BURMISTER'S PROBLEM EXTENDED TO A MICROSTRUCTURED LAYER

摘要

The problem of calculating the displacement and stress field in a layered elastic system loaded on its surface by a certain pressure distribution often arises in engineering analysis and design, in a number of scientific areas ranging from mechanical engineering to soil mechanics and materials science. The solution of such a problem is very important and was first introduced by Biot (1935) but later it was Burmister who presented a complete solution for the stresses and displacements in a general two layer elastic system in which the lower layer is not necessarily rigid (Burmister 1943; Burmister et al. 1944). His results found great application in the field of civil engineering but nowadays can be extended to the technology of barrier, multilayered and/or functionally coatings. Furthermore, due to the ease of manufacturing and assembly, coatings with micro- or even nano-thickness are pursued by manufacturers as hybrid materials for multifunctional devices but as manufacturing scales reduce progressively, the material microstructure itself can play an important role and size effects can be dominant upon the macroscopic mechanical response of the layer/coating. In this study we focus on the loading of a microstructural layer by a normal point load and we present the corresponding Green's functions by extending the solutions suggested by Burmister et al. in order to introduce into the generated displacement and stress fields the effect of the microstructural characteristics of the layer. In order to incorporate the layer material microstructural characteristics we use an effective generalized continuum theory, that is the couple-stress elasticity, in which the material microstructure is introduced constitutively through a length scale. The presented results suggest deviation from those suggested by Burmister et al. in the context of classical elasticity for a layer of finite thickness as well as from those suggested by Gourgiotis and Zisis (2016) in the context of couple stress elasticity for a half-plane.