XFEM Analysis of the Effect of Fiber Volume Fraction on Crack Propagation in Fiber Reinforced Ceramic Matrix Composites

The mechanism of crack propagation near the interphase plays an important role in the failure procedure of ceramic matrix composites. The aim of this paper is to examine the effect of fiber volume fraction on the crack propagation. A three-phase unit-cell model was used to describe the micro-structure of fiber reinforced ceramic matrix composites. The interphase between fiber and matrix was modelled as a finite-thickness cylinder around the fiber. The extended finite element method (XFEM) was applied to simulate the crack initiation and propagation in the unit-cell. Without the definition of crack path priori, the approach based on XFEM can simulate the arbitrary path of cracks growing in the composites. The crack patterns and stress-strain responses for different fiber volume fraction were compared. In particular, the penetration/deflection mechanism when the matrix cracks grow to the interphase was analysed and compared. The results show that the initiation and propagation of the secondary crack/cracks were significantly influenced by the fiber volume fraction.


Introduction
Owning to the excellent thermo-mechanical properties, the continuous fiber reinforced ceramic matrix composites (CMCs) are promising structural candidates for advanced high-performance structure, such as turbine engine, hypersonic aero craft, gas cooled fast reactor, and fusion reactor [1][2][3].In these applications, the components made of CMCs should have ability to operate at high-temperature and maintain their strength and toughness. To this end, the CMCs consist of a ceramic matrix and a reinforcing fiber and this combination can lead to a composite with superior properties to the constituents. Therefore, the understanding of the connection between composite behaviors and constituents characteristics has an important role in tailoring the properties of CMCs components.
It is now well demonstrated that the mechanical behavior of fiber reinforced CMCs is strongly dependent on the properties of the fiber/matrix interface [4][5][6]. In a properly designed CMCs, interphase can arrest, deflect and branch the propagating cracks that have initiated at outer or void surfaces of the matrix. The most common interphase is pyrolytic carbon (PyC) or boron nitride (BN). During further loading process, the deflected cracks in the interphase enables energy dissipation through interface sliding and consequently allows a pseudo-ductile failure behavior in CMCs [7]. Therefore, it is necessary to study the crack propagation in the interphase of CMCs. Many researches had been performed in order to capture the mechanism of crack deflection in CMCs.
Previous authors had studied the case where the crack deflects at an interface between fiber and matrix using micromechanics. Cook et al. [8] predicted the crack deflection at interfaces using a stress criterion considering the interfacial debonding caused by the approaching crack tip into consideration. Gupta et al. [9] determined the crack propagation using the ratio of maximum normal stress for the deflected and penetrated crack. He et al. [10] developed a deflection criterion by comparing the energy release rates for the deflected and penetrated crack. Within these studies, conditions for crack deflection were established by comparing a toughness or a strength ratio to a critical ratio depending on the elastic mismatch between the two constituents. Most of these studies cannot consider the structure and properties of the interphase with a finite thickness, they treated the interphase as an interface with zero thickness.
A few attempts aimed at modeling the propagation of a matrix crack through numerical technology. By numerical method, the thickness of interphase between fiber and matrix can be considered. Carrere et al. [11] computed strain energy release rates using the finite element method (FEM) at the penetrated or deflected crack tip of CMCs with a finite interphase thickness. The extension of a penetrated or deflected crack was pre-defined. Liu et al. [12] combined the virtual crack closure technique (VCCT) and a finite element model to predict the competition between the matrix crack deflection and penetration in C/C composites. They numerically investigated the effect of interphase by considering the thickness of interphase or not. Therefore, a few numerical studies based on FEM have discussed the influence of a finite interphase on the crack propagation of composites. But, it is still a great challenge to model the deflected and branched cracks in the interphase of CMCs.
In general, the approaches based on classical finite element method such as VCCT, are powerful numerical tools for cracking simulation [13]. These approaches usually possess high requirement of modeling skills and computing resource because of mesh refinement at the crack tip and pre-defined crack path [14]. To avoid these disadvantages, the extended finite element method (XFEM) has been developed and revealed to be a useful numerical tool for strength and fracturing problems. In the past decades, the XFEM has been combined with the level-set method (LSM) and provided another methodology for simulation of crack propagation involving interface of dissimilar materials. Several works [15][16][17] had been performed to study numerically the cracking process of bi-material interfaces in composites using XFEM. Recently, Braginsky et al. [18] simulated the crack deflection in CMCs using the extended finite element method (XFEM). The XFEM are widely used to simulate the crack propagation along an arbitrary path. They discussed the effects of relative stiffness and strength of interphase on the crack propagation. It is obviously that a similar methodology can be used to analysis the influence of other characteristics of microstructure.
As mentioned, phenomena of vital importance in CMCs are the mechanism of crack propagation, particularly the alternative of penetration/deflection at the interface between fiber and matrix. It has been well demonstrated that the mechanical behaviors of CMCs are strongly dependent on the interface debond process, i.e., the deflection of matrix crack at interface. Theoretical analyses dealing with the crack propagation near the fiber/matrix interface are essential to understanding how and what extent the mesoscopic characteristic, such as mecro-geometry and constituents properties, influence the macroscopic structural performance of the CMCs.
After reviewing the main approaches for the classical problem of crack deflection at fiber/matrix interface in CMCs, we intend to develop a simulation method: (1) considering the actual thickness of interphase, and (2) allowing the initiation of crack. To this end, the XFEM was applied to simulate the crack propagation in a unit-cell of unidirectional fiber reinforced CMCs with a finite interphase thickness. The propagation process of a primary matrix crack were simulated and the effect of fiber volume fraction, V f is discussed. A clear understanding of this mechanism of crack propagation at the interphase and the influence of related factor, e.g. V f , is necessary for the optimal designs of CMCs.

Analysis methodology
According to the concept of representative volume element (RVE), the unit-cell model composed of a single fiber composite cylinder with axisymmetric boundary conditions is often employed to study the mechanical behavior of fiber/matrix interface. Here, this unit-cell was used to model a representative SiC/SiC composites with BN interphase. Consider a unit-cell with cylindrical reinforcements of circular cross-section (fiber) surrounded by cylindrical interphase and embedded in a matrix material ( Figure 1). Table 1 lists the basic properties, e.g. elastic modulus and Poisson's ratio, of each constituent. The matrix, fiber and interphase are characterized as isotropic elastic constitutive law. An annulus pre-crack normal to the fiber/matrix interface was incorporated into the matrix in the unit-cell. Multiple matrix cracking is not taken into account here as the present study consider the early stage of crack propagation. To simulate failure procedure under tensile in the fiber direction, strain-controlled loading with uniform displacements was applied at the both ends of cylindrical unit-cell. Considering the unit-cell as an axisymmetric problem, the loading and boundary conditions are illustrated in Figure  1. All of the simulations were performed using the commercially software ABAQUS. The response of a pre-crack unit-cell model under quasi-static tensile loading was analyzed by the XFEM implemented in this software. Axisymmetric elements were chosen for this type of analysis. Table 1 lists the basic parameters, initial stress and fracture energy, for crack simulations. In ABAQUS, the additional parameter damage stabilization and tolerance is necessary to aid convergence of the XFEM solution. These parameters of each constituent are also listed in Table 1 according to references [18] and [19]. To setup the simulation using XFEM, a maximum principal stress criterion is chosen to predict the crack initiation, and a cohesive law is applied to govern the crack propagation. Accordingly, the values of this initiation stress (Tables 1) are referred to as the strength of each constituent. For the cohesive law, the cohesive stiffness defined by the fracture energy is degraded once the initiation criterion is met.
In general, the approaches based on classical finite element method such as VCCT, are powerful numerical tools for cracking simulation. However, their use in the case where secondary cracks initiates and propagates is very difficult. The apparent lack of such studies is possibly due to numerical challenges associated with the prediction of crack path. The FEM has been widely used to model cracks with pre-defined locations and lengths. However, modeling crack initiation is challenging due to the fact that the FEM does not efficiently handle moving interfaces and discontinuities (such as cracks) because of the need for re-meshing. The approaches based on VCCT has been used to simulate crack propagation by inserting cohesive elements along potential crack paths. Therefore, the problem of re-meshing may be avoided if the crack path is assumed to be known a priori, but in general, the simulation of crack propagation remains a challenge. According to the present study, the propagation of primary crack and the initiation of secondary cracks can be simulated by using XFEM. Using this technology, there is no need to re-mesh the model after crack propagation and the crack can initiate and propagate along an arbitrary path. It is clear that the pre-define propagation path for cracking is no longer necessary for the simulation based on XFEM. So, the crack deflection mechanism can be modeled without assumptions of deflect position. Therefore, the XFEM is suitable for the simulation of crack propagation within CMCs which contains complex mechanism. But, the implementation of XFEM in ABAQUS employed here does not allow for crack coalescence. This is a major limitation when the matrix crack and secondary cracks attempt to interact and coalesce. For all the simulations performed here, results were only considered up to the moment when the cracks grew to within two elements of each other just prior to pending crack coalescence.

Analysis methodology
This work concentrates on the influence of fiber volume fraction on the crack propagation in fiber reinforced CMCs. Several numerical studies were performed with different unit-cells. Table 2 lists the geometry parameters of the unit-cells for the studied SiC/SiC composites. The length of unit-cell is chosen as L=10R f to ensure a uniform remote axial strain. The thickness of BN interphase is assumed to be 0.5μmas the interphase thickness is usually <100 nm for multilayer interphase and <1 μm for monolayer interphase. In order to study the effect of fiber volume fraction on the crack propagation, a set of values for the fiber volume fraction, V f , is selected between 10% and 50%. The radius of unitcell, R c , can be calculated accordingly.  Figure 2 shows a typical procedure of crack propagation in the unit-cell of SiC/SiC composites with a V f =20%. As can be seen, the primary matrix crack grown toward the interphase as the increasing load. was extrac nce of these ed by the mi F in ce in re 6 shows th the SiC/SiC t fiber volum initiate withi e matrix wit ted in the int end of the tion of secon xamine the d th of second the variety ent V f . As ca arriving at th he initiated l e secondary c nd to initiate ated in the in k patterns is p wer fiber vol it-cell was ap he primary c o secondary Figure 5. A as the fiber v As a displac ted to calcu parameters icrostructure   Figure 4 ent is direct ult of the dif e, i.e. the un tress contour ccurs for V f = n in Figure 3 The major n the interpha se. This is th