复合弹性材料中非线性波的传播

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 important problem in the surface wave theory in elasticity. Our focuses are the global existence of classical solutions to the corresponding problem and the formation and propagation of singularties. These problems are long standing and have not been solved completely to our knowledge.

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。

复合弹性材料是弹性力学中非常重要的研究对象，非线性波在复合弹性材料中的传播规律的研究不仅具有重要的理论意义，而且在工程、航空和医学等诸多领域具有非常重要的应用价值. 本项目主要研究非线性波在具有预应力的复合弹性材料中的传播，该问题由相应的双曲型方程组的初边值问题所刻画，由此可以构造该问题的分片光滑的弱解，并对所构造弱解的结构稳定性展开研究，给出非线性波传播规律较为深入的刻画. 同时，本项目对一类非经典的非线性弹性材料进行研究，该材料的本构方程由一个应变可表示为应力的函数关系式给出，它不属于经典的柯西和格林弹性材料，主要对该非线性弹性材料杆的运动方程组作了研究，考察相应双曲型方程组 Riemann 问题的弱解及柯西问题周期解的爆破现象，为实际相关应用提供理论依据. 此外，本项目还研究了其他有关问题，主要包括 Schwarzschild 时空中相对论弦的运动、等熵 Chaplygin 气体的平面波解和若干包含间断的精确解、平面曲线的双曲平均曲率流的自相似解、一类 Keller-Segel 型交叉扩散方程组的周期解和柯西问题解的逐点估计、调和分析中奇异积分算子及其生成的交换子在某些函数空间上的有界性以及一类 Hele-Shaw-Cahn-Hilliard 两相流方程组整体弱解的研究，得到若干有意义的研究结果.