Implicit gradient elasticity based on laplacians of stress and strain

真实材料表现出一种亚结构（微观结构），可以通过材料内部长度特征来解决。在许多应用中，微观结构对宏观响应有很大的影响。这种现象在静力学中以材料响应的加强效应的形式出现，在动力学中以非经典色散关系的形式出现。这些例子包括微观范围内的薄膜、纳米管、微传感器、粘接接头等，宏观范围内的土板响应、土中隧道衬砌、毛细效应等。所观察到的现象可以用梯度弹性的本构律来充分解释。在这个项目的第一部分，它已经表明，所谓的隐式梯度弹性模型可以导出为微形态弹性的特殊情况或在非常规热力学的框架内。在第二部分中，我们通过模拟孔隙弹性材料的行为并结合隐式梯度弹性来继续研究隐式梯度弹性，并讨论了几个例子。这些例子涉及静力学问题（静单轴拉伸载荷下的直棱柱杆、伯努利梁的弯曲、界面不完美的平面圆形非均匀性研究、薄膜裂纹的有限元分析）和动力学问题（平面波的传播、波的反射和散射、模型的因果关系）。这些问题将有助于隐式梯度弹性与标准梯度弹性以及与经典弹性的比较。

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.