Extended Radial Point Interpolation Method for crack analysis in orthotropic media

Orthotropic materials are particular type of anisotropic materials; In contrast with isotropic materials, their properties depend on the direction in which they are measured. Orthotropic composite materials and their structures have been extensively used in a wide range of engineering applications. Studies on their physical behaviors under in-work loading conditions are essential. In this present, we apply an extended meshfree radial point interpolation method (XRPIM) for analyzing crack behaviour in 2D orthotropic materials models. The thin plate spline (TPS) radial basis function (RBF) is used for constructing the RPIM shape functions. Typical advantages of using RBF are the satisfaction of the Kronecker’s delta property and the high-order continuity. To calculate the stress intensity factors (SIFs), Interaction integral method with orthotropic auxiliary fields are used. Numerical examples are performed to show the accuracy of the approach; the results are compared with available refered results. Our numerical experiments have shown a very good performance of the present method.


INTRO DUCTIO N
Orthotropic composite materials and their structures are used widely in various fields in engineering. One of the most preeminent property of composite is the high strength to weight ratio in comparison with conventional engineering materials. In many cases, orthotropic composites are fabricated in thin plate forms which are so susceptible to fault. A typical fault in composite structure is cracking due to inperfection in fabrication process or hard working conditions such as overload, fatigue, corrosion and so on. For the reason that, crack behavior of orthotropic materials has become an interesting study subject.
In the analytical field, there are some important results early given by Sih et al [1], Bowie et al [2], Tupholme et al [3], Barnet et al [4] and Kuo and Bogy [5]. They forcused on finding out the singular fields such as stress and displacement at near crack tip in anisotropic models. More recent contributions can be listed in Nobile et al [6,7] and Carloni et al [8,9].
There are several numerical studies that have performed to obtain the fracture behavior of orthotropic materials such as the extended finite element method (XFEM) [10,11,12]. In XFEM, the finite element approximation is enriched with Heaviside function for crack face and appropriate functions extracted from the analytical solutions for a crack tip near field. Moreover, the element free Galerkin method (EFG) [13] has been applied for fracture analysis of composite by Ghorashi et al [14]. In this aproach, the support domain is modified to involve the discontinuity at the crack face and the singularity at the crack tip. Unlike the FEM, a set of scattered nodes is used to model the domain in the meshfree methods. Since no finite element mesh is required in the approximation, meshfree methods are very suitable for modeling crack growth problems [15,16,17,18].
In this work, we present an extended meshfree Galerkin method based on the radial point interpolation method (XRPIM) associated with the vector level set method for modeling the crack problem in orthotropic materials under static and dynamic loading conditions. To calculate the SIFs, the dynamic form of interaction integral formulation for homogeneous orthotropic materials is taken. Several numerical examples including static, dynamic SIFs calculation are performed and investigated to highlight the accuracy of the proposed method.

FRACTURE M ECHANICS FO R RTHOTRO PIC M ATERIALS
The linear elastic stress-strain relations can be written as  ε Cσ (1) where σ , ε are linear stress and strain vector respectivily and C is the fourth-order compliance tensor, in 2D, C can be defined as: For a plane stress state, with , 1, 2, 6 i j  , C can be simplified into: For a plane strain state, C can be written as: Consider an anisotropic cracked body subjected to arbitrary forces with general boundary conditions as shown in Fig. 1 where , k k p q are defined by

Meshless XRPIM 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 [20] and     (11) where r is the distance from x to the crack tip

Discrete equations
Substituting the approximation (9) into the well-known weak form for solid problem, using the meshless procedure, a linear system of equation can be written as  Ku F (12) with K being the stiffness matrix, respectively, and F being the vector of force, they can be defined by 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 orthotropic material can be adopted [21]     ,1 1 , 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: ( [10] with 1925 nodes and FEM solution given by Aliabadi [22]. The plot in Fig. 3 show the comparison and it can be see that the single mode

Cantilever orthotropic plate under shear stress
In this example, a cantilever rectangular plate made of orthotropic material with an edge crack at left side is considered. The plate is subjected to a shear loading at the top edge. Dimension, load and boundary condition are display in Fig. 6. The orthotropic material properties are the same with the previous example.   The plots in Fig. 7 show the mixed-mode values of stress intensity factor with respect to various orthotropic angle from -90 0 to 90 0 . The obtained results from the proposed XRIM approach are compared with EFG solutions given by Ghorashi et al [14] and FEM solutions from Chu and Hong [23]. A very close agreement is acquired.

Orthotropic plate with central slant crack
The last example studies a rectangular orthotropic plate with a slanted crack at center. As shown in Fig. 8 [24] 0