## Abstract

The dynamic response of a homogeneous half-space, with a traction-free surface, is considered within the framework of non-local elasticity. The focus is on the dominant effect of the boundary layer on overall behaviour. A typical wavelength is assumed to considerably exceed the associated internal lengthscale. The leading-order long-wave approximation is shown to coincide formally with the ‘local’ problem for a half-space with a vertical inhomogeneity localized near the surface. Subsequent asymptotic analysis of the inhomogeneity results in an explicit correction to the classical boundary conditions on the surface. The order of the correction is greater than the order of the better-known correction to the governing differential equations. The refined boundary conditions enable us to evaluate the interior solution outside a narrow boundary layer localized near the surface. As an illustration, the effect of non-local elastic phenomena on the Rayleigh wave speed is investigated.

## 1. Introduction

Analysis of non-local elastic phenomena is of major interest for various advanced applications including micro- and nanomechanics (e.g. [1–3]). Non-local elasticity is a particularly powerful and appropriate theory for investigating properties of solids with impurities, dislocations and granular microstructure. The fundamental concepts underpinning contemporary non-local continuum models were developed in a series of well-known papers by Kroner [4], Eringen [5], Eringen & Edelen [6]; see also [7] and references therein. The state of art has been presented by a number of authors throughout the area's scientific development [8–13]. The latter paper addresses Piola's important contribution, not widely known for a long time to a broad international audience, see also references to Piola's original papers in Dell'Isola *et al.* [13].

Among other recent publications on the subject, we mention papers by Di Paola & Zingales [14], Di Paola *et al.* [15], Zingales [16], Schwartz *et al.* [17], Benvenuti & Simone [18], Abdollahi & Boroomand [19,20], dealing with various analytical and numerical aspects of non-local elasticity. Here, we also cite publications developing novel micromechanical approaches known as ‘structured deformations’ (e.g. [21–23]).

Non-local models (e.g. [24–26]) are oriented to the investigation of the distant interaction between small material particles, assuming that the stress at a reference point is dependent upon the entire strain field in the body. The associated constitutive relations are usually expressed through integral operators involving internal sizes which characterize microstructure. As a rule (e.g. [27]), the long-wave limit of the non-local elasticity relations is identical to its classical counterpart. We also remark that a number of non-local elasticity predictions are in good agreement with lattice dynamics, including the regions near the boundaries of the body [28].

In spite of the numerous publications, the fundamental effect of boundaries on the implementation of non-local elasticity concepts has not yet been properly addressed. The key point is that the intervals of integration corresponding to the above-mentioned operators, expressing non-local constitutive relations, are dependent on the distance from a reference point to the boundary [25]. This results in boundary layers corresponding to localized non-homogeneous stress and strain fields. In this paper, we fill the gap in tackling the influence of boundary layers on overall dynamic behaviour. Although several authors emphasized the crucial role of boundary layers (e.g. [29,20]), we are not aware of any related asymptotic developments.

As an example, we consider an elastic half-space governed by the non-local equations given in Eringen [25], see §2. For the sake of definiteness, we assume that the non-local behaviour is modelled by an exponential kernel involving a small internal lengthscale. In §3, we proceed with a long-wave asymptotic scheme, originating from Goldenveizer *et al.* [30] and later developed by, for example, Dai *et al.* [31] and Aghalovyan [32]. Within the framework of these studies, the characteristic wavelength is assumed to be much greater than a typical microscale parameter. We begin by reducing the original non-local problem to a formulation which is identical to the classical problem for an elastic half-space with a vertical inhomogeneity localized near the surface. The effect of the inhomogeneity can be reduced to effective boundary conditions imposed at a near-surface interface. In this case, we can only asymptotically evaluate the interval, yielding the location of the interface. A better option seems to be a transformation of the effective conditions to refined boundary conditions along the surface of a homogeneous half-space. This approach is exploited in §4, enabling us to evaluate the interior stress and strain outside the narrow boundary layer. In §5, the refined boundary conditions are applied to calculate the non-local correction to the Rayleigh surface wave. The order of this correction exceeds that of the correction established in Eringen [25] associated with the non-local differential equations of motion.

## 2. Equations of non-local linear elasticity

In this section, we use as our starting point the equations of non-local elasticity, (e.g. [25]). For a homogeneous isotropic elastic solid, we therefore have (2.1)–(2.5) below:
*u*_{β}, *β*=1,2,3, the components of the displacement vector, *ρ* volume density, *t* time and
*s*_{αβ} and *σ*_{αβ} are the non-local and classical stress tensors, respectively, considered at time *t*, **x**=(*x*_{1},*x*_{2},*x*_{3}) is a reference point, *V* the domain occupied by the body, *K*(**x**,*a*) the so-called non-local modulus, and *a* is an internal characteristic length, e.g. lattice parameter or granular distance. Throughout the paper we assume that the internal size *a* is asymptotically small in comparison with a typical wavelength. This long-wave assumption provides the validity of the adapted non-local model for bounded domains as it follows on, in particular, from lattice dynamics [28]; for further details, see concluding remarks.

The function *K* in (2.2) is normalized over three-dimensional space, so that
*e*_{αβ} is the linear elastic strain tensor, *δ*_{αβ} the Kronecker's delta, and *λ* and *μ* are the Lamé constants.

For the sake of definiteness, we specify the three-dimensional exponential non-local modulus in the same way as Eringen [25], thus
*σ*_{αβ} in Taylor series about the reference point **x**′=**x**, assuming as before that the typical wavelength characterizing the classical stress field is much greater than the internal size *a*. Thus, we establish from (2.7) that
*a* which is specific for a half-space. Such term does not appear in the case of three-dimensional space.

The formulae (2.2), taking into account (2.4) and keeping the leading-order term in (2.9), may be presented as
*h* (figure 1), where *h*≫*a*. This strong inequality justifies the validity of non-local theory on the scale of layer thickness. As a rule, the asymptotic error of the one-term expansion in (2.9) is *O*(*a*/*h*). It is less than *O*(*a*/*h*) only provided that the associated local field is uniform in *x*_{3}.

Along the interface *x*_{3}=*h*, to within an exponentially small error, erfc (−*h*/*a*)=2 and, consequently,
*s*_{αβ} tend to their local analogues *σ*_{αβ} in (2.9).

In the case of a thin layer of thickness much smaller than a macroscale wavelength, its effect on the substrate may be incorporated by deriving effective boundary conditions using well-known asymptotic methodology (see for example [31,32] and references therein).

## 3. Asymptotic analysis of a vertically inhomogeneous thin layer

Let us consider a thin, vertically inhomogeneous layer of thickness *h*≪ℓ, where ℓ is a typical wavelength with *ε*=*h*/ℓ assumed to be a small geometric parameter. Equations (2.1) and (2.10) in the previous section are formally identical to the classical ‘local’ equations. They can be rewritten as
*i*≠*j*=1,2 and Einstein's summation convention is not employed. The variable wave speeds in (3.2), inspired by (2.11), are given by

The traction-free boundary conditions at the surface of the layer *x*_{3}=0 are given by
*x*_{3}=*h* requiring that
*v*_{n}=*v*_{n}(*x*_{1},*x*_{2},*t*) denotes the prescribed displacements in the substrate, *n*=1,2,3.

We now adapt the asymptotic approach developed by Goldenveizer *et al.* [30], Dai *et al.* [31] and Aghalovyan [32] in order to express the stresses *s*_{3n} along the interface *x*_{3}=*h* in terms of the prescribed substrate displacements *v*_{n}. To begin, we scale the original variables as follows:
*c*_{2}=*c*_{2}′(*h*), and also define the dimensionless quantities
*V* is the maximum displacement amplitude and all quantities with an asterisk are assumed to be of the same asymptotic order.

The equations of motion (3.1) and constitutive relations (3.2) can now be rewritten as
*κ*_{m}′=*c*_{m}′(*x*_{3})/*c*_{2}, *m*=1,2.

It is convenient to express _{2} from (3.9)_{4}, having

The boundary conditions (3.4) and (3.5) become

Next, we expand the displacements and stresses in asymptotic series in terms of the previously specified small parameter *ε*, and thus introduce
_{2} and (3.14)_{3} with respect to *η*, and taking into account the appropriate boundary conditions (3.15), we may establish that
_{2}

We finally integrate (3.13)_{1}, using (3.14)_{2} and (3.16), and then satisfy (3.15)_{2}, to establish that
*s*_{3i} and *s*_{33} may be obtained from (3.19) and (3.17) in the form

In what follows we also use the formula for other components of the non-local stress tensor, which are given by
*x*_{3}=*h*, may be expressed through the substrate displacements, yielding
*κ*_{m}′=*c*_{m}′/*c*_{2}, *m*=1,2 as above.

## 4. Refined boundary conditions

For a vertically inhomogeneous layer with the elastic moduli given by (2.11), we obtain
*a*≪*h*, to within an exponentially small error may be presented in the forms
*κ*=*c*_{2}/*c*_{1}. Thus, the stresses along the interface *x*_{3}=*h* may be presented as

Inspection of (4.2)_{1} shows that taking non-local elastic properties into account results in an asymptotic correction of the relative asymptotic order *O*(*a*/*h*). On the other hand, this correction must be greater than the truncation error *O*(*ε*) related to the asymptotic derivation of formulae (3.22) in §3. Thus, we arrive at the double strong inequality
*O*(*a*/*h*) correction associated with (4.2). To this end, we recall that at leading order, the stresses *s*_{3i} and *s*_{33} are expressed in terms of the stresses *s*_{ii} and the displacements *u*_{n} in §3, see (3.13). In this case, the local stresses *σ*_{ii} and *σ*_{ij} corresponding to their non-local counterparts *s*_{ii} and *s*_{ij} in formula (3.21), following from the dimensionless formula (3.13)_{1}, are given by
*O*(*a*/*h*) contribution of the second term in the expansion (2.9) vanishes after differentiation with respect to *x*_{3}. For the same reason, the inertial terms in (3.13) will also not make a *O*(*a*/*h*) contribution to non-local stresses.

It is clear that outside a narrow near-surface layer, all non-local stresses, to within the asymptotic error less than *O*(*a*/*h*), coincide with their local analogues (see (2.9)–(2.12)). Therefore, over the interior domain *x*_{3}≥*h*, we may proceed with a classical problem with constant coefficients, subject to the boundary conditions
*s*_{3n} are given by (4.2).

We are now in a position to formulate an inverse problem for a thin homogeneous elastic layer within the classical framework. A crucial aspect that now needs addressing concerns the boundary conditions to be imposed on the surface *x*_{3}=0 so that the stresses *σ*_{3i} and *σ*_{33}, at the interface *x*_{3}=*h*, satisfy the conditions (4.5). Let the boundary conditions at the surface of the layer *x*_{3}=0 be given by
*p*_{n} are the sought for surface stresses.

The asymptotic solution of the classical elastodynamic equations for a thin homogeneous layer, subject to the boundary conditions (4.5) and (4.6) along the faces *x*_{3}=0 and *x*_{3}=*h*, is presented in Dai *et al.* [31]. The formulae for the stresses of interest at *x*_{3}=*h* may be written as
*σ*_{3n} in (4.5) and (4.7), we have
*x*_{3}=0, become
*x*_{3}≫*a*), the half-space motion is governed by the elastodynamic equations with constant moduli *λ* and *μ*, subject to the boundary conditions (4.9). The last formulae involve *O*(*a*/ℓ) correction, where ℓ is a typical macroscale size as described above. This is greater than the *O*(*a*^{2}/ℓ^{2}) correction in the differential equations of non-local elasticity [25].

## 5. Rayleigh surface wave

As an illustration, we consider the effect of non-local elastic behaviour on surface wave propagation in the case of plane strain, in which ∂/∂*x*_{2}≡0, *u*_{m}=*u*_{m}(*x*_{1},*x*_{3}), *m*=1,3, and *u*_{2}=0. Accordingly, the two boundary conditions, following directly from (4.9), become
*φ* and *ψ*, are given by
*Δ* is the two-dimensional Laplacian.

We first look for travelling wave solutions of the form
*c* is the phase speed and in which the attenuation coefficients are given by
*θ*=*a*/ℓ=*ak*/2*π*≪1 is a small parameter, *γ*=*c*/*c*_{2} and *R*(*γ*) is the Rayleigh denominator, i.e.
*γ* as an asymptotic series in the small parameter *θ*, with
*R*(*γ*), about *γ*=*γ*_{0}, is given by
*γ*_{0} is the normalized classical Rayleigh wave speed, i.e. *R*(*γ*_{0})=0. Then, on substituting (5.8) and (5.9) into (5.7), we readily obtain
*θ*=*ak*/2*π*, as before.

We remark that the constructed correction, originating from the refined boundary conditions (5.1), exceeds the correction in Eringen [25], associated with the ‘non-local terms’ within the differential equations of motion.

Numerical results are presented in figure 2. The classical Rayleigh root *γ*_{0} and the ‘non-local’ root *γ* in (5.11) are plotted as function of the small parameter *θ* for the value of Poisson ratio *ν*=0.25. For this scenario, the coefficient (5.10) takes the value *γ*_{1}=−0.37, while its ‘local’ counterpart is *γ*_{0}=0.92. The effect of non-local phenomena decreases the Rayleigh wave speed due to low values of the Lamé parameters, denoting the stiffness of the system, near the surface, see (2.11).

## 6. Concluding remarks

An asymptotic treatment of the non-local boundary value problem under consideration demonstrates the primary importance of analysing the peculiarities of near-surface behaviour. It has been established that the effect of the associated boundary layer may be incorporated just by refining the boundary conditions in classical elasticity. In particular, the refined boundary conditions (4.9) involve an explicit correction to their classical counterparts; this arises by taking into account non-local phenomena.

The linear elastodynamic equations, subject to the derived boundary conditions on the free surface of a homogeneous half-space, enable us to determine the interior stress and strain fields outside a narrow near-surface layer, with thickness satisfying the asymptotic inequality (4.3). As an illustration, *O*(*a*/ℓ) non-local correction to the Rayleigh surface wave speed was calculated. This correction is greater than *O*(*a*^{2}/ℓ^{2}) correction associated with the non-local equations of motion in Eringen [25].

We recall that approximate nature of non-local models originates from truncation of homogenization procedures, including asymptotic homogenization for periodic structures (e.g. [33,34]), underlying the associated macroscale relations. In this case, the truncation error for the classical boundary conditions should be of the same order as the deviation from the uniform microscale variation of the sought for solution. The latter might be expected to be negligible in comparison with *O*(*a*/ℓ) correction suggested in the paper. In particular, it is *O*(*a*^{2}/ℓ^{2}) for a range of periodic lattices [35]. This issue certainly merits a thorough consideration.

We remark that the proposed approach is not merely restricted to the exponential kernel (2.6) studied in this paper. We envisage similar non-local effects for a range of kernels having the same asymptotic behaviour at small internal scales. The results obtained may also readily be extended to non-locally elastic solids with a boundary of arbitrary shape. Investigation of elastic waveguides, including beams, plates and shells, with the boundary conditions of the form (4.9) imposed on the free faces would also be of obvious interest. This would in fact seem to be a natural generalization of the above-mentioned example for the Rayleigh surface wave.

The general asymptotic scheme presented in §3 may also seemingly have potential applications outside the area of non-local elasticity. Firstly, we note applications for solids with localized near-surface inhomogeneities, such as functionally graded structures (see for example the review by Birman & Byrd [36]). There is also the possibility of adapting this scheme for long-wave dynamic analysis of vertically inhomogeneous foundations (see Muravskii [37] and references therein).

## Authors' contributions

All authors developed the asymptotic approach and wrote the paper. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

PhD studies of R.C. were partly supported by Keele University. The support is gratefully acknowledged.

## Acknowledgements

The authors greatly appreciate valuable discussions with Dr D. A. Prikazchikov.

- Received November 17, 2015.
- Accepted January 11, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.