An interaction integral method for evaluating T-stress for two-dimensional crack problems using the extended radial point interpolation method

27 th , 2015) ABSTRACT: The so-called T-stress, or second term of the William (1957) series expansion for linear elastic crack-tip fields, has found many uses in fracture mechanics applications. In this paper, an interaction integral method for calculating the T-stress for two-dimensional crack problems using the extended radial point interpolation method (XRPIM) is presented. Typical advantages of RPIM shape function are the satisfactions of the Kronecker’s delta property and the high-order continuity. The T-stress can be calculated directly from a path independent interaction integral entirely based on the J-integral by simply the auxiliary field. Several benchmark examples in 2D crack problem are performed and compared with other existing solutions to illustrate the correction


ABSTRACT:
The so-called T-stress, or second term of the William (1957) series expansion for linear elastic crack-tip fields, has found many uses in fracture mechanics applications. In this paper, an interaction integral method for calculating the T-stress for two-dimensional crack problems using the extended radial point interpolation method (XRPIM) is presented. Typical advantages of RPIM shape function are the satisfactions of the Kronecker's delta property and the high-order continuity. The T-stress can be calculated directly from a path independent interaction integral entirely based on the J-integral by simply the auxiliary field. Several benchmark examples in 2D crack problem are performed and compared with other existing solutions to illustrate the correction of the presented approach.

INTRO DUCTIO N
The fracture behavior of cracked structures is dominated mainly by the near-tip stress field. In linear-elastic fracture mechanics interest is focused mostly on stress intensity factors (SIFs) which describe the singular stress field ahead of a crack tip and govern fracture of a specimen when a critical stress intensity factor is reached. The usefulness of crack tip parameters representing the singular stress field was shown very early by numerous investigations. Nevertheless, there is experimental evidence that also the stress contributions acting over a longer distance from the crack tip may affect fracture mechanics properties [1,2]. The constant stress contribution (first "higher-order" term of the Williams stress expansion, denoted as the T-stress term [3]) is the next important parameter. Several researchers [4, 5, 6 and 7] have shown that the T-stress, in addition to the K or J-integral, provides an effective two-parameter characterization of plane strain elastic crack-tip fields in a variety of crack configurations and loading conditions. In special cases, the T-stress may be advantageous to know also higher coefficients of the stress series expansion. In order to calculate the T-stress, researchers have used several techniques such as the stress substitution method [1], the variational method [8], the Eshelby J-integral method [9], the Betti-Rayleigh reciprocal theorem [10,11] and the interaction integral method [10,12]. Among these method, the last three method are based on path-independent integral and the T-Stress can be caculated using data remote from crack-tip, so the result is achieved higher accuracy compared to the other method.
For a few idealized cases, analytical solutions for T-stress are available. However, for practical problems involving finite geometries with complex loading, numerical methods need to be employed. Chuin-Shan Chen et al (2001) applied a p-version finite element method to compute the T-stress [10]. In 2003, Glaucio H. Paulino and Jeong-Ho Kim presented a new approach to compute the T-stress in funtionally graded materials (FGMs) based on the interaction integral method, in combination with the finite element method [13]. In 2004, Alok Sutradhar and Glaucio H. Paulino used Symmetric Galerkin boundary element method (SBEM) for calculating T-stress and SIFs [14].
During the past two decades, the so-called meshless or meshfree methods have developed, and their applications in solving many engineering problems have proved their applicability. 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 analyzing crack problems [15,16,17,18].
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 evaluating T-stress for two-dimensional crack problems. To calculate the T-stress, the interaction integral formulation for homogeneous materials is used. Several numerical examples T-stress 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. If the crack faces are assumed to be tractionfree, the weak-form obtained for this elastostatic problem can be written as where u are the vectors of displacements, 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 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  Ku F (5) with K being the stiffness matrices, 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.

M-integral formulation
The path-independent J-integral [20] is defined as where W is strain energy density given by and j n denotes the outward normal vector to the contour  .
After some mathematical transformations, the interaction integral can be written as

Auxiliary fields for T-stress
The auxiliary fields are judiciously chosen for the interaction integral depending on the nature of the problem to be solved. Since the Tstress is a constant stress that is parallel to the crack, the auxiliary stress and displacement fields are chosen due to a point force f in the 1 x direction (locally), applied to the tip of a semiinfinite crack in an infinite homogeneous body, as shown in Fig. 3. cos , sin The corresponding auxiliary displacements are [22]  

Determination of T-stress
By considering the auxiliary field in Eq. (11) , a simple expression for the T-stress in terms of the interaction integral M, the point force for the auxiliary field f , and material properties , E  can be obtained.

Edge crack plate under tensile loading
In the first example, a rectangular plate with an edge crack is considered. The plate is subjected to a tensile stress 1   as shown in Fig. 4 ) are compared with Symmetric Galerkin boundary element method solution given by Sutradhar and Paulino [14], FEM solution given by Chuin-Shan Chen et al [10] and Feet T et al [23].  Because of the symmetry of geometry and load, a half model is consider with the symmetry boundary condition. A distribution of 25 50  nodes is used for the XRPIM model. Table 3 shows the comparison between the XRPIM results and other solutions with a good agreement.   Figure 6. Inclined edge crack plate This problem was solved by Kim and Paulino using FEM with interaction integral [24]. Moreover, Sutradhar and Paulino [14] used symmetric Galerkin boundary element method to get solution for this problem. In this work, the mixed mode values of normalized SIF and T-Stress are calculated using XRPIM to compare with available reference results as shown in Table  4, which indicates good agreement.

CONSLUSION
The interaction integral method applied to two-dimensional crack problems to evaluate Tstress using the XRPIM has been presented.
Three numerical examples in which the T-stress are evaluated by means of the M-integral. The numerical results obtained are good agreement with known results from the references. The presented approach has shown several advantages and it is promising to be extended to more complicated problems such as computation Tstress and SIFs, crack propagation problems in functionally graded materials.