Fracture analysis of functionally graded materials by a BEM

https://doi.org/10.1016/j.compscitech.2007.08.029Get rights and content

Abstract

In this paper, crack analysis in two-dimensional (2D), continuously nonhomogeneous, isotropic and linear elastic functionally graded materials (FGMs) is presented. For this purpose, a boundary element method (BEM) based on a boundary-domain integral equation formulation is developed. An exponential variation with spatial variables is assumed for Young’s modulus of the FGMs, while a constant Poisson’s ratio is considered. Fundamental solutions for homogeneous, isotropic and linear elastic solids are applied in the formulation. To avoid displacement gradients in the domain integral, normalized displacements are introduced. By using the radial integration method, the domain integral is transformed into boundary integrals over the global boundary. The normalized displacements in the domain integral are approximated by a combination of radial basis functions and polynomials in terms of global coordinates, which leads to a meshless scheme. Special attention of the analysis is devoted to the computation of the most important crack-tip characterizing parameters of cracked FGMs, namely the stress intensity factors. To show the effects of the material gradation on the stress intensity factors, numerical examples are presented and discussed.

Introduction

In recent years, a new class of composite materials, the so-called functionally graded materials (FGMs), attracted many research interests in materials and engineering sciences [1], [2]. FGMs are advantageous over classical homogeneous materials with only one material constituent, because FGMs consist of more material constituents and they combine the desirable properties of each constituent. As a representative example for FGMs, we just mention the metal/ceramic FGMs, which are compositionally graded from a ceramic phase to a metal phase. Metal/ceramic FGMs can incorporate advantageous properties of both ceramics and metals such as the excellent heat, wear, and corrosion resistances of ceramics and the high strength, high toughness, good machinability and bonding capability of metals without severe internal thermal stresses. However, ceramics have a brittle nature, and microcracks or crack-like defects are often induced in the fabrication process or under the in-service loading conditions. Thus, fracture and fatigue analysis of FGMs is an important research issue to the design, optimization, and novel engineering applications of FGMs. For cracked FGMs with general geometry and loading conditions, advanced numerical methods have to be applied, because of the high mathematical complexity of the corresponding governing partial differential equations with variable coefficients, and because the most available analytical methods can be successfully applied to cracked FGMs only with very simple geometry and loading conditions. In this context, we just mention the singular integral equation method [3], [4], [5], [6], [7], the classical finite element method (FEM) [8], [9], [10], [11], [12], [13], [14], [15], the graded finite element method [16], [17], [18], [19], the extended finite element method (XFEM) [20], the element-free Galerkin method (EFG) [21], [22], the boundary integral equation method (BIEM) or boundary element method (BEM) [23], [24], [25], [26], [27], and the meshless Petrov–Galerkin method (MLPG) [28], [29], [30], [31].

Although the BEM has been successfully applied to homogeneous, isotropic and linear elastic solids for many years, its application to FGMs is yet very limited due the fact that the corresponding fundamental solutions or Green’s functions for general FGMs are either not available or mathematically too complex [32], [33]. The nonhomogeneous nature of FGMs prohibits an easy construction and implementation of fundamental solutions for general FGMs.

In this paper, crack analysis in 2D, continuously nonhomogeneous, isotropic and linear elastic FGMs is presented. For this purpose, a boundary-domain integral equation formulation is applied. For simplicity, an exponential variation of Young’s modulus and constant Poisson’s ratio are assumed. Fundamental solutions for homogeneous, isotropic and linear elastic solids are applied in the present formulation, which results in a boundary-domain integral equation formulation due to the materials nonhomogeneity. To avoid displacement gradients in the domain integral, normalized displacements are introduced. The radial integration method of Gao [34], [35] is applied to convert the arising domain integral into boundary integrals over the global boundary of the cracked solids. Basis functions consisting of a combination of radial basis functions and polynomials in terms of global coordinates are used to approximate the normalized displacements in the domain integral. In this manner, a meshless scheme is obtained, which requires only conventional boundary discretization and additional interior nodes instead of cells or meshes. An advantage of the present BEM is that it is easy to implement and can be easily incorporated into an existing BEM code for homogeneous, isotropic and linear elastic solids. Special attention of the analysis is devoted to the investigation of the material gradation on the stress intensity factors. Numerical examples for cracks parallel and perpendicular to the material gradation are presented and discussed.

Section snippets

Boundary-domain integral equations

We consider 2D, continuously nonhomogeneous, isotropic and linear elastic FGMs. In the absence of body forces, the equilibrium equations are given byσij,j=0,where σij represents the stress tensor, a comma after a quantity represents spatial derivatives and repeated indexes denote summation. It is assumed that the Young’s modulus E(x) of the FGMs depends on Cartesian coordinates while Poisson’s ratio ν is constant. In this case, the elasticity tensor Cijkl(x) can be written asCijkl(x)=μ(x)Cijkl0,

Numerical solution procedure

The boundary discretization of the boundary-domain integral equations (12) can be done by following the usual procedure applied in the conventional BEM. A key issue to the numerical solution procedure is how to compute the domain integral in (12). Compared to the classical mesh- or cell-integration method, a transform of the domain integral in (12) into boundary integrals over the global boundary is advantageous. In this paper, the radial integration method (RIM) of Gao [34], [35] is used for

Computation of stress intensity factors

Since the asymptotic crack-tip field for continuously nonhomogenous, isotropic and linear elastic solids has the same structure as that for homogeneous, isotropic and linear elastic solids [8], the stress intensity factors can be computed by using the following relationsKIKII=2μtipκ+12πru2(r,π)u1(r,π),κ=3-4ν,plane strain,(3-ν)/(1+ν),plane stress,where r is the radial coordinate with the origin at the crack-tip, u1(r,π) and u2(r,π) are the crack-face displacements, μtip is the shear modulus at

Numerical results

In the first numerical example, we consider an edge crack parallel to the material gradation in a rectangular FGM plate, which is subjected to a uniform tensile loading as depicted in Fig. 2. The geometry of the cracked plate is described by: plate width b=10, plate length 2h=30 and crack-length a=0.4b. The material gradation in the x1-direction parallel to the crack is described by an exponential lawE(x1)=E0eαx1,α=1blnEwE0,with E0 and Ew being the Young’s modulus at the left and the right side

Conclusions

In this paper, fracture analysis of cracked FGMs by a BEM is presented. By using fundamental solutions for homogeneous, isotropic and linear elastic solids, a boundary-domain integral equation formulation is obtained. Normalized displacements are introduced to avoid displacement gradients in the domain integral. A numerical solution procedure is developed to solve the boundary-domain integral equations. The domain integral is transformed into boundary integrals over the global boundary of the

Acknowledgement

Support by the German Research Foundation (DFG) under the project number ZH 15/10-1 is gratefully acknowledged.

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