Refined boundary conditions on the free surface of an elastic half-space taking into account non-local effects

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:

$$s_{\alpha \beta ,\alpha }=\rho \frac{{\mathrm{\partial }}^2{u}_{\beta }}{\mathrm{\partial }{t}^2},$$ 2.1

with u_β, β=1,2,3, the components of the displacement vector, ρ volume density, t time and

$$s_{\alpha \beta }(\mathbf{\text{x}})={\int }_{V}K(|{\mathbf{\text{x}}}^{\prime }-\mathbf{\text{x}}|,a){\sigma }_{\alpha \beta }({\mathbf{\text{x}}}^{\prime })\hspace{0.17em}\mathrm{d}v({\mathbf{\text{x}}}^{\prime }),$$ 2.2

where s_αβ and σ_αβ are the non-local and classical stress tensors, respectively, considered at time t, x=(x₁,x₂,x₃) 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

$${\int }_{V_{\mathrm{\infty }}}K(|{\mathbf{\text{x}}}^{\prime }|,a)\hspace{0.17em}\mathrm{d}v({\mathbf{\text{x}}}^{\prime })=1.$$ 2.3

The two equations (2.1) and (2.2) are accompanied by

$${\sigma }_{\alpha \beta }=\lambda e_{\gamma \gamma }{\delta }_{\alpha \beta }+2\mu e_{\alpha \beta }$$ 2.4

and

$$e_{\alpha \beta }=\frac{1}{2}(\frac{\mathrm{\partial }{u}_{\alpha }}{\mathrm{\partial }{x}_{\beta }}+\frac{\mathrm{\partial }{u}_{\beta }}{\mathrm{\partial }{x}_{\alpha }}),$$ 2.5

where 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

$$K(|\mathbf{\text{x}}|,a)=\frac{1}{{\pi }^{3/2}{a}^3}\exp [-\frac{\mathbf{\text{x}}\cdot \mathbf{\text{x}}}{a^2}],$$ 2.6

where in the case of a half-space $-\mathrm{\infty }<x_1<\mathrm{\infty }$, $-\mathrm{\infty }<x_2<\mathrm{\infty }$ and $0\le x_3<\mathrm{\infty }$, (2.2) becomes

$$s_{\alpha \beta }(\mathbf{\text{x}})=\frac{1}{{\pi }^{3/2}{a}^3}{\int }_0^{\mathrm{\infty }}\hspace{0.17em}\mathrm{d}{x}_3^{\prime }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hspace{0.17em}\mathrm{d}{x}_1^{\prime }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hspace{0.17em}\mathrm{d}{x}_2^{\prime }\exp [-\frac{{({\mathbf{\text{x}}}^{\prime }-\mathbf{\text{x}})}^2}{a^2}]{\sigma }_{\alpha \beta }({\mathbf{\text{x}}}^{\prime }).$$ 2.7

Let us now expand the stresses 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

$$\begin{array}{rl}{s}_{\alpha \beta }(\mathbf{\text{x}})& =\frac{1}{a\sqrt{\pi }}\{{\sigma }_{\alpha \beta }(\mathbf{\text{x}}){\int }_0^{\mathrm{\infty }}\exp [-\frac{{(x_3^{\prime }-x_3)}^2}{a^2}]\mathrm{d}{x}_3^{\prime }+\frac{\mathrm{\partial }{\sigma }_{\alpha \beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_3}{\int }_0^{\mathrm{\infty }}(x_3^{\prime }-x_3)\exp [-\frac{{(x_3^{\prime }-x_3)}^2}{a^2}]\mathrm{d}{x}_3^{\prime }\}\\ & +\dots ,\end{array}$$ 2.8

which on integration yields

$$s_{\alpha \beta }(\mathbf{\text{x}})=\frac{{\sigma }_{\alpha \beta }(\mathbf{\text{x}})}{2}\text{ erfc }(-\frac{x_3}{a})+\frac{a}{2\sqrt{\pi }}\frac{\mathrm{\partial }{\sigma }_{\alpha \beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_3}\exp [-\frac{x_3^2}{a^2}]+\dots ,$$ 2.9

where $\text{ erfc }(x)=(2/\sqrt{\pi }){\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2}\hspace{0.17em}\mathrm{d}t$. Here, we keep only a linear term in 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

$$s_{\alpha \beta }={\lambda }^{\prime }{e}_{\gamma \gamma }{\delta }_{\alpha \beta }+2{\mu }^{\prime }{e}_{\alpha \beta },$$ 2.10

where

$$\begin{array}{rl}{\lambda }^{\prime }& =\frac{1}{2}\text{ erfc}(-\frac{x_3}{a})\lambda \\ \text{and}{\mu }^{\prime }& =\frac{1}{2}\text{ erfc}(-\frac{x_3}{a})\mu .\end{array}\}$$ 2.11

We remark that the non-local problem (2.1), (2.10)–(2.11) is thus formally equivalent to the classical (‘local’) problem for a vertically inhomogeneous elastic half-space. This problem may in fact be reduced to that of analysis of a homogeneous elastic substrate coated by a vertically inhomogeneous layer of a certain thickness 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 x₃.

Figure 1.

A homogeneous substrate coated by a vertically inhomogeneous layer of thickness h; a≪h≪ℓ.

Along the interface x₃=h, to within an exponentially small error, erfc (−h/a)=2 and, consequently,

$${\lambda }^{\prime }(h)=\lambda \text{and}{\mu }^{\prime }(h)=\mu ,$$ 2.12

and the non-local stresses 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

$$\begin{array}{rl}\frac{\mathrm{\partial }{s}_{ii}}{\mathrm{\partial }{x}_i}+\frac{\mathrm{\partial }{s}_{ij}}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{s}_{3i}}{\mathrm{\partial }{x}_3}& =\rho \frac{{\mathrm{\partial }}^2{u}_i}{\mathrm{\partial }{t}^2}\\ \text{and}\frac{\mathrm{\partial }{s}_{3i}}{\mathrm{\partial }{x}_i}+\frac{\mathrm{\partial }{s}_{3j}}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{s}_{33}}{\mathrm{\partial }{x}_3}& =\rho \frac{{\mathrm{\partial }}^2{u}_3}{\mathrm{\partial }{t}^2},\end{array}\}$$ 3.1

and

$$\begin{array}{rl}{s}_{ij}& =\rho c_2^{{\hspace{0.17em}}^{\prime }2}(x_3)(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}),\\ s_{ii}& =\rho c_1^{{\hspace{0.17em}}^{\prime }2}(x_3)\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_i}+\rho (c_1^{{\hspace{0.17em}}^{\prime }2}(x_3)-2c_2^{{\hspace{0.17em}}^{\prime }2}(x_3))(\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_3}{\mathrm{\partial }{x}_3}),\\ s_{3i}& ={\sigma }_{i3}=\rho c_2^{{\hspace{0.17em}}^{\prime }2}(x_3)(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_3}+\frac{\mathrm{\partial }{u}_3}{\mathrm{\partial }{x}_i})\\ \text{and}{s}_{33}& =\rho c_1^{{\hspace{0.17em}}^{\prime }2}(x_3)\frac{\mathrm{\partial }{u}_3}{\mathrm{\partial }{x}_3}+\rho (c_1^{{\hspace{0.17em}}^{\prime }2}(x_3)-2c_2^{{\hspace{0.17em}}^{\prime }2}(x_3))(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_i}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_j}),\end{array}\}$$ 3.2

where 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

$$c_1^{\prime }(x_3)=\sqrt{\frac{{\lambda }^{\prime }(x_3)+2{\mu }^{\prime }(x_3)}{\rho }}\text{and}{c}_2^{\prime }(x_3)=\sqrt{\frac{{\mu }^{\prime }(x_3)}{\rho }}.$$ 3.3

The traction-free boundary conditions at the surface of the layer x₃=0 are given by

$$s_{3n}=0\text{at }{x}_3=0,$$ 3.4

with continuity of displacement along the interface x₃=h requiring that

$$u_n=v_n\text{at }{x}_3=h,$$ 3.5

where v_n=v_n(x₁,x₂,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 x₃=h in terms of the prescribed substrate displacements v_n. To begin, we scale the original variables as follows:

$${\xi }_i=\frac{x_i}{\ell },\eta =\frac{x_3}{h}\text{and}\tau =\frac{t{c}_2}{\ell },$$ 3.6

where c₂=c₂′(h), and also define the dimensionless quantities

$$u_n^{\ast }=\frac{1}{V}{u}_n,v_n^{\ast }=\frac{1}{V}{v}_n$$

and

$$s_{ij}^{\ast }=\frac{\ell }{\mu V}{s}_{ij},s_{ii}^{\ast }=\frac{\ell }{\mu V}{s}_{ii},s_{3n}^{\ast }=\frac{{\ell }^2}{\mu hV}{s}_{3n},$$ 3.7

where 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

$$\begin{array}{rl}& \frac{\mathrm{\partial }{s}_{ii}^{\ast }}{\mathrm{\partial }{\xi }_i}+\frac{\mathrm{\partial }{s}_{ij}^{\ast }}{\mathrm{\partial }{\xi }_j}+\frac{\mathrm{\partial }{s}_{3i}^{\ast }}{\mathrm{\partial }\eta }=\frac{{\mathrm{\partial }}^2{u}_i^{\ast }}{\mathrm{\partial }{\tau }^2}\\ \text{and}& \frac{\mathrm{\partial }{s}_{33}^{\ast }}{\mathrm{\partial }\eta }+\epsilon (\frac{\mathrm{\partial }{s}_{3i}^{\ast }}{\mathrm{\partial }{\xi }_i}+\frac{\mathrm{\partial }{s}_{3j}^{\ast }}{\mathrm{\partial }{\xi }_j})=\frac{{\mathrm{\partial }}^2{u}_3^{\ast }}{\mathrm{\partial }{\tau }^2},\end{array}\}$$ 3.8

and

$$\begin{array}{rl}{s}_{ij}^{\ast }& ={\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(\frac{\mathrm{\partial }{u}_i^{\ast }}{\mathrm{\partial }{\xi }_j}+\frac{\mathrm{\partial }{u}_j^{\ast }}{\mathrm{\partial }{\xi }_i}),\\ \epsilon s_{ii}^{\ast }& =({\kappa }_1^{{\hspace{0.17em}}^{\prime }2}-2{\kappa }_2^{{\hspace{0.17em}}^{\prime }2})\frac{\mathrm{\partial }{u}_3^{\ast }}{\mathrm{\partial }\eta }+\epsilon ({\kappa }_1^{{\hspace{0.17em}}^{\prime }2}\frac{\mathrm{\partial }{u}_i^{\ast }}{\mathrm{\partial }{\xi }_i}+({\kappa }_1^{{\hspace{0.17em}}^{\prime }2}-2{\kappa }_2^{{\hspace{0.17em}}^{\prime }2})\frac{\mathrm{\partial }{u}_j^{\ast }}{\mathrm{\partial }{\xi }_j}),\\ {\epsilon }^2{s}_{3i}^{\ast }& ={\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(\frac{\mathrm{\partial }{u}_i^{\ast }}{\mathrm{\partial }\eta }+\epsilon \frac{\mathrm{\partial }{u}_3^{\ast }}{\mathrm{\partial }{\xi }_i})\\ \text{and}{\epsilon }^2{s}_{33}^{\ast }& ={\kappa }_1^{{\hspace{0.17em}}^{\prime }2}\frac{\mathrm{\partial }{u}_3^{\ast }}{\mathrm{\partial }\eta }+\epsilon ({\kappa }_1^{{\hspace{0.17em}}^{\prime }2}-2{\kappa }_2^{{\hspace{0.17em}}^{\prime }2})(\frac{\mathrm{\partial }{u}_i^{\ast }}{\mathrm{\partial }{\xi }_i}+\frac{\mathrm{\partial }{u}_j^{\ast }}{\mathrm{\partial }{\xi }_j}),\end{array}\}$$ 3.9

with κ_m′=c_m′(x₃)/c₂, m=1,2.

It is convenient to express $\mathrm{\partial }{u}_3^{\ast }/\mathrm{\partial }\eta$ in (3.9)₂ from (3.9)₄, having

$$s_{ii}^{\ast }=4{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(1-\frac{{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}}{{\kappa }_1^{{\hspace{0.17em}}^{\prime }2}})\frac{\mathrm{\partial }{u}_i^{\ast }}{\mathrm{\partial }{\xi }_i}+2{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(1-\frac{2{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}}{{\kappa }_1^{{\hspace{0.17em}}^{\prime }2}})\frac{\mathrm{\partial }{u}_j^{\ast }}{\mathrm{\partial }{\xi }_j}+\epsilon (1-\frac{2{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}}{{\kappa }_1^{{\hspace{0.17em}}^{\prime }2}})s_{33}^{\ast }.$$ 3.10

The boundary conditions (3.4) and (3.5) become

$$\begin{array}{rl}{s}_{3n}^{\ast }& =0\text{at }\eta =0\\ \text{and}{u}_n^{\ast }& =v_n^{\ast }\text{at }\eta =1.\end{array}\}$$ 3.11

Next, we expand the displacements and stresses in asymptotic series in terms of the previously specified small parameter ε, and thus introduce

$$\left(\begin{array}{c}{u}_n^{\ast }\\ s_{ii}^{\ast }\\ s_{ij}^{\ast }\\ s_{3i}^{\ast }\\ s_{33}^{\ast }\end{array}\right)=\left(\begin{array}{c}{u}_n^{(0)}\\ s_{ii}^{(0)}\\ s_{ij}^{(0)}\\ s_{3i}^{(0)}\\ s_{33}^{(0)}\end{array}\right)+\epsilon \left(\begin{array}{c}{u}_n^{(1)}\\ s_{ii}^{(1)}\\ s_{ij}^{(1)}\\ s_{3i}^{(1)}\\ s_{33}^{(1)}\end{array}\right)+\cdots$$ 3.12

Substitution of these expressions into equations (3.8)–(3.10), and boundary conditions (3.11) results, at leading order, in the following equations:

$$\begin{array}{rl}& \frac{\mathrm{\partial }{s}_{ii}^{(0)}}{\mathrm{\partial }{\xi }_i}+\frac{\mathrm{\partial }{s}_{ij}^{(0)}}{\mathrm{\partial }{\xi }_j}+\frac{\mathrm{\partial }{s}_{3i}^{(0)}}{\mathrm{\partial }\eta }=\frac{{\mathrm{\partial }}^2{u}_i^{(0)}}{\mathrm{\partial }{\tau }^2}\\ \text{and}& \frac{\mathrm{\partial }{s}_{33}^{(0)}}{\mathrm{\partial }\eta }=\frac{{\mathrm{\partial }}^2{u}_3^{(0)}}{\mathrm{\partial }{\tau }^2},\end{array}\}$$ 3.13

and

$$\begin{array}{rl}{s}_{ij}^{(0)}& ={\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(\frac{\mathrm{\partial }{u}_i^{(0)}}{\mathrm{\partial }{\xi }_j}+\frac{\mathrm{\partial }{u}_j^{(0)}}{\mathrm{\partial }{\xi }_i}),\\ s_{ii}^{(0)}& =4{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(1-\frac{{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}}{{\kappa }_1^{{\hspace{0.17em}}^{\prime }2}})\frac{\mathrm{\partial }{u}_i^{(0)}}{\mathrm{\partial }{\xi }_i}+2{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(1-\frac{2{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}}{{\kappa }_1^{{\hspace{0.17em}}^{\prime }2}})\frac{\mathrm{\partial }{u}_j^{(0)}}{\mathrm{\partial }{\xi }_j}\\ \text{and}\frac{\mathrm{\partial }{u}_n^{(0)}}{\mathrm{\partial }\eta }& =0,\end{array}\}$$ 3.14

together with

$$\begin{array}{rl}{u}_n^{(0)}& =v_n^{\ast }\text{at }\eta =1\\ \text{and}{s}_{3n}^{(0)}& =0\text{at }\eta =0.\end{array}\}$$ 3.15

On integrating (3.13)₂ and (3.14)₃ with respect to η, and taking into account the appropriate boundary conditions (3.15), we may establish that

$$u_n^{(0)}=v_n^{\ast }$$ 3.16

and

$$s_{33}^{(0)}=\eta \frac{{\mathrm{\partial }}^2{v}_3^{\ast }}{\mathrm{\partial }{\tau }^2}.$$ 3.17

Now we obtain from (3.14)₂

$$s_{ii}^{(0)}=4{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(1-\frac{{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}}{{\kappa }_1^{{\hspace{0.17em}}^{\prime }2}})\frac{\mathrm{\partial }{v}_i^{\ast }}{\mathrm{\partial }{\xi }_i}+2{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(1-\frac{2{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}}{{\kappa }_1^{{\hspace{0.17em}}^{\prime }2}})\frac{\mathrm{\partial }{v}_j^{\ast }}{\mathrm{\partial }{\xi }_j}.$$ 3.18

We finally integrate (3.13)₁, using (3.14)₂ and (3.16), and then satisfy (3.15)₂, to establish that

$$\begin{array}{rl}{s}_{3i}^{(0)}& =\eta \frac{{\mathrm{\partial }}^2{v}_i^{\ast }}{\mathrm{\partial }{\tau }^2}-\frac{{\mathrm{\partial }}^2{v}_i^{\ast }}{\mathrm{\partial }{\xi }_j^2}{\int }_0^{\eta }{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}\hspace{0.17em}\mathrm{d}{\eta }^{\prime }-4\frac{{\mathrm{\partial }}^2{v}_i^{\ast }}{\mathrm{\partial }{\xi }_i^2}{\int }_0^{\eta }{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(1-\frac{{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}}{{\kappa }_1^{{\hspace{0.17em}}^{\prime }2}})\mathrm{d}{\eta }^{\prime }\\ & -\frac{{\mathrm{\partial }}^2{v}_j^{\ast }}{\mathrm{\partial }{\xi }_i\mathrm{\partial }{\xi }_j}{\int }_0^{\eta }{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(3-\frac{4{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}}{{\kappa }_1^{{\hspace{0.17em}}^{\prime }2}})\mathrm{d}{\eta }^{\prime }.\end{array}$$ 3.19

In terms of the original variables, the expressions for the stresses s_3i and s₃₃ may be obtained from (3.19) and (3.17) in the form

$$\begin{array}{rl}{s}_{3i}& =\rho [x_3\frac{{\mathrm{\partial }}^2{u}_i}{\mathrm{\partial }{t}^2}-c_2^2\frac{{\mathrm{\partial }}^2{u}_i}{\mathrm{\partial }{x}_j^2}{\int }_0^{x_3}{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}\hspace{0.17em}\mathrm{d}{x}_3^{\prime }-\rho c_2^2\frac{{\mathrm{\partial }}^2{u}_i}{\mathrm{\partial }{x}_i^2}{\int }_0^{x_3}4{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(1-\frac{{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}}{{\kappa }_1^{{\hspace{0.17em}}^{\prime }2}})\mathrm{d}{x}_3^{\prime }\\ & -c_2^2\frac{{\mathrm{\partial }}^2{u}_j}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}{\int }_0^{x_3}{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(3-\frac{4{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}}{{\kappa }_1^{{\hspace{0.17em}}^{\prime }2}})\mathrm{d}{x}_3^{\prime }]\\ \text{and}{s}_{33}& =\rho x_3\frac{{\mathrm{\partial }}^2{u}_3}{\mathrm{\partial }{t}^2},\end{array}\}$$ 3.20

where now $u_n=V{u}_n^{(0)}$.

In what follows we also use the formula for other components of the non-local stress tensor, which are given by

$$\begin{array}{rl}{s}_{ii}& =2\rho c_2^{{\hspace{0.17em}}^{\prime }2}[2(1-\frac{c_2^{{\hspace{0.17em}}^{\prime }2}}{c_1^{{\hspace{0.17em}}^{\prime }2}})\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_i}+(1-\frac{2c_2^{{\hspace{0.17em}}^{\prime }2}}{c_1^{{\hspace{0.17em}}^{\prime }2}})\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_j}]\\ \text{and}{s}_{ij}& =\rho c_2^{{\hspace{0.17em}}^{\prime }2}(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}).\end{array}\}$$ 3.21

The stresses at the interface, x₃=h, may be expressed through the substrate displacements, yielding

$$\begin{array}{rl}{s}_{3i}& =\rho [h\frac{{\mathrm{\partial }}^2{v}_i}{\mathrm{\partial }{t}^2}-c_2^2\frac{{\mathrm{\partial }}^2{v}_i}{\mathrm{\partial }{x}_j^2}{\int }_0^h{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}\hspace{0.17em}\mathrm{d}{x}_3^{\prime }-\rho c_2^2\frac{{\mathrm{\partial }}^2{v}_i}{\mathrm{\partial }{x}_i^2}{\int }_0^{h}4{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(1-\frac{{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}}{{\kappa }_1^{{\hspace{0.17em}}^{\prime }2}})\mathrm{d}{x}_3^{\prime }\\ & -\hspace{0.17em}{c}_2^2\frac{{\mathrm{\partial }}^2{v}_j}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}{\int }_0^h{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(3-\frac{4{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}}{{\kappa }_1^{{\hspace{0.17em}}^{\prime }2}})\mathrm{d}{x}_3^{\prime }]\\ \text{and}{s}_{33}& =\rho h\frac{{\mathrm{\partial }}^2{v}_3}{\mathrm{\partial }{t}^2},\end{array}\}$$ 3.22

where κ_m′=c_m′/c₂, m=1,2 as above.

4. Refined boundary conditions

For a vertically inhomogeneous layer with the elastic moduli given by (2.11), we obtain

$$c_m^{{\hspace{0.17em}}^{\prime }2}(x_3)=\frac{1}{2}{c}_m^2\text{ erfc }(-\frac{x_3}{a}),m=1,2,$$ 4.1

where, see (2.12),

$$c_1=\sqrt{\frac{\lambda +2\mu }{\rho }}\text{and}{c}_2=\sqrt{\frac{\mu }{\rho }}.$$

Consequently, the integrals in (3.22), under the assumption a≪h, to within an exponentially small error may be presented in the forms

$$\begin{array}{rl}{\int }_0^h{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}\hspace{0.17em}\mathrm{d}{x}_3^{\prime }& =h(1-\frac{1}{2\sqrt{\pi }}\frac{a}{h}),\\ {\int }_0^{h}4{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(1-\frac{{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}}{{\kappa }_1^{{\hspace{0.17em}}^{\prime }2}})\mathrm{d}{x}_3^{\prime }& =4h(1-{\kappa }^2)(1-\frac{1}{2\sqrt{\pi }}\frac{a}{h})\\ \text{and}{\int }_0^h{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}(3-\frac{4{\kappa }_2^{{\hspace{0.17em}}^{\prime }2}}{{\kappa }_1^{{\hspace{0.17em}}^{\prime }2}})\mathrm{d}{x}_3^{\prime }& =h(3-4{\kappa }^2)(1-\frac{1}{2\sqrt{\pi }}\frac{a}{h}),\end{array}$$

where κ=c₂/c₁. Thus, the stresses along the interface x₃=h may be presented as

$$\begin{array}{rl}{s}_{3i}& =\rho h[\frac{{\mathrm{\partial }}^2{v}_i}{\mathrm{\partial }{t}^2}-c_2^2\{\frac{{\mathrm{\partial }}^2{v}_i}{\mathrm{\partial }{x}_j^2}+4(1-{\kappa }^2)\frac{{\mathrm{\partial }}^2{v}_i}{\mathrm{\partial }{x}_i^2}+(3-4{\kappa }^2)\frac{{\mathrm{\partial }}^2{v}_j}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}\}\\ & +\hspace{0.17em}{c}_2^2\frac{a}{2h\sqrt{\pi }}\{\frac{{\mathrm{\partial }}^2{v}_i}{\mathrm{\partial }{x}_j^2}+4(1-{\kappa }^2)\frac{{\mathrm{\partial }}^2{v}_i}{\mathrm{\partial }{x}_i^2}+(3-4{\kappa }^2)\frac{{\mathrm{\partial }}^2{v}_j}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}\}]\\ \text{and}{s}_{33}& =\rho h\frac{{\mathrm{\partial }}^2{v}_3}{\mathrm{\partial }{t}^2}.\end{array}\}$$ 4.2

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

$$a\ll h\ll \sqrt{a\ell },$$ 4.3

underlying equations (4.2). We also need to show that the accuracy of the leading-order approximation (2.10) is consistent with the O(a/h) correction associated with (4.2). To this end, we recall that at leading order, the stresses s_3i and s₃₃ 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

$$\begin{array}{rl}{\sigma }_{ii}& =2\rho c_2^2[2(1-{\kappa }^2)\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_i}+(1-2{\kappa }^2)\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_j}]\\ \text{and}{\sigma }_{ij}& =\rho c_2^2(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}).\end{array}\}$$ 4.4

These stresses are uniform across the thickness. As a result, the expected O(a/h) contribution of the second term in the expansion (2.9) vanishes after differentiation with respect to x₃. 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 x₃≥h, we may proceed with a classical problem with constant coefficients, subject to the boundary conditions

$${\sigma }_{3n}=s_{3n}\text{at}\hspace{0.17em}{x}_3=h,$$ 4.5

where 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 x₃=0 so that the stresses σ_3i and σ₃₃, at the interface x₃=h, satisfy the conditions (4.5). Let the boundary conditions at the surface of the layer x₃=0 be given by

$${\sigma }_{3n}=p_n,$$ 4.6

where 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 x₃=0 and x₃=h, is presented in Dai et al. [31]. The formulae for the stresses of interest at x₃=h may be written as

$$\begin{array}{rl}{\sigma }_{3i}& =\rho h[\frac{{\mathrm{\partial }}^2{v}_i}{\mathrm{\partial }{t}^2}-c_2^2\{\frac{{\mathrm{\partial }}^2{v}_i}{\mathrm{\partial }{x}_j^2}+4(1-{\kappa }^2)\frac{{\mathrm{\partial }}^2{v}_i}{\mathrm{\partial }{x}_i^2}+(3-4{\kappa }^2)\frac{{\mathrm{\partial }}^2{v}_j}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}\}]+p_i\\ \text{and}{\sigma }_{33}& =\rho h\frac{{\mathrm{\partial }}^2{v}_3}{\mathrm{\partial }{t}^2}+p_3.\end{array}\}$$ 4.7

Next, on equating the stresses σ_3n in (4.5) and (4.7), we have

$$\begin{array}{rl}{p}_i& =\rho c_2^2\frac{a}{2\sqrt{\pi }}\{\frac{{\mathrm{\partial }}^2{v}_i}{\mathrm{\partial }{x}_j^2}+4(1-{\kappa }^2)\frac{{\mathrm{\partial }}^2{v}_i}{\mathrm{\partial }{x}_i^2}+(3-4{\kappa }^2)\frac{{\mathrm{\partial }}^2{v}_j}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}\}\\ \text{and}{p}_3& =0.\end{array}\}$$ 4.8

Thus, the refined boundary conditions, at x₃=0, become

$$\begin{array}{rl}{\sigma }_{3i}& =\rho c_2^2\frac{a}{2\sqrt{\pi }}\{\frac{{\mathrm{\partial }}^2{v}_i}{\mathrm{\partial }{x}_j^2}+4(1-{\kappa }^2)\frac{{\mathrm{\partial }}^2{v}_i}{\mathrm{\partial }{x}_i^2}+(3-4{\kappa }^2)\frac{{\mathrm{\partial }}^2{v}_j}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}\}\\ \text{and}{\sigma }_{33}& =0.\end{array}\}$$ 4.9

Outside a narrow boundary layer (x₃≫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(a²/ℓ²) 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 ∂/∂x₂≡0, u_m=u_m(x₁,x₃), m=1,3, and u₂=0. Accordingly, the two boundary conditions, following directly from (4.9), become

$$\begin{array}{rl}{\sigma }_{31}& =\frac{2a}{\sqrt{\pi }}\rho c_2^2(1-{\kappa }^2)\frac{{\mathrm{\partial }}^2{u}_1}{\mathrm{\partial }{x}_1^2}\\ \text{and}{\sigma }_{33}& =0,\end{array}\}$$ 5.1

and the equations of motion, in terms of wave potentials φ and ψ, are given by

$$\mathrm{\Delta }\phi -\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{t}^2}=0$$ 5.2

and

$$\mathrm{\Delta }\psi -\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{t}^2}=0,$$ 5.3

where Δ is the two-dimensional Laplacian.

We first look for travelling wave solutions of the form

$$\begin{array}{rl}\phi & =A\hspace{0.17em}{\mathrm{e}}^{-r{x}_3+\mathrm{i}k(x_1-ct)}\\ \text{and}\psi & =B\hspace{0.17em}{\mathrm{e}}^{-q{x}_3+\mathrm{i}k(x_1-ct)},\end{array}\}$$ 5.4

where c is the phase speed and in which the attenuation coefficients are given by

$$r=k\sqrt{1-\frac{c^2}{c_1^2}}\text{and}q=k\sqrt{1-\frac{c^2}{c_2^2}}.$$

The displacements may now be expressed in terms of potentials, thus yielding

$$\begin{array}{rl}{u}_1& ={\phi }_{,1}+{\psi }_{,3}=(\mathrm{i}kA\hspace{0.17em}{\mathrm{e}}^{-r{x}_3}-qB\hspace{0.17em}{\mathrm{e}}^{-q{x}_3})\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}k(x_1-ct)}\\ \text{and}{u}_3& ={\phi }_{,3}-{\psi }_{,1}=(-rA\hspace{0.17em}{\mathrm{e}}^{-r{x}_3}-\mathrm{i}kB\hspace{0.17em}{\mathrm{e}}^{-q{x}_3})\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}k(x_1-ct)}.\end{array}\}$$ 5.5

Next, on substituting (5.5) into the boundary conditions (5.1), we obtain, after taking into account the plane strain forms of (2.4) and (2.5), that

$$\begin{array}{rl}& [2-\frac{c^2}{c_2^2}]A+\left[2\mathrm{i}\sqrt{1-\frac{c^2}{c_2^2}}\right]B=0\\ \text{and}& [-\mathrm{i}(\frac{2ak}{\sqrt{\pi }}({\kappa }^2-1)+2\sqrt{1-\frac{c^2}{c_1^2}})]A\\ & +[(2-\frac{c^2}{c_2^2})+\frac{2ak}{\sqrt{\pi }}({\kappa }^2-1)\sqrt{1-\frac{c^2}{c_2^2}}]B=0.\end{array}\}$$ 5.6

The condition for existence of a non-trivial solution of (5.6) yields

$$R(\gamma )-4\sqrt{\pi }\theta ({\kappa }^2-1){\gamma }^2\sqrt{1-{\gamma }^2}=0,$$ 5.7

where θ=a/ℓ=ak/2π≪1 is a small parameter, γ=c/c₂ and R(γ) is the Rayleigh denominator, i.e.

$$R(\gamma )={(2-{\gamma }^2)}^2-4\sqrt{(1-{\gamma }^2)(1-{\kappa }^2{\gamma }^2)}.$$

We may now expand γ as an asymptotic series in the small parameter θ, with

$$\gamma ={\gamma }_0+\theta {\gamma }_1+\cdots .$$ 5.8

In this case, the Taylor series expansion of R(γ), about γ=γ₀, is given by

$$R(\gamma )=R({\gamma }_0)+R^{\prime }({\gamma }_0)(\gamma -{\gamma }_0)+\cdots ,$$ 5.9

where γ₀ is the normalized classical Rayleigh wave speed, i.e. R(γ₀)=0. Then, on substituting (5.8) and (5.9) into (5.7), we readily obtain

$${\gamma }_1=\frac{4\sqrt{\pi }({\kappa }^2-1){\gamma }_0^2\sqrt{1-{\gamma }_0^2}}{R^{\prime }({\gamma }_0)}$$ 5.10

and, consequently,

$$\gamma ={\gamma }_0+\theta \frac{4\sqrt{\pi }({\kappa }^2-1){\gamma }_0^2\sqrt{1-{\gamma }_0^2}}{R^{\prime }({\gamma }_0)}+\dots ,$$ 5.11

where θ=ak/2π, 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 γ₀ 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 γ₁=−0.37, while its ‘local’ counterpart is γ₀=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).

Figure 2.

Effect of non-local phenomena on Rayleigh wave speed. (Online version in colour.)

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(a²/ℓ²) 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(a²/ℓ²) 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.