Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1863714, 9 pages

https://doi.org/10.1155/2017/1863714

## A New and Efficient Boundary Element-Free Method for 2-D Crack Problems

^{1}School of Water Conservancy and Environment, Zhengzhou University, Zhengzhou 450001, China^{2}School of Civil Engineering and Architecture, Zhongyuan University of Technology, Zhengzhou 450007, China

Correspondence should be addressed to Jinchao Yue and Yuzhou Sun

Received 4 September 2016; Revised 14 December 2016; Accepted 24 January 2017; Published 21 February 2017

Academic Editor: Elisa Francomano

Copyright © 2017 Jinchao Yue et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An efficient boundary element-free method is established for 2-D crack problems by combining a pair of boundary integral equations and the moving-least square approximation. The displacement boundary integral equation is collated on the on-crack boundary, and a new traction boundary integral equation is applied on the crack surface without the separate consideration of the upper and lower sides. In virtue of integration by parts, only singularity in order is involved in the integral kernels of new traction boundary integral equation, which brings convenience to the numerical implementation. Meanwhile, the integration by parts produces the new variables, the displacement density, and displacement dislocation density, and they are the coexisting unknowns along with the displacement and displacement dislocation. With the high-order continuity of the moving-least square approximation, these new variables are directly approximated with the nodal displacement or displacement dislocation, and the final system of equations contains the unknowns of nodal displacements and displacement dislocations only. The boundary element-free computational scheme is established, and several examples show the efficiency and flexibility of the proposed method.

#### 1. Introduction

In comparison with the domain-type methods such as finite element method (FEM), the boundary integral equation (BIE) method show some special advantages for the fracture mechanics problems because it only requires the discretization of the general boundary and crack surface rather than the domain. But the traditional displacement BIE cannot directly be applied to crack bodies because the geometrical overlapping of the upper and lower crack surfaces leads to an indeterminacy of equations [1, 2]. To overcome this difficulty, a direct way is to derive the traction BIE of cracked bodies [3–6], but the kernels of the traction BIE involve hypersingular terms and their evaluation is very arduous. A different approach is the dual BIE method, in which the displacement BIE is applied to the outside boundary and to one side of crack surface, and the traction BIE is applied to another side of crack surface [7–9]. To reduce the number of equations in the final system matrix, Pan and Amadei proposed a new pair of BIEs so that the displacement BIE is collocated only on the no-crack boundary, and the traction BIE is applied only on one side of crack surface [10, 11]. But the evaluation of hypersingular integrals is still a vital task in the above methods. Actually, the hypersingular integral can be avoided by using integration by parts, and this skill has been used to derive a regular new traction BIE by Chau and Wang [12]. This skill has also been applied to the anisotropic medium and bimaterials by Sun et al. [13–15]. In this paper, we propose that an alternative computational scheme by combining the new traction BIE and displacement BIE, that is, the displacement BIE, is collocated on the no-crack boundary, and the traction BIE only with the singular integral kernel in order of is collocated along the entire crack surface (no need for the separate discretization of the upper and lower crack surfaces).

In the derivation of the new traction BIE, two new variables, the displacement density and displacement dislocation density, are introduced, and they are the basic coexisting unknowns with the displacement and displacement dislocation in the proposed computational scheme. So the new technique is needed to efficiently bridge these unknowns in the numerical simulation. Mesh-free/element-free methods emerge from the nonlocal interpolation/approximation techniques such as the moving-least square (MLS) approximation [16] that has no dependence on the elements. The domain-type element-free methods, such as element-free Galerkin method [17–19], can conveniently model the crack discontinuity with the enriched nodal degrees of freedom [20–22]. With the enlightenment of element-free methods, the extended FEM method has been proposed for the crack problems through the embedding of displacement jumps within elements [23–25]. BIE can also be combined with the nonlocal approximations to generate the boundary-type element-free methods, for example, the called boundary node method [26, 27] and the called boundary element-free method [28–30], in which the unknowns are approximated with the MLS approximation [26], improving MLS approximation [28–30] or Shepard and Taylor interpolation [27]. Another advantage of element-free methods is that the shape function stratifies the higher-order continuity automatically, and the derivative of shape function can be constructed easily, which brings convenience to the higher-order continuity problems [31–33]. For example, for the strain gradient structures, the strain gradient can be approximated directly with nodal displacement, and nodal displacements are the only unknowns in the final system matrix [33]. This advantage encourages us to establish a boundary element-free method to implement the numerical simulation, in which the displacement density and displacement dislocation density are directly approximated with the nodal displacement and displacement dislocation.

This paper will firstly present the new traction BIE only with singularity in order by using integration by parts [12] and then combine it to the displacement BIE to construct a pair of BIEs for 2-D crack problems. The displacement BIE is collocated on the on-crack boundary, and the unknowns are the nodal displacements and displacement dislocations. The traction BIE is collocated on crack surface, and the unknown displacement density and displacement dislocation density are directly approximated with the nodal displacements and displacement dislocations, so that the final system of equations contains the nodal displacements and displacement dislocations as the only unknowns. Since the shape functions are digitally known in the present method and the singular integrals must be evaluated in a pure numerical scheme, the method of Torino [34] is employed to efficiently evaluate the Cauchy integrals.

#### 2. New Boundary Integral Equations

Consider small displacement in two-dimensional, homogeneous, isotropic, and linear-elastic solids, the displacement, strain, and stress are, respectively, denoted as , , and (). The governing equations are given aswhere is the body force; ;* G* is the shear modulus; equals for plane strain and for plane stress with being Poisson’s ratio. Equation (2) can also be written as

The fundamental solutions and are the th displacement and traction at the field point caused by a unit point force along the th direction at a source point , and they can be expressed as follows [1, 2, 36]:where is the Kronecker delta; is the distance between points and ; is the unit normal along the boundary.

Consider a crack embedding in the elastic body as Figure 1, if its upper and lower surfaces are viewed as being geometrically overlapping, that is, , the Somigliana formula expresses the displacement at an interior point in terms of the traction and displacement on the boundary point [1, 12]:where is the arc length along the boundary or crack surface ; is the sum of the tractions acting on the upper and lower crack surfaces; is the difference of the displacements between the upper and lower crack surfaces.