MPa to ITM = 8.67 TPa⁻¹ at the unloading stress σ_unloading = 58.6 MPa (i.e., A–B₂ in Figure 4b), and then decreases to ITM = 7.69 TPa⁻¹ at the valley stress σ_min = zero MPa (i.e., B₂–C₂ in Figure 4b); the reloading ITM increases with reloading stress due to internal damage of interface debonding and new sliding, i.e., from ITM = 4 TPa⁻¹ at the valley stress σ_min = zero MPa to ITM = 7.52 TPa⁻¹ at the reloading stress σ_reloading = 137 MPa (i.e., A–B₂ in Figure 4c), and increases slowly to ITM = 7.69 TPa⁻¹ at the tensile peak stress σ_max = 189 MPa (i.e., B₂–C₂ in Figure 4c); the unloading ICSR increases with unloading stress due to the increase of the interface counter slip length, i.e., from ICSR = zero at the peak stress σ_max = 189 MPa to ICSR = 0.09 at the unloading stress σ_unloading = 110 MPa (i.e., A–B₂ in Figure 4d), and decreases to ICSR = zero the valley stress σ_min = zero MPa (i.e., B₂–C₂ in Figure 4d); and the reloading INSR increases with reloading stress due to the increase of the interface new slip length, i.e., from INSR = zero at the valley stress σ_min = zero MPa to INSR = 0.214 at the reloading stress σ_reloading = 164 MPa (i.e., A–B₂ in Figure 4e), and then remains constant till the peak stress σ_max = 189 MPa (i.e., B₂–C in Figure 4e).

Table 4. Hysteresis loops parameters of cross-ply SiC/SiC composite considering closure of matrix fragmentation.

Items	σmax = 189 MPa	σmax = 206 MPa
η	0.214	0.35
σtr_unloading/(MPa)	52	34
σtr_reloading/(MPa)	137	172
Unloading Peak ITM/(TPa−1)	8.67	10.32
Reloading Peak ITM/(TPa−1)	7.69	9.68
Unloading Peak ICSR	0.09	0.137
Reloading Peak INSR	0.214	0.35

shows the hysteresis loops related parameters of cross-ply SiC/SiC composite under σ_max = 189 and 206 MPa. When the peak stress increases from σ_max = 189 to 206 MPa, the interface debonding ratio increases from η = 0.214 to 0.35; the unloading transition stress decreases from σ_{tr_unlooading} = 52 to 34 MPa; the unloading ITM increases from ITM = 8.67 to 10.32 TPa⁻¹; the unloading peak ICSR increases from 0.09 to 0.137; the reloading transition stress increases from σ_{tr_relooading} = 137 to 172 MPa; the reloading ITM increases from ITM = 7.69 to 9.68 TPa⁻¹; and the reloading peak INSR increases from 0.214 to 0.35.

3.4. Prediction of Tensile and Fatigue Hysteresis Loops of 2D Plain-Woven SiC/SiC Composite

Smith [35] performed experimental study on the tensile and cyclic loading/unloading hysteresis behavior of 2D plain-woven SiC/SiC composite at room temperature. The composite was fabricated using the CVI+MI method at temperature near 1400 °C. The Sylramic^TM fibers were coated with a Boron Nitride (BN) interphase.

Figure 5 shows experimental and predicted tensile stress-strain curves. The composite exhibited obvious nonlinear behavior and fractured at approximately σ_UTS = 430 MPa with the failure strain of ε_f = 0.36%.

Figure 6 show the experimental and predicted stress-strain hysteresis loops, ITM, and ICSR/INSR versus stress curves under σ_max = 360 MPa. Theoretical predicted hysteresis loops considering closure of matrix fragmentation agreed with experimental data, as shown in Figure 6a.

(a)

Under σ_max = 360 MPa, when the closure of matrix fragmentation is not considered, the unloading ITM increases with unloading stress due to internal damage of interface debonding and counter sliding, i.e., from ITM = 4.59 TPa⁻¹ at the peak stress σ_max = 360 MPa to ITM = 5.63 TPa⁻¹ at the unloading transition stress σ_{tr_unloading} = 91 MPa (i.e., A–B₁ in Figure 6b), and then increases to ITM = 6.04 TPa⁻¹ at the valley stress σ_min = zero MPa (i.e., B₁–C₁ in Figure 6b); the reloading ITM increases with reloading stress due to internal damage of interface debonding and new sliding, i.e., from ITM = 4.59 TPa⁻¹ at the valley stress σ_min = zero MPa to ITM = 5.63 TPa⁻¹ at the reloading transition stress σ_{tr_reloading} = 269 MPa (i.e., A–B₁ in Figure 6c), and increases to ITM = 6.04 TPa⁻¹ at the tensile peak stress σ_max = 360 MPa (i.e., B₁–C₁ in Figure 6c); the unloading ICSR increases with unloading stress due to the increase of the interface counter slip length, i.e., from ICSR = zero at the tensile peak stress σ_max = 360 MPa to ICSR = 0.15 at the unloading transition stress σ_{tr_unloading} = 91 MPa (i.e., A–B₁ in Figure 6d), and remains constant till the valley stress σ_min = zero MPa (i.e., B₁–C₁ in Figure 6d); and the reloading INSR increases with reloading stress due to the increase of the interface new slip length, i.e., from INSR = zero at the valley stress σ_min = zero MPa to INSR = 0.15 at the reloading transition stress σ_{tr_reloading} = 269 MPa (i.e., A–B₁ in Figure 6e), and then remains constant till the tensile peak stress σ_max = 360 MPa (i.e., B₁–C in Figure 6e).

(b)

When the closure of matrix fragmentation is considered, the unloading ITM increases with unloading stress due to internal damage of interface debonding and counter sliding, i.e., from ITM = 4.63 TPa⁻¹ at the tensile peak stress σ_max = 360 MPa to ITM = 7.08 TPa⁻¹ at the unloading stress σ_unloading = 54 MPa (i.e., A–B₂ in Figure 6b), and then decreases to ITM = 6.87 TPa⁻¹ at the valley stress σ_min = zero MPa (i.e., B₂–C₂ in Figure 6b); the reloading ITM increases with reloading stress due to internal damage of interface debonding and new sliding, i.e., from ITM = 4.58 TPa⁻¹ at the valley stress σ_min = zero MPa to ITM = 6.67 TPa⁻¹ at the reloading stress σ_reloading = 269 MPa (i.e., A–B₂ in Figure 6c), and increases slowly to ITM = 6.88 TPa⁻¹ at the tensile peak stress σ_max = 360 MPa (i.e., B₂–C₂ in Figure 6c); the unloading ICSR increases with unloading stress due to the increase of the interface counter slip length, i.e., from ICSR = zero at the peak stress σ_max = 360 MPa to ICSR = 0.082 at the unloading stress σ_unloading = 169 MPa (i.e., A–B₂ in Figure 6d), and decreases to ICSR = zero the valley stress σ_min = zero MPa (i.e., B₂–C₂ in Figure 6d); and the reloading INSR increases with reloading stress due to the increase of the interface new slip length, i.e., from INSR = zero at the valley stress σ_min = zero MPa to INSR = 0.15 at the reloading stress σ_reloading = 307 MPa (i.e., A–B₂ in Figure 6e), and then remains constant till the peak stress σ_max = 360 MPa (i.e., B₂–C in Figure 6e).

Table 5. Hysteresis loops parameters of 2D plain-woven SiC/SiC composite considering closure of matrix fragmentation.

Items	σmax = 312 MPa	σmax = 360 MPa	σmax = 410 MPa
η	0.084	0.15	0.23
σtr_unloading/(MPa)	138	91	41
σtr_reloading/(MPa)	174	269	369
Unloading Peak ITM/(TPa−1)	6.32	7.08	8.01
Reloading Peak ITM/(TPa−1)	6.12	6.87	7.73
Unloading Peak ICSR	0.054	0.08	0.098
Reloading Peak INSR	0.084	0.15	0.23

shows the hysteresis loops parameters under σ_max = 312, 360, and 410 MPa. When the tensile peak stress increases from σ_max = 312 to 410 MPa, the interface debonding ratio increases from η = 0.084 to 0.23; the unloading transition stress decreases from σ_{tr_unlooading} = 138 to 41 MPa; the unloading ITM increases from ITM = 6.32 to 8.01 TPa⁻¹; the unloading peak ICSR increases from 0.054 to 0.098; the reloading transition stress increases from σ_{tr_relooading} = 174 to 369 MPa; the reloading ITM increases from ITM = 6.12 to 7.73 TPa⁻¹; and the reloading peak INSR increases from 0.084 to 0.23.

3.5. Comparative Analysis

Under tensile and fatigue loading, composite’s internal damage evolution depends on the fiber’s preform and volume fraction. For minicomposite, unidirectional, cross-ply and 2D plain-woven SiC/SiC composites with different fiber volume fraction, it can be found that the composite’s unloading and reloading peak ITMs decrease with increasing fiber’s volume faction.

▪

For mini-SiC/SiC composite with the fiber’ volume of V_f = 0.43, when the tensile peak stress increases from σ_max = 890 to 1078 MPa, the unloading peak ITM increases from ITM = 3.58 to 5.18 TPa⁻¹; the reloading ITM increases from ITM = 3.486 to 4.87 TPa⁻¹.

▪

For cross-ply SiC/SiC composite with fiber’s volume of V_f = 0.25, when the peak stress increases from σ_max = 189 to 206 MPa, the unloading ITM increases from ITM = 8.67 to 10.32 TPa⁻¹; the reloading ITM increases from ITM = 7.69 to 9.68 TPa⁻¹.

The composite’s interface debonding and slip parameters, i.e., the interface debonding ratio (η), the interface counter slip ratio (ICSR) and interface new slip ratio (INSR) decrease with increasing fiber’s volume fraction.

▪

For mini-SiC/SiC composite with the fiber’ volume of V_f = 0.43, when the tensile peak stress increases from σ_max = 890 to 1078 MPa, the interface debonding ratio increases from η = 0.075 to 0.38; the unloading peak ICSR increases from 0.07 to 0.17; and the reloading peak INSR increases from 0.075 to 0.38.

▪

For unidirectional SiC/SiC composite with fiber’s volume of V_f = 0.25, when the peak stress increases from σ_max = 415 to 449 MPa, the interface debonding ratio increases from η = 0.14 to 0.295; the unloading peak ICSR increases from 0.11 to 0.1888; and the reloading peak INSR increases from 0.14 to 0.295.

4. Summary and Conclusions

In this paper, micromechanical constitutive models were developed to analyze the effects of matrix fragmentation and closure on tensile and fatigue behavior of fiber-reinforced CMCs. Relationships between composite’s tensile and fatigue response, matrix fragmentation and closure and related interface debonding and fiber’s failure were established. Experimental matrix fragmentation density, tensile curves, and fatigue hysteresis loops of mini, unidirectional, cross-ply, and 2D plain-woven SiC/SiC composites are predicted.

(1)

The Matrix fragmentation density changes with increasing or decreasing tensile stress and affects the tensile nonlinear strain and the interface debonding ratio.

(2)

The closure of matrix fragmentation affects the fatigue hysteresis loops. Upon unloading, the inverse tangent modulus increases to the peak value, and then decreases with unloading stress, and the interface counter slip ratio increases to the peak value, and then decreases to zero with unloading stress; upon reloading, the inverse tangent modulus increases with reloading stress, and the interface new slip ratio increases slowly during initial stage of reloading, then increases to the peak value, and remains constant till peak stress.

(3)

Theoretical predicted matrix fragmentation density, tensile nonlinear curves, and the fatigue hysteresis loops agreed with experimental data.

There is inherent variation across samples tested in CMCs, however, the paper focused on the micromechanical constitutive models for tensile and fatigue behavior of CMCs without considering the inherent variation for different tested samples. The author will consider the inherent variation for different samples tested in CMCs in the further study.