Anti-plane Surface Waves in Materials with Surface Energy
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Surface wave is a wave whose amplitude decays exponentially with the distance from surface.
By anti-plane surface wave, we mean the surface wave when anti-shear is realized.
Surface energy and surface stresses play an important role for material behavior at the nanoscale; see, e.g., Duan et al. (2008), Wang et al. (2011), Javili et al. (2013), and Eremeyev (2016). In particular, they are responsible for size effect at the nanoscale as well as for significant changes in effective (apparent) properties of nanostructured materials. In addition, in materials with surface energy and surface stresses may even exist new phenomena which are absent within the classic continuum models. As an example of such phenomenon, one can consider propagation of anti-plane surface waves in media with surface stresses.
Following Eremeyev et al. (2016), we discuss here the propagation of anti-plane surface waves in an elastic half-space taking into account the surface stresses within the linear Gurtin–Murdoch model of surface elasticity (Gurtin and Murdoch, 1975, 1978).
Anti-plane Motions in the Bulk
Boundary Conditions Within the Surface Elasticity
Here we derived and discussed dispersion relations for anti-plane surface waves within the linear Gurtin–Murdoch model of surface elasticity. Let us note that similar waves exist also within other models possessing surface energy. It is worth to mention here the strain-gradient elasticity; see, e.g., Vardoulakis and Georgiadis (1997), Georgiadis et al. (2000), and Gourgiotis and Georgiadis (2015). The comparison of the Gurtin–Murdoch model with the Toupin–Mindlin of strain-gradient elasticity was performed by Eremeyev et al. (2019). Almost the same waves exist within the lattice dynamics (Eremeyev and Sharma, 2019). For anti-plane waves in media with surface stresses, we refer also to (Eremeyev, 2019), where stress- and strain-gradient surface elasticity models were compared through the phenomenon of surface anti-plane waves.
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