1 Introduction

Elasticity, an attractive field of study, has gathered the keen interest of numerous researchers due to its numerical applications in modern industries. Even though the classical linear theory of elasticity is most accurate when the physical phenomena of interest occurs on a scale significantly larger than the material’s internal characteristic length, it possesses a notable limitation in that it lacks an internal length scale. Shortcomings become evident in scenarios such as the singular stress field at the tip of a crack and the non-dispersive nature of wave propagation, see [1]. To address and extend the applicability of the classical theory of elasticity to micro- and nanostructures, several alternative theories have been proposed. These include micropolar elasticity [2,3,4,5], couple stress theory, strain gradient elasticity theory [6,7,8] and the non-local theory of elasticity [9,10,11,12]. After the introduction of the theory of non-local elasticity, which states that the stress at any given point within a continuous body depends not only on the strain at that specific point but also on the strain fields across the entire body, the study of non-local elastic phenomena has gained significant importance. Recent papers, including [13,14,15], have provided valuable insights and further advancements in this area. Micro- and nanomechanics [16, 17] and nanotechnology [18, 19] can be presented as advanced applications of non-local elasticity.

As mentioned above, materials exhibit size-dependent behavior at the micro- and nanoscale. Consequently, the size effect on material properties plays a crucial role in their mechanical behavior. In addition to the previously mentioned theories, new theories have emerged in recent times. One such widely used non-classical continuum theory is the modified couple stress theory proposed in [7]. This theory introduces equilibrium of the moment of couples as an additional equation for the couple stresses and finds applications in analyzing dynamic behavior of Bernoulli–Euler and Timoshenko microbeam models [20, 21], and Kirchhoff and Mindlin microplate models [22, 23]. The modified strain gradient elasticity theory, which is one of the higher-order continuum theories, has been proposed in [24]. This theory has introduced new equilibrium equations that were specifically designed to govern the behavior of higher-order strain gradients. The modified strain gradient theory then has been employed to investigate the bending, buckling and longitudinal vibration of microsized beams [25,26,27,28], and to develop models for microplates [29,30,31].

The generalization of classical surface waves to non-local materials has emerged as another significant subject of study. Researchers have focused on extending the understanding of surface wave phenomena in the context of non-local materials, exploring the characteristics and behaviors that arise in such systems. The propagation of Rayleigh-type surface waves in non-local micropolar elastic solid half-space has been examined in a paper by [32]. The paper explores the existence conditions for Rayleigh-type surface waves in this non-local framework. Similarly, the propagation of Love-type waves in a non-local elastic layer with voids resting over a non-local elastic solid half-space with voids has been studied by [33]. The research highlights that two fronts of Love-type surface waves can travel with distinct speeds in this system. Furthermore, the application of non-local elasticity has been extended to the study of Stoneley waves in non-local orthotropic elastic half-spaces [34]. Additionally, the investigation of non-local shear interfacial waves between two non-local elastic half-spaces is explored in another publication [35]. These studies contribute to the understanding of interfacial wave phenomena in non-local elastic materials.

Due to the limitations of the applicability of integro-differential equations, Eringen [36] introduced an equivalent differential non-local model which transforms the original integral non-local theory into a differential form. Although subsequent investigations have revealed inconsistencies within this model, as highlighted in [37], the theory may still be valid for the limit of the well-posed two-phase theory applied to a non-local beam problem, see [38].

Nowadays, the research generally focuses on developing asymptotic formulations for non-local structures, which provide to express the governing equations and boundary conditions in terms of the local stresses. Asymptotic analysis of an elastic half-space with a vertically inhomogeneous non-local thin layer has been investigated in [39]. In this study, a general asymptotic scheme has been proposed, which ensures that all the formulations are expressed in terms of local quantities. Additionally, a non-local asymptotic theory for thin elastic plates has been presented in [40]. The general formulations have been performed for an integral with an exponential non-local kernel across the thickness. Furthermore, Kaplunov [41] have analyzed the Rayleigh-type wave solution in non-local elasticity via the asymptotic model proposed aforementioned papers.

Despite the large number of publications, the effect of boundaries on the application of non-local elasticity concepts has not been adequately studied. The research in this area has generally focused on the significant role of boundary layers, see [42, 43]. The asymptotic approaches have been presented in [35, 37, 39,40,41] to address the impact of boundary layers on the overall dynamic behavior. However, it is worth noting that the existing asymptotic models, formulating the non-local problem in terms of local stresses, mostly deal with plane strain and anti-plane problems. Therefore, the main objective of this paper is to develop an asymptotic formulation for boundary value problems in a non-local elastic half-space. In this context, the constitutive integral relation given by [36], then refined by [37], is taken into consideration. For the sake of simplicity, the non-locality of the elastic medium is considered only along the vertical axis. Consequently, the non-local stresses can be expressed through the integrals with an exponential kernel which is a function of the vertical variable. With the help of integro-differential equations presented in [36], the governing equations and boundary conditions can be written in differential forms. A similar approach, as presented in [35, 37, 41], is then employed to derive an asymptotic formulation that allows expressing all the formulations in terms of the local stresses. As an example, the derived asymptotic formulation is implemented for analysis of the Rayleigh surface waves in plane strain problem. The results reveal that the non-local effect causes a correction to the Rayleigh surface waves, which is further illustrated for certain numerical values of Poisson’s ratio.

2 Basic equations and constitutive relations

We consider an elastic half-space (\(-\infty< x_1,x_2<\infty \), \(0\le x_3<\infty \)) where non-locality is exclusively observed along \(x_3\) axis, see Fig. 1. The governing equations of motion in terms of the non-local stresses may be written as

$$\begin{aligned} \begin{aligned}&\frac{\partial s_{ii}}{\partial x_i}+\frac{\partial s_{ij}}{\partial x_j}+\frac{\partial s_{i3}}{\partial x_3}=\rho \frac{\partial ^2 u_i}{\partial t^2},\\&\frac{\partial s_{3i}}{\partial x_i}+\frac{\partial s_{3j}}{\partial x_j}+\frac{\partial s_{33}}{\partial x_3}=\rho \frac{\partial ^2 u_3}{\partial t^2}, \end{aligned} \end{aligned}$$
(1)

where \(\rho \) is mass density, \(u_n\) are components of the displacement and \(s_{mn}\) are non-local stress components given by

$$\begin{aligned} s_{mn}=\frac{1}{2a}\int \limits _0^\infty e^{-\frac{|x_3-x_3'|}{a}}\sigma _{mn}(x_1,x_2,x'_3,t)\textrm{d}x'_3, \end{aligned}$$
(2)

see [36, 37]. Throughout the paper, unless otherwise stated, \(i\ne j=1,2\) and \(m,n=1,2,3\). In equation (2), the internal character a is a small parameter in comparison with a typical wavelength l. Moreover, the constitutive relations for local stresses \(\sigma _{mn}\) are given by

$$\begin{aligned} \begin{aligned}&\sigma _{ii}=(\lambda +2\mu ) \frac{\partial u_i}{\partial x_i}+\lambda \left( \frac{\partial u_j}{\partial x_j}+\frac{\partial u_3}{\partial x_3}\right) ,\quad \sigma _{ij}=\mu \left( \frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right) ,\\&\sigma _{33}=(\lambda +2\mu ) \frac{\partial u_3}{\partial x_3}+\lambda \left( \frac{\partial u_i}{\partial x_i}+\frac{\partial u_j}{\partial x_j}\right) ,\quad \sigma _{3i}=\mu \left( \frac{\partial u_3}{\partial x_i}+\frac{\partial u_i}{\partial x_3}\right) ,\\ \end{aligned} \end{aligned}$$
(3)

where \(\lambda \) and \(\mu \) are elastic moduli. Here and throughout the paper, no summation convention rule is employed.

Fig. 1
figure 1

A non-local elastic half-space

Full size image

The boundary conditions on the surface \(x_3=0\) are prescribed as

$$\begin{aligned} s_{3i}={{f_i}},\quad s_{33}={{f_3}}, \end{aligned}$$
(4)

where \(f_n=f_n(x_1,x_2,t).\)

Differentiating Eq. (2) with respect to \(x_3\), similar to [35, 37], we obtain

$$\begin{aligned} a^2\frac{\partial ^2 s_{mn}}{\partial x_3^2}-s_{mn}=-\sigma _{mn},\quad m,n=1,2,3. \end{aligned}$$
(5)

Substituting (5) in Eq. (2), we derive extra conditions on the surface \(x_3=0\)

$$\begin{aligned} s_{mn}- a \frac{\partial s_{mn}}{\partial x_3}=0,\quad m,n=1,2,3. \end{aligned}$$
(6)

3 Asymptotic analysis

We, now, apply an asymptotic approach to express the formulated non-local problem in terms of the local stresses. Considering that the internal characteristic length is assumed to be significantly smaller than a typical wavelength, l, we may introduce a natural small parameter given by

$$\begin{aligned} \eta =\frac{a}{l}\ll 1, \end{aligned}$$
(7)

allowing an asymptotic analysis of the formulated problem. Let us specify dimensionless variables by

$$\begin{aligned} \xi _i=\frac{x_i}{l},\quad \zeta _p=\frac{x_3}{l},\quad \zeta _{q}=\frac{x_3}{a},\quad \tau =\frac{c_{2}}{l}t, \end{aligned}$$
(8)

where \(\zeta _p \) and \( \zeta _{q} \) are slow and fast variables signifying the variation of the unknown quantities with respect to the vertical coordinate \(x_3\) and \( c_{2}=(\mu /\rho )^{1/2} \) is the transverse wave speed. We also define the dimensionless quantities as

$$\begin{aligned} \sigma _{mn}= \mu \sigma ^*_{mn}, \hspace{5.0pt}s_{mn}= \mu s^*_{mn}, \hspace{5.0pt}u_n=l u^{*}_n, \hspace{5.0pt}\end{aligned}$$
(9)

where all starred quantities are assumed of the same asymptotic order.

Let us now split the non-local stresses into slow and fast components \(p^*_{mn}\) and \(q^*_{mn}\), respectively, i.e.,

$$\begin{aligned} \begin{aligned}&s^*_{ii}=p^*_{ii}+q^{*}_{ii},\qquad s^*_{ij}=p^*_{ij}+q^{*}_{ij}\\&s^*_{3i}=p^*_{3i}+\eta q^{*}_{3i},\quad s^*_{33}=p^*_{33}+\eta ^2 q^{*}_{33}, \end{aligned} \end{aligned}$$
(10)

where \(p^{*}_{mn}= p^{*}_{mn}(\xi _i,\xi _j, \zeta _p,\tau )\) and \(q^{*}_{mn}= q^{*}_{mn}(\xi , \xi _j,\zeta _{q},\tau )\), see [40]. Therefore, governing Eqs. (1), (5) and boundary conditions (4), (6) can be expressed in terms of the new variables, incorporating both slow and fast quantities, as follows:

$$\begin{aligned}{} & {} \begin{aligned}&\frac{\partial p^{*}_{ii}}{\partial \xi _i}+\frac{\partial p^{*}_{ij}}{\partial \xi _j}+\frac{\partial p^{*}_{3i}}{\partial \zeta _p}=\frac{\partial ^2 u^{*}_i}{\partial \tau ^2},\\&\frac{\partial p^{*}_{3i}}{\partial \xi _i}+\frac{\partial p^{*}_{3j}}{\partial \xi _j}+\frac{\partial p^{*}_{33}}{\partial \zeta _p}=\frac{\partial ^2 u^{*}_3}{\partial \tau ^2}, \end{aligned} \end{aligned}$$
(11)
$$\begin{aligned}{} & {} \begin{aligned}&\frac{\partial q^{*}_{ii}}{\partial \xi _i}+\frac{\partial q^{*}_{ij}}{\partial \xi _j}+\frac{\partial q^{*}_{3i}}{\partial \zeta _p}=0,\\&\frac{\partial q^{*}_{3i}}{\partial \xi _i}+\frac{\partial q^{*}_{3j}}{\partial \xi _j}+\frac{\partial q^{*}_{33}}{\partial \zeta _q}=0, \end{aligned} \end{aligned}$$
(12)
$$\begin{aligned}{} & {} \begin{aligned}&p^*_{ii}-\eta ^2\frac{\partial ^2p^*_{ii}}{\partial \zeta _p^2}=\sigma ^*_{ii},\quad p^*_{ij}-\eta ^2\frac{\partial ^2p^*_{ij}}{\partial \zeta _p^2}=\sigma ^*_{ij} \\&p^*_{3i}-\eta ^2\frac{\partial ^2p^*_{3i}}{\partial \zeta _p^2}=\sigma ^*_{3i},\quad p^*_{33}-\eta ^2\frac{\partial ^2p^*_{33}}{\partial \zeta _p^2}=\sigma ^*_{33} \end{aligned} \end{aligned}$$
(13)
$$\begin{aligned}{} & {} \begin{aligned}&q^*_{ii}-\frac{\partial ^2 q^*_{ii}}{\partial \zeta _q^2}=0,\quad q^*_{ij}-\frac{\partial ^2 q^*_{ij}}{\partial \zeta _q^2}=0 \\&q^*_{3i}-\frac{\partial ^2 q^*_{3i}}{\partial \zeta _q^2}=0,\quad q^*_{33}-\frac{\partial ^2 q^*_{33}}{\partial \zeta _q^2}=0, \end{aligned} \end{aligned}$$
(14)

subject to boundary conditions

$$\begin{aligned} \begin{aligned}&\left. p^{*}_{ii}\right| _{\zeta _p=0}+\left. q^{*}_{ii}\right| _{\zeta _q=0}=\left. \frac{\partial q^{*}_{ii}}{\partial \zeta _q}\right| _{\zeta _q=0}+\eta \left. \frac{\partial p^{*}_{ii}}{\partial \zeta _p}\right| _{\zeta _p=0},\\&\left. p^{*}_{ij}\right| _{\zeta _p=0}+\left. q^{*}_{ij}\right| _{\zeta _q=0}=\left. \frac{\partial q^{*}_{ij}}{\partial \zeta _q}\right| _{\zeta _q=0}+\eta \left. \frac{\partial p^{*}_{ij}}{\partial \zeta _p}\right| _{\zeta _p=0},\\&\left. p^{*}_{3i}\right| _{\zeta _p=0}+\eta \left. q^{*}_{3i}\right| _{\zeta _q=0}={{f^*_i}},\\&\left. p^{*}_{33}\right| _{\zeta _p=0}+\eta ^2 \left. q^{*}_{33}\right| _{\zeta _q=0}={{f^*_3}}. \end{aligned} \end{aligned}$$
(15)

Let us now expand all the dimensionless quantities into asymptotic series in terms of small parameter \(\eta \) as

$$\begin{aligned} \left( \begin{array}{c}u^*_n \\ p^*_{mn} \\ q^*_{mn}\\ \sigma ^*_{mn}\end{array}\right) =\left( \begin{array}{c}u^{(0)}_n \\ p^{(0)}_{mn} \\ q^{(0)}_{mn}\\ \sigma ^{(0)}_{mn}\end{array}\right) +\eta \left( \begin{array}{c}u^{(1)}_n \\ p^{(1)}_{mn} \\ q^{(1)}_{mn}\\ \sigma ^{(1)}_{mn}\end{array}\right) +\cdots . \end{aligned}$$
(16)

3.1 Leading-order approximation

On substituting the asymptotic expansions (16) into Eqs. (11)–(15) and keeping only the terms with the suffix (0) , we arrive at

$$\begin{aligned}{} & {} \begin{aligned}&\frac{\partial p^{(0)}_{ii}}{\partial \xi _i}+\frac{\partial p^{(0)}_{ij}}{\partial \xi _j}+\frac{\partial p^{(0)}_{3i}}{\partial \zeta _p}=\frac{\partial ^2 u^{(0)}_i}{\partial \tau ^2},\\&\frac{\partial p^{(0)}_{3i}}{\partial \xi _i}+\frac{\partial p^{(0)}_{3j}}{\partial \xi _j}+\frac{\partial p^{(0)}_{33}}{\partial \zeta _p}=\frac{\partial ^2 u^{(0)}_3}{\partial \tau ^2}, \end{aligned} \end{aligned}$$
(17)
$$\begin{aligned}{} & {} \begin{aligned}&\frac{\partial q^{(0)}_{ii}}{\partial \xi _i}+\frac{\partial q^{(0)}_{ij}}{\partial \xi _j}+\frac{\partial q^{(0)}_{3i}}{\partial \zeta _q}=0,\\&\frac{\partial q^{(0)}_{3i}}{\partial \xi _i}+\frac{\partial q^{(0)}_{3j}}{\partial \xi _j}+\frac{\partial q^{(0)}_{33}}{\partial \zeta _q}=0, \end{aligned}\end{aligned}$$
(18)
$$\begin{aligned}{} & {} p^{(0)}_{ii}=\sigma ^{(0)}_{ii},\; p^{(0)}_{ij}=\sigma ^{(0)}_{ij},\; p^{(0)}_{3i}=\sigma ^{(0)}_{3i},\; p^{(0)}_{33}=\sigma ^{(0)}_{33}, \end{aligned}$$
(19)
$$\begin{aligned}{} & {} \begin{aligned}&q^{(0)}_{ii}-\frac{\partial ^2 q^{(0)}_{ii}}{\partial \zeta _q^2}=0,\quad q^{(0)}_{ij}-\frac{\partial ^2 q^{(0)}_{ij}}{\partial \zeta _q^2}=0 \\&q^{(0)}_{3i}-\frac{\partial ^2 q^{(0)}_{3i}}{\partial \zeta _q^2}=0,\quad q^{(0)}_{33}-\frac{\partial ^2 q^{(0)}_{33}}{\partial \zeta _q^2}=0, \end{aligned} \end{aligned}$$
(20)

and boundary conditions

$$\begin{aligned}&\begin{aligned}&\left. p^{(0)}_{ii}\right| _{\zeta _p=0}+\left. q^{(0)}_{ii}\right| _{\zeta _q=0}=\left. \frac{\partial q^{(0)}_{ii}}{\partial \zeta _q}\right| _{\zeta _q=0},\\&\left. p^{(0)}_{ij}\right| _{\zeta _p=0}+\left. q^{(0)}_{ij}\right| _{\zeta _q=0}=\left. \frac{\partial q^{(0)}_{ij}}{\partial \zeta _q}\right| _{\zeta _q=0},\\ \end{aligned} \end{aligned}$$
(21)
$$\begin{aligned}&\begin{aligned}&\left. p^{(0)}_{3i}\right| _{\zeta _p=0}={{f^*_i}},\\&\left. p^{(0)}_{33}\right| _{\zeta _p=0}={{f^*_3}}. \end{aligned} \end{aligned}$$
(22)

Combining (17) and (19), we have

$$\begin{aligned} \begin{aligned}&\frac{\partial \sigma ^{(0)}_{ii}}{\partial \xi _i}+\frac{\partial \sigma ^{(0)}_{ij}}{\partial \xi _j}+\frac{\partial \sigma ^{(0)}_{3i}}{\partial \zeta _p}=\frac{\partial ^2 u^{(0)}_i}{\partial \tau ^2},\\&\frac{\partial \sigma ^{(0)}_{3i}}{\partial \xi _i}+\frac{\partial \sigma ^{(0)}_{3j}}{\partial \xi _j}+\frac{\partial \sigma ^{(0)}_{33}}{\partial \zeta _p}=\frac{\partial ^2 u^{(0)}_3}{\partial \tau ^2}. \end{aligned} \end{aligned}$$
(23)

The solutions of Eq. (20) may easily be obtained as

$$\begin{aligned} \begin{aligned}&q^{(0)}_{ii}=Q^{(0)}_{ii}(\xi _i,\xi _j,\tau )e^{- \zeta _{q}},\; q^{(0)}_{ij}=Q^{(0)}_{ij}(\xi _i,\xi _j,\tau )e^{- \zeta _{q}},\\&q^{(0)}_{3i}=Q^{(0)}_{3i}(\xi _i,\xi _j,\tau )e^{- \zeta _{q}},\; q^{(0)}_{33}=Q^{(0)}_{33}(\xi _i,\xi _j,\tau )e^{- \zeta _{q}}, \end{aligned} \end{aligned}$$
(24)

where the coefficients \( Q_{mn}^{(0)} \) are, due to (18), related by

$$\begin{aligned} Q^{(0)}_{3i}=\frac{\partial Q^{(0)}_{ii}}{\partial \xi _i}+\frac{\partial Q^{(0)}_{ij}}{\partial \xi _j},\quad Q^{(0)}_{33}=\frac{\partial Q^{(0)}_{3i}}{\partial \xi _i}+\frac{\partial Q^{(0)}_{3j}}{\partial \xi _j}. \end{aligned}$$
(25)

Employing (21), together with (19), we arrive at

$$\begin{aligned} Q^{(0)}_{ii}=-\left. \frac{1}{2} \sigma ^{(0)}_{ii}\right| _{\zeta _p=0},\quad Q^{(0)}_{ij}=-\left. \frac{1}{2} \sigma ^{(0)}_{ij}\right| _{\zeta _p=0}. \end{aligned}$$
(26)

Therefore, the coefficients \(Q^{(0)}_{3i}\) and \(Q^{(0)}_{33}\) may be expressed from Eqs. (25) and (26) as

$$\begin{aligned} Q^{(0)}_{3i}=-\frac{1}{2}\left. \left( \frac{\partial \sigma ^{(0)}_{ii}}{\partial \xi _i}+\frac{\partial \sigma ^{(0)}_{ij}}{\partial \xi _j}\right) \right| _{\zeta _p=0},\quad Q^{(0)}_{33}=-\frac{1}{2}\left. \left( \frac{\partial ^2 \sigma ^{(0)}_{ii}}{\partial \xi _i^2}+2\frac{\partial ^2 \sigma ^{(0)}_{ij}}{\partial \xi _i\partial \xi _j}+\frac{\partial ^2 \sigma ^{(0)}_{jj}}{\partial \xi _j^2}\right) \right| _{\zeta _p=0}. \end{aligned}$$
(27)

Finally, the leading-order fast quantities can be written in terms of local stresses as

$$\begin{aligned}{} & {} q^{(0)}_{ii}=-\frac{1}{2}\left. \sigma ^{(0)}_{ii}\right| _{\zeta _{p}=0}e^{-\zeta _q},\quad q^{(0)}_{ij}=-\frac{1}{2}\left. \sigma ^{(0)}_{ij}\right| _{\zeta _{p}=0}e^{-\zeta _q},\nonumber \\{} & {} q^{(0)}_{3i}=-\frac{1}{2}\left. \left( \frac{\partial \sigma ^{(0)}_{ii}}{\partial \xi _i}+\frac{\partial \sigma ^{(0)}_{ij}}{\partial \xi _j}\right) \right| _{\zeta _p=0}\hspace{-10.0pt}e^{-\zeta _q},\; q^{(0)}_{33}=-\frac{1}{2}\left. \left( \frac{\partial ^2 \sigma ^{(0)}_{ii}}{\partial \xi _i^2}+2\frac{\partial ^2 \sigma ^{(0)}_{ij}}{\partial \xi _i\partial \xi _j}+\frac{\partial ^2 \sigma ^{(0)}_{jj}}{\partial \xi _j^2}\right) \right| _{\zeta _p=0}\hspace{-10.0pt}e^{-\zeta _q}. \end{aligned}$$
(28)

3.2 First-order approximation

At the first-order approximation, the following equations are obtained by retaining the terms with the suffix (1)

$$\begin{aligned}{} & {} \begin{aligned}&\frac{\partial p^{(1)}_{ii}}{\partial \xi _i}+\frac{\partial p^{(1)}_{ij}}{\partial \xi _j}+\frac{\partial p^{(1)}_{3i}}{\partial \zeta _p}=\frac{\partial ^2 u^{(1)}_i}{\partial \tau ^2},\\&\frac{\partial p^{(1)}_{3i}}{\partial \xi _i}+\frac{\partial p^{(1)}_{3j}}{\partial \xi _j}+\frac{\partial p^{(1)}_{33}}{\partial \zeta _p}=\frac{\partial ^2 u^{(1)}_3}{\partial \tau ^2}, \end{aligned} \end{aligned}$$
(29)
$$\begin{aligned}{} & {} \begin{aligned}&\frac{\partial q^{(1)}_{ii}}{\partial \xi _i}+\frac{\partial q^{(1)}_{ij}}{\partial \xi _j}+\frac{\partial q^{(1)}_{3i}}{\partial \zeta _q}=0,\\&\frac{\partial q^{(1)}_{3i}}{\partial \xi _i}+\frac{\partial q^{(1)}_{3j}}{\partial \xi _j}+\frac{\partial q^{(1)}_{33}}{\partial \zeta _q}=0, \end{aligned} \end{aligned}$$
(30)
$$\begin{aligned}{} & {} p^{(1)}_{ii}=\sigma ^{(1)}_{ii},\; p^{(0)}_{ij}=\sigma ^{(1)}_{ij},\; p^{(1)}_{3i}=\sigma ^{(1)}_{3i},\; p^{(1)}_{33}=\sigma ^{(1)}_{33}, \end{aligned}$$
(31)
$$\begin{aligned}{} & {} \begin{aligned}&q^{(1)}_{ii}-\frac{\partial ^2 q^{(1)}_{ii}}{\partial \zeta _q^2}=0,\quad q^{(1)}_{ij}-\frac{\partial ^2 q^{(1)}_{ij}}{\partial \zeta _q^2}=0, \\&q^{(1)}_{3i}-\frac{\partial ^2 q^{(1)}_{3i}}{\partial \zeta _q^2}=0,\quad q^{(1)}_{33}-\frac{\partial ^2 q^{(1)}_{33}}{\partial \zeta _q^2}=0, \end{aligned} \end{aligned}$$
(32)

with boundary conditions

$$\begin{aligned}{} & {} \begin{aligned}&\left. p^{(1)}_{ii}\right| _{\zeta _p=0}+\left. q^{(1)}_{ii}\right| _{\zeta _q=0}=\left. \frac{\partial p^{(0)}_{ii}}{\partial \zeta _p}\right| _{\zeta _p=0}+\left. \frac{\partial q^{(1)}_{ii}}{\partial \zeta _q}\right| _{\zeta _q=0},\\&\left. p^{(1)}_{ij}\right| _{\zeta _p=0}+\left. q^{(1)}_{ij}\right| _{\zeta _q=0}=\left. \frac{\partial p^{(0)}_{ij}}{\partial \zeta _p}\right| _{\zeta _p=0}+\left. \frac{\partial q^{(1)}_{ij}}{\partial \zeta _q}\right| _{\zeta _q=0}, \end{aligned} \end{aligned}$$
(33)
$$\begin{aligned}{} & {} \begin{aligned}&\left. p^{(1)}_{3i}\right| _{\zeta _p=0}=-\left. q^{(0)}_{3i}\right| _{\zeta _p=0}=\frac{1}{2}\left. \left( \frac{\partial \sigma ^{(0)}_{ii}}{\partial \xi _i}+\frac{\partial \sigma ^{(0)}_{ij}}{\partial \xi _j}\right) \right| _{\zeta _p=0},\\&\left. p^{(1)}_{33}\right| _{\zeta _p=0}=0. \end{aligned} \end{aligned}$$
(34)

Taking into consideration (31) with (29), we obtain

$$\begin{aligned} \begin{aligned}&\frac{\partial \sigma ^{(1)}_{ii}}{\partial \xi _i}+\frac{\partial \sigma ^{(1)}_{ij}}{\partial \xi _j}+\frac{\partial \sigma ^{(1)}_{3i}}{\partial \zeta _p}=\frac{\partial ^2 u^{(1)}_i}{\partial \tau ^2},\\&\frac{\partial \sigma ^{(1)}_{3i}}{\partial \xi _i}+\frac{\partial \sigma ^{(1)}_{3j}}{\partial \xi _j}+\frac{\partial \sigma ^{(1)}_{33}}{\partial \zeta _p}=\frac{\partial ^2 u^{(1)}_3}{\partial \tau ^2}. \end{aligned} \end{aligned}$$
(35)

Equation (32), as before, may readily be solved to give

$$\begin{aligned} \begin{aligned}&q^{(1)}_{ii}=Q^{(1)}_{ii}(\xi _i,\xi _j,\tau )e^{- \zeta _{q}},\; q^{(1)}_{ij}=Q^{(1)}_{ij}(\xi _i,\xi _j,\tau )e^{- \zeta _{q}},\\&q^{(1)}_{3i}=Q^{(1)}_{3i}(\xi _i,\xi _j,\tau )e^{- \zeta _{q}},\; q^{(1)}_{33}=Q^{(1)}_{33}(\xi _i,\xi _j,\tau )e^{- \zeta _{q}}, \end{aligned} \end{aligned}$$
(36)

where the relationship between the coefficients \(Q_{mn}^{(1)}\) can be derived from Eq. (30) as

$$\begin{aligned} Q^{(1)}_{3i}=\frac{\partial Q^{(1)}_{ii}}{\partial \xi _i}+\frac{\partial Q^{(1)}_{ij}}{\partial \xi _j},\quad Q^{(1)}_{33}=\frac{\partial Q^{(1)}_{3i}}{\partial \xi _i}+\frac{\partial Q^{(1)}_{3j}}{\partial \xi _j}. \end{aligned}$$
(37)

Using (36) with (37) in boundary condition (33), we deduce that

$$\begin{aligned} Q^{(1)}_{ii}=\frac{1}{2}\left. \left( \frac{\partial \sigma ^{(0)}_{ii}}{\partial \zeta _p}-\sigma ^{(1)}_{ii}\right) \right| _{\zeta _p=0},\quad Q^{(1)}_{ij}=\frac{1}{2}\left. \left( \frac{\partial \sigma ^{(0)}_{ij}}{\partial \zeta _p}-\sigma ^{(1)}_{ij}\right) \right| _{\zeta _p=0}, \end{aligned}$$
(38)

and

$$\begin{aligned} \begin{aligned}&Q^{(1)}_{3i}=\frac{1}{2}\left. \left( \frac{\partial ^2 \sigma ^{(0)}_{ii}}{\partial \xi _i\partial \zeta _p}+\frac{\partial ^2 \sigma ^{(0)}_{ij}}{\partial \xi _j\partial \zeta _p}-\frac{\partial \sigma ^{(1)}_{ii}}{\partial \xi _i}-\frac{\partial \sigma ^{(1)}_{ij}}{\partial \xi _j}\right) \right| _{\zeta _p=0},\\&Q^{(1)}_{33}=\frac{1}{2}\left. \left( \frac{\partial ^3 \sigma ^{(0)}_{ii}}{\partial \xi _i^2\partial \zeta _p}+2\frac{\partial ^3 \sigma ^{(0)}_{ij}}{\partial \xi _i\partial \xi _j\partial \zeta _p}+\frac{\partial ^3 \sigma ^{(0)}_{jj}}{\partial \xi _j^2\partial \zeta _p}-\frac{\partial ^2 \sigma ^{(1)}_{ii}}{\partial \xi _i^2}-2\frac{\partial ^2 \sigma ^{(1)}_{ij}}{\partial \xi _i\partial \xi _j}-\frac{\partial ^2 \sigma ^{(1)}_{jj}}{\partial \xi _j^2}\right) \right| _{\zeta _p=0}. \end{aligned} \end{aligned}$$
(39)

Consequently, it follows from (36), (38) and (39) that

$$\begin{aligned} \begin{aligned}&q^{(1)}_{ii}=\frac{1}{2}\left. \left( \frac{\partial \sigma ^{(0)}_{ii}}{\partial \xi _p}-\sigma ^{(1)}_{ii}\right) \right| _{\zeta _p=0}e^{-\zeta _q},\quad q^{1}_{ij}=\frac{1}{2}\left. \left( \frac{\partial \sigma ^{(0)}_{ij}}{\partial \xi _j}-\sigma ^{(1)}_{ij}\right) \right| _{\zeta _p=0}e^{-\zeta _q},\\&q^{(1)}_{3i}=\frac{1}{2}\left. \left( \frac{\partial ^2 \sigma ^{(0)}_{ii}}{\partial \xi _i\partial \zeta _p}+\frac{\partial ^2 \sigma ^{(0)}_{ij}}{\partial \xi _j\partial \zeta _p}-\frac{\partial \sigma ^{(1)}_{ii}}{\partial \xi _i}-\frac{\partial \sigma ^{(1)}_{ij}}{\partial \xi _j}\right) \right| _{\zeta _p=0}e^{-\zeta _q},\\&q^{(1)}_{33}=\frac{1}{2}\left. \left( \frac{\partial ^3 \sigma ^{(0)}_{ii}}{\partial \xi _i^2\partial \zeta _p}+2\frac{\partial ^3 \sigma ^{(0)}_{ij}}{\partial \xi _i\partial \xi _j\partial \zeta _p}+\frac{\partial ^3 \sigma ^{(0)}_{jj}}{\partial \xi _j^2\partial \zeta _p}-\frac{\partial ^2 \sigma ^{(1)}_{ii}}{\partial \xi _i^2}-2\frac{\partial ^2 \sigma ^{(1)}_{ij}}{\partial \xi _i\partial \xi _j}-\frac{\partial ^2 \sigma ^{(1)}_{jj}}{\partial \xi _j^2}\right) \right| _{\zeta _p=0} e^{-\zeta _q}. \end{aligned} \end{aligned}$$
(40)

3.3 Second-order approximation

By following the same approach as in the preceding sections and retaining the terms with the suffix (2), we obtain

$$\begin{aligned}{} & {} \begin{aligned}&\frac{\partial p^{(2)}_{ii}}{\partial \xi _i}+\frac{\partial p^{(2)}_{ij}}{\partial \xi _j}+\frac{\partial p^{(2)}_{3i}}{\partial \zeta _p}=\frac{\partial ^2 u^{(2)}_i}{\partial \tau ^2},\\&\frac{\partial p^{(2)}_{3i}}{\partial \xi _i}+\frac{\partial p^{(2)}_{3j}}{\partial \xi _j}+\frac{\partial p^{(2)}_{33}}{\partial \zeta _p}=\frac{\partial ^2 u^{(2)}_3}{\partial \tau ^2}, \end{aligned} \end{aligned}$$
(41)
$$\begin{aligned}{} & {} \begin{aligned}&\frac{\partial q^{(2)}_{ii}}{\partial \xi _i}+\frac{\partial q^{(2)}_{ij}}{\partial \xi _j}+\frac{\partial q^{(2)}_{3i}}{\partial \zeta _q}=0,\\&\frac{\partial q^{(2)}_{3i}}{\partial \xi _i}+\frac{\partial q^{(2)}_{3j}}{\partial \xi _j}+\frac{\partial q^{(2)}_{33}}{\partial \zeta _q}=0, \end{aligned} \end{aligned}$$
(42)
$$\begin{aligned}{} & {} \begin{aligned}&p^{(2)}_{ii}-\frac{\partial ^2 p^{(0)}_{ii}}{\partial \zeta _p^2}=\sigma ^{(2)}_{ii},\quad p^{(2)}_{ij}-\frac{\partial ^2 p^{(0)}_{ij}}{\partial \zeta _p^2}=\sigma ^{(2)}_{ij},\\&p^{(2)}_{3i}-\frac{\partial ^2 p^{(0)}_{3i}}{\partial \zeta _p^2}=\sigma ^{(2)}_{3i},\quad p^{(2)}_{33}-\frac{\partial ^2 p^{(0)}_{33}}{\partial \zeta _p^2}=\sigma ^{(2)}_{33}, \end{aligned} \end{aligned}$$
(43)
$$\begin{aligned}{} & {} \begin{aligned}&q^{(2)}_{ii}-\frac{\partial ^2 q^{(2)}_{ii}}{\partial \zeta _q^2}=0,\quad q^{(2)}_{ij}-\frac{\partial ^2 q^{(2)}_{ij}}{\partial \zeta _q^2}=0, \\&q^{(2)}_{3i}-\frac{\partial ^2 q^{(2)}_{3i}}{\partial \zeta _q^2}=0,\quad q^{(2)}_{33}-\frac{\partial ^2 q^{(2)}_{33}}{\partial \zeta _q^2}=0, \end{aligned} \end{aligned}$$
(44)

and boundary conditions

$$\begin{aligned}&\begin{aligned}&\left. p^{(2)}_{ii}\right| _{\zeta _p=0}+\left. q^{(2)}_{ii}\right| _{\zeta _q=0}=\left. \frac{\partial p^{(1)}_{ii}}{\partial \zeta _p}\right| _{\zeta _p=0}+\left. \frac{\partial q^{(2)}_{ii}}{\partial \zeta _q}\right| _{\zeta _q=0},\\&\left. p^{(2)}_{ij}\right| _{\zeta _p=0}+\left. q^{(2)}_{ij}\right| _{\zeta _q=0}=\left. \frac{\partial p^{(1)}_{ij}}{\partial \zeta _p}\right| _{\zeta _p=0}+\left. \frac{\partial q^{(2)}_{ij}}{\partial \zeta _q}\right| _{\zeta _q=0}, \end{aligned} \end{aligned}$$
(45)
$$\begin{aligned}&\begin{aligned}&\left. p^{(2)}_{3i}\right| _{\zeta _p=0}=\left. -q^{(1)}_{3i}\right| _{\zeta _q=0}=-\frac{1}{2}\left. \left( \frac{\partial ^2 \sigma ^{(0)}_{ii}}{\partial \xi _i\partial \zeta _p}+\frac{\partial ^2 \sigma ^{(0)}_{ij}}{\partial \xi _j\partial \zeta _p}-\frac{\partial \sigma ^{(1)}_{ii}}{\partial \xi _i}-\frac{\partial \sigma ^{(1)}_{ij}}{\partial \xi _j}\right) \right| _{\zeta _p=0},\\&\left. p^{(1)}_{33}\right| _{\zeta _p=0}=\left. -q^{(0)}_{33}\right| _{\zeta _q=0}=\frac{1}{2}\left. \left( \frac{\partial ^2 \sigma ^{(0)}_{ii}}{\partial \xi _i^2}+2\frac{\partial ^2 \sigma ^{(0)}_{ij}}{\partial \xi _i\partial \xi _j}+\frac{\partial ^2 \sigma ^{(0)}_{jj}}{\partial \xi _j^2}\right) \right| _{\zeta _p=0}. \end{aligned} \end{aligned}$$
(46)

After substituting Eq. (43) into Eq. (41) and considering Eq. (17), we arrive at

$$\begin{aligned} \begin{aligned}&\frac{\partial \sigma ^{(2)}_{ii}}{\partial \xi _i}+\frac{\partial \sigma ^{(2)}_{ij}}{\partial \xi _j}+\frac{\partial \sigma ^{(2)}_{3i}}{\partial \zeta _p}=\frac{\partial ^2 u^{(2)}_i}{\partial \tau ^2}-\eta ^2\frac{\partial ^4 u^{(0)}_i}{\partial \zeta _p^2\partial \tau ^2},\\&\frac{\partial \sigma ^{(2)}_{3i}}{\partial \xi _i}+\frac{\partial \sigma ^{(2)}_{3j}}{\partial \xi _j}+\frac{\partial \sigma ^{(2)}_{33}}{\partial \zeta _p}=\frac{\partial ^2 u^{(2)}_3}{\partial \tau ^2}-\eta ^2\frac{\partial ^4 u^{(0)}_3}{\partial \zeta _p^2\partial \tau ^2}. \end{aligned} \end{aligned}$$
(47)

The solutions to Eq. (44) are once again represented as

$$\begin{aligned} \begin{aligned}&q^{(2)}_{ii}=Q^{(2)}_{ii}(\xi _i,\xi _j,\tau )e^{- \zeta _{q}},\; q^{(2)}_{ij}=Q^{(2)}_{ij}(\xi _i,\xi _j,\tau )e^{- \zeta _{q}},\\&q^{(2)}_{3i}=Q^{(2)}_{3i}(\xi _i,\xi _j,\tau )e^{- \zeta _{q}},\; q^{(2)}_{33}=Q^{(2)}_{33}(\xi _i,\xi _j,\tau )e^{- \zeta _{q}}, \end{aligned} \end{aligned}$$
(48)

where

$$\begin{aligned} \begin{aligned}&Q^{(2)}_{ii}=\frac{1}{2}\left. \left( -\frac{\partial ^2\sigma ^{(0)}_{ii}}{\partial \zeta _p^2}+\frac{\partial \sigma ^{(1)}_{ii}}{\partial \zeta _p}-\sigma ^{(2)}_{ii}\right) \right| _{\zeta _p=0},\\&Q^{(2)}_{ij}=\frac{1}{2}\left. \left( -\frac{\partial ^2\sigma ^{(0)}_{ij}}{\partial \zeta _p^2}+\frac{\partial \sigma ^{(1)}_{ij}}{\partial \zeta _p}-\sigma ^{(2)}_{ij}\right) \right| _{\zeta _p=0}, \end{aligned} \end{aligned}$$
(49)

and

$$\begin{aligned} \begin{aligned} Q^{(1)}_{3i}&=\frac{1}{2}\left. \left( -\frac{\partial ^3 \sigma ^{(0)}_{ii}}{\partial \xi _i\partial \zeta _p^2}-\frac{\partial ^3 \sigma ^{(0)}_{ij}}{\partial \xi _j\partial \zeta _p^2}+\frac{\partial ^2\sigma ^{(1)}_{ii}}{\partial \xi _i \partial \zeta _p}+\frac{\partial ^2 \sigma ^{(1)}_{ij}}{\partial \xi _j \partial \zeta _p}-\frac{\partial \sigma ^{(2)}_{ii}}{\partial \xi _i}-\frac{\partial \sigma ^{(2)}_{ij}}{\partial \xi _j }\right) \right| _{\zeta _p=0},\\ Q^{(1)}_{33}&=\frac{1}{2}\left. \left( -\frac{\partial ^4 \sigma ^{(0)}_{ii}}{\partial \xi _i^2\partial \zeta _p^2}-2\frac{\partial ^4 \sigma ^{(0)}_{ij}}{\partial \xi _i\partial \xi _j\partial \zeta _p^2}-\frac{\partial ^4 \sigma ^{(0)}_{jj}}{\partial \xi _j^2\partial \zeta _p^2}+\frac{\partial ^3 \sigma ^{(1)}_{ii}}{\partial \xi _i^2\partial \zeta _p}+2\frac{\partial ^3 \sigma ^{(1)}_{ij}}{\partial \xi _i\partial \xi _j\partial \zeta _p}\right. \right. \\&\quad \left. \left. +\frac{\partial ^3 \sigma ^{(0)}_{jj}}{\partial \xi _j^2\partial \zeta _p}-\frac{\partial ^2 \sigma ^{(2)}_{ii}}{\partial \xi _i^2}-2\frac{\partial ^2 \sigma ^{(2)}_{ij}}{\partial \xi _i\partial \xi _j}-\frac{\partial ^2 \sigma ^{(2)}_{jj}}{\partial \xi _j^2}\right) \right| _{\zeta _p=0}. \end{aligned} \end{aligned}$$
(50)

Taking into account leading-, next- and second-order equations (23), (35), (47), in terms of the original variables, we arrive at the governing equations given by

$$\begin{aligned} \begin{aligned}&\frac{\partial \sigma ^{(*)}_{ii}}{\partial \xi _i}+\frac{\partial \sigma ^{(*)}_{ij}}{\partial \xi _j}+\frac{\partial \sigma ^{(*)}_{3i}}{\partial \zeta _p}=\frac{\partial ^2 u^{(*)}_i}{\partial \tau ^2}-\eta ^2\frac{\partial ^4 u^{(*)}_i}{\partial \zeta _p^2\partial \tau ^2},\\&\frac{\partial \sigma ^{(*)}_{3i}}{\partial \xi _i}+\frac{\partial \sigma ^{(*)}_{3j}}{\partial \xi _j}+\frac{\partial \sigma ^{(*)}_{33}}{\partial \zeta _p}=\frac{\partial ^2 u^{(*)}_3}{\partial \tau ^2}-\eta ^2\frac{\partial ^4 u^{(*)}_3}{\partial \zeta _p^2\partial \tau ^2}, \end{aligned} \end{aligned}$$
(51)

subject to boundary conditions, which are obtained by combining leading-, next- and second-order equations for \(p_{3i}\), \(p_{33}\), \(q_{3i}\) and \(p_{33}\),

$$\begin{aligned} \begin{aligned}&\sigma ^{*}_{3i}-\frac{\eta }{2}\left( \frac{\partial \sigma ^{*}_{ii} }{\partial \xi _i}+\frac{\partial \sigma ^{*}_{ij} }{\partial \xi _j}\right) +\frac{\eta ^2}{2}\left( \frac{\partial ^2 \sigma ^{*}_{ii} }{\partial \xi _i \partial \zeta _p}+\frac{\partial ^2 \sigma ^{*}_{ij} }{\partial \xi _j \partial \zeta _p }+2\frac{\partial ^2 \sigma ^{*}_{3i} }{\partial \zeta _p ^2}\right) \\&\quad -\frac{\eta ^3}{2}\left( \frac{\partial ^3 \sigma ^{*}_{ii} }{\partial \xi _i \partial \zeta _p^2}+\frac{\partial ^3 \sigma ^{*}_{ij} }{\partial \xi _j \partial \zeta _p^2 }\right) ={{f^*_i}},\\&\sigma ^{*}_{33}-\frac{\eta ^2}{2}\left( \frac{\partial ^2 \sigma ^{*}_{ii} }{\partial \xi _i^2}+2\frac{\partial ^2 \sigma ^{*}_{ij} }{\partial \xi _i\partial \xi _j}+\frac{\partial ^2 \sigma ^{*}_{jj} }{\partial \xi _j^2}-2\frac{\partial ^2 \sigma ^{*}_{33} }{\partial \zeta _p^2}\right) \\&\quad +\frac{\eta ^3}{2}\left( \frac{\partial ^3 \sigma ^{*}_{ii} }{\partial \xi _i^2 \partial \zeta _p}+2\frac{\partial ^3 \sigma ^{*}_{ij} }{\partial \xi _i \partial \xi _j \partial \zeta _p}+\frac{\partial ^3 \sigma ^{*}_{jj} }{\partial \xi _j^2 \partial \zeta _p^2 }\right) ={{f^*_3}}. \end{aligned} \end{aligned}$$
(52)

In the original dimensional form, the boundary value problem formulated above can be rewritten as

$$\begin{aligned} \begin{aligned}&\frac{\partial \sigma _{ii}}{\partial x_i}+\frac{\partial \sigma _{ij}}{\partial x_j}+\frac{\partial \sigma _{3i}}{\partial x_3}=\rho \left( \frac{\partial ^2 u_i}{\partial t^2}-a^2\frac{\partial ^4 u_i}{\partial x_3^2\partial t^2}\right) ,\\&\frac{\partial \sigma _{3i}}{\partial x_i}+\frac{\partial \sigma _{3j}}{\partial x_j}+\frac{\partial \sigma _{33}}{\partial x_3}=\rho \left( \frac{\partial ^2 u_3}{\partial t^2}-a^2\frac{\partial ^4 u_3}{\partial x_3^2\partial t^2}\right) , \end{aligned} \end{aligned}$$
(53)

with

$$\begin{aligned} \begin{aligned}&\sigma _{3i}-\frac{a}{2}\left( \frac{\partial \sigma _{ii} }{\partial x_i}+\frac{\partial \sigma _{ij} }{\partial x_j}\right) +\frac{a^2}{2}\left( \frac{\partial ^2 \sigma _{ii} }{\partial x_i \partial x_3}+\frac{\partial ^2 \sigma _{ij} }{\partial x_j \partial x_3 }+2\frac{\partial ^2 \sigma _{3i} }{\partial x_3^2}\right) \\&\quad -\frac{a^3}{2}\left( \frac{\partial ^3 \sigma _{ii} }{\partial x_i \partial x_3^2}+\frac{\partial ^3 \sigma _{ij} }{\partial x_j \partial x_3^2 }\right) ={{f^*_i}},\\&\sigma _{33}-\frac{a^2}{2}\left( \frac{\partial ^2 \sigma _{ii} }{\partial x_i^2}+2\frac{\partial ^2 \sigma _{ij} }{\partial x_i\partial x_j}+\frac{\partial ^2 \sigma _{jj} }{\partial x_j^2}-2\frac{\partial ^2 \sigma _{33} }{\partial x_3^2}\right) \\&\quad +\frac{a^3}{2}\left( \frac{\partial ^3 \sigma _{ii} }{\partial x_i^2 \partial x_3}+2\frac{\partial ^3 \sigma _{ij} }{\partial x_i \partial x_j \partial x_3}+\frac{\partial ^3 \sigma _{jj} }{\partial x_j^2 \partial x_3^2 }\right) ={{f^*_3}}, \end{aligned} \end{aligned}$$
(54)

at \(x_3=0\).

4 Rayleigh surface waves

To investigate the influence of non-local elastic behavior on the propagation of surface waves, we focus on the plane strain case, where \(\partial /\partial x_2\equiv 0\), the displacements \(u_n\) are functions of \(x_1\) and \(x_3\) \((u_n = u_n(x_1, x_3))\) for \(n =1,3\), while the displacement \(u_2\) is zero. In this case, by neglecting the \(O(a^3)\) terms, the equations of motion (53) and boundary conditions (54) for the traction-free surface, meaning \(f_n=0\) at \(x_3=0\), can be simplified to

$$\begin{aligned} \begin{aligned}&\frac{\partial \sigma _{11}}{\partial x_1}+\frac{\partial \sigma _{31}}{\partial x_3}=\rho \left( \frac{\partial ^2 u_1}{\partial t^2}-a^2\frac{\partial ^4 u_1}{\partial x_3^2\partial t^2}\right) ,\\&\frac{\partial \sigma _{31}}{\partial x_i}+\frac{\partial \sigma _{33}}{\partial x_3}=\rho \left( \frac{\partial ^2 u_3}{\partial t^2}-a^2\frac{\partial ^4 u_3}{\partial x_3^2\partial t^2}\right) , \end{aligned} \end{aligned}$$
(55)

and

$$\begin{aligned} \begin{aligned}&\sigma _{31}-\frac{a}{2}\frac{\partial \sigma _{11} }{\partial x_1}+\frac{a^2}{2}\left( \frac{\partial ^2 \sigma _{11} }{\partial x_1 \partial x_3}+2\frac{\partial ^2 \sigma _{31} }{\partial x_3^2}\right) =0,\\&\sigma _{33}-\frac{a^2}{2}\left( \frac{\partial ^2 \sigma _{11} }{\partial x_1^2}-2\frac{\partial ^2 \sigma _{33} }{\partial x_3^2}\right) =0, \end{aligned} \end{aligned}$$
(56)

at \(x_3=0\), respectively.

It is well known that the components of the displacement can be expressed through the wave potentials \(\varphi \) and \(\psi \) as

$$\begin{aligned} u_1=\frac{\partial \varphi }{\partial x_1}+\frac{\partial \psi }{\partial x_3},\quad u_3=\frac{\partial \varphi }{\partial x_3}-\frac{\partial \psi }{\partial x_1}. \end{aligned}$$
(57)

Substituting the expressions of local stresses in terms of the displacements (3) into (55) and (56) and taking into account (57), we obtain

$$\begin{aligned} \begin{aligned}&c_1^2\nabla ^2 \varphi -\left( 1-a^2\frac{\partial ^2}{\partial x_3^2}\right) \frac{\partial ^2\varphi }{\partial t^2}=0,\\&c_2^2\nabla ^2 \psi -\left( 1-a^2\frac{\partial ^2}{\partial x_3^2}\right) \frac{\partial ^2\psi }{\partial t^2}=0, \end{aligned} \end{aligned}$$
(58)

and boundary conditions at \(x_3=0\)

$$\begin{aligned} \begin{aligned}&\mu \left[ 2\frac{\partial ^2\varphi }{\partial x_1 \partial x_3}+\frac{\partial ^2\psi }{\partial x_3^2}-\frac{\partial ^2\psi }{\partial x_1^2}\right] -\frac{a}{2}\left[ (\lambda +2\mu )\frac{\partial ^3\varphi }{\partial x_1^3}+\lambda \frac{\partial ^3\varphi }{\partial x_1 \partial x_3^2}+2\mu \frac{\partial ^3\psi }{\partial x_1^2\partial x_3}\right] \\&\quad +\frac{a^2}{2}\left[ (\lambda +2\mu )\left( \frac{\partial ^4\varphi }{\partial x_1^3\partial x_3}+\frac{\partial ^4\varphi }{\partial x_1 \partial x_3^3}\right) +\mu \left( \frac{\partial ^4\psi }{\partial x_1^4}+\frac{\partial ^4\psi }{\partial x_1^2\partial x_3^2}\right) \right] =0,\\&\lambda \frac{\partial ^2\varphi }{\partial x_1^2}+(\lambda +2\mu )\frac{\partial ^2\varphi }{\partial x_3^2}-2\mu \frac{\partial ^2\psi }{\partial x_1 \partial x_3}+a^2\left[ (\lambda +2\mu )\frac{\partial ^4\varphi }{\partial x_3^4}-\frac{\lambda +2\mu }{2}\frac{\partial ^4\varphi }{\partial x_1^4}\right. \\&\quad \left. +\frac{\lambda }{2}\frac{\partial ^4\varphi }{\partial x_1^2\partial x_3^2}-2\mu \frac{\partial ^4\psi }{\partial x_1\partial x_3^3}-\mu \frac{\partial ^4\psi }{\partial x_1^3\partial x_3}\right] =0, \end{aligned} \end{aligned}$$
(59)

where \(\nabla ^2=\partial ^2/\partial x_1^2+\partial ^2/\partial x_3^2\), \(c_1\) and \(c_2\) are longitudinal and transverse wave speeds defined by

$$\begin{aligned} c_1=\sqrt{\frac{\lambda +2\mu }{\rho }},\quad c_2=\sqrt{\frac{\mu }{\rho }}. \end{aligned}$$
(60)

The solutions of Eq. (58) are sought in the form of a traveling harmonic wave, i.e.,

$$\begin{aligned} \varphi =A e^{ik(x_1-ct)-r_1x_3},\quad \psi =B e^{ik( x_1-ct)-r_2x_3}, \end{aligned}$$
(61)

where k is wave number, c is wave speed, and \(r_1\) and \(r_2\) are attenuation coefficients ensuring decay away from the surface \(x_3=0\). These coefficients can be determined by substituting (61) into (58) as

$$\begin{aligned} r_1=\frac{k\sqrt{1-c^2/c_1^2}}{\sqrt{1-a^2k^2 c^2/c_1^2}},\quad r_2=\frac{k\sqrt{1-c^2/c_2^2}}{\sqrt{1-a^2k^2 c^2/c_2^2}}. \end{aligned}$$
(62)

We, now, assume that \(\varepsilon =a k\) is a small parameter, see [36]. On expanding (62) in small \(\varepsilon \) and ignoring terms of order \(O(\varepsilon ^3)\), we find that

$$\begin{aligned} r_1=kP(\kappa \gamma ), \quad r_2=k P(\gamma ), \end{aligned}$$
(63)

where

$$\begin{aligned} P(\alpha )=\sqrt{1-\alpha ^2}+\frac{1}{2}\alpha ^2 \sqrt{1-\alpha ^2}\varepsilon ^2,\; \kappa =c_2/c_1,\; \gamma =c/c_2. \end{aligned}$$
(64)

Now, inserting (61) into the boundary conditions (59) and considering (62), we arrive at the following set of linear equations in A and B:

$$\begin{aligned}{} & {} \left\{ -4iP(\kappa \gamma )-\varepsilon i\left[ \left( \frac{1}{\kappa ^2}-2\right) P(\kappa \gamma )-\frac{1}{\kappa ^2}\right] +\varepsilon ^2 iP(\kappa \gamma )\left[ \frac{1}{\kappa ^2}-\left( \frac{1}{\kappa ^2}+2\right) P^2(\kappa \gamma )\right] \right\} A \nonumber \\{} & {} \quad \qquad +2\left\{ 1+P^2(\gamma )-\varepsilon P(\gamma )+\varepsilon ^2P^4(\gamma )\right\} B=0,\nonumber \\{} & {} \qquad \left\{ \frac{1}{\kappa ^2}P^2(\kappa \gamma )+2-\frac{1}{\kappa ^2}+\varepsilon ^2 \left[ \frac{1}{\kappa ^2}P^4(\kappa \gamma )-\frac{1}{2}\left( \frac{1}{\kappa ^2} -2\right) P^2(\kappa \gamma )-\frac{1}{2\kappa ^2}\right] \right\} A\nonumber \\{} & {} \qquad \quad +\left\{ 2 i P(\gamma )+\varepsilon ^2 i P(\gamma ) \left( 2P^2(\gamma )-1\right) \right\} B=0. \end{aligned}$$
(65)
Fig. 2
figure 2

Comparison of classical Rayleigh root, \(\gamma _0\) and its non-local counterpart, \(\gamma \), with respect to \(\varepsilon \) for \(\nu =0.25\)

Full size image

Neglecting all terms higher than \(O(\varepsilon ^2)\), the simultaneous equations possess non-trivial solutions provided that the associated determinant vanishes, i.e.,

$$\begin{aligned} R(\gamma )-\varepsilon S_1(\gamma )+\varepsilon ^2 S_2(\gamma )=0, \end{aligned}$$
(66)

where leading-order term, \(R(\gamma )\), is the classical Rayleigh equation, i.e.,

$$\begin{aligned} R(\gamma )=\left( 2-\gamma ^2\right) ^2-4\sqrt{\left( 1-\gamma ^2\right) \left( 1-\kappa ^2\gamma ^2\right) }, \end{aligned}$$
(67)

and

$$\begin{aligned} \begin{aligned}&S_1(\gamma )=2\left( \kappa ^2-1\right) \gamma ^2\sqrt{1-\gamma ^2},\\&S_2(\gamma )=R(\gamma )+\frac{2\kappa ^2+1}{2}\gamma ^2\left( \gamma ^2-1\right) + 3\gamma ^2\sqrt{\left( 1-\gamma ^2\right) \left( 1-\kappa ^2\gamma ^2\right) }. \end{aligned} \end{aligned}$$
(68)

Following the procedure outlined in [37], we, first, expand \(\gamma \) as an asymptotic series in the small parameter \(\varepsilon \), given by

$$\begin{aligned} \gamma =\gamma _0+\varepsilon \gamma _1+\varepsilon ^2\gamma _2+\cdots . \end{aligned}$$
(69)

Then, the Taylor expansion of \(R(\gamma )\) about \(\gamma =\gamma _0\) becomes

$$\begin{aligned} R(\gamma )=R(\gamma _0)+R'(\gamma _0)(\gamma -\gamma _0)+\cdots , \end{aligned}$$
(70)

where \(\gamma _0\) is the normalized classical Rayleigh wave speed, i.e., \(R(\gamma _0)=0\). Inserting (69) and (70) into (66), it may be readily obtained that

$$\begin{aligned} \gamma _1=\frac{S_1(\gamma _0)}{R'(\gamma _0)},\quad \gamma _2=-\frac{S_2(\gamma _0)+S_3(\gamma _0)\gamma _1}{R'(\gamma _0)}, \end{aligned}$$
(71)

where

$$\begin{aligned} S_3(\gamma )=\frac{2\gamma \left( 3\gamma ^2-2\right) \left( \kappa ^2-1\right) }{\sqrt{1-\gamma ^2}} \end{aligned}$$
(72)

Therefore, all the coefficients associated with the corrections to the Rayleigh wave speed (69) have been determined.

In what follows, we present numerical results for the non-local wave speed, \(\gamma \). In Fig. 2, the classical Rayleigh wave root \(\gamma _0\) and the non-local root \(\gamma \) are plotted against the parameter \(\varepsilon \) specifically for \(\nu =0.25\). It can be observed from this figure that as the non-local effect decreases, the speed of the non-local surface wave approaches the classical Rayleigh wave speed. Conversely, when the non-local effect increases the obtained speed decreases and deviates from the classical wave speed (see [39] for more details).

Figure 3 displays the behavior of non-local root with respect to the small parameter \(\varepsilon \), with calculations performed for the different values of \(\nu \). It can be easily seen from this figure that the obtained root for the non-local Rayleigh wave speed exhibits similar profile to the results presented in [36].

Fig. 3
figure 3

Variation of the non-local root \(\gamma \) versus \(\varepsilon \) for various values of \(\nu \)

Full size image

In Table 1, the values of the non-local root are presented for various small parameter \(\epsilon \) and Poisson’s ratio \(\nu \). It can be observed from the table that for the small values of \(\varepsilon \) the obtained root, \(\gamma \), is very close to the values presented in [36]. This comparison indicates that the obtained results in the current study are in good agreement with the findings from the reference. The similarity between the values suggests that the proposed approach in this paper effectively captures the non-local behavior and provides reliable results, consistent with the established knowledge in the field.

Table 1 Values of non-local root \(\gamma \) for \(\varepsilon \) and \(\nu \)
Full size table

5 Conclusions

In this paper, boundary value problems in a non-local elastic half-space are considered. The non-locality of the half-space is assumed to be only in the vertical coordinate. The problem is formulated using Eringen’s theory, which involves constitutive integral relations with a one-dimensional exponential kernel. An asymptotic approach enabling to express equations in terms of local counterparts is developed by following the similar procedure given in the previous studies, e.g., see [35, 37, 39]. Derived formulations include explicit correction terms to the classical governing equations and boundary conditions. It can be easily seen from the formulations that the leading-order approximation coincides with the local problem for the half-space. To validate the modeling, Rayleigh surface waves in a plane strain problem are considered. The dispersion relation for this problem is found to include the Rayleigh equation as well as additional terms arising from the non-local effects.

The propagation of Rayleigh waves in non-local media has numerous applications in surface wave theory. In particular, the analysis of Rayleigh waves in an elastic lattice attached to gyroscopic spinners has been conducted in [44], which holds significant relevance in various engineering fields, including the design of energy splitters. The proposed model has the potential to analyze functionally graded structures [45] and structures with vertically inhomogeneous foundations [46]. Additionally, the proposed model can be generalized for other types of kernels given in [36]. The refined formulation also has potential to be extended to non-local elastic solids with different types of boundary conditions. Waveguides including elastic plates, shells and beams with the presented non-local boundary conditions might also be studied.