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Interface stiffness transfer method for elastic multi-layered structures

DOI：

 作者 单位 邮编 王超 清华大学 100084 刘天云* 清华大学 100084

针对弹性层状结构问题，本文提出一种界面刚度传递求解新方法。基于弹性层的Love通解，引入状态向量间刚度矩阵，得到弹性层界面刚度传递矩阵方程，即一种Riccati矩阵方程，以及半无限弹性层解析的界面刚度矩阵。界面刚度传递矩阵法通过分解传递矩阵将半无限底层界面刚度自下而上传递至顶层，再根据顶层表面边界条件建立与求解边界面对称刚度平衡方程。本文方法保留了经典传递矩阵方法的优点，自然克服了指数增长计算问题，特别还为最优控制问题的Riccati方程提供了一种新解法。数值算例验证了界面刚度传递法。

This paper presents an analytical method, namely interface stiffness transfer method, for evaluating the responses of multilayered elastic structures. Based on the Love function and general solutions, the stiffness matrix relationship of the displacement-stress state vectors is introduced to obtain the interface stiffness transfer matrix equation between adjacent layers, which is also an algebraic Riccati matrix equation. When the elastic layer is a half-space, an explicit solution is obtained directly for the interface stiffness matrix. The interfacial stiffness transfer matrix method starts from the bottom layer with a known stiffness, and then deals with one layer at a time until the uppermost layer is reached, resulting in the interface stiffness of the multilayered structure. Finally, by solving the symmetric equilibrium equations of the boundary conditions, the displacement-stress state vector of arbitrary layer is obtained. This method keeps the advantages of the classical transfer matrix method, but naturally excludes its exponential growth terms. In particular, the proposed method is also a powerful candidate for efficiently solving the algebraic Riccati equation for the optimal control problems. Numerical examples shows the properties of the interface stiffness transfer method.
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