Implicit gradient elasticity based on laplacians of stress and strain

真实的材料表现出可以通过特征材料内部长度来描述的子结构（微观结构）。在许多应用中，微结构可以强烈地影响宏观响应。这种现象已被观察到在静态的形式在材料响应的硬化效应和动态的形式的非经典色散关系。这些例子包括，除其他外，在微观范围内的薄膜，纳米管，微型传感器，粘合剂接头等，并在宏观范围内的地球板，隧道衬砌的土壤，毛细效应等的响应。观察到的现象可以充分解决梯度弹性本构定律。在这个项目的第一部分，它已被证明，所谓的隐式梯度弹性模型，可以推导出无论是作为微观弹性的特殊情况下，或在非传统的热力学框架内。在第二部分中，我们继续调查隐式梯度弹性模型结合隐式梯度弹性多孔弹性材料的行为，并讨论了几个例子。例子涉及的问题在静态（直棱柱杆静态单轴拉伸载荷下，弯曲的伯努利梁，研究平面圆形不均匀性与不完善的界面，有限元分析的裂纹在一个薄膜）和动态（平面波的传播，反射和散射波，因果关系的模型）。这些问题将有助于将隐式梯度弹性与标准梯度弹性和经典弹性进行比较。

Real materials exhibit a substructure (microstructure) which can be addressed by characteristic material internal length. The microstructure can strongly effect the macroscopic response in many applications. Such phenomena have been observed in statics in form of a stiffening effect in the material response and in dynamics in form of non-classical dispersion relations. The examples cover, among others, in the microscopic range thin films, nanotubes, micro sensors,adhesive joints etc. and in the macroscopic range the response of earth plates, lining of tunnels in soil, capillarity effects, etc. . The observed phenomena can be addressed adequately by constitutive laws of gradient elasticity. In the first part of this project it has been shown that models of so-called implicit gradient elasticity can be derived either as particular cases of micromorphic elasticity or within the framework of non-conventional thermodynamics. In the second part we continue the investigations about implicit gradient elasticity by modelling poroelastic material behaviour combined with implicit gradient elasticity and by discussing several examples. The examples concern problems in statics (straight prismatic bar under static uniaxial tensile loading, bending of Bernoulli-beams, study of plane circular inhomogeneity with imperfect interface, FE-analysis of a crackin a thin film) and in dynamics (propagation of plane waves, reflection and scattering of waves, causality of the models). These problems will help to compare implicit gradient elasticity with standard Gradient elasticity and with classical elasticity.