Mathematical Modelling of Nonlinear Waves in Layered Elastic Waveguides with Inhomogeneities

拟议的多学科项目位于三个主要领域的交叉：非线性波的数学理论，固体力学和材料和结构的无损检测。该项目的目的是发展一种非线性波在分层弹性波导中传播的数学理论，其中扩展的不均匀表示损伤/分层。这将使用不同复杂程度的数学模型以及广泛的分析方法和数值模拟来实现。将对理论预测进行实验验证，以验证数学理论。传统上，流体和固体中的非线性波由不同的研究人员研究。然而，越来越清楚的是，对于看似不同的问题，数学方法中有许多相似之处。在非均匀介质中的非线性波领域最著名的问题之一是表面重力波在光滑的海底地形上传播的动力学的经典流体力学问题。人们已经用各种形式研究了这个问题，包括单色波、浅水孤子和波包。本课题旨在对具有扩展界面不均匀的非完全粘结层状弹性波导中的非线性波过程进行理论和实验研究(模拟粘结不良区域)，这在某种意义上可以被认为是上述经典问题的类比。事实上，我们可以假设各层之间的耦合沿着分层的波导缓慢变化(模拟各层之间的内聚力因缺陷的存在而减弱的区域)。然后，我们可以使用表面重力波在可变地形上传播的问题和方法的变分公式来描述非线性波(如应变孤子、单色波或波包)的缓慢演化。特别是，给定应变孤子的初始速度，我们期望能够找到足以导致孤子传输失败的损伤区的特征。这最后一个效应可能用于层状结构的无损检测，利用非线性波。请注意，尽管一些经典的流体和固体力学问题已经发展了类似的数学方法，但它们在所描述的非线性弹性问题中的应用将是全新的和非常重要的。

The proposed multi-disciplinary project lies at the intersection of three major areas: Mathematical Theory of Nonlinear Waves, Solid Mechanics and Non-destructive Testing of Materials and Structures. The aim of the project is to develop a mathematical theory of nonlinear waves propagating in layered elastic waveguides with extended inhomogeneities representing damage/delamination. This will be achieved using mathematical models of different complexity and the application of a wide range of analytical methods and numerical simulations. Theoretical predictions will be tested experimentally to verify the mathematical theory. Traditionally, nonlinear waves in fluids and solids are studied by different communities of researchers. However, it becomes increasingly clear that there are many analogies in the mathematical approaches to seemingly different problems. One of the best known problems in the area of nonlinear waves in inhomogeneous media is the classical fluid mechanics problem of the dynamics of surface gravity waves propagating over smooth bottom topography. This problem has been studied in various formulations, including monochromatic waves, shallow-water solitons and wave packets. The present project aims at the theoretical and experimental study of nonlinear wave processes in imperfectly bonded layered elastic waveguides with extended interfacial inhomogeneities (modelling poorly bonded areas), which can be considered, in a sense, as an analogue of the classical problem described above. Indeed, we can assume that coupling between the layers varies slowly along a layered waveguide (modelling the areas where the cohesive forces between the layers are weakened by the presence of defects). We can then use the variational formulation of the problem and methods developed for surface gravity waves propagating over variable topography to describe the slow evolution of a nonlinear wave (such as a strain soliton, monochromatic wave or a wave packet). In particular, given the initial velocity of the strain soliton, we expect to be able to find the characteristics of the damaged region sufficient to cause the propagation failure of the soliton. This last effect can be potentially used for the non-destructive testing of layered structures, using nonlinear waves.Note that although similar mathematical methods have already been developed for some classical fluid and solid mechanics problems, their application to the described problem of nonlinear elasticity will be completely new and highly nontrivial.