Fracture analysis of functionally graded materials by a BEM
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 bywhere 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 of the FGMs depends on Cartesian coordinates while Poisson’s ratio is constant. In this case, the elasticity tensor can be written as
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 relationswhere r is the radial coordinate with the origin at the crack-tip, and are the crack-face displacements, 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 , plate length and crack-length . The material gradation in the -direction parallel to the crack is described by an exponential lawwith and 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.
References (40)
Fracture mechanics of functionally graded materials
Compos Eng
(1995)- et al.
The mixed mode crack problem in a nonhomogeneous elastic medium
Eng Fract Mech
(1994) - et al.
The crack problem for nonhomogeneous materials under antiplane shear loading – A displacement based formulation
Int J Solids Struct
(2001) - et al.
Cracks in functionally graded materials
Int J Solids Struct
(1997) - et al.
Crack deflection in functionally graded materials
Int J Solids Struct
(1997) - et al.
Numerical analysis of crack-tip fields in functionally graded materials with a crack normal to the elastic gradient
Int J Solids Struct
(2000) - et al.
Experimental investigation of the quasi-static fracture of functionally graded materials
Int J Solids Struct
(2000) - et al.
T-stress, mixed-mode stress intensity factors, and crack initiation angles in functionally graded materials: a unified approach using the interaction integral method
Comp Meth Appl Mech Eng
(2003) - et al.
On the computation of mixed-mode stress intensity factors in functionally graded materials
Int J Solids Struct
(2002) - et al.
Mesh-free analysis of cracks in isotropic functionally graded materials
Eng Fract Mech
(2003)
Boundary element analysis of crack problems in functionally graded materials
Int J Solids Struct
Effects of material gradients on transient dynamic mode III SIFs in a FGM
Int J Solids Struct
Antiplane crack analysis of a functionally graded material by a BIEM
Comp Mater Sci
An advanced numerical method for computing elastodynamic fracture parameters in functionally graded materials
Comp Mater Sci
A meshless local boundary integral equation method for dynamic anti-plane shear crack problem in functionally graded materials
Eng Anal Bound Elem
The radial integration method for evaluation of domain integrals with boundary-only discretization
Eng Anal Bound Elem
Some recent results and proposals for the use of radial basis functions in the BEM
Eng Anal Bound Elem
Fundamentals of functionally graded materials
The crack problem for a nonhomogeneous plane
J Appl Mech
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