非线性和线性弹性动力变换隐身

本文讨论了几何框架下的非线性和线性弹性动力变换问题。特别要注意的是，隐形变换既不是空间变换，也不是坐标系(坐标)的参考变换;隐身变换将各向同性均匀弹性体的边值问题(虚问题)映射到各向异性非均匀弹性体的边值问题，该弹性体的孔被要设计的斗篷包围(物理问题)。虚拟身体有一个理想的机械反应，而物理身体被设计成通过隐形变形来模仿斗篷外面的相同反应。我们证明了非线性弹性动力变换隐身是不可能的，而非线性弹性静力变换隐身对于特殊变形是可能的，例如，具有圆柱形或球形腔体的径向变形。在经典线性弹性动力学的情况下，与文献中先前的观察一致，我们表明斗篷中的弹性常数不是完全对称的;它们不具有微小的对称性。我们证明了无论孔和斗篷的形状如何，弹性动力变换隐形都是不可能的。结果表明，小对大理论，即相对于预应力结构的线性化弹性，也不允许变换隐蔽。然而，对于平面内变形，圆柱孔的弹性动力隐身是可能的，而对于反平面变形，则不可能。 We next show that for a cavity of any shape elastodynamic transformation cloaking cannot be achieved for linear gradient elastic solids either; similar to classical linear elasticity the balance of angular momentum is the obstruction to transformation cloaking. We finally prove that transformation cloaking is not possible for linear elastic generalized Cosserat solids in dimension two for any shape of the hole and the cloak. In particular, in dimension two transformation cloaking cannot be achieved in linear Cosserat elasticity. We also show that transformation cloaking for a spherical cavity covered by a spherical cloak is not possible in the setting of linear elastic generalized Cosserat elasticity. We conjecture that this result is true for a cavity of any shape.