Abstract:
The aim of this lecture is to discuss the applications of surface elasticity determination for effective properties of materials, and for some related phenomena as surface wave propagation. Here, in addition to the constitutive relations in the bulk, constitutive relations at the surface are independently introduced. Nowadays the most popular models of surface elasticity relates to the models by Gurtin and Murdoch [1, 2] and by Steigmann and Ogden [3, 4]. Some other models are also known in the literature, which can describe surface/interface related phenomena, see e.g. [5-7].
First, we discuss some useful surface elasticity models. From a physical point of view, surface elasticity models correspond to an elastic solid with an elastic membrane or shell or another 2D continuum attached to its boundary. The corresponding boundary dynamic boundary conditions are derived at the smooth parts of the boundary as well as its edges and corner points. Let us underline that these conditions also include dynamic terms. As a result, we have here a dynamic generalization of the Laplace-Young equation as 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 the previous case, where surface stresses are slightly changing the material properties, there is another example of the essential influence of surface properties. This example relates to the propagation of anti-plane surface waves. We discuss some peculiarities of the wave propagation.
References:[1]. Gurtin ME, Murdoch AI. 1975 A continuum theory of elastic material surfaces. Arch. Ration. Mech. Analysis. 57, 291-323.
[2]. Gurtin ME, Murdoch AI. 1978 Surface stress in solids. Int. J. Solids Struct. 14, 431-440.
[3]. Steigmann DJ, Ogden RW. 1997 Plane deformations of elastic solids with intrinsic boundary. Proc. Roy. Soc. A. 453, 853-877.
[4]. Steigmann DJ, Ogden RW. 1999 Elastic surface-substrate interactions. Proc. Roy. Soc. A. 455, 437-474.
[5]. Duan HL, Wang J, Karihaloo BL. 2008 Theory of elasticity at the nanoscale. Adv. Appl. Mech. 42, pp. 1-68. Elsevier.
[6]. Wang J, Huang Z, Duan H, Yu S, Feng X, Wang G, Zhang W, Wang T. 2011 Surface stress effect in mechanics of nanostructured materials. Acta Mech. Solida Sin. 24, 52-82.
[7]. Eremeyev VA. 2016 On effective properties of materials at the nano- and microscales considering surface effects. Acta Mechanica 227, 29-42.
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