Mathematical Problems in Engineering

Volume 2015, Article ID 851548, 22 pages

http://dx.doi.org/10.1155/2015/851548

## Determination of the Stress State of a Piecewise Homogeneous Elastic Body with a Row of Cracks on an Interface Surface Subject to Antiplane Strains with Inclusions at the Tips

Institute of Mechanics, National Academy of Sciences, 24 b M. Baghramian Avenue, 0019 Yerevan, Armenia

Received 6 January 2015; Accepted 11 March 2015

Academic Editor: Christopher Pretty

Copyright © 2015 Ali Golsoorat Pahlaviani and Suren Mkhitaryan. 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 stress state of a bimaterial elastic body that has a row of cracks on an interface surface is considered. It is subjected to antiplane deformations by uniformly distributed shear forces acting on the horizontal sides of the body. The governing equations of the problem, the stress intensity factors, the deformation of the crack edges, and the shear stresses are derived. The solution of the problem via the Fourier sine series is reduced to the determination of a singular integral equation (SIE) and consequently to a system of linear equations. In the end, the problem is solved in special cases with inclusions. The results of this paper and the previously published results show that the used approach based on the Gauss-Chebyshev quadrature method can be considered as a generalized procedure to solve the collinear crack problems in mode I, II, or III loadings.

#### 1. Introduction

Stress analysis near a fracture in an elastic material is one of the most explored topics in solid mechanics. Calculation and stress analysis of engineering structures, particularly their connections and the determination of the stress and strain distribution fields of cracked bodies, have received attention from numerous investigators in recent years. The stress intensity factors, shear stresses, and crack opening displacements are major concepts that must be determined. The stress intensity factor is an important parameter that denotes the magnitude of the stress singularity. The singular order is a single real value, for example, 0.5 for a crack in a homogeneous material. The singular order of general interface corners may be real or complex. An asymptotic stress near the tip of a sharp interfacial corner is generally singular as a result of a mismatch between the materials’ elastic constants.

In this paper, the problem of a piecewise homogeneous rectangular prismatic elastic body in an antiplane strain state due to antiplane forces is discussed. The plate is composed of two bonded dissimilar materials with a number of arbitrary collinear cracks on their interface surface. The aim of this work is the derivation of analytical expressions for the stress intensity factors of the cracks and the presentation of a new mathematical-numerical approach to solve singular integral equations related to the beginning of stresses at the tips of cracks in antiplane deformations; thus, a numerical method to calculate the SIFs of an interface crack between dissimilar materials is developed. Calculation of the SIFs for bimaterial plates in dimensionless form is conducted for several cases: a row of cracks, one and two cracks, and one crack with inclusions at the tips. The objectives of the present study are (i) to present a new method for calculating SIFs of interfacial cracks subject to antiplane loading and (ii) to investigate the influence of inclusion moduli on SIFs to reduce SIFs in cracked bodies and to direct us towards a method for repairs.

The existence of three-dimensional effects at cracks has been known for many years, but understanding has been limited, and for some situations, it still is. Understanding improved when the existence of corner point singularities and their implications became known for straight through-the-thickness cracks [1, 2].

It has been known for a long time that shear and antiplane fracture modes are coupled. This means that shear or antiplane loading of an elastic plate with a through-the-thickness crack generates a coupled three-dimensional antiplane or shear singular stress state, respectively. These singular stress states (or coupled fracture modes) are currently largely ignored in theoretical and experimental investigations as well as in standards and failure assessment codes of structural components, in which it is implicitly assumed that the intensities of these modes as well as other three-dimensional effects are negligible in comparison with the stress fields generated by the primary modes (modes I, II, and III) [3, 4].

The theoretical bases of fracture stresses are discussed in the literature [5–7], and the conclusions of numerous studies and investigations on the derivation of SIFs are categorized in [8–10]. Most of these studies were performed on homogeneous plates. It is proven that, under certain circumstances, the three-dimensional governing equations of elasticity can be reduced to a system where a biharmonic equation and a harmonic equation have to be simultaneously satisfied. The former provides the solution of the corresponding plane problem, while the latter provides the solution of the corresponding out-of-plane shear problem [11]. On the other hand, a mixed fracture mode under antiplane loading may also occur. This coupled fracture mode represents one of three-dimensional phenomena that are currently largely ignored in numerical simulations and failure assessments of structural components weakened by cracks. It arises due to the boundary conditions on the plate-free surfaces, which negate the transverse shear stress components corresponding to classical mode III. Instead, a new singular stress state in addition to the well-known 3D corner singularity is generated. This singular stress state can affect or contribute significantly to the fracture initiation conditions [12, 13].

Inclusions and cavities are also important in understanding the mechanical behavior of structures and are studied in several papers, for example, 2D linear elastic materials [14] and antiplane shear cracks [15]. Photoelasticity and finite element methods have also been employed to study the interaction between collinear cracks, and good agreement was found [16]. Photoelasticity is very helpful in investigating the stress state near inclusions. The results show that the singular stress field predicted by the linear elastic solution for an inclusion can be generated in reality with great accuracy [17]. The inclusions form a thin material that constituted a rigid line inclusion, embedded in a linear elastic body to produce an inhomogeneous stress state. The experiments fully validate the stress state calculated for an elastic plate [18].

The mechanical behavior of thin inclusions is fundamental to the design of composite materials. It is realized that, for a given geometry and boundary condition, depends on the gradation of both the modulus of elasticity and Poisson’s ratio [19]. The cracked sandwich plate twist specimen is viable to characterize mode III fracture [20].

The stress intensity factors can be calculated by a path-independent h-integral and through the virtual crack closure-integral method (VCCM) for numerical implementation [21]. The present study is aimed at investigating the stress state of a piecewise homogeneous elastic body which has a row of collinear cracks in mode III. The numerical procedure based on the loading Gauss-Chebyshev quadrature method is applied. This approach can be used for other multicrack problems or more complicated types of loading [22, 23].

At the end of this paper, we discuss the influence of the bimaterial nature of the body, the distance and geometry of the cracks, and the presence of inclusions at the tips on the characteristics of antiplane shear stresses and deformations.

#### 2. Derivation of the Singular Integral Equation for the General Form of the Problem

A piecewise homogeneous rectangular body in the Cartesian coordinate system* Oxyz* is considered as shown in Figure 1.