Zhang Pu, Parnell William J
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK.
Proc Math Phys Eng Sci. 2017 Apr;473(2200):20160865. doi: 10.1098/rspa.2016.0865. Epub 2017 Apr 5.
Soft phononic crystals have the advantages over their stiff counterparts of being flexible and reconfigurable. Normally, the band gaps of soft phononic crystals will be modified after deformation due to both geometric and constitutive nonlinearity. Indeed these are important properties that can be exploited to tune the dynamic properties of the material. However, in some instances, it may be that one wishes to deform the medium while retaining the band gap structure. A special class of soft phononic crystals is described here with band gaps that are independent or almost-independent of the imposed mechanical deformation, which enables the design of phononic crystals with robust performance. This remarkable behaviour originates from transformation elasticity theory, which leaves the wave equation and the eigenfrequencies invariant after deformation. The necessary condition to achieve such a property is that the Lagrangian elasticity tensor of the hyperelastic material should be constant, i.e. independent of deformation. It is demonstrated that incompressible neo-Hookean materials exhibit such a unique property. Semilinear materials also possess this property under special loading conditions. Phononic crystals composed of these two materials are studied theoretically and the predictions of invariance, or the manner in which the response deviates from invariance, are confirmed via numerical simulation.
软声子晶体相较于其刚性对应物具有柔性和可重构的优点。通常，由于几何和本构非线性，软声子晶体的带隙在变形后会发生改变。实际上，这些都是可用于调节材料动态特性的重要特性。然而，在某些情况下，可能希望在保持带隙结构的同时使介质变形。这里描述了一类特殊的软声子晶体，其带隙与施加的机械变形无关或几乎无关，这使得能够设计出具有稳健性能的声子晶体。这种显著的行为源于变换弹性理论，该理论使波动方程和本征频率在变形后保持不变。实现这种特性的必要条件是超弹性材料的拉格朗日弹性张量应为常数，即与变形无关。结果表明，不可压缩的新胡克材料具有这种独特的特性。半线性材料在特殊加载条件下也具有此特性。对由这两种材料组成的声子晶体进行了理论研究，并通过数值模拟证实了不变性的预测，或者响应偏离不变性的方式。
