Surface Elasticity for Applications to Material Modelling at Small Scales

Abstract

We discuss the various applications of the surface elasticity to determination of effective properties of materials and some related phenomena as surface wave propagation. In the frame of surface elasticity in addition to the constitutive relations in the bulk we independently introduce constitutive relations at the surface. Nowadays the most popular models of surface elasticity relates to the models by Gurtin and Murdoch and by Steigmann and Ogden. First we discuss some useful surface elasticity models. The corresponding boundary dynamic boundary conditions are derived at the smooth parts of the boundary as well as at edges and corner points. Let us underline that these conditions include also dynamic terms. As a result, we have here a dynamic generalization of the Laplace-Young equation known from the theory of capillarity. Second, we discuss the influence of the surface stresses at the effective stiffness parameters of layered plates and shallow shells. For small deformations we derived the exact formulae for modified tangent and bending stiffness parameters of the plates and shells. The influence of residual surface stresses is also discussed. Unlike to previous case where surface stresses are slightly changing the material properties, there is another example of essential influence of surface properties. This example relates to the propagation of anti-plane surface waves. We discuss some peculiarities of the wave propagation.

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