Mathematical Problems in Engineering

Volume 2015, Article ID 831419, 4 pages

http://dx.doi.org/10.1155/2015/831419

## Antiplane Problem of Periodically Stacked Parallel Cracks in an Infinite Orthotropic Plate

School of Mechanical & Electronic Engineering, Zhongyuan University of Technology, Zhengzhou 450007, China

Received 14 May 2014; Accepted 16 July 2014

Academic Editor: Oluwole Daniel Makinde

Copyright © 2015 Long Wu and Pengcheng Du. 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

The antiplane problem of the periodic parallel cracks in an infinite linear elastic orthotropic composite plate is studied in this paper. The antiplane problem is turned into the boundary value problem of partial differential equation. By constructing proper Westergaard stress function and using the periodicity of the hyperbolic function, the antiplane problem of the periodic parallel cracks degenerates into an algebra problem. Using the complex variable function method and the undetermined coefficients method, as well as with the help of boundary conditions, the boundary value problem of partial differential equation can be solved, and the analytic expressions for stress intensity factor, stress, and displacement near the periodical parallel cracks tip are obtained. When the cracks spacing tends to infinity, the antiplane problem of the periodic parallel cracks degenerates into the case of the antiplane problem of a single central crack.

#### 1. Introduction

Composite materials are a very promising class of structural materials and widely used in many fields. Defects in the composite materials are easier to cause singular stress and cracks. However, periodic crack is the important model to study the problem of multiple cracks. For simplicity, we can consider the agminate cracks as periodic cracks ideally. And the research on periodic cracks problem contributes to making an intensive understanding of failure mechanism of composite materials; therefore, it is very important to study the periodic cracks problem.

Over the past few decades, the antiplane problem of periodic cracks was investigated by many researchers. For example, by using Fourier transforms method, Erdogan, Ozturk, Chen, and Ding [1–4] studied the antiplane problem in functionally graded materials containing a periodic array of collinear cracks. By using Laplace transform and Fourier transform, Wang and Mai [5] analyzed the dynamic antiplane problem of periodic parallel cracks in an infinite functionally graded material, and the stress intensity factors were obtained. By using distributed dislocation method, Pak and Goloubeva [6] studied the antiplane problem in piezoelectric materials containing a periodic array of parallel cracks. The stress and the electric displacement intensity factors were obtained. By using complex variable function method, Tong et al. [7] studied the antiplane problem in piezoelectric materials containing a doubly periodic cracks of unequal size, and a closed form solution of stress intensity factor was obtained. By using the method of conformal mapping, Hao and Wu [8, 9] considered the antiplane problem on parallel periodical cracks of finite length starting from the interface of two half-planes, and the stress intensity factor was obtained. By using the complex variable function method and the undetermined coefficients method, Lekhnitskii [10] studied the antiplane problem of collinear periodic cracks in an infinite orthotropic fiber reinforcement composite plate, and the analytic expressions for stress intensity factors, stress field, and displacement field of the collinear periodic cracks tip were achieved.

The antiplane problem of the periodic parallel cracks in an infinite linear elastic orthotropic composite plate is studied in this paper. The antiplane problem is turned into the boundary value problem of partial differential equation. By constructing proper Westergaard stress function and using the periodicity of the hyperbolic function, the antiplane problem of the periodic parallel cracks degenerates into an algebra problem. The analytic expressions for stress intensity factor, stress, and displacement near the periodical parallel cracks tip are obtained.

#### 2. Mechanical Model

As seen in Figure 1, we consider an infinite linear elastic orthotropic composite plate with periodic parallel cracks of mode . The crack length is , the crack spacing is , and the antiplane shear force is .