Extended radial point interpolation method for dynamic crack analysis in functionally graded materials

Functionally graded materials (FGMs) have been widely used as advanced materials characterized by variation in properties as the dimension varies. Studies on their physical responses under in-serve or external loading conditions are necessary. Especially, crack behavior analysis for these advanced material is one of the most essential in engineering. In this present, an extended meshfree radial point interpolation method (RPIM) is applied for calculating static and dynamic stress intensity factors (SIFs) in functionally graded materials. Typical advantages of RPIM shape function are the satisfactions of the Kronecker’s delta property and the high-order continuity. To assess the static and dynamic stress intensity factors, non-homogeneous form of interaction integral with the nonhomogeneous asymptotic near crack tip fields is used. Several benchmark examples in 2D crack problem are performed such as static and dynamic crack parameters calculation. The obtained results are compared with other existing solutions to illustrate the correction of the presented approach.

properties in one or more specified directions [1,2]. In recent years, the FGMs hold promising for applications that require extra high material performance [3]. For example, FGMs are used in thermal protection systems because they evolve the advantage of typical ceramics such as heat and corrosion resistance and typical of metal such as stiffness and mechanical strength. FGMs can be applied to generate thermal barrier coating for space applications, thermal-electric and piezoelectric devices, optical materials with graded reflective indices, bone and dental implants in medicine and so on. In many cases, FGMs structure are brittle and prone to cracking due to hard working conditions such as overload, vibration, fatigue, and so on. For the reason that, crack behaviors of such FGMs has become an interesting study subject.
In this work, we focus on fracture behaviors of FGMs under static and dynamic loading. There are several analytical and also numerical studies that have been performed to obtain the fracture behavior of FGMs structures. Delale and Erdogan et al considered the stress field at crack tip in FGM which has the same square root singularity as that in the homogenous materials [4]. In 1987, Eischen et al present his mixed-mode crack analysis in non-homogenous materials using finite element method (FEM) [5]. Gu P. et al (1999) used domain J-integral to calculate the crack tip field of FGM [6]. In 2002, Kim and Paulino used FEM to calculate the mixed-mode SIFs in FGMs with some modifies for pathindependent integral [7]. In 2005, Menouillard et al applied extended finite element method (XFEM) to calculate mixed-mode stress intensity factors for graded materials [8]. In the next year, Song et al applied FEM to compute the dynamic SIFs for heterogeneous materials [9]. In 2007, Kim and Paulino performed crack propagation problems in FGMs using XFEM [10]. Recently, in the last year, Chiong et at presented the scaled boundary FEM using polygon element for dynamic SIFs calculation for FGMs [11].
Over the ensuing decades, the so-called meshless or meshfree methods have developed. Different from FEM, meshfree methods do not require a mesh connect data points of the simulation domain. Since no finite mesh is required in the approximation, meshfree methods are very suitable for modeling crack growth problems [12,13,14,15]. There are a few studies about meshless method for fracture problems in FGMs in recent years. Rao and Rahman (2003) used EFG method for calculating SIFs in isotropic FGMs [16]. In 2006, Sladek et al applied meshless local Petrov-Galerkin method to evaluate fracture parameters for crack problems in FGM [17]. In 2009, Koohkan et al presented a new technique with J-integral to calculate the SIF values for FGM crack problems [18].
In this study, we propose an extended meshfree method based on the radial point interpolation method (XRPIM) associated with the vector level set method for modeling the crack problem in functionally graded materials under static and dynamic loading conditions. To calculate the SIFs, the dynamic form of interaction integral formulation for nonhomogeneous materials is used. Several numerical examples including static and dynamic SIFs calculation are performed and investigated to highlight the accuracy of the proposed method.

Weak-form formulation
Consider a 2D solid with domain  and bounded by  , the initial crack face is denoted by boundary C  , the body is subjected to a body force b and traction t on t  as depicted in Fig.   1. The weak-form obtained for this elastodynamic problem can be written as where , u u  are the vectors of displacements and acceleration, σ and ε are stress and strain tensors, respectively. These unknowns are functions of location and time:

Meshless X-RPIM discretization and vector level set method
Base on the extrinsic enrichment technique, the displacement approximation is rewritten in terms of the signed distance function f and the distance from the crack tip as follow: where I  is the RPIM shape functions [19] and where r is the distance from x to the crack

Discrete equations
Substituting the approximation (2) into the well-known weak form for solid problem (1), using the meshless procedure, a linear system of equation can be written as with , M K being the mass and stiffness matrices, respectively, and F being the vector of force, they can be defined by t T T where Φ is the vector of enriched RPIM shape functions; the displacement gradient matrix B must be calculated appropriately dependent upon enriched or non-enriched nodes.

J-INTEGRAL FOR DYNAMIC SIFS IMPLEMENTATION
The dynamic stress intensity factors are important parameters, and they are used to calculate the positive maximum hoop stress to evaluate dynamic crack propagation properties. The dynamic form of J-integral for nonhomogeneous materials is written as [9]     ,1 is strain energy density; q is a weight function, changing from 1 q  near a crack-tip and 0 q  at the exterior boundary of the J domain.
In this paper, the interaction integral technique is applied to extract SIFs. After some mathematical transformations, the path independent integration can be written as The stress intensity factors can then be evaluated by solving a system of linear algebraic equations: ( [20] and XFEM solution given by Dolbow and Gosz [21].   and x2 coordinates as follows: SIFs start to increase. The amplitude of the mode-I SIF is much larger than that of the mode-II SIF.

Center crack FGM plate under dynamic tensile loading
The last example deals with a center crack FGM plate that has the same geometry and load condition with the one in 5.2. section. However, in this problem, as shown in Fig. 6, the material distribution is different from the previous case in

CONSLUSION
An extended radial point interpolation method (XRPIM) has been proposed for static and dynamic cracks analysis in functionally graded models. This method is convenient in treating the Dirichlet boundary conditions because of the RPIM shape functions satisfying the Kronecker's delta property. Three numerical examples are investigated with different material models and crack modes. The obtained solutions show a good agreement of between the presented method and the references. The presented approach has shown several advantages and it is promising to be extended to more complicated problems such as dynamic crack propagation problems for functionally graded materials.