# Anti-plane Surface Waves in Materials with Surface Energy

**DOI:**https://doi.org/10.1007/978-3-662-53605-6_171-1

- 51 Downloads

## Synonyms

## Definitions

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.

## Introduction

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

*x*

_{3}≤ 0. Here

*x*

_{1},

*x*

_{2}, and

*x*

_{3}are the Cartesian coordinates with corresponding unit base vectors

**i**

_{k},

*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)

*t*is time. With (1) we get the formula for the gradient of the displacement vector

*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*.

*ω*is a circular frequency, and \(i=\sqrt {-1}\) is the imaginary unit. With (3) Eq. (2) transforms into

*x*

_{3}→−

*∞*solution

*k*is a wavenumber, U

_{0}is a constant, and

*κ*is given by the relation

*c*

_{T}is the phase velocity of transverse waves in the bulk (Achenbach, 1973).

## Boundary Conditions Within the Surface Elasticity

_{0}= 0. So in this case,

**u**=

**0**and anti-plane surface waves do not exist.

*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

*μ*

_{s}is a surface shear modulus. As a result, the generalized Young–Laplace equation (7) reduces to

_{0}≠0 we get the dispersion relation (Eremeyev et al., 2016)

*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*=

*ω*∕

*k*and characteristic dynamic wavenumber

*p*=

*ρ*∕

*m*we transform (10) into dimensionless form

*k*is real if and only if

*c*lies in the range

*c*

_{s}= 0.75

*c*

_{T}and \(\bar {k}=k/p\). Dispersion curve starts at the point (0,

*c*

_{T}) with the horizontal tangent. Then

*c*(

*k*) →

*c*

_{s}when

*k*→

*∞*. So the surface stresses are more pronounced for short waves as it should be.

## Conclusions

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.

## Cross-References

## References

- Achenbach J (1973) Wave Propagation in elastic solids. North Holland, AmsterdamzbMATHGoogle Scholar
- Duan HL, Wang J, Karihaloo BL (2008) Theory of elasticity at the nanoscale. In: Van der Giessen E, Aref H (eds) Advances in applied mechanics, vol 42. Elsevier, Burlington, pp 1–68Google Scholar
- Eremeyev VA (2016) On effective properties of materials at the nano-and microscales considering surface effects. Acta Mech 227(1):29–42MathSciNetCrossRefGoogle Scholar
- Eremeyev VA (2019) Surface elasticity models: comparison through the condition of the anti-plane surface wave propagation. In: "Altenbach H, Öchsner A (eds) State of the art and future trends in material modeling. Advanced structured materials, vol 100. Springer, Cham, pp 113–124Google Scholar
- Eremeyev VA, Sharma BL (2019) Anti-plane surface waves in media with surface structure: discrete vs. continuum model. Int J Eng Sci 143:33–38MathSciNetCrossRefGoogle Scholar
- Eremeyev VA, Rosi G, Naili S (2016) Surface/interfacial anti-plane waves in solids with surface energy. Mech Res Commun 74:8–13CrossRefGoogle Scholar
- Eremeyev VA, Cloud MJ, Lebedev LP (2018) Applications of tensor analysis in continuum mechanics. World Scientific, New JerseyCrossRefGoogle Scholar
- Eremeyev VA, Rosi G, Naili S (2019) Comparison of anti-plane surface waves in strain-gradient materials and materials with surface stresses. Math Mech Solids 24:2526–2535. https://doi.org/10.1177/1081286518769960 MathSciNetCrossRefGoogle Scholar
- Georgiadis H, Vardoulakis I, Lykotrafitis G (2000) Torsional surface waves in a gradient-elastic half-space. Wave Motion 31(4):333–348MathSciNetCrossRefGoogle Scholar
- Gourgiotis P, Georgiadis H (2015) Torsional and {SH} surface waves in an isotropic and homogenous elastic half-space characterized by the Toupin–Mindlin gradient theory. Int J Solids Struct 62(0):217–228CrossRefGoogle Scholar
- Gurtin ME, Murdoch AI (1975) A continuum theory of elastic material surfaces. Arch Ration Mech Anal 57(4):291–323MathSciNetCrossRefGoogle Scholar
- Gurtin ME, Murdoch AI (1978) Surface stress in solids. Int J Solids Struct 14(6):431–440CrossRefGoogle Scholar
- Javili A, McBride A, Steinmann P (2013) Thermomechanics of solids with lower-dimensional energetics: on the importance of surface, interface, and curve structures at the nanoscale. A unifying review. Appl Mech Rev 65(1):010802Google Scholar
- Lebedev LP, Cloud MJ, Eremeyev VA (2010) Tensor analysis with applications in mechanics. World Scientific, New JerseyCrossRefGoogle Scholar
- Vardoulakis I, Georgiadis HG (1997) SH surface waves in a homogeneous gradient-elastic half-space with surface energy. J Elast 47(2):147–165MathSciNetCrossRefGoogle Scholar
- 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: 52–82CrossRefGoogle Scholar