# Conditions for matrix crack deflection at an interface in ceramic matrix composites

Eric Martin, Piet W.M. Peters, Dominique Leguillon, Jean-Michel Quenisset

► To cite this version:

Eric Martin, Piet W.M. Peters, Dominique Leguillon, Jean-Michel Quenisset. Conditions for matrix
crack deflection at an interface in ceramic matrix composites. Materials Science and Engineering: A,
1998, 250 (2), pp.291-302. 10.1016/S0921-5093(98)00604-2. hal-05527216

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# Conditions for matrix crack deflection at an interface in ceramic matrix composites

E. Martin<sup>a,*</sup>, P.W.M. Peters<sup>b</sup>, D. Leguillon<sup>c</sup>, J.M. Quenisset<sup>a</sup>

This paper describes a numerical approach developed to simulate the mechanism of matrix crack deflection at the fibre/matrix interface in brittle matrix composites. For this purpose, the fracture behaviour of a unit cell (microcomposite) consisting of a single fibre surrounded by a cylindrical tube of matrix was studied with the help of a finite element model. A fracture mechanics approach was used to design a criterion for deflection at the fibre/matrix interface of an annular crack present in the matrix. The analysis of the fracture behaviour of SiC/SiC and SiC/glass ceramics microcomposites shows that the introduction of a low modulus and low toughness interfacial layer at the fibre/matrix interface (e.g. a carbon coating) greatly favours matrix crack deflection at the interphase/fibre interface.

Keywords: Ceramic matrix composites; Crack deflection

## 1. Introduction

Fibre reinforced ceramic matrix composites (CMCs) are promising candidates for structural applications in aerospace where high mechanical performance is required in combination with high temperature resistance [1]. The main objective of the efforts concerning the development of these composite systems has been to improve toughness and strain at failure. Examination of the failure process at the microstructural level indicates that the first step is matrix cracking [2]. When a crack further propagates through the matrix and intercepts the fibre, two competing effects have to be considered. On the one hand, the matrix crack can propagate through the fibre and trigger a brittle failure of the composite. On the other hand, the matrix crack can be deflected along the fibre/matrix (F/M) interface and continue to propagate along it in a benign manner. Moreover, matrix crack deflection at the F/M interface is a prerequisite for the activation of various toughening mechanisms like fibre bridging and frictional fibre pull-out [3]. For instance, a thin layer (interphase) of a

soft material introduced at the F/M interface has been shown to promote matrix crack deflection [4]. Optimal design of high toughness CMCs cannot be efficiently achieved without a clear understanding of the role of the various interfaces and interphases constituting the interfacial zone on this mechanism of crack deflection.

Several investigators have studied the configuration of a crack lying perpendicular to and penetrating into a plane interface [5,6]. The competition between the deflection and the penetration mode is analysed by comparing the energy release rates related to each crack path. Upon an incremental loading, the condition for crack deflection can be expressed as follows:

$$ \frac{\mathcal{G}_1^c}{\mathcal{G}_2^c} < \frac{\mathcal{G}_d}{\mathcal{G}_p} \qquad (1) $$

where $\mathcal{G}_1^c$ and $\mathcal{G}_2^c$ are the toughness of the interface and the toughness of the penetrated material, respectively, also $\mathcal{G}_d$ and $\mathcal{G}_p$ are the energy release rate for deflection and penetration, respectively. Numerical methods are required to evaluate the ratio $\mathcal{G}_d/\mathcal{G}_p$ which only depends on the elastic constants of the bimaterial and is modified by the presence of residual stresses [7,8]. Nevertheless, it must be pointed out that the previous studies do not account for the extension and propaga-

<footnote><sup>a</sup> Laboratoire de Génie Mécanique, IUT A, Université de Bordeaux, F-33405 Talence, France</footnote>

<footnote><sup>b</sup> Institute for Materials Research, DLR, D-51140 Köln, Germany</footnote>

<footnote><sup>c</sup> Laboratoire de Modélisation en Mécanique, CNRS URA 229, Université Pierre et Marie Curie, Tour 66, 4 place Jussieu, F-75252 Paris Cedex 05, France</footnote>

<footnote>* Corresponding author.</footnote>
tion of the matrix crack and thus neglect any toughness mismatch between the bimaterial components.

The approach presented in this paper analyses the behaviour of a composite cylinder assemblage (micro-composite) when an annular matrix crack develops during the loading. The criterion for crack deflection at the F/M interface is based on the evaluation of differences between the energy release rates related to alternative crack paths. Applying this micro-mechanical model to analyse the fracture behaviour of SiC/SiC and SiC/glass ceramic microcomposites shows the influence of the presence of a carbon interphase at the F/M interface.

## 2. Microcomposite model

The geometry of the microcomposite is schematically represented in Fig. 1(a). It consists of a single fibre (Young's modulus $E_f$ and Poisson's ratio $v_f$) of radius $R_f$ and infinite length surrounded by a concentric cylinder of matrix (Young's modulus $E_m$ and Poisson's ratio $v_m$) whose inner and outer radius are $R_f$ and $R_f/\sqrt{V_f}$, respectively, where $V_f$ is the fibre volume fraction. An annular matrix crack is introduced in the plane $z=0$ as depicted in Fig. 1(b). Upon loading two modes of crack extension can be induced by the presence of the F/M interface: fibre cracking (Fig. 1(c)) or interface cracking (Fig. 1(d)). The mode of deformation is axisymmetric so that the non-zero stress and displacement compo-

nents $\sigma_{rr}$, $\sigma_{\theta\theta}$, $\sigma_{zz}$, $\sigma_{rz}$, $u_r$ and $u_z$ only depend on $r$ and $z$. Two boundary conditions are considered on the outer cylindrical surface. Assuming a stress free state (Type I condition) is appropriate to model the behaviour of coated fibres. Forcing the radial displacement of the outer surface to be equal to its value far above the matrix crack everywhere (Type II condition) accounts for the lateral restraint from the surrounding material that may prevail in internal parts of a composite. Residual stresses arise from thermal expansion mismatch between fibre and matrix as the microcomposite is cooled down after processing ($\Delta T$ denotes the difference between the operating and the manufacturing temperatures). Each constituent of the microcomposite is assumed to behave like a linear elastic and brittle material.

The considerable difficulty involved in the analytical description of the failure behaviour of the microcomposite leads to a preference for the use of a numerical method of analysis. The finite element library MODULEF is used as the analysis tool and the numerical scheme is described in Appendix A [9].

## 3. Fracture mechanics approach

The fracture behaviour of the microcomposite is described with the help of the principles of fracture mechanics. The propagation of the matrix crack of length $a$ in a brittle solid becomes possible when the

<figure><img src="image_2.png" /><figcaption>Fig. 1. The geometry of the microcomposite.</figcaption></figure>
energy release rate $\mathcal{G}_m$ equals or exceeds the critical value $\mathcal{G}_m^c$ as indicated by the following expression:

$$ \mathcal{G}_m(a, \sigma) \geq \mathcal{G}_m^c \qquad (2) $$

where $\sigma$ is the loading stress applied on the cracked microcomposite. In the absence of residual stresses, the energy release rate in the matrix can be expressed as follows:

$$ \mathcal{G}_m(a, \varepsilon) = A_m(a)\varepsilon^2 E_c^2 \qquad (3) $$

where $A_m(a)$ depends on geometric and material parameters, $E_c$ is the longitudinal Young's modulus of the non-damaged cell and $\varepsilon$ is the axial applied strain.

In the vicinity of the F/M interface ($a$ close to $a_i$), the parameter $A_m(a)$ can be written as [10]:

$$ A_m(a) = K_m \left( \frac{\Delta a}{a} \right)^{2\lambda - 1} \qquad (4) $$

where $K_m$ is a stress intensity factor which is only related to geometric parameters, $\Delta a$ is the distance of the crack tip to the interface, $\lambda$ is the displacement singularity exponent when the crack tip is at the interface.

The value of $\lambda$ differs from 1/2 and only depends on Young's modulus and Poisson's ratio of the components of the bimaterial [11]. The discontinuous change in the order of the singularity as $\Delta a$ becomes equal to zero implies that the energy release rate is either infinite or zero in the case of a strong ($\lambda < 1/2$) or a weak ($\lambda > 1/2$) singularity. The crack is thus predicted to reach the interface at infinite load or at zero load. To avoid this non-satisfactory result, a characteristic length $\delta_c$ is introduced to define the conditions of propagation of a crack in the neighbourhood of an interface. It is assumed here that the characteristic length $\delta_c$ is independent of the considered interface. Eq. (2) must then be written as:

$$ \mathcal{G}_m(a, \delta_c, \varepsilon) \geq \mathcal{G}_m^c. \qquad (5) $$

The matrix crack will reach the F/M interface at length $a_i$ (Fig. 2(a)) if the following condition is satisfied:

$$ \mathcal{G}_m(a_i, \delta_c, \varepsilon) = A_m(a_i - \delta_c)\varepsilon^2 E_c^2 \geq \mathcal{G}_m^c \qquad (6) $$

Then, penetration (Fig. 2(b)) or deflection (Fig. 2(c)), respectively, at the F/M interface will be predicted if:

$$ \begin{aligned} \mathcal{G}_f(a_i, \delta_c, \varepsilon) &= A_f(a_i + \delta_c)\varepsilon^2 E_c^2 \geq \mathcal{G}_f^c \\ \mathcal{G}_i(a_i, \delta_c, \varepsilon) &= A_i(a_i + \delta_c)\varepsilon^2 E_c^2 \geq \mathcal{G}_i^c \end{aligned} \qquad (7) $$

where $\mathcal{G}_f^c$ and $\mathcal{G}_i^c$ are the fibre and interface toughness.

The numerical scheme described in Appendix A is used to compute the values of the normalised energy release rate $A$. They are represented in Fig. 3 as a function of the crack length for different values of the ratio of the matrix modulus over the fibre modulus when the matrix crack propagates normally to the F/M

<figure><img src="image_3.png" /><figcaption>Fig. 2. Definition of the conditions of propagation of the matrix crack near the F/M interface making use of the characteristic length $\delta_c$: (a) the matrix crack reaches the F/M interface (b) the matrix crack penetrates the fibre (c) the matrix crack is deflected at the F/M interface.</figcaption></figure>

interface. In this computation Poisson's ratio mismatch is ignored ($v_f = v_m = 0.2$).

### 3.1. Propagation of the matrix crack in the absence of residual stresses

Using Eq. (3), the critical strain $\varepsilon^i$ for extension of the matrix crack of initial length $a_0$ ($a_0 < a_i$) is given by:

$$ \varepsilon^i = \frac{1}{E_c} \left( \frac{\mathcal{G}_m^c}{A_m(a_0)} \right)^{1/2} \qquad (8) $$

<figure><img src="image_4.png" /><figcaption>Fig. 3. Variation of the normalised energy release rate $A$ (a) versus the crack length when the matrix crack propagates normally to the F/M interface in the absence of residual stresses ($V_f = 0.4$, Type I boundary conditions, $v_f = v_m = 0.2$).</figcaption></figure>
After extension, the condition of matrix crack growth (stable or unstable) is expressed by the value of $(\partial A_m/\partial a)_a = a_0$. As shown by Fig. 3, $A_m(a)$ is an increasing function when $E_m \ge E_f$. In this case, the propagation of the matrix crack toward the interface is unstable. When the matrix modulus is smaller than the fibre one, $A_m(a)$ decreases near the interface and a more stable propagation can be expected. In this case the local maximum of $A_m(a)$ allows for the definition of length $a_{min}$ by:

$$A_m(a_{min}) = A_m(a_i - \delta_c) \quad \text{with } a_{min} < a_i - \delta_c. \qquad (9)$$

When the length of the initial matrix crack exceeds $a_{min}$, the F/M interface will not be reached unless the applied load is increased so that:

$$A_m(a_i - \delta_c)(\varepsilon^c)^2 E_c^2 = A_m(a_{min})(\varepsilon^c)^2 E_c^2 = \mathcal{G}_m^c. \qquad (10)$$

Finally, the critical strain $\varepsilon^c$ which leads to the extension and the propagation of the matrix crack up to the F/M interface can be defined by:

$$\varepsilon^c = \frac{1}{E_c} \left( \frac{\mathcal{G}_m^c}{A_m^*(a_0)} \right)^{\frac{1}{2}}$$
with $A_m^*(a_0) = A_m(a_0)$ if $E_m \ge E_f$ \\
$A_m^*(a_0) = A_m(a_0)$ if $E_m < E_f$ and $a_0 < a_{min}$ \\
$A_m^*(a_0) = A_m(a_{min})$ if $E_m < E_f$ and $a_0 \ge a_{min}$ \\
(11)

## 3.2. Conditions for deflection of the matrix crack at the F/M interface in the absence of residual stresses

Eqs. (7) and (11) allow the evaluation of the corresponding energy release rates for the two competing crack paths at the F/M interface. The energy release rate for penetration is given by:

$$\mathcal{G}_f(a_i, \delta_c, \varepsilon^c) = A_f(a_i + \delta_c)(E_c \varepsilon^c)^2 = \frac{A_f(a_i + \delta_c)}{A_m^*(a_0)} \mathcal{G}_m^c \qquad (12)$$

while the energy release rate for deflection is given by:

$$\mathcal{G}_i(a_i, \delta_c, \varepsilon^c) = A_i(a_i + \delta_c)(E_c \varepsilon^c)^2 = \frac{A_i(a_i + \delta_c)}{A_m^*(a_0)} \mathcal{G}_m^c \qquad (13)$$

The favoured path orientation at the F/M interface is assumed to be that which maximizes the decrease in total system free energy [12]. This implies the comparison of the parameters $\Delta_f(a_i, \delta_c, \varepsilon^c)$ and $\Delta_i(a_i, \delta_c, \varepsilon^c)$ which give the rate of energy decrease of the system after penetration and deflection at the interface:

$$\Delta_f(a_i, \delta_c, \varepsilon^c) = \mathcal{G}_f(a_i, \delta_c, \varepsilon^c) - \mathcal{G}_f^c \qquad (14)$$

$$\Delta_i(a_i, \delta_c, \varepsilon^c) = \mathcal{G}_i(a_i, \delta_c, \varepsilon^c) - \mathcal{G}_i^c. \qquad (15)$$

The condition for crack deflection at the interface can be written as:

$$\Delta_i(a_i, \delta_c, \varepsilon^c) > \Delta_f(a_i, \delta_c, \varepsilon^c). \qquad (16)$$

Whether $\Delta_f \ge 0$ and then Eq. (16) must hold or $\Delta_f < 0$ and then $\Delta_i \ge 0$ is sufficient leads to the following conditions for crack deflection at the F/M interface in absence of residual stresses:

$$\frac{\mathcal{G}_f^c}{\mathcal{G}_m^c} \le \frac{A_f(a_i + \delta_c)}{A_m^*(a_0)}, \\
\frac{\mathcal{G}_i^c}{\mathcal{G}_m^c} \le \frac{A_i(a_i + \delta_c)}{A_m^*(a_0)} - \left[ \frac{A_f(a_i + \delta_c)}{A_m^*(a_0)} - \frac{\mathcal{G}_f^c}{\mathcal{G}_m^c} \right] \qquad (17)$$

$$\frac{\mathcal{G}_f^c}{\mathcal{G}_m^c} > \frac{A_f(a_i + \delta_c)}{A_m^*(a_0)}, \qquad \frac{\mathcal{G}_i^c}{\mathcal{G}_m^c} \le \frac{A_i(a_i + \delta_c)}{A_m^*(a_0)}. \qquad (18)$$

It must be noted that the case of the incremental loading of a crack touching the interface is obtained when crack propagation is first prohibited at the F/M interface ($\Delta_f(a_i, \delta_c, \varepsilon^c) < 0$ and $\Delta_i(a_i, \delta_c, \varepsilon^c) < 0$) so that the load must be increased to propagate the crack. Deflection at the F/M interface is predicted if $\Delta_f(a_i, \delta_c, \varepsilon^c) < 0$ and $\Delta_i(a_i, \delta_c, \varepsilon^c) = 0$. The related condition is:

$$\frac{\mathcal{G}_i^c}{\mathcal{G}_f^c} \le \frac{A_i(a_i + \delta_c)}{A_f(a_i + \delta_c)} \qquad (19)$$

which is similar to Eq. (1) and can be deduced by combining the two expressions in Eq. (18).

A direct consequence of Eq. (17) is that crack deflection is prohibited if the following condition is fulfilled:

$$\frac{\mathcal{G}_f^c}{\mathcal{G}_m^c} \le \frac{A_f(a_i + \delta_c) - A_i(a_i + \delta_c)}{A_m^*(a_0)}. \qquad (20)$$

Eq. (20) means that the fracture toughness of the fiber is so much lower than the fracture toughness of the matrix that the competition between the two modes of cracking at the F/M interface will inevitably favour the penetration of the fiber. Fig. 4(a) makes clear that Eq. (20) is enhanced when the modulus of the matrix is greater than the modulus of the fiber and when the initial crack length is small.

In contrast, Eq. (18) shows that crack deflection is promoted if the fracture toughness of the fiber is greater than that of the matrix and if the interfacial toughness is small enough. Fig. 4(b) illustrates the relevant conditions when the components of the micro-composite exhibit identical moduli.

The set of Eqs. (17) and (18) can also be used to determine the influence of the presence of an interphase (Young's modulus $E_p$, Poisson's ratio $v_p$ and thickness $e_p$) on the conditions for matrix crack deflection. As illustrated by Fig. 5(a), it is assumed that an interphase is introduced between the fibre and the matrix which are taken to be identical materials. The toughness of each interface (matrix/interphase and interphase/fibre) is also assumed to be identical. Fig. 5(b) shows the influence of the presence of the interphase on the values of the normalised energy release rate $A$. As already
<figure><img src="image_5.png" /><figcaption>Fig. 4. Conditions for matrix crack propagation at the F/M interface in absence of residual stresses ($V_f = 0.4$, Type I boundary conditions, $v_f = v_m = 0.2$). (a) When the fibre toughness is lower than the matrix toughness crack deflection is prohibited if the ratio is inferior to the plotted ratio as required by Eq. (20). (b) When the fibre toughness is greater than the matrix toughness crack deflection is predicted if the plotted conditions are fulfilled as required by Eq. (18).</figcaption></figure>

observed, this parameter increases (decreases) when the crack approaches an interface with a more rigid material (with a more compliant material). If the interphase toughness ($\mathcal{G}_p^c$) is greater than that of the matrix ($\mathcal{G}^c$), Eq. (18) with $a_i = a_{i1}$ provides the required conditions for crack deflection at the first interface between the matrix and interphase. For the opposite case ($\mathcal{G}_p^c < \mathcal{G}^c$) Eq. (17) shows that the matrix crack is predicted to penetrate the interphase. The crack will reach the second interface ($a_i = a_{i2}$) if the interphase toughness is small enough as required by the following condition:

$$ \frac{\mathcal{G}_p^c}{\mathcal{G}^c} < \frac{A_p(a_{i2} - \delta_c)}{A_m^*(a_0)} \qquad (21) $$

Crack deflection is then predicted if Eq. (18) is fulfilled which implies that:

$$ \frac{A_f(a_{i2} + \delta_c)}{A_m^*(a_0)} < 1 \quad \text{and} \quad \frac{\mathcal{G}_i^c}{\mathcal{G}^c} < \frac{A_i(a_{i2} + \delta_c)}{A_m^*(a_0)} \qquad (22) $$

Eqs. (21) and (22) are plotted versus the normalised initial crack length in Fig. 6 when the value of the ratio $E_p/E$ is 0.1. They demonstrate that the insertion of an interphase with a low modulus and a low toughness is able

to promote matrix crack deflection at the interphase/fibre interface.

### 3.3. Influence of residual stresses

When the cracked microcomposite is submitted to a uniform cooling ($\Delta T < 0$) as a consequence of the manufacturing process, the energy release rate for the matrix cracking is [13,14]:

$$ \mathcal{G}_m(a) = E_c^2(\sqrt{A_m(a)}\epsilon - \sqrt{B_m(a)}\alpha_c\Delta T)^2 \qquad (23) $$

where $\alpha_c$ is the longitudinal thermal expansion coefficient of the non-damaged microcomposite. The parameter $B_m(a)$ depends on the thermoelastic coefficients of the components of the microcomposite and is evaluated with the help of the numerical scheme described in Appendix

<figure><img src="image_6.png" /><figcaption>Fig. 5. (a) The geometry of the microcomposite after the interposition of an interphase between the fibre and the matrix. (b) Influence of an interphase on the variation of the normalised energy release rate versus the crack length when the crack propagates normally to the F/M interface. Residual stresses are not taken into account and $e_p/R_f = 6.7\%$, $E_f = E_m$. $A_h(a)$ is the normalized energy release rate in absence of the interphase (the matrix and the fibre are assumed to be identical materials, $V_f = 0.4$, Type I boundary conditions, $v_f = v_m = 0.2$).</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 6. Conditions for matrix crack deflection at the interphase/fibre interface in the absence of residual stresses. Crack deflection is predicted if the plotted conditions are fulfilled ($V_f = 0.4$, Type I boundary conditions, $v_f = v_m = v_p = 0.2$, $E_f = E_m = E$, $E_p/E = 0.1$, $e_p/R_f = 6.7\%$).</figcaption></figure>

A by preventing any displacement of the ends of the microcomposite during cooling so that:

$$ \varepsilon = 0 \quad \text{and} \quad \mathcal{G}_m(a) = B_m(a)(E_c \alpha_c \Delta T)^2 \qquad (24) $$

Fig. 7 shows the value of the parameter $B_m(a)$ in the matrix of a microcomposite whose components exhibit identical Young's moduli but different thermal expansion coefficients denoted $\alpha_f$ in the fiber and $\alpha_m$ in the matrix. As expected, the tensile ($\alpha_m > \alpha_f$) or compressive ($\alpha_m < \alpha_f$) longitudinal residual stress in the matrix increases or decreases the value of $B_m(a)$ comparatively to the value of $A_m(a)$. It must be noted that $A_m(a) = B_m(a)$ if $\alpha_m = \alpha_f$. In order to reduce the influence of thermal residual stresses on the energy release rate, a strain has to be applied as given by:

$$ \varepsilon \gg \sqrt{\frac{B_m(a)}{A_m(a)}} |\alpha_c \Delta T| \quad \text{and} \quad \mathcal{G}_m(a) \approx A_m(a) (E_c \varepsilon)^2 \qquad (25) $$

Nevertheless, the tensile longitudinal stress which devel-

<figure><img src="image_8.png" /><figcaption>Fig. 7. The normalised energy release rate $B(a)$ versus the crack length for various values of the thermal expansion mismatch for the matrix crack propagating normally to the F/M interface ($V_f = 0.4$, Type I boundary conditions, $v_f = v_m = 0.2$, $E_f = E_m$).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8. The normalised decrease in temperature required to extend a matrix crack of initial length $a_0$ during cooling for different values of the thermal expansion mismatch (the ends of the microcomposite are free during cooling, $V_f = 0.4$, Type I boundary conditions, $v_f = v_m = 0.2$, $E_f = E_m$).</figcaption></figure>

ops in the matrix during cooling when the thermal expansion coefficient of the matrix is greater than that of the fibre is able to initiate matrix cracking. Assuming the ends of the microcomposite are not fixed during cooling, the relevant conditions are given by:

$$ \varepsilon = \alpha_c \Delta T \quad \text{and} \quad \mathcal{G}_m(a_0) = \mathcal{G}_m^c. \qquad (26) $$

Combining Eqs. (23) and (26) allows the deduction of the critical decrease in temperature which will initiate the matrix crack extension.

$$ \alpha_m > \alpha_f \quad \text{and} \quad \Delta T^c = \frac{\sqrt{\mathcal{G}_m^c}}{E_c \alpha_c (\sqrt{B_m(a_0)} - \sqrt{A_m(a_0)})} \qquad (27) $$

As expected the matrix crack growth is promoted by a large thermal expansion mismatch and by a large initial crack length. This aspect is illustrated in Fig. 8 which plots the normalised value of $\Delta T^c$ when the microcomposite components have identical Young's moduli but different thermal expansion coefficients.

The presence of residual stresses complicates the investigation of conditions for deflection of a matrix crack after its propagation towards the F/M interface. The critical strain at initiation of matrix crack growth is given by:

$$ [\sqrt{A_m(a_0)} \varepsilon^i(a_0) - \sqrt{B_m(a_0)} \alpha_c \Delta T]^2 E_c^2 = \mathcal{G}_m^c. \qquad (28) $$

This expression must be used to evaluate the condition of stability of crack propagation. Further the parameters $\Delta_f(a_i, \delta_c, \varepsilon^c)$ and $\Delta_i(a_i, \delta_c, \varepsilon^c)$ have to be compared to determine the crack path at the F/M interface. Conditions for matrix crack deflection are simplified if the matrix crack is assumed to be directly located at the interface and if the ends of the microcomposite are fixed ($\varepsilon = 0$) during the cooling process. If the matrix crack propagates during the temperature change, deflection at the interface takes place if:

$$ \mathcal{G}_i(a_i, \delta_c) = \mathcal{G}_i^c $$
<figure><img src="image_10.png" /><figcaption>Fig. 9. Critical ratios which define the conditions for matrix crack deflection at the F/M interface taking into account the residual stresses when the matrix crack is located at the interface ($a_0 = a_i$, $V_f = 0.4$, Type I boundary conditions, $v_f = v_m = 0.2$, $E_f = E_m$).</figcaption></figure>

$$ \mathcal{G}_f(a_i, \delta_c) < \mathcal{G}_f^c \quad (29) $$

which leads to a condition equivalent to Eq. (19):

$$ \frac{\mathcal{G}_f^c}{\mathcal{G}_f^c} < \frac{B_i(a_i + \delta_c)}{B_f(a_i + \delta_c)}. \quad (30) $$

The ratio $B_i(a_i + \delta_c)/B_f(a_i + \delta_c)$ plotted versus the thermal expansion mismatch is presented in Fig. 9 for the case that the components of the microcomposite exhibit identical moduli and Poisson's ratios. In this case, matrix crack deflection is promoted when $\alpha_m < \alpha_f$. If the matrix crack is not initiated during the cooling process, it is necessary to increase the applied mechanical loading to propagate the crack. As demonstrated in Appendix B, matrix crack deflection is observed if:

$$ \frac{\mathcal{G}_f^c}{\mathcal{G}_f^c} < \frac{A_i(a_i + \delta_c)}{A_f(a_i + \delta_c)}(1 + X)^2 \\ \text{with } X = \frac{E_c \alpha_c \Delta T}{\sqrt{\mathcal{G}_f^c}} \sqrt{B_f(a_i + \delta_c)} \times y \\ y = \left[ 1 - \sqrt{\frac{A_f(a_i + \delta_c) B_i(a_i + \delta_c)}{A_i(a_i + \delta_c) B_f(a_i + \delta_c)}} \right]. \quad (31) $$

Crack deflection will be enhanced by thermal residual stresses if $X > 0$ which is observed when $\alpha_m > \alpha_f$ as shown in Fig. 9 in the case of a microcomposite having components with identical moduli and Poisson's ratios.

<table><caption>Table 1
Data for the SiC/SiC and SiC/glass ceramic materials</caption><thead><tr><th></th><th>Nicalon fibre</th><th>SiC matrix</th><th>CAS matrix</th><th>LAS matrix</th></tr></thead><tbody><tr><td>Modulus (GPa)</td><td>200</td><td>400</td><td>100</td><td>100</td></tr><tr><td>Poisson's ratio</td><td>0.12</td><td>0.2</td><td>0.3</td><td>0.3</td></tr><tr><td>Thermal expansion coefficient (10<sup>-6</sup> K<sup>-1</sup>)</td><td>2.9</td><td>4.6</td><td>5</td><td>0.5</td></tr><tr><td>Toughness (J m<sup>-2</sup>)</td><td>15</td><td>5–25</td><td>20–40</td><td>20–40</td></tr></tbody></table>

## 4. Results and discussion

The fracture mechanics approach described in the previous section is now used to determine the conditions for matrix crack deflection at the F/M interface in unidirectional CMCs. The considered microcomposites consist of a silicon carbide fibre (Nicalon NLM 202) of radius 7.5 µm surrounded by a cylinder of matrix made of (i) silicon carbide deposited by a chemical vapour deposition process or (ii) glass ceramic materials (CAS and LAS) produced by a liquid route. Each component is supposed to be isotropic and the relevant thermoelastic coefficients are tabulated in Table 1 [15–18]. The toughness of each matrix material is assumed to lie within a given range [19–21] while the fibre toughness is taken to be 15 J m<sup>-2</sup> [22]. The fibre volume fraction is $V_f = 0.4$ and the value of the manufacturing temperature is 1000°C so that $\Delta T = -1000°C$. Type II conditions are applied on the external cylindrical surface to model a microcomposite embedded within a unidirectionally reinforced composite.

### 4.1. Conditions for matrix crack deflection in SiC/SiC and SiC/glass ceramic microcomposite

The numerical method detailed in Appendix A was used to estimate the values of the normalised energy release rates A and B for each system. Assuming that the ends of the microcomposite are not fixed during the cooling step, the initial thermomechanical strain is $\epsilon_0 = \alpha_c \Delta T$. It was verified that the induced thermal residual stresses could not initiate any annular matrix crack of length $a_0$ ($a_0 < a_i$). The mechanical applied strain at initiation of a matrix crack with length $a_0$ is thus $\epsilon^i(a_0) - \epsilon_0$ which is plotted for each system in Fig. 10(a) and (b). Computations indicate that in every case the crack propagation towards the F/M interface is unstable so that $\epsilon^i(a_0)$ is the appropriate value of the strain to be used to evaluate the parameters $\Delta_f(a_i, \delta_c, \epsilon^c)$ and $\Delta_i(a_i, \delta_c, \epsilon^c)$ which are required to define the conditions for crack deflection as defined by Eq. (16).

In the case of a SiC/glass ceramic microcomposite, Fig. 11 makes clear that the parameter $\Delta_f(a_i, \delta_c, \epsilon^c)$ is always positive. Using Eq. (16), matrix crack deflection is predicted if:

$$ \mathcal{G}_f^c - (\mathcal{G}_f(\epsilon^c(a_0), a_i, \delta_c) - \mathcal{G}_f(\epsilon^c(a_0), a_i, \delta_c)) > \mathcal{G}_f^c \quad (32) $$
<figure><img src="image_11.png" /><figcaption>Fig. 10. Mechanical strain required to propagate the matrix crack of initial length $a_0$ ($V_f = 0.4$, Type II boundary conditions): (a) SiC/SiC microcomposite (b) SiC/glass ceramic microcomposite (c) SiC/C/glass ceramic microcomposite. In this case the matrix crack reaches the carbon interphase/fibre interface.</figcaption></figure>

which implies as a necessary condition:

$$ \mathcal{G}_f^c - (\mathcal{G}_f(\varepsilon^c(a_0), a_i, \delta_c) - \mathcal{G}_i(\varepsilon^c(a_0), a_i, \delta_c)) > 0 \quad (33) $$

Fig. 11 shows that this last condition is not fulfilled which hinders any possibility of matrix crack deflection in SiC/glass ceramic microcomposites as a result of a higher value of the matrix toughness compared with the fibre toughness.

The same results are obtained for the SiC/SiC micro-composite when $\mathcal{G}_m^c = 25 \text{ J m}^{-2}$. Nevertheless, decreasing matrix toughness provides some possibility for matrix deflection as the parameter $\Delta_f(a_i, \delta_c, \varepsilon^c)$ turns negative. The conditions for matrix crack deflection

which are indicated in Table 2 require a large initial matrix defect and a low interfacial toughness. A few conditions for matrix crack deflections are thus predicted in such microcomposite systems and thus the benefit of the presence of an interphase between the fibre and the matrix has now to be analyzed.

## 4.2. Conditions for matrix crack deflection in SiC/C/SiC and SiC/C/glass ceramic microcomposites

A carbon interphase is interposed between the fibre and the matrix of the previous microcomposites. The selected thermoelastic constants of this interphase
<figure><img src="image_12.png" /><figcaption>Fig. 11. Critical parameters defined by Eqs. (16) and (33) to evaluate the conditions for matrix crack deflection at the F/M interface in SiC/glass ceramic microcomposite ($V_f = 0.4$, Type II boundary conditions).</figcaption></figure>

which is supposed to be isotropic are given in Table 3 [23]. The value of the interphase toughness $\mathcal{G}_p^c$ is unknown and could measure from 0.1 to 100 J m$^{-2}$ depending on its microstructure [24]. It is assumed here that the value of the interphase toughness $\mathcal{G}_p^c$ is lower than half the value of the fibre toughness ($\mathcal{G}_p^c < 7.5$ J m$^{-2}$). Following the analysis already described in the previous section, the investigation of the conditions for the matrix crack deflection reveals that:

1. the presence of the carbon interphase induces a decrease of the mechanical applied strains at the extension of the matrix crack as it is shown in Fig. 10(c) for the SiC/glass ceramic systems,

2. as a consequence of the low values of the modulus and the toughness of the interphase, the matrix crack is predicted to cross the first interface between the matrix and the interphase,

3. in spite of the decrease in the value of the energy release rate within the interphase, the low value of the interphase toughness allows the crack to reach the second interface between the interphase and the fibre,

<table><caption>Table 2
Description of the conditions for matrix crack deflection at the F/M interface in SiC/SiC microcomposite</caption><thead><tr><th>Matrix toughness</th><th>\mathcal{G}_m^c = 5 J m^{-2}\</th><th>\mathcal{G}_m^c = 10 J m^{-2}\</th><th>\mathcal{G}_m^c = 25 J m^{-2}\</th></tr></thead><tbody><tr><td>Initial crack length condition</td><td>$\frac{a_0}{a_i} > 0.68$</td><td>$\frac{a_0}{a_i} > 0.865$</td><td>—</td></tr><tr><td>Interfacial toughness condition</td><td>$\mathcal{G}_i^c < 0.3$ J m$^{-2}$</td><td>$\mathcal{G}_i^c < 1$ J m$^{-2}$</td><td>—</td></tr></tbody></table>

<table><caption>Table 3
Data for the carbon interphase</caption><thead><tr><th></th><th>Carbon interphase</th></tr></thead><tbody><tr><td>Modulus (GPa)</td><td>30</td></tr><tr><td>Poisson's ratio</td><td>0.25</td></tr><tr><td>Thermal expansion coefficient (10<sup>-6</sup> K<sup>-1</sup>)</td><td>2</td></tr><tr><td>Toughness (J m<sup>-2</sup>)</td><td>—</td></tr></tbody></table>

4. crack deflection is predicted when fibre penetration is prohibited as a consequence of the low value of the energy release rate after growing through the interphase and in case the value of the interfacial toughness is low enough.

Table 4 summarizes the predicted conditions leading to matrix crack deflection at the interface between the interphase and the fibre for the different material systems. The results include the minimum value of the initial crack length and the maximum value of the interfacial toughness as a function of the matrix toughness for two values of the interphase thickness. A comparison with the results of the previous section shows that the presence of the carbon interphase enhances the possibility for matrix crack deflection provided the value of the toughness of the fibre/interphase interface is low enough (i.e. within the range 0.2–2.8 J m$^{-2}$). Moreover, increasing the thickness of the interphase facilitates matrix crack deflection. These numerical predictions given here should be correlated with experimental observations.

Several investigators observed the deflection of the matrix cracks at the interface between the fibre and the interphase in SiC/C/SiC, SiC/BN/SiC and SiC/BN/glass ceramics materials [25–28]. Microstructural analysis of the interfacial zone shows the presence of an additional thin (a few nanometers thick) and anisotropic layer between the fibre and the quasi isotropic interphase [29]. The presence of such an anisotropic layer induces a strong fracture toughness anisotropy and provides the required condition for a low value of the interfacial toughness between the fibre and the interphase. This was clearly demonstrated by indentation fracture tests performed on anisotropic materials of the β-alumina group suggesting ratios of fracture toughness $\mathcal{G}_i^c/\mathcal{G}_f^c$ as low as 0.01 [30]. Nevertheless, matrix crack deflection can also be observed within the interphase [25,31]. Other investigators showed that the crack path can be correlated with the disordered microstructure of the interphase including strongly anisotropic zones [32] or defects which attract a crack [33]. An enhancement of matrix crack deflection due to the presence of defects at the matrix/interphase interface was shown in a SiC/C/ glass ceramic material as a result of a discontinuous string of Niobium carbide grains [34].
## 5. Conclusion

The study of the fracture behaviour of a microcom-
posite allows for the determination of the conditions for
matrix crack deflection at the F/M interface. An annular
crack is first assumed to propagate in the matrix and to
reach the interface. It is then necessary to define an
interface characteristic length to analyse the interaction
of the matrix crack with the interface. The use of the
principles of fracture mechanics shows that matrix crack
deflection is enhanced if the initial location of the matrix
crack is close to the F/M interface and if the matrix
toughness is smaller than the fibre toughness. In the
presence of residual stresses, the possibility for matrix
crack deflection in SiC/SiC and SiC/glass ceramics micro-
composites is reduced due to the higher value of the
matrix toughness compared with the fibre toughness.
Introducing a low modulus and a low toughness inter-
phase between the fibre and the matrix facilitates crack
deflection at the interface between the interphase and the

fibre provided the interfacial toughness is low enough.

## Appendix A. Numerical scheme

Due to symmetry in the geometry and the boundary
conditions the finite element calculations are performed
only on one quadrant of the microcomposite as shown
in Fig. A1. Linear axisymmetric elements are used
throughout the analysis. Although it is difficult to model
the singular behaviour of the stress fields in the neigh-
bouring of the crack tip by using linear interpolations,
very fine meshes were used to limit inaccuracy.

The size $\delta a$ of the smallest element which is immedi-
ately adjacent to the crack tip is chosen to be less than
the ratio $d/50$ where $d$ is the distance between the crack
tip and the nearest interface or boundary. When the
crack approaches an interface, the smallest value of $d$
which defines the characteristic length $\delta_c$ is taken as

<table><caption>Table 4
Description of conditions for matrix crack deflection at the carbon interphase/fibre interface in SiC/SiC and SiC/glass ceramics microcomposites</caption><thead><tr><th>Matrix toughness</th><th>G<sub>m</sub><sup>c</sup> = 20 J m<sup>-2</sup></th><th>G<sub>m</sub><sup>c</sup> = 30 J m<sup>-2</sup></th><th>G<sub>m</sub><sup>c</sup> = 40 J m<sup>-2</sup></th></tr></thead><tbody><tr><td><strong>SiC/C/CAS</strong></td><td></td><td></td><td></td></tr><tr><td>e<sub>p</sub> = 0.5 µm</td><td></td><td></td><td></td></tr><tr><td>Initial crack length condition</td><td>a<sub>0</sub>/a<sub>i1</sub> ≥ 0.56</td><td>a<sub>0</sub>/a<sub>i1</sub> ≥ 0.72</td><td>a<sub>0</sub>/a<sub>i1</sub> ≥ 0.86</td></tr><tr><td>Interfacial toughness condition</td><td>G<sub>i</sub><sup>c</sup> &lt; 0.6 J m<sup>-2</sup></td><td>G<sub>i</sub><sup>c</sup> &lt; 1.1 J m<sup>-2</sup></td><td>G<sub>i</sub><sup>c</sup> &lt; 1.6 J m<sup>-2</sup></td></tr><tr><td><strong>SiC/C/CAS</strong></td><td></td><td></td><td></td></tr><tr><td>e<sub>p</sub> = 1 µm</td><td></td><td></td><td></td></tr><tr><td>Initial crack length condition</td><td>a<sub>0</sub>/a<sub>i1</sub> ≥ 0.52</td><td>a<sub>0</sub>/a<sub>i1</sub> ≥ 0.68</td><td>a<sub>0</sub>/a<sub>i1</sub> ≥ 0.8</td></tr><tr><td>Interfacial toughness condition</td><td>G<sub>i</sub><sup>c</sup> &lt; 0.5 J m<sup>-2</sup></td><td>G<sub>i</sub><sup>c</sup> &lt; 1 J m<sup>-2</sup></td><td>G<sub>i</sub><sup>c</sup> &lt; 1.5 J m<sup>-2</sup></td></tr><tr><td><strong>SiC/C/LAS</strong></td><td></td><td></td><td></td></tr><tr><td>e<sub>p</sub> = 0.5 µm</td><td></td><td></td><td></td></tr><tr><td>Initial crack length condition</td><td>a<sub>0</sub>/a<sub>i1</sub> ≥ 0.73</td><td></td><td></td></tr><tr><td>Interfacial toughness condition</td><td>G<sub>i</sub><sup>c</sup> &lt; 1.7 J m<sup>-2</sup></td><td></td><td></td></tr><tr><td><strong>SiC/C/LAS</strong></td><td></td><td></td><td></td></tr><tr><td>e<sub>p</sub> = 1 µm</td><td></td><td></td><td></td></tr><tr><td>Initial crack length condition</td><td>a<sub>0</sub>/a<sub>i1</sub> ≥ 0.74</td><td>a<sub>0</sub>/a<sub>i1</sub> ≥ 0.89</td><td></td></tr><tr><td>Interfacial toughness condition</td><td>G<sub>i</sub><sup>c</sup> &lt; 1.9 J m<sup>-2</sup></td><td>G<sub>i</sub><sup>c</sup> &lt; 2.8 J m<sup>-2</sup></td><td></td></tr><tr><td><strong>Matrix toughness</strong></td><td>G<sub>c</sub><sup>m</sup> = 5 J m<sup>-2</sup></td><td>G<sub>c</sub><sup>m</sup> = 10 J m<sup>-2</sup></td><td>G<sub>c</sub><sup>m</sup> = 25 J m<sup>-2</sup></td></tr><tr><td><strong>SiC/C/SiC</strong></td><td></td><td></td><td></td></tr><tr><td>e<sub>p</sub> = 0.5 µm</td><td></td><td></td><td></td></tr><tr><td>Initial crack length condition</td><td>a<sub>0</sub>/a<sub>i1</sub> ≥ 0.39</td><td>a<sub>0</sub>/a<sub>i1</sub> ≥ 0.58</td><td>a<sub>0</sub>/a<sub>i1</sub> ≥ 0.83</td></tr><tr><td>Interfacial toughness condition</td><td>G<sub>i</sub><sup>c</sup> &lt; 0.3 J m<sup>-2</sup></td><td>G<sub>i</sub><sup>c</sup> &lt; 0.7 J m<sup>-2</sup></td><td>G<sub>i</sub><sup>c</sup> &lt; 1.9 J m<sup>-2</sup></td></tr><tr><td><strong>SiC/C/SiC</strong></td><td></td><td></td><td></td></tr><tr><td>e<sub>p</sub> = 1 µm</td><td></td><td></td><td></td></tr><tr><td>Initial crack length condition</td><td>a<sub>0</sub>/a<sub>i1</sub> ≥ 0.34</td><td>a<sub>0</sub>/a<sub>i1</sub> ≥ 0.53</td><td>a<sub>0</sub>/a<sub>i1</sub> ≥ 0.78</td></tr><tr><td>Interfacial toughness condition</td><td>G<sub>i</sub><sup>c</sup> &lt; 0.2 J m<sup>-2</sup></td><td>G<sub>i</sub><sup>c</sup> &lt; 0.5 J m<sup>-2</sup></td><td>G<sub>i</sub><sup>c</sup> &lt; 1.7 J m<sup>-2</sup></td></tr></tbody></table>
<figure><img src="image_13.png" /><figcaption>Fig. A1. Boundary conditions used for the finite element calculation.</figcaption></figure>

$\delta_c = a_i/50$ or $\delta_c = e_p/50$ in case of the presence of an interphase. Half the length $L$ of the microcomposite is chosen as $L = 10 R_f$. This value was selected to simulate a structure with an infinite length by comparing the far field stress given by the finite element solution with the classical Lamé solution for a damage free concentric cylinder model [35]. The resulting mesh typically contains a number of nodes varying from 5000 to 10000.

The energy release rate $\mathcal{G}(a)$ is computed with the help of the crack closure method which states that the energy necessary for a crack extension $\delta a$ is equal to the work required to close the crack to its original length. The evaluation of the energy release rate thus requires the resolution of two cases corresponding to the different crack lengths $a$ and $a + \delta a$. The adequacy of the mesh refinement was evaluated by performing a convergence study of a homogeneous cylinder with a concentric crack and comparing with an analytical solution [36].

The normalised energy release rate $A(a)$ and $B(a)$ are computed as follows: (1) $A(a) = \mathcal{G}(a)/\sigma^2$ where $\sigma$ is the axial stress which results from the axial displacement imposed on the microcomposite (2) $B(a) = \mathcal{G}(a)/\sigma^2$ where $\sigma$ is the axial stress which results from the thermal load $\Delta T$ applied on the microcomposite with fixed ends ($u_z = 0$).

## Appendix B. Conditions for crack deflection at the F/M interface under incremental loading in the presence of residual stresses

The matrix crack is considered to meet the F/M interface. The energy release rates for fibre penetration and crack deflection are respectively:

$$ \mathcal{G}_f(a_i, \delta_c, \varepsilon) = \left[ \sqrt{A_f(a_i + \delta_c)}\varepsilon - \sqrt{B_f(a_i + \delta_c)}\alpha_c\Delta T \right]^2 E_c^2 \quad (B.1) $$

$$ \mathcal{G}_f(a_i, \delta_c, \varepsilon) = \left[ \sqrt{A_i(a_i + \delta_c)}\varepsilon - \sqrt{B_i(a_i + \delta_c)}\alpha_c\Delta T \right]^2 E_c^2 \quad (B.2) $$

The conditions for crack deflection under incremental loading are:

$$ \mathcal{G}_i(a_i, \delta_c) = \mathcal{G}_i^c \quad (B.3) $$

$$ \mathcal{G}_f(a_i, \delta_c, \varepsilon) < \mathcal{G}_f^c. \quad (B.4) $$

Eq. (B.3) gives:

$$ \varepsilon = \frac{1}{\sqrt{A_i(a_i + \delta_c)}} \left[ \frac{\sqrt{\mathcal{G}_i^c}}{E_c} + \sqrt{B_i(a_i + \delta_c)}\alpha_c\Delta T \right]. \quad (B.5) $$

Whereas Eq. (B.4) leads to:

$$ \sqrt{\mathcal{G}_i^c} + E_c \alpha_c \Delta T \left[ \sqrt{B_i(a_i + \delta_c)} - \sqrt{\frac{B_f(a_i + \delta_c) A_i(a_i + \delta_c)}{A_f(a_i + \delta_c)}} \right] < \sqrt{\frac{A_f(a_i + \delta_c)}{A_f(a_i + \delta_c)} \mathcal{G}_f^c. \quad (B.6) $$

After some manipulations, the previous expression becomes:

$$ \frac{\mathcal{G}_i^c}{\mathcal{G}_f^c} < \frac{A_i(a_i + \delta_c)}{A_f(a_i + \delta_c)}(1 - X)^2 $$

with

$$ X = \frac{E_c \alpha_c \Delta T}{\sqrt{\mathcal{G}_f^c}} \sqrt{B_f(a_i + \delta_c)} \left[ 1 - \sqrt{\frac{A_f(a_i + \delta_c) B_f(a_i + \delta_c)}{A_i(a_i + \delta_c) B_f(a_i + \delta_c)}} \right]. \quad (B.7) $$

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