Interfacial Crack Problem of a Class of Spliced Materials

The interfacial crack problem of a class of spliced materials is discussed. Using plane elastic complex variable method and integral equation theory, one method of solving the complex stress functions is given.


Introduction
Composite materials are widely used in engineering practice. During the long-term use of composite materials, phenomena such as holes and cracks will occur, causing damage to composite materials. Cracking problem of composite materials, especially the problem of interfacial crack has been the topic of considerable research during the past decades, for instance [1] [2] [3] [4].
However, to the best of our knowledge, various types of interfacial crack problems have not been fully studied. The aim of this paper is to study the stress state of one splicing problem of a strip and a half-plane of isotropic materials with interfacial cracks. By employing plane elastic complex variable method and theory of boundary value problems for analytic functions, using proper decomposition of the functions and integral transformation, the problem is reduced to a singular integral equation of normal type. The existence and uniqueness of solution were proved. Further, using the integral equation theory, the closed form solution of stress function was given.
For definiteness and simplicity, we will only discuss the first fundamental problem and the case of a single crack.

Formulation of Problems and Stress Function
Let the elastic body occupy the lower half-plane Z − . The elastic body spliced by two dissimilar isotropic materials, one is a strip S + ( 0, , ty line) on X-axis. Besides, the stresses and the angle of rotation at infinity is also given. Obviously, the principal vectors of the external stresses on r ± and on X is Our problem is how to determine the complex potentials describing the stress state of elastic body.
is the complex stress functions for the elastic body.
Because the elastic body is in balance state, without loss of generality, we can assume the principle vectors of stress on both X and r is zero, and no stresses or rotation at infinity. Thus, the stress functions ( ) According to the conditions known above and the theory of elastic complex variable [5], our problem boils down to the following boundary value problem with functions ( ) , .

Solving Boundary Value Problem
For solving the above boundary value problem (2.1)-(2.4), we introduce a new function:  Using Plemelj's integral formula, from Equations (3.2) and (3.3), we get According to Equation (3.5), we can define two holomorpic functions of the entire lower half plane ( 0 Imz < ), which continuous to X-axis as following: Multiplying both sides of Equation (3.7) and its conjugating equation by ( ) x Imz i x z ⋅ < π − respectively, and then integrating along the X-axis. Considering the previously given conditions and Cauchy's integral theorem, we get Substituting Equations (3.9) and (3.10) into Equation (2.2), either for positive or negative boundary value, we obtain the same equation as following:  In fact, the latter corresponds to the case where there are no stress on r and X, no stresses or rotation at infinity and 0 is a solution of this homogeneous equation. By the uniqueness theorem for elastic problems [5], It is easily verified that the index of the singular integral equation Equations

Discussion of Integral Equations
In practice, it is rather difficult to determine C from Equation (3.13). In fact, we usually only need to know stress distribution of the elastic body. Therefore, it is enough to know only ∈ . For this purpose, differentiating Equations (3.11) and (3.12) respectively, we obtain one singular integral equation as following: Equations (4.1) and (4.2) constitute a singular integral equation on L without constant C.
We will give the solution of Equations (4.1) and (4.2) below. It is notice when