Possibilities for formulating the integral equation of the curved rigid line problem in an infinite plate are discussed and summarized. The relation between the obtained integral equations is also analyzed. A solution strategy for one type of the resulting integral equation is proposed. The basic idea is an approximation of the jump function of the resultant force along the rigid line by a polynomial multiplied by a weight function. The zero resultant force condition around the rigid line is thus satisfied automatically, and the Cauchy singular integral involved in the integral equation can be integrated in a closed form. A technique for evaluating the rotation of the rigid line is also investigated. After considering the interaction effect between the curved rigid lines, a system of integral equations for double rigid lines is also proposed. Finally, several numerical examples with the calculated stress singularity coefficients are given.
where ax,o, ay,o and trxy,O denote the stress at a point in the front of the rigid line tip at a distance r.
More recently, rigid line problems have been a topic of considerable research. The stress distribution near the rigid line tip as well as some solution techniques were discussed in [4,5]. A mixed boundary value problem for an elastic half-plane containing the rigid line was solved by a rational mapping technique, and the definition of SSC and the asymptotic expression of the stress components near the rigid line tip was given in [2]. The property of J-integral for a path around the rigid line tip has been found [6].
It is seen that the works published in this field were mainly limited to the case of an elastic medium containing a rigid straight line. Meantime, very few works have been carried out in the field for investigating the curved rigid line problem in plane elasticity. Some proposals for formulating the singular integral equation of the curved rigid line problem were suggested [7]. In addition, regularization of the obtained singular integral equation was discussed and some numerical solutions were obtained [8].
In this paper, several possibilities for formulating the integral equation of the curved rigid line problem in an infinite plate are discussed and summarized. After considering some possibilities for the choice of the unknown function and the right hand term (abbreviated as RHT in the following description) in the integral equation, three types of integral equation for the curved rigid line problem are obtained. The first two types belong to Cauchy singular integral equations, and the third type belongs to weakly singular integral equations with Log kernel.
It is well known that the main difficulty encountered in the solution of the curved rigid line problem is to evaluate the Cauchy singular integral. For example, the available quadrature rules are mainly limited to the case of the singular integrals along a straight line [9]. Therefore, very few solutions in this field have been carried out. The aim of this paper is to find out all the possibilities of formulating the integral equation and to develop the solution technique for the curved rigid line problem.
Generally speaking, the resultant force function has a jump along the curved rigid line. Therefore, it is natural to let the jump of the resultant force take a form of some polynomial multiplied by a weight function. In this case, advantage can be seen from two aspects. First, the equilibrium condition of forces applied on the line can be satisfied automatically. Secondly, the Cauchy singular integral involved in the equation can be evaluated in a closed form. As a result