带裂纹非均匀各向异性材料的双周期弹性焊接第一基本问题

First Fundamental Plane Elasticity Welding Problem of Inhomogeneous Anisotropic Materials with A Doubly-periodic Set of Cracks

双周期弹性问题作为构建各向异性损伤理论的基础问题,是弹性和断裂力学理论的重要研究课题.利用复变函数理论提出并讨论两种各向异性材料组成的无限板的平面弹性第一基本问题,板内含有的双周期分布裂纹群以及焊接界面都假设是任意光滑的曲线.运用Lekhnitskii各向异性板的复变函数理论,将求解该平面弹性问题划归为寻求满足对应边值问题的解析函数;然后构造Sherman变换得到解析函数的广义表达式;进一步利用广义Plemelj公式将问题转化为一组正则型奇异积分方程的解,并在数学上严格证明积分方程的唯一可解性.

Doubly periodic plane elasticity problem is very useful in elasticity and anisotropic elastic materials. In this paper, the first fundamental plane elasticity welding problem is investigated . Employing the complex variable function method, solving this problem is transferred into seeking analytic functions which it certain boundary value problems. Further, under some general restrictions, Sherman＇ s transform for doubly-periodic elasticity theory of homogeneous anisotropic materials to that of inhomogeneous materials is developed. Then, using the general representation for thesolution, the boundary value problem is reduced to a normal type singular integral equation with a Weierstrass zeta kernel along the boundary of cracks and welding interface, and which s established.