Anti-plane surface waves in media with surface structure: discrete vs. continuum model
aa r X i v : . [ phy s i c s . c l a ss - ph ] J un Anti-plane surface waves in media with surfacestructure: discrete vs. continuum model
Victor A. Eremeyev a,b, ∗ , Basant Lal Sharma c a Faculty of Civil and Environmental Engineering, Gda´nsk University of Technology,ul. Gabriela Narutowicza 11/12 80-233 Gda´nsk, Poland b Don State Technical University, Gagarina sq., 1, 344000 Rostov on Don, Russia c Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur208016, India
Abstract
We present a comparison of the dispersion relations derived for anti-plane sur-face waves using the two distinct approaches of the surface elasticity vis-a-visthe lattice dynamics. We consider an elastic half-space with surface stressesdescribed within the Gurtin–Murdoch model, and present a formulation of itsdiscrete counterpart that is a square lattice half-plane with surface row of parti-cles having mass and elastic bonds different from the ones in the bulk. As bothmodels possess anti-plane surface waves we discuss similarities between the con-tinuum and discrete viewpoint. In particular, in the context of the behaviourof phase velocity, we discuss the possible characterization of the surface shearmodulus through the parameters involved in lattice formulation.
Keywords: lattice dynamics, surface elasticity, surface waves, anti-planeshear, Gurtin–Murdoch model
1. Introduction
Recent advances in nanotechnology have resulted in growing interest to theapplication of discrete and continuum models for the description and under-standing of the phenomena at the nanoscale. In particular, the surface elasticity ∗ Corresponding author. Tel.: +48 58 3471891
Email addresses: [email protected] (Victor A. Eremeyev), [email protected] (Basant Lal Sharma )
Preprint submitted to International Journal of Engineering Science June 18, 2019 odel proposed by Gurtin and Murdoch (1975, 1978) and its further extensions have found significant applications to the modelling of material behaviour at thenanoscale, see, e.g., Duan et al. (2008); Wang et al. (2011); Javili et al. (2013);Eremeyev (2016). Indeed, surface elasticity has been found very useful in thedescription of such phenomena due to a prominent size-effect observed for thenano-structured materials. As in the general framework of continuum mechan- ics, a key problem of the material description, also within the Gurtin–Murdochmodel, is the determination of the additional material parameters such as sur-face elastic moduli and surface mass density. A straightforward experimentalapproach to their measurement has been presented by Cuenot et al. (2004);Jing et al. (2006); Xu et al. (2017) but it requires rather complex techniques as well as some additional assumptions concerning the material behaviour and theused model. An alternative approach uses the numerical technique of molec-ular dynamics simulations, see, e.g., Miller and Shenoy (2000); Shenoy (2005),where the surface elastic moduli are determined from direct atomistic simula-tions. Let us note that the lattice dynamics as described by Brillouin (1946); Born and Huang (1985) provides the possibility to solve various dynamical prob-lems involving waves with attention focussed on the influence of microstructurein as much detail as possible, including even the surface microstructure, seeSlepyan (2002); Mishuris et al. (2007, 2009); Porubov and Andrianov (2013);Sharma (2017b); Porubov et al. (2018) and the references therein. The aim of this paper is to characterize the material parameters used inthe linear Gurtin–Murdoch model through the lattice model parameters. Tothis end we consider the anti-plane surface waves and compare the dispersionrelations derived within the continuum and discrete model.The paper is organized as follows. First, following Eremeyev et al. (2016) in Section 2, we briefly review the anti-plane surface waves in an elastic half-spaceassuming the presence of surface stresses within the linear Gurtin–Murdochmodel. The dispersion relation is derived. In Section 3 using the techniquedescribed by Sharma (2015a,b, 2017a) we find dispersion relation for a squarelattice with one surface row of particles which masses and bonds stiffness are
2. Gurtin–Murdoch model of surface elasticity
Let us consider a three-dimensional elastic half-space y ≤
0, where x , y , z areCartesian coordinates, and i , j , k are the unit basis vectors, see Fig. 1(1). On thefree boundary y = 0 we assume the action of surface stresses described withinthe model of surface elasticity by Gurtin and Murdoch (1975, 1978). Within thelinear Gurtin-Murdoch model it has been shown earlier that anti-plane surfacewaves can be constructed, see Eremeyev et al. (2016). For anti-plane deforma-tion, given by the displacement u ( x, y, t ) = u ( x, y, t ) k , the equation of motionfor x ∈ R , y < µ (cid:18) ∂ ∂x + ∂ ∂y (cid:19) u ( x, y, t ) = ρ ¨ u ( x, y, t ) , (1)while for the boundary y = 0 we have µ ∂∂y u ( x, y, t ) = µ s ∂ ∂x u ( x, y, t ) − ρ s ¨ u ( x, y, t ) , (2)where µ is the shear modulus, ρ is the mass density, µ s and ρ s are the surfaceshear modulus and mass density, respectively. For infinitesimal anti-plane mo- tions within the Gurtin–Murdoch model, given µ and ρ for the bulk, one alsoneeds to find the two surface parameters that is µ s and ρ s .Considering an anti-plane surface wave of the form u ( x, y, t ) = u exp( ikx − iωt ) exp( γy ) , (3)where ω is the circular frequency, k is the wave number, u is the amplitude,and i is the imaginary unit, we find the dispersion relation µγ ( k, ω ) = ρ s ω − µ s k , γ = γ ( k, ω ) ≡ (cid:18) k − ω c T (cid:19) / , (4)where c T = p µ/ρ is the shear wave speed, see Eremeyev et al. (2016) for details.Introducing the phase velocity c = ω/k and the characteristic length r = ρ s /ρ ,3Sfrag replacements
1) 2) a aa αK MKmM y = 0 y = 0 y = − y = − xy z xy z ii jj kk x = − x = 0 x = 1 Figure 1: 1) Half-space with surface stresses and 2) a square lattice with different particles atthe surface. we transform (4) into the dimensionless form c c T = c s c T + 1 | k | r s − c c T , (5)where c s = p µ s /ρ s is the shear wave speed in the thin film associated withthe Gurtin–Murdoch model. The solution of (5) exists if and only if c is in therange c s < c ≤ c T . In addition, we have the following properties c (0) = c T and c ( k ) → c s at k → ∞ .
3. Lattice Model
Following the technique by Brillouin (1946); Mishuris et al. (2009); Sharma(2015a, 2017a) let us consider a square lattice which occupies half-space y ≤ x ∈ Z , y ≤ y ∈ Z . The lattice mostly consists ofidentical particles of mass M connected to each other by linearly elastic bonds(springs) of stiffness K . In order to model surface tension we assume that the4ree surface y = 0 is constituted by particles with masses mM and bonds withspring constant αK , whereas m and α are dimensionless parameters. The anti-plane displacement of a particle, indexed by its lattice coordinates x ∈ Z , y ∈ Z ,is denoted by u x , y . Herein and after, let Z denote the set of integers. The motionequation for square lattice is given by M ¨ u x , y = K (cid:0) u x +1 , y + u x − , y + u x , y +1 + u x , y − − u x , y (cid:1) (6)for x ∈ Z , y < y ∈ Z , see, e.g. Sharma (2017a). On the free surface that isfor x ∈ Z , y = 0 we have mM ¨ u x , y = αK (cid:0) u x +1 , y + u x − , y − u x , y (cid:1) + K (cid:0) u x , y − − u x , y (cid:1) . (7)Let us consider the discrete analogue of the surface wave form (3), i.e., u x , y = u exp( iξ x − iωt ) exp( η y ) , (8)where ξ is the discrete wave number, ξ ∈ ( − π, π ), and η is assumed to bepositive. It is found that ω and η satisfy the two equations − M ω = K (2 cos ξ + 2 cosh η − , (9) − mM ω = αK (2 cos ξ −
2) + K (exp( − η ) − . (10)Motivated by a continuum context (Sharma, 2015a), let M = ρa , K = µa, (11)where ρ and µ are the mass density and shear modulus introduced in Section 2.Then from (9) and (10) we get ω = c T a (4 − ξ − η ) , (12) ω =2 αc T ma (1 − cos ξ ) + c T ma (1 − exp( − η )) . (13)These two equations result in a dispersion relation for the surface waves onsquare lattice half-plane with surface structure. As ξ and η play a role of k and γ , respectively, Eqs. (12) and (13) are the discrete analogues of the dispersion relation (4) for the elastic half-space with surface stresses.5 . Comparison of surface wave dispersion In order to compare the dispersion relation (5) with (12) and (13) we sub-stitute ξ = ka and consider k in the range k ∈ [0 , π/a ]. The typical dispersioncurves is shown in Fig. 2. All curves start from the point (0 , c T ) with a hori-zontal tangent c = c T . Two horizonal dashed lines correspond to c = c T and c = c s , respectively. Curves c = c GM ( k ) and c = c lm ( k ) present the solutions of(5) for the Gurtin–Murdoch model and of (12) and (13) for the lattice model.The dashed blue curve in Fig. 2 corresponds to the equation c = c o ( k ) ≡ c T (cid:12)(cid:12) sin (cid:0) ka (cid:1)(cid:12)(cid:12) ka , (14)which gives the phase velocity c o for an infinite square lattice (Brillouin, 1946;Sharma, 2017a). Here we used the following values of material parameters: c T = 1, c s = √ . r = 0 .
005 for continuum model and a = 0 . α = 0 . m = 0 . c GM (0) = c T = c lm (0) , (15)which constitutes the first correspondence between continuum and discrete model.From (15) we get the relation c T = r µρ = a r KM , which is consistent with assumption (11). So for long wave approximation ( k ≈
0) we have good coincidence between both discrete and continuum model.Clearly, while keeping m and α constant as a → plane surface wave as a continuum limit of the discrete model, since in this casewe recover an elastic half-space for which it is well known that such waves donot exist (Achenbach, 1973). Hence, to capture the behaviour of the Gurtin–Murdoch model one needs to apply an appropriate scaling for m and α .We propose the following scaling law α = 1 a µ s µ , m = 1 a ρ s ρ . (16)6Sfrag replacements c T c s kc GM c o c lm c π/a c = c GM ( k ) is the phase velocity for the Gurtin–Murdoch model given by (5) c = c lm ( k ) is the phase velocity for the lattice model given by (12) and (13) c = c o ( k ) is the phase velocity for an infinite square lattice given by (14) Figure 2: Phase velocity vs. wave number for discrete and continuum model.
With (16) we find that the surface bond stiffness becomes constant as a → αK = µ s , whereas the mass of surface particles mM = ρ s a . As a results, for c s we have c s = r µ s ρ s = r αKmM a = r αm c T . (17)Thus, the scaling law (16) establishes the second correspondence between con- tinuum and discrete model or, more precisely, between lattice model with surfaceparticles different from the ones in the bulk and the Gurtin–Murdoch model ofsurface elasticity. Using (16) in Fig. 3 we present the dispersion relations for a = 0 .
001 and a = 0 . c T c T c s c s kk c GM c GM c lm c lm c o c o cc π/aπ/a Figure 3: Phase velocity vs. wave number for discrete and continuum model for a = 0 . a = 0 . Let us note that the relations between the linear Gurtin–Murdoch model andthe lattice model in a certain sense similar to relations between surface elasticityand the Toupin–Mindlin linear strain gradient elasticity. Indeed, both theoriespossess surface energy and the corresponding dispersion relations for anti-planesurface waves are qualitatively similar for both models (Eremeyev et al., 2018).The relations between material parameters of these models can be obtainedfrom the equations c GM (0) = c T = c T M (0) , lim k →∞ c GM ( k ) = c s = lim k →∞ c T M ( k ) , where c T M = c T M ( k ) is the phase velocity for the Toupin–Mindlin constitutiverelations. Nevertheless, there is difference in decay with the depth, so theircorrespondence is not straightforward as in presented case here. In addition, forthe discrete model c lm ( k ) is defined for the finite range of k . Conclusions For anti-plane surface waves, we demonstrate the essential similarity betweendispersion relations derived within both discrete and continuum model of asurface structure. We consider a square semi-infinite lattice with a surfacerow of particles which properties are different from ones in the bulk, and thelinear Gurtin–Murdoch model of surface elasticity. These different models can a . Acknowledgments V.A.E. acknowledges the support of the Government of the Russian Federa-tion (contract No. 14.Z50.31.0046). B.L.S. acknowledges the support of SERBMATRICS grant MTR/2017/000013.
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