Implicit gradient elasticity based on laplacians of stress and strain

真实材料表现出一个子结构(微结构)，它可以通过特征材料内部长度来处理。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刚性效应的形式被观察到，在动力学中以非经典色散关系的形式被观察到。在微观范围内，这些例子包括薄膜、纳米管、微型传感器、胶接接头等；在宏观范围内，例子包括接地板的响应、土壤中隧道的衬砌、毛细效应等。观察到的现象可以用梯度弹性本构定律来充分解释。在这个项目的第一部分，已经证明了所谓的隐式梯度弹性模型可以作为微形态弹性的特殊情况或在非常规热力学的框架内得到。在第二部分中，我们通过将多孔弹性材料的行为与隐式梯度弹性相结合来建模，并讨论了几个例子，从而继续对隐式梯度弹性的研究。这些例子涉及静力学(静力单轴拉伸载荷作用下的直柱杆、Bernoulli梁的弯曲、含不理想界面的平面圆形不均匀研究、薄膜裂纹的有限元分析)和动力学(平面波的传播、波的反射和散射、模型的因果关系)中的问题。这些问题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.