论文标题
弹性板中的转换
Transformation Cloaking in Elastic Plates
论文作者
论文摘要
在本文中，我们制定了具有平面内和平面外位移的Kirchhoff-love板和弹性板的弹性动力转化的问题。一个掩盖的转换将各向同性和均匀弹性板（虚拟问题）的边界值问题映射到各向异性和不均匀的弹性板的边界问题，其孔被设计为斗篷（物理问题）。对于Kirchhoff-love板，虚拟板的（平面外）控制方程将转换为物理板的方程式，直到一个未知的标量场。这样一来，人们找到了物理板的初始应力和初始切向身体力，以及一组我们称之为掩盖兼容方程的约束。这些约束涉及掩盖转换，未知标量场和虚拟板的弹性常数。据注意，掩盖图需要满足斗篷外边界和孔表面上的某些条件。特别是，披风图需要将斗篷的外边界固定到三阶。假设有一个通用的径向掩盖图，我们表明在Kirchoff-love板上掩盖了一个圆形孔。掩盖兼容性方程和盖映射需要满足的边界条件是掩盖的障碍。接下来，放宽纯弯曲假设，在面积和面外位移存在下弹性板的转换掩盖问题被制定。在这种情况下，有两组管理方程需要在掩饰地图下同时转换。我们表明，对于一般的径向隐身图，无法掩盖圆形孔。掩盖的兼容性方程和盖映射需要满足阻塞掩盖所需的边界条件。
In this paper we formulate the problem of elastodynamic transformation cloaking for Kirchhoff-Love plates and elastic plates with both the in-plane and out-of-plane displacements. A cloaking transformation maps the boundary-value problem of an isotropic and homogeneous elastic plate (virtual problem) to that of an anisotropic and inhomogeneous elastic plate with a hole surrounded by a cloak that is to be designed (physical problem). For Kirchhoff-Love plates, the (out-of-plane) governing equations of the virtual plate is transformed to those of the physical plate up to an unknown scalar field. In doing so, one finds the initial stress and the initial tangential body force for the physical plate, along with a set of constraints that we call cloaking compatibility equations. These constraints involve the cloaking transformation, the unknown scalar field, and the elastic constants of the virtual plate. It is noted that the cloaking map needs to satisfy certain conditions on the outer boundary of the cloak and the surface of the hole. In particular, the cloaking map needs to fix the outer boundary of the cloak up to the third order. Assuming a generic radial cloaking map, we show that cloaking a circular hole in Kirchoff-Love plates is not possible; the cloaking compatibility equations and the boundary conditions that the cloaking map needs to satisfy are the obstruction to cloaking. Next, relaxing the pure bending assumption, the transformation cloaking problem of an elastic plate in the presence of in-plane and out-of-plane displacements is formulated. In this case, there are two sets of governing equations that need to be simultaneously transformed under the cloaking map. We show that cloaking a circular hole is not possible for a general radial cloaking map; the cloaking compatibility equations and the boundary conditions that the cloaking map needs to satisfy obstruct cloaking.
