Extended meshless moving Kriging method for crack propagation analyzing in orthotropic media

 Abstract—orthotropic composite material is the particular type of anisotropic materials and their products have been extensively used in a wide range of engineering applications. Study on mechanical behaviors of such materials under working conditions is very essential. In this study, an extended meshfree moving Kriging interpolation method (namely as XMK) is presented for crack analyzing in 2D orthotropic materials models. The Gaussian function is used for constructing the moving Kriging shape functions. Typical advantages of the MK shape function are the high-order continuity and the satisfaction of the Kronecker’s delta property. To calculate the stress intensity factors (SIFs), interaction integral method is used with orthotropic auxiliary fields. Several numerical tests including static SIFs calculating and crack propagation predicting are performed to verify the accuracy of the present approach. The obtained results are compared with available refered results and they have shown a very good performance of the present method.


Extended meshless moving Kriging method for crack propagation analyzing in orthotropic media
Nguyen Thanh Nha, Nguyen Ngoc Minh, Bui Quoc Tinh, Truong Tich Thien *  Abstract-orthotropic composite material is the particular type of anisotropic materials and their products have been extensively used in a wide range of engineering applications. Study on mechanical behaviors of such materials under working conditions is very essential. In this study, an extended meshfree moving Kriging interpolation method (namely as X-MK) is presented for crack analyzing in 2D orthotropic materials models. The Gaussian function is used for constructing the moving Kriging shape functions. Typical advantages of the MK shape function are the high-order continuity and the satisfaction of the Kronecker's delta property. To calculate the stress intensity factors (SIFs), interaction integral method is used with orthotropic auxiliary fields. Several numerical tests including static SIFs calculating and crack propagation predicting are performed to verify the accuracy of the present approach. The obtained results are compared with available refered results and they have shown a very good performance of the present method.

INTRODUCTION
n recent decades, orthotropic composite materials are used widely in various fields in engineering such as automobile, aerospace and civil industries, etc. One of the most advance property of composite is the strength per weight ratio of these materials is higher than other conventional engineering materials. In many cases, orthotropic composites are fabricated in thin plate or thin shell forms which are so easy to fault. Moreover, fiber enforced composites are so brittle and usually have linear elastic crack behavior without or with very little plasticity. For that reason, linear elastic crack behavior of orthotropic materials has become a very attracting study topic.
There are some important analytical solutions for othortropic crack models 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 found out the singular fields such as displacement and stress near crack tip zone in anisotropic models. More recent contributions can be listed in Nobile et al [6,7] and Carloni et al [8,9]. However, analytical formulations cannot be applied to practical problems that have complex geometries and loading conditions. In the numerical fields, the extended finite element method (XFEM) has shown a very good capability in analyzing of fracture behavior of orthotropic materials, some typical publications can be listed in [10][11][12][13][14]. 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) [15] has been applied for fracture analysis of I composite by Ghorashi et al [16]. 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, meshfree method uses a set of scattered nodes to model the domain and approximate the field variables. Because no finite element or mesh is required in the approximation, meshfree methods are very suitable for modeling crack growth problems [17][18][19][20].
In this work, an extended meshfree Galerkin method based on the moving Kriging interpolation method (X-MK) associated with the vector level set method is presented for modeling the crack problem in orthotropic materials. To calculate the SIFs, the interaction integral formulation for orthotropic materials is taken. Several numerical examples including static SIFs calculation and crack propagation angle prediction are performed and the obtained results are compared to the solutions given by other methods to verify the accuracy of the proposed method.

Linear elastic behavior of orthotropic material
In orthotropic material, the linear elastic stressstrain relations can be written as  ε Cσ (1) Where σ , ε are linear stress and strain vectors, respectivily and C is the fourth-order compliance tensor. For plane stress problem, C can be defined as:

Crack behavior of orthotropic material
Consider an orthotropic cracked body subjected to arbitrary forces with general boundary conditions as shown in Fig. 1 It was proved by Lekhnitskii

Criterion for crack growth direction
In orthotropic material, the crack growth direction is predicted based on the maximum hoop stress criterion [27]. This criterion means that the crack tends to propagate in the direction where the hoop stress   is maximum. Moreover, diferent from isotropic material that has only one fracture toughness value in every direction, in orthotropic case, the fracture toughness is given by 1 To apply this criterion for crack propagation in orthotropic model that have general crack angle and material orientation, the formulation is generalized as [28] where mat  is the material orientation and  is the crack angle. The value of  that makes the expression (6) get maximum is the crack growth direction.

X-MK FORMULATION FOR CRACK
PROBLEM.

The moving Kriging shape function
According to Gu et al. [22], the approximation of where () ux is the vector of nodal displacements; () px is the vector of m polynomial basis functions the vector of n correlation functions. The vector of basis functions can be chosen as linear functions The matrixes A (3×n) and B (n×n) are determined by   where I is an unit matrix, matrix P of the basis functions and correlation matrix R are given in detail in [22] In this study, the Gaussian function is used as correlation function  8) is to eliminate the effect of the correlation coeffiction in the Gaussian correlation function in [22].

Meshless X-MK discretization and vector
In Eq. (9), b W denotes the set of nodes whose support contains the point x and is bisected by the crack line and S W is the set of nodes whose support contains the point x and is slit by the crack line and contains the crack tip. ,

I Ij
 are additional variables in the variational formulation [18].

Discrete equations
Applying the meshless procedure [23] by substituting the approximation (9) into the wellknown weak form for solid problem, a linear system of equation can be written as  Ku F (14) 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 MK shape functions; the displacement gradient matrix B must be calculated appropriately dependent upon enriched or non-enriched nodes [20].

STRESS INTENSITY FACTORS CALCULATION FOR ORTHOTROPIC
MODELS.
The stress intensity factors are important parameters in linear elastic fracture mechanics, they are used to evaluate the status of crack and predict the angle of crack propagation.
In this paper, the interaction integral derived from the path independent J-integral is used to extract the SIFs for orthotropic model [13]. The path independent integration can be written as , aux aux ij i u  are real and auxiliary states of stress and derivative of displacement respectively. The weight function q is defined in [13] The stress intensity factors can then be evaluated by solving a system of linear algebraic equations:  Figure 3. Square orthotropic plate with center crack The results for normalized mode-I SIF 0 / I K K a   are given in Table 1. The obtained X-MK results are compared to the solutions given by other methods such as FEM [24], XFEM [12], X-RPIM [20] and EFG [16].  [24] 11702 0.997 XFEM [12] 4278 1.020 In Table 1, the values of DOFs in X-MK and X-RPIM are assumed for the full models. Practically, authors only use 800 nodes (1600 dofs) for the symmetric model. The numerical results of the SIFs indicated that the proposed X-MK method gives acceptable solution with fewer DOFs than others.
To investigate the effect of the dimensionless size of support domain, various values of sd  are considered and reported in Table 2. It can be seen that the optimum values for this size coefficient are from 1.9 to 2.1.

Orthotropic plate with edge crack under shear stress
In the second example, a cantilever orthotropic plate with an edge crack is considered as shown in Fig. 4. The plate is subjected to a shear stress at the top edge. Dimension, load and boundary condition are display in Fig. 4  . A distribution of 20 40  scatter nodes is used in this plane stress analysis.