Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Anti-plane Surface Waves in Materials with Surface Energy

  • Victor A. EremeyevEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_171-1



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

Let us consider the deformation of an elastic half-space x3 ≤ 0. Here x1, x2, and x3 are the Cartesian coordinates with corresponding unit base vectors ik, k = 1, 2, 3; see Fig. 1. Hereinafter we use the direct (coordinate-free) tensor calculus as described in Lebedev et al. (2010) and Eremeyev et al. (2018). For anti-plane strains, the vector of displacements takes the following simple form (Achenbach, 1973)
$$\displaystyle \begin{aligned} \mathbf u= u(x_2,x_3, t)\mathbf i_1, \end{aligned} $$
where t is time. With (1) we get the formula for the gradient of the displacement vector
$$\displaystyle \begin{aligned} \nabla\mathbf u= u_{,\alpha}\mathbf i_\alpha \otimes \mathbf i_3 =\nabla u\otimes \mathbf i_3,\end{aligned}$$
where ⊗ denotes the dyadic product. Hereafter for brevity we used the notation \(u_{,\alpha }=\frac {\partial u}{\partial x_\alpha }\), and Greek indices take values 2, 3.
Fig. 1

Elastic half-space and an anti-plane surface wave

In what follows we restrict ourselves by isotropic linear materials. So for the anti-plane shear deformation (1), the equation of motion can be reduced to the wave equation (Achenbach, 1973)
$$\displaystyle \begin{aligned} \rho \ddot{ u} = \mu \varDelta u, \end{aligned} $$
where Δu = u,22 + u,33 is the 2D Laplace operator, ρ is a mass density, μ is a shear modulus, and the upper dot stands for derivative with respect to t.
Assuming steady-state behavior, we consider a solution of (2) in the form
$$\displaystyle \begin{aligned} u = U(x_2,x_3) \exp(-\mathrm{i}\omega t), \end{aligned} $$
where U is an amplitude, ω is a circular frequency, and \(i=\sqrt {-1}\) is the imaginary unit. With (3) Eq. (2) transforms into
$$\displaystyle \begin{aligned} \mu \varDelta U = -\rho \omega^2 U. \end{aligned} $$
It has decaying at x3 →− solution
$$\displaystyle \begin{aligned} U = U_{0} \exp(\kappa x_3) \exp(\mathrm{i} k x_2), \end{aligned} $$
where k is a wavenumber, U0 is a constant, and κ is given by the relation
$$\displaystyle \begin{aligned}\ \kappa=\kappa(k,\omega)\equiv \sqrt{k^2 - \frac{\omega^2}{c_T^2} },\quad c_T=\sqrt{\frac{\mu}{\rho}}.\end{aligned}$$
Here cT is the phase velocity of transverse waves in the bulk (Achenbach, 1973).

Boundary Conditions Within the Surface Elasticity

Within the classic linear elasticity, the boundary condition for anti-plane strain at a free surface takes the form
$$\displaystyle \begin{aligned} \mu u_{,3}\big|{}_{x_3=0}=0. \end{aligned} $$
Substituting obtained solution (5) into (6), one gets
$$\displaystyle \begin{aligned}\mu U_{0} \kappa \exp(\mathrm{i} k x_2)=0,\end{aligned}$$
which results only in U0 = 0. So in this case, u = 0 and anti-plane surface waves do not exist.
Unlike classic elasticity for a free surface with surface stresses we get more complex boundary condition. It is the so-called generalized Young–Laplace equation, which for anti-plane deformations transforms into
$$\displaystyle \begin{aligned} \mu u_{,3}\big|{}_{x_3=0}= \tau{,_2}-m \ddot{u}, \end{aligned} $$
where τ is a surface stress and m is the surface mass density as introduced by Gurtin and Murdoch (1978). Within the Gurtin–Murdoch model here, τ takes the form
$$\displaystyle \begin{aligned} \tau = \mu_s u_{,2}\big|{}_{x_3=0} , \end{aligned} $$
where μs is a surface shear modulus. As a result, the generalized Young–Laplace equation (7) reduces to
$$\displaystyle \begin{aligned} \mu u_{,3}\big|{}_{x_3=0}= -m\ddot{u}+\mu_s u_{,22}\big|{}_{x_3=0}. \end{aligned} $$
Substituting now (5) into (9), we obtain the relation
$$\displaystyle \begin{aligned} \mu U_{0} \kappa \exp(i k x_2)=&\, \omega^2 U_{0} \kappa \exp(i k x_2)\\ &- \mu_s k^2 U_{0} \kappa \exp(i k x_2),\end{aligned} $$
from which assuming that U0≠0 we get the dispersion relation (Eremeyev et al., 2016)
$$\displaystyle \begin{aligned} m\omega^2-\mu_s k^2=\mu \sqrt{k^2-\frac{\omega^2}{c_T^2}}. \end{aligned} $$
This equation relates ω and k, so if (10) is fulfilled, we have a nontrivial solution, that is, a surface anti-plane wave. Introducing the phase velocity c = ωk and characteristic dynamic wavenumber p = ρm we transform (10) into dimensionless form
$$\displaystyle \begin{aligned} \frac{c^2}{c_T^2} = \frac{ c_s^2 }{c_T^2} +\frac{p}{|k|} \sqrt{1-\frac{c^2}{c_T^2}} , \end{aligned} $$
where \(c_s=\sqrt {{\mu }_s/m}\) is the shear wave velocity in the thin film associated with the Gurtin–Murdoch model. Obviously, wavenumber k is real if and only if c lies in the range
$$\displaystyle \begin{aligned} c_s <c\le c_T, \quad c>c_s. \end{aligned} $$
The latter equation means that the surface antiplane waves exist in the case when the surface film is softer than material in the bulk as in the case of the Love waves (Achenbach, 1973).
A typical dispersion curve is shown in Fig. 2. Here cs = 0.75cT and \(\bar {k}=k/p\). Dispersion curve starts at the point (0, cT) with the horizontal tangent. Then c(k) → cs when k →. So the surface stresses are more pronounced for short waves as it should be.
Fig. 2

Dispersion curve for the Gurtin–Murdoch model. Here we assumed that cs = 3∕4cT and \(\bar {k}=k/p\)


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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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Civil and Environmental EngineeringGdańsk University of TechnologyGdańskPoland

Section editors and affiliations

  • Victor A. Eremeyev
    • 1
  1. 1.Faculty of Civil and Environmental EngineeringGdańsk University of TechnologyGdańskPoland