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.
