复合弹性材料中非线性波的传播

项目名称： 复合弹性材料中非线性波的传播

项目编号： No.11301006

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 黄守军

作者单位： 安徽师范大学

项目金额： 22万元

中文摘要： 复合弹性材料是弹性力学中非常重要的研究对象，在工程、航空和医学等诸多领域有着十分广泛的应用。本项目研究复合弹性材料中非线性波的传播，主要包括以下三个方面的内容：（1）对于由线性和非线性材料组成的半无限长复合弹性杆，研究当杆端受到外部冲击时，非线性波在复合弹性杆中的传播，它对应的数学问题是双曲型方程组的初边值问题。我们将构造该问题的整体分片光滑解，研究非线性波之间的相互作用、预应变对波传播的影响以及所构造解的结构稳定性等问题；（2）研究上述复合弹性杆在外部冲击作用下线性杆和非线性杆的分离问题，特别是弹性杆的分离机制及其与所受冲击的依赖关系；（3）研究半空间的超弹性材料中非线性Rayleigh波的整体存在性问题，它是弹性材料表面波理论中一个非常重要的公开问题。我们将着重讨论经典解的整体存在性以及奇性的形成和传播。

中文关键词： 双曲守恒律；复合弹性材料；非线性波；整体解；爆破

英文摘要： The research on composite elastic materials plays a very important role in the study of elastic mechanics and has many various applications in engeneering,aeronautics and medical sciences etc. This project aims to study the propagation of nonlinear waves in composite elastic materials, mainly including the following three aspects: (1) for a composite elastic bar composed of a linearly elastic bar with finite length and a semi-infinite nonlinearly elastic bar, when its end is subjected to an impulsive impact, what is the propagation of nonlinear waves in the bar. It is well-known that this problem can be described by an initial-boundary value problem for hyperbolic differential equations. We shall construct the piecewise smooth solutions and study the interactions of nonlinear waves and how does the initial pre-stress affect the propagation of nonlinear waves. The structure stabilities will be also investigated; (2) under what conditions on the initial data and strength of the impulsive impact, the aforementioned composite bar is able to separate from each other. At the same time, we utilize the theory on hyperbolic partial differential equations to explore the mechanism for this problem; (3)we shall study the long time propagation of nonlinear Rayleigh waves in half-space hyperelastic materials, which is a very

英文关键词： hyperbolic conservation law；composite elastic material；nonlinear wave；global solution；blowup