Complex Variable Solutions for Inhomogeneous and Laminated Elastic Plates
Complex Variable Solutions for Inhomogeneous and Laminated Elastic Plates
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DOI：
10.1177/1081286505036417
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发表时间：
2005-10
影响因子：
2.6
通讯作者：
A. H. England;A. Spencer
中科院分区：
文献类型：
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作者：
A. H. England;A. Spencer
以前的论文中已经开发了一种程序，用于构造非均匀各向同性线弹性材料板（不一定很薄）中线弹性方程的精确解，其中弹性模量以任何指定方式取决于垂直于板平面的坐标。其基本思想是，经典薄板或经典层合理论方程（二维理论）的任何解，通过直接替换，都会生成非均匀材料的三维弹性方程的解。在本文中，我们根据复势来阐述该理论。结果表明，非均匀材料中的位移和应力可以用四个复势来表示，这四个复势是 ze = x + iy 的解析函数。应力和力矩合力的表达式是根据这些复势导出的。作为示例，我们解决无限板中的双轴应力问题以及无限板中和半无限板边界处的面内点力问题。我们还制定了半平面的一般边值问题，并开发了一个程序，根据相应平面应变问题的复势来确定非均匀板的复势。
A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical thin plate or classical laminate theory equations (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material. In this paper we formulate this theory in terms of complex potentials. It is shown that the displacement and stress in the inhomogeneous material can be expressed in terms of four complex potentials that are analytic functions of ζ = x + iy. Expressions for stress and moment resultants are derived in terms of these complex potentials. As examples we solve the problems of biaxial stress in an infinite plate and of an in-plane point force in an infinite plate and at the boundary of a semiinfinite plate. We also formulate the general boundary-value problem for a half-plane and develop a procedure to determine the complex potentials for an inhomogeneous plate in terms of the complex potentials for the corresponding plane-strain problem.