Dispersive behavior of high frequency Rayleigh waves propagating on an elastic half space

Abstract

When the wavelength of Rayleigh wave is comparable with nanometers, Rayleigh wave will become dispersive. Such an interesting phenomenon cannot be predicted by the classical theory of elastodynamics. In order to reveal the internal mechanism and influencing factors of the dispersion, a model of Rayleigh wave propagating on an elastic half space is established and analyzed by a new theory of surface elastodynamics, in which the surface effect characterized by both the surface energy density and surface inertia is introduced. Two intrinsic nano-length scales, including the ratio of bulk surface energy density to bulk shear modulus and the ratio of surface mass density to bulk mass density, are achieved. It is found that when the wavelength of Rayleigh wave is comparable with the two intrinsic nano-lengths, the surface effect becomes significant. As a result, dispersion of Rayleigh wave happens and even two Rayleigh waves with different wave speeds may appear. Furthermore, it is found that the effect of surface energy density would enhance the wave speed, while that of surface inertia would reduce it. With the increase of wavelength, both effects gradually disappear and the Rayleigh wave speed degenerates to the classical one. The results of this paper are not only helpful to understand the dispersive mechanism of elastic waves, but also helpful for the fine design and measurement of nanowave devices.

Graphic abstract

Introduction

Rayleigh wave, a kind of surface wave confined closely to the surface of a solid medium, has widespread applications in the field of applied physics and engineering, such as seismology, nondestructive evaluation, signal processing in electronic systems and surface acoustic wave sensors [1]. The classical elastodynamics declares that the main feature of Rayleigh wave propagating in a semi-infinite isotropic medium is non-dispersive, which means that its wave speed is independent of its wave frequency (or wave number) and only depends on the material properties of the solid medium [2].

However, a dispersive behavior of Rayleigh wave occurs at high frequencies of sub-THz and THz (1 THz = 1 × 1012 Hz), of which the corresponding wavelength is at the nanoscale [3, 4]. Such a phenomenon is attributed to the surface effect of solid mediums, which is induced by the inhomogeneous property at nanoscales near the material surface [5, 6]. When the scale of inhomogeneity near the material surface is comparable with the nanometer wavelength of Rayleigh wave, the propagation behavior of Rayleigh wave will be influenced. However, the classical elastodynamics lacks of any surface material parameter [2], which cannot predict the dispersive behavior of Rayleigh wave. It becomes a critical issue to characterize the surface property of solid mediums and the surface effect on the mechanical behavior, when the characteristic length of a system achieves nanometer.

In 1970s, Gurtin and Murdoch [7] proposed a theory of surface elastostatics. Surface effect is considered, in which the surface is regarded as a material different from the inner one and abides by a linearly elastic constitutive relation. The residual surface stress induced by surface relaxation and the surface elastic constants, as two key material parameters, characterize the surface property, which will further influence the mechanical property of the whole solid system. Comparison with the classical elastostatics [8] shows that a surface-induced traction in terms of the surface stress is additionally introduced into the stress boundary conditions. Based on the theory of surface elastostatics [8], many static problems of the nanostructured materials have been investigated [9, 10].

In order to predict the dynamic behavior of solid mediums with surface effect, Gurtin and Murdoch [11, 12] further extended the static theory of surface elasticity to a dynamic one. The equation of motion and dynamic stress boundary conditions are established. In contrast to the classical theory of dynamics, a surface inertial force in terms of surface mass density (the surface mass per unit area) is included in the boundary conditions, except for the surface-induced traction. Consequently, the residual surface stress, surface elastic constants as well as surface mass density are additionally introduced in the theory of surface elastodynamics. Typical problems, including vibration of nanostructures and propagation of elastic waves in nanomaterials, have been carried out based on the Gurtin–Murdoch theory (G–M theory) of surface elastodynamics [13,14,15,16,17,18].

Among the above typical dynamic problems, propagation of Rayleigh wave is a primary one. Using the G–M theory of surface elastodynamics, Murdoch [12] studied the problem of a plane Rayleigh wave propagating in an isotropic elastic half space, in which an unusual dispersive phenomenon of Rayleigh waves was found due to the surface effect. Chandrasekharaiah [19] further generalized Murdoch’s work to an extended case for arbitrary surface waves. Velasco and Garciamoliner [6] and Krylov [20] considered the problem of surface waves propagating in solids with structural inhomogeneities or planar defects near the surface. Anti-plane waves propagating in solids with surface effect was discussed by Eremeyev et al. [21, 22]. Enzevaee and Shodja [3] analyzed the effect of crystallographic structure on the propagation behavior of Rayleigh waves in single crystal structures. He and Zhao [4] developed a finite element model to simulate the influence of surface effect on Rayleigh waves, in which contributions of surface curvature and thickness were emphasized. In addition, Georgiadis and Velgaki [23] investigated the dispersion of Rayleigh waves in materials with both micro-structure and couple-stress effects.

A common finding is that Rayleigh waves at high frequencies would become dispersive, which is due to the surface effect. Furthermore, it is found that the dispersion of Rayleigh waves is closely related to the sign of surface elastic constants [3, 4]. However, it is still a problem of how to achieve the surface elastic constants in the G–M theory, which cannot be measured experimentally. Almost all the adopted parameters in existing literatures come from the molecular dynamics (MD) simulations [24, 25]. Even in the MD simulations, many problems still open, such as how many atom layers can be regarded as a surface in the numerical model, whether the surface elastic constants depends on the size of the numerical model, what is the physical meaning of a negative surface elastic constant, etc.

In order to avoid the above problems, an alternative static theory of surface elasticity was proposed by Chen and Yao [26], in which the bulk surface energy density (the surface energy density of the bulk material) and surface relaxation parameter, as two critical material parameters, were adopted to characterize the surface effect, instead of the surface elastic constants and residual surface stress in the G–M model. Both the bulk surface energy density and surface relaxation parameter have clear physical meanings and can be easily found from material handbooks or experimental measurements [27,28,29,30]. The static theory of surface elasticity has been well used to analyze many typical problems including surface effect [31,32,33,34].

Based on the static theory of surface elasticity [26], Jia et al. [35] further proposed an alternative theory of surface elastodynamics, in which the equation of motion and dynamic stress boundary conditions were well formulated. Compared with the classical elastodynamics [2], a surface-induced traction related to the surface energy density and a surface inertial force related to the surface mass density are additionally introduced in the stress boundary conditions. Based on the newly developed theory of surface elastodynamics [35], surface effect on the propagation behavior of Rayleigh waves is analyzed in the present study.

Brief introduction of the theory of surface elastodynamics

Imagine a solid with a surface consisting of idealized crystal structures, as shown in Fig. 1. The whole deformation process can be divided into three parts: the initial (or reference) configuration, the relaxed configuration due to spontaneous surface relaxation and the current configuration induced by external loadings. The equation of motion inside the solid can be formulated in the current configuration as [8],

Fig. 1
figure1

Schematic of a solid with a surface consisting of idealized crystal structures, in which the lattice lengths in two bond directions of a surface unit cell are shown

$${\varvec{\sigma}} \cdot \nabla + {\varvec{f}} = \rho \user2{u^{\prime\prime}}\quad ({\text{in V}}\!-\! {\text{S}}),$$
(1)

where \({{\varvec{\upsigma}}}\) is the bulk Cauchy stress tensor; \({\varvec{u}}\) is the displacement vector and \(\user2{u^{\prime\prime}}\) denotes the second order time derivative of \({\varvec{u}}\). \({\varvec{f}}\) is the body force and \(\rho\) is the bulk mass density (mass density of bulk materials). \(\nabla\) is a spatial gradient operator in the current configuration. \({\text{V}}\) and \({\text{S}}\) represent the volume and surface of the solid, respectively.

Different from the static theory of surface elasticity [26], a surface inertial force related to the surface mass density is introduced in the stress boundary condition in the theory of surface elastodynamics [35], in addition to the surface-induced traction \({\varvec{\gamma}}\) related to the surface energy density. The stress boundary condition in the theory of surface elastodynamics is expressed as,

$$ {\varvec{p}} - {\varvec{\gamma}} - {{\varvec{\upsigma}}} \cdot {\varvec{n}} = \rho_{0} \varvec{u^{\prime\prime}}\quad \left( {\text{on S}} \right), $$
(2)

with

$$ \begin{gathered} \varvec{\gamma } = \varvec{\gamma }_{t} + \gamma _{n} \varvec{n}, \hfill \\ \varvec{\gamma }_{t} = \frac{{\nabla _{s} \phi _{0} }}{{J_{s} }} - \frac{{\phi _{0} \left( {\nabla _{s} J_{s} } \right)}}{{J_{s}^{2} }},\;\gamma _{n} \varvec{n} = \frac{{\phi _{0} \left( {\varvec{n} \cdot \nabla _{s} } \right)\varvec{n}}}{{J_{s} }}, \hfill \\ \end{gathered} $$
(3)

where \({\varvec{p}}\) denotes the external load vector, \(\phi_{0}\) is the Lagrangian surface energy density [26]. \({\varvec{\gamma}}_{t}\) and \(\gamma_{n}\) are the tangential and normal components of \({\varvec{\gamma}}\), respectively. \(\rho_{0}\) is the surface mass density of the two-dimensional surface \(S\) of the solid [3, 6, 12]. \({\varvec{n}}\) is a normal vector of the surface \({\text{S}}\); \(\nabla_{s}\) is a surface gradient operator. \(J_{s}\) is a Jacobean determinant characterizing the surface deformation from the reference configuration to the current one.

Based on the surface lattice model, the Lagrangian surface energy density \(\phi_{0}\) of a solid with a surface consisting of idealized crystal structures can be explicitly expressed as [26]

$$ \begin{aligned} & \phi _{0}{\,\,\,\,}= \phi _{0}^{{{\text{stru}}}} + \phi _{0}^{{{\text{chem}}}} , \\ & \phi _{0}^{{{\text{stru}}}} = \frac{{E_{b} }}{{2\sin \beta }}\sum\limits_{{i = 1}}^{2} {a_{{0i}} \eta _{i} \{ [3 + (\lambda _{i} + \lambda _{i} \varepsilon _{{si}} )^{{ - m}} - 3(\lambda _{i} + \lambda _{i} \varepsilon _{{si}} )]} \\ & \quad \quad\qquad \times [\lambda _{i}^{2} \varepsilon _{{si}}^{2} + (\lambda _{i} - 1)^{2} + 2\lambda _{i} (\lambda _{i} - 1)\varepsilon _{{si}} ]\} , \\ & \phi _{0}^{{{\text{chem}}}} = \phi _{{0{\text{b}}}} \left( {1 - w_{1} \frac{{D_{0} }}{D}} \right), \\ & \eta _{1} = {{a_{{01}} } / {a_{{02}} }},\quad \eta _{2} = {{a_{{02}} } / {a_{{01}} }}, \\ & \lambda _{i} = a_{{ri}} /a_{{0i}} ,\quad \varepsilon _{{si}} = (a_{i} - a_{{ri}} )/a_{{ri}} ,\quad i = 1,2, \\ \end{aligned} $$
(4)

where \(\phi_{0}\) consists of a structural part \(\phi_{0}^{{{\text{stru}}}}\) and a chemical part \(\phi_{0}^{{{\text{chem}}}}\). \(E_{{\text{b}}}\) denotes the bulk Young’s modulus (Young’s modulus of the bulk material). \(\phi_{{0{\text{b}}}}\), \(\lambda_{i}\) are the surface energy density of the bulk material and the size-dependent surface relaxation parameter, respectively. The two surface material parameters can be obtained from material manual and experimental measurements [27, 30, 36, 37]. \(\varepsilon_{si}\) is the surface strain induced only by external loadings. \(D_{0}\) describes an intrinsic size, which equals \(3d_{{\text{a}}}\) for nanoparticles, nanowires and \(2d_{{\text{a}}}\) for nanofilms with \(d_{{\text{a}}}\) as the atomic diameter. \(D\) is a characteristic length of nanomaterials and \(w_{1}\) an empirical parameter governing the size-dependent \(\phi_{0}^{{{\text{chem}}}}\). \(m\) equals 4 for alloys or compounds and 1 for pure metals.

Rayleigh waves propagating on an elastic half space

Governing equations

Consider an elastic half space with the x-axis lying along the initially flat surface and the z-axis pointing into the interior. Rectangular coordinate system \(\left( {x,y,z} \right)\) with an origin \(O\) is attached on the surface. Here, a plane strain problem in the \(xOz\) plane, corresponding to a Rayleigh wave propagating in the x-direction on a two-dimensional elastic half space, is studied, in which both the surface energy effect and the surface initial effect are included.

In the absence of body force, according to Eq.  (1), the equation of motion of a plane strain problem can be written as,

$$\frac{{\partial \sigma_{x} }}{\partial x} + \frac{{\partial \tau_{xz} }}{\partial z} = \rho \frac{{\partial^{2} u}}{{\partial t^{2} }},\quad \frac{{\partial \tau_{xz} }}{\partial x} + \frac{{\partial \sigma_{z} }}{\partial z} = \rho \frac{{\partial^{2} w}}{{\partial t^{2} }},$$
(5)

where \(\sigma_{x}\), \(\sigma_{z}\) denote the normal stresses in the x-axis and z-axis directions, respectively. \(\tau_{xz}\) is the component of shear stress. \(u\) and \(w\) are the displacements in the x-axis and z-axis directions, respectively. \(\rho\) is the density of the bulk material.

With an assumption of infinitesimal deformation, the non-zero strain components \(\varepsilon_{x}\), \(\varepsilon_{z}\) and \(\gamma_{xz}\) can be written as

$$ \varepsilon_{x} = \frac{\partial u}{{\partial x}},\quad \varepsilon_{z} = \frac{\partial w}{{\partial z}},\quad \gamma_{xz} = \frac{\partial u}{{\partial z}} + \frac{\partial w}{{\partial x}}. $$
(6)

The constitutive equations of the linearly elastic medium are

$$\begin{aligned} &\varepsilon_{x} = \frac{1}{2\mu }\left[ {\left( {1 - \nu } \right)\sigma_{x} - \nu \sigma_{z} } \right],\quad \varepsilon_{z} = \frac{1}{2\mu }\left[ {\left( {1 - \nu } \right)\sigma_{z} - \nu \sigma_{x} } \right],\\ &` \gamma_{xz} = \frac{{\tau_{xz} }}{\mu }, \end{aligned}$$
(7)

where \(\mu = {{E_{{\text{b}}} } \mathord{\left/ {\vphantom {{E_{{\text{b}}} } {\left[ {2\left( {1 + \nu } \right)} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {2\left( {1 + \nu } \right)} \right]}}\) and \(\nu\) is Poisson’s ratio of the bulk medium.

Substituting Eqs. (6) and (7) into Eq. (5) yields

$$ \left\{ \begin{gathered} \left( {\lambda + \mu } \right)\frac{\partial \theta }{{\partial x}} + \mu \nabla^{2} u = \rho \frac{{\partial^{2} u}}{{\partial t^{2} }}, \hfill \\ \left( {\lambda + \mu } \right)\frac{\partial \theta }{{\partial z}} + \mu \nabla^{2} w = \rho \frac{{\partial^{2} w}}{{\partial t^{2} }}, \hfill \\ \end{gathered} \right. $$
(8)

where \(\lambda = {{E_{{\text{b}}} \nu } \mathord{\left/ {\vphantom {{E_{{\text{b}}} \nu } {\left[ {\left( {1 + \nu } \right)\left( {1 - 2\nu } \right)} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\left( {1 + \nu } \right)\left( {1 - 2\nu } \right)} \right]}}\), \(\mu = {{E_{{\text{b}}} } \mathord{\left/ {\vphantom {{E_{{\text{b}}} } {\left[ {2\left( {1 + \nu } \right)} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {2\left( {1 + \nu } \right)} \right]}}\) are the Lamé moduli of the bulk medium. \(\theta = {{\partial u} \mathord{\left/ {\vphantom {{\partial u} {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}} + {{\partial w} \mathord{\left/ {\vphantom {{\partial w} {\partial z}}} \right. \kern-\nulldelimiterspace} {\partial z}}\) is the volumetric strain and \(\nabla^{2} = {{\partial^{2} } \mathord{\left/ {\vphantom {{\partial^{2} } {\partial x^{2} }}} \right. \kern-\nulldelimiterspace} {\partial x^{2} }} + {{\partial^{2} } \mathord{\left/ {\vphantom {{\partial^{2} } {\partial z^{2} }}} \right. \kern-\nulldelimiterspace} {\partial z^{2} }}\) is the Laplace operator in the plane deformation.

Without the externally applied traction, according to Eq. (2), the dynamic stress boundary conditions of the half space can be written as,

$$ \left\{ \begin{gathered} \left. {\sigma_{z} = - \gamma_{n} + \rho_{0} \frac{{\partial^{2} w}}{{\partial t^{2} }}} \right|_{z = 0} , \hfill \\ \left. {\tau_{xz} = \gamma_{x} + \rho_{0} \frac{{\partial^{2} u}}{{\partial t^{2} }}} \right|_{z = 0} , \hfill \\ \end{gathered} \right. $$
(9)

where \(\rho_{0}\) is the surface mass density of the surface \(xOy\). According to Eq. (3), the normal and tangential surface-induced tractions \(\gamma_{n}\), \(\gamma_{x}\) imposed on the surface can be explicitly expressed as a function of the Lagrangian surface energy density \(\phi_{0}\),

$$ \left\{ \begin{gathered} \left. {\gamma_{n} = \frac{{\phi_{0} }}{{J_{{\text{s}}} }}\frac{{\partial^{2} w}}{{\partial x^{2} }}} \right|_{z = 0} , \hfill \\ \left. {\gamma_{x} = \frac{1}{{J_{{\text{s}}} }}\frac{{\partial \phi_{0} }}{\partial x} - \frac{{\phi_{0} }}{{J_{{\text{s}}}^{2} }}\frac{{\partial J_{{\text{s}}} }}{\partial x}} \right|_{z = 0} . \hfill \\ \end{gathered} \right. $$
(10)

The general formula of surface energy density \(\phi_{0}\) is Eq. (4) for a solid with a surface consisting of idealized crystal structures. For simplicity, the initial configuration of the elastic half space is assumed to be an un-deformed face-centered-cubic (fcc) metallic material with a (100) surface, where a local coordinate system (1, 2) coincides with the global Cartesian coordinate system \(xOy\). As a result, we have \(m = 1\), \(\beta = 90^\circ\) and the initial lattice length in two bond directions \(a_{01} = a_{02} = a_{0}\). The surface relaxation parameters in both bond directions vanish, i.e., \(\lambda_{1} = \lambda_{2} \approx 1\). As a result, the chemical part of \(\phi_{0}\) is reduced to \(\phi_{0}^{{{\text{chem}}}} = \phi_{{0\text{b}}}\). The surface strains should be \(\varepsilon_{{{\text{s}}1}} = \left. {\varepsilon_{x} } \right|_{z = 0}\) and \({\upvarepsilon }_{{{\text{s2}}}} { = 0}\) under the plane strain assumption. Using the method of Taylor’s expansion and ignoring the high-order strain terms (\(n > 2\)), the Lagrangian surface energy density \(\phi_{0}\) of an elastic half space could be expressed as

$$\phi_{0} = \phi_{{0{\text{b}}}} + \frac{{\sqrt 2 E_{{\text{b}}} a_{0} }}{8}\left. {\varepsilon_{x}^{2} } \right|_{z = 0}.$$
(11)

Equation (11) provides an alternative surface elastic constitutive relation depending on the bulk surface energy density, instead of the commonly-used linear elastic one between surface stress and surface strain in Murdoch’s model [12].

Substituting Eq. (11) into Eq. (10) and noting that \(J_{{\text{s}}} = 1 + \varepsilon_{x}\) yield

$$ \left\{ \begin{aligned} &\gamma _{{\text{n}}} = \phi _{{0{\text{b}}}} \left( {1 - \varepsilon _{x} } \right)\frac{{\partial ^{2} w}}{{\partial x^{2} }}, \hfill \\& \gamma _{x} = - \phi _{{{\text{b}}}} \left( {1 - \chi \varepsilon _{x} } \right)\frac{{\partial ^{2} u}}{{\partial x^{2} }}, \hfill \\ \end{aligned} \right. $$
(12)

where \(\chi = 2 + {{\sqrt 2 E_{{\text{b}}} a_{0} } \mathord{\left/ {\vphantom {{\sqrt 2 E_{{\text{b}}} a_{0} } {(4\phi_{{0{\text{b}}}} }}} \right. \kern-\nulldelimiterspace} {(4\phi_{{0{\text{b}}}} }})\) is a dimensionless parameter related to the material properties. With the assumption of infinitesimal deformation, terms of \(\varepsilon_{x}\) and \(\chi \varepsilon_{x}\) in Eq. (12) can be approximately omitted [38]. Then, combining Eqs. (6), (7) and (12), the dynamic stress boundary conditions in Eq. (9) can be rewritten as

$$ \left\{ \begin{gathered} \lambda \frac{\partial u}{{\partial x}} + \left( {\lambda + 2\mu } \right)\frac{\partial w}{{\partial z}}{ = } - \phi_{{0{\text{b}}}} \frac{{\partial^{2} w}}{{\partial x^{2} }} + \rho_{0} \frac{{\partial^{2} w}}{{\partial t^{2} }}, \hfill \\ \mu \left( {\frac{\partial u}{{\partial z}} + \frac{\partial w}{{\partial x}}} \right) = - \phi_{{0{\text{b}}}} \frac{{\partial^{2} u}}{{\partial x^{2} }} + \rho_{0} \frac{{\partial^{2} u}}{{\partial t^{2} }}. \hfill \\ \end{gathered} \right. $$
(13)

It is worth noting that, compared with Murdoch’s model considering the surface effect [12], the present model has two differences: the first is that the surface-induced traction in boundary conditions is obtained as a function of the surface energy density, as shown in Eq. (10), in contrast to that related to the surface stress in Murdoch’s model. The second is the surface material parameters related to surface effect. For an elastic half space, the effect of surface relaxation tends to vanish, which is physically reasonable [38]. Consequently, only the bulk surface energy density \(\phi_{{0{\text{b}}}}\) related to the surface energy effect and the surface mass density \(\rho_{0}\) related to the surface inertial effect are involved to characterize the surface effect. The previously-used surface elastic constants and residual surface stress in the existing model [12] are no longer involved. Combining Eqs. (6)–(8) and Eq. (13) yields the governing equations for the present model of Rayleigh waves propagating on an elastic half space with surface effect.

Solution of the Rayleigh wave speed

Similar to the method of the classical elastodynamics [2], the two non-zero displacement components \(u\) and \(w\) can be divided into two parts,

$$ \left\{ \begin{gathered} u = u_{1} + u_{2} , \hfill \\ w = w_{1} + w_{2} , \hfill \\ \end{gathered} \right. $$
(14)

with

$$ \left\{ \begin{gathered} u_{1} = A{\text{e}}^{ - Skz} \cos \left[ {k\left( {x - ct} \right)} \right], \hfill \\ u_{2} = B{\text{e}}^{ - Tkz} \cos \left[ {k\left( {x - ct} \right)} \right], \hfill \\ w_{1} = - AS{\text{e}}^{ - Skz} \sin \left[ {k\left( {x - ct} \right)} \right], \hfill \\ w_{2} = - BT^{ - 1} {\text{e}}^{ - Tkz} \sin \left[ {k\left( {x - ct} \right)} \right], \hfill \\ \end{gathered} \right.\quad \left\{ \begin{gathered} S = \left( {1 - s^{2} } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} , \hfill \\ T = \left( {1 - \tau^{2} s^{2} } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} , \hfill \\ s = {c \mathord{\left/ {\vphantom {c {c_{{\text{s}}} ,}}} \right. \kern-\nulldelimiterspace} {c_{{\text{s}}} ,}} \hfill \\ \tau = {{c_{{\text{s}}} } \mathord{\left/ {\vphantom {{c_{{\text{s}}} } {c_{{\text{p}}} ,}}} \right. \kern-\nulldelimiterspace} {c_{{\text{p}}} ,}} \hfill \\ \end{gathered} \right.\quad \left\{ \begin{gathered} k > 0, \hfill \\ c < c_{{\text{s}}} , \hfill \\ 0 < s < 1, \hfill \\ \end{gathered} \right. $$
(15)

where \(u_{1}\), \(w_{1}\) are the irrotational displacements and \(u_{2}\), \(w_{2}\) are the isovolumic displacements. \(c_{{\text{p}}} = \sqrt {{{\left( {\lambda + 2\mu } \right)} \mathord{\left/ {\vphantom {{\left( {\lambda + 2\mu } \right)} \rho }} \right. \kern-\nulldelimiterspace} \rho }}\), \(c_{{\text{s}}} = \sqrt {{\mu \mathord{\left/ {\vphantom {\mu \rho }} \right. \kern-\nulldelimiterspace} \rho }}\) denote the longitudinal and shear wave speeds, respectively. \(k\) and \(c\) denote the Rayleigh wave number and Rayleigh wave speed, respectively. The coefficients \(A\) and \(B\) can be determined by the dynamic stress boundary conditions in Eq. (13). Obviously, it can be easily proved that Eqs. (14) and (15) have already satisfied the equation of motion shown in Eq. (8).

Substituting Eqs. (14) and (15) into Eq. (13) yields

$$ \left\{ \begin{gathered} \left[ {T^{2} \left( {\lambda + 2\mu } \right) - \lambda + \left( {\phi_{{0{\text{b}}}} - \rho_{0} c^{2} } \right)Tk} \right]A + \left[ {2\mu + \left( {\phi_{{0{\text{b}}}} - \rho_{0} c^{2} } \right)S^{ - 1} k} \right]B = 0, \hfill \\ \left[ {2T\mu + \left( {\phi_{{0{\text{b}}}} - \rho_{0} c^{2} } \right)k} \right]A + \left[ {S\mu + S^{ - 1} \mu + \left( {\phi_{{0{\text{b}}}} - \rho_{0} c^{2} } \right)k} \right]B = 0. \hfill \\ \end{gathered} \right. $$
(16)

For a non-trivial solution, the determinant of parameters \(A\) and \(B\) should be zero, which further yields

$$C^{2} \left( {1 - TS} \right)\left( {l_{1} k} \right)^{2} + s^{2} \left( {TC + CS} \right)l_{1} k - g\left( s \right) = 0$$
(17)

with the following definitions

$$ \left\{ \begin{gathered} g\left( s \right) = \left( {2 - s^{2} } \right)^{2} - 4ST, \hfill \\ C = 1 - {{l_{2} s^{2} } \mathord{\left/ {\vphantom {{l_{2} s^{2} } {l_{1} .}}} \right. \kern-\nulldelimiterspace} {l_{1} .}} \hfill \\ \end{gathered} \right. $$
(18)

In the above equation, \(l_{1} = {{\phi_{{0{\text{b}}}} } \mathord{\left/ {\vphantom {{\phi_{{0{\text{b}}}} } \mu }} \right. \kern-\nulldelimiterspace} \mu }\), \(l_{2} = {{\rho_{0} } \mathord{\left/ {\vphantom {{\rho_{0} } \rho }} \right. \kern-\nulldelimiterspace} \rho }\) are two intrinsic length scales, both of which achieve nanoscales for most metallic materials [3, 12, 25]. The former \(l_{1}\) characterizes the surface energy effect, while the latter \(l_{2}\) reflects the surface inertial effect. When the Rayleigh wavelength (\({1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}\)) is at a macro-scale, the item \(l_{1} k\) in Eq. (17) tends to vanish. Then, Eq. (17) would reduce to the classical case, i.e., \(g\left( s \right) = 0\) [2], which can give the classical normalized Rayleigh wave speed as \(s_{0} \approx {{\left( {0.862 + 1.14\nu } \right)} \mathord{\left/ {\vphantom {{\left( {0.862 + 1.14\nu } \right)} {\left( {1 + \nu } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + \nu } \right)}}\). However, when the Rayleigh wavelength shrinks to a nanometer scale, the surface effect cannot be ignored, inducing a dispersive Rayleigh wave.

It is noted that a similar equation shown in Eq. (17) was also obtained by Murdoch [12] based on the G–M model. The main difference between the present equation and that achieved by Murdoch [12] is that three intrinsic length scales related to the residual surface stress, surface elastic constants and surface mass density, were involved in the Murdoch’s model.

Results and discussion

Case 1: \(l_{1} \ge l_{2}\)

If \(l_{1} \ge l_{2}\), \(C\left( s \right)\) is always positive for all \(s \in \left( {0,1} \right)\). Solutions of Eq. (17) can be obtained as

$$ k = - \frac{{s^{2} \left( {T + S} \right)}}{{2Cl_{1} \left( {1 - TS} \right)}} \pm \frac{{\sqrt {s^{4} \left( {T + S} \right)^{2} + 4\left( {1 - TS} \right)g\left( s \right)} }}{{2Cl_{1} \left( {1 - TS} \right)}}. $$
(19)

The relation between the normalized Rayleigh wave speed \({c \mathord{\left/ {\vphantom {c {c_{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {c_{{\text{s}}} }}\) and the wave number \(k\) is plotted in Fig. 2 with a fixed \(\nu = 0.25\). The classical result without the surface effect is also given for comparison. Due to the lack of any surface material parameters in the classical model [2], the classical Rayleigh wave speed keeps to be a constant (normalized one \(s_{0} = 0.9194\)) independent of the intrinsic length scales \(l_{1}\) and \(l_{2}\). However, when the surface effect is considered, it is found that for a relatively small wave number (a large wavelength or a small wave frequency), the surface effect is very weak and the Rayleigh wave is non-dispersive with a speed of the classical Rayleigh wave. With an increasing wave number, the Rayleigh wave speed gradually increases, which demonstrates that the surface effect starts to play a role and yields a dispersive Rayleigh wave. Specifically, with a fixed wave number and a determined \(l_{2}\), a larger \(l_{1}\) leads to an improved wave speed. With a fixed wave number and a determined \(l_{1}\), a larger \(l_{2}\) would reduce the wave speed. It indicates that the surface energy effect and the surface inertial effect show an opposite influence on the dispersive behavior of Rayleigh waves. An interesting phenomenon is that a Rayleigh wave may be transformed into a wave with a shear wave speed \(c_{{\text{s}}}\) or another unknown one due to the surface effect when the so-called Rayleigh wave number is large enough.

Fig. 2
figure2

Relation between the normalized Rayleigh wave speed \({c \mathord{\left/ {\vphantom {c {c_{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {c_{{\text{s}}} }}\) and the wave number \(k\) in the case of \(l_{1} \ge l_{2}\)

Case 2: \(l_{1} < l_{2}\)

If \(l_{1} < l_{2}\), \(C\left( s \right)\) has a zero point at \(s = s_{1} = \sqrt {{{l_{1} } \mathord{\left/ {\vphantom {{l_{1} } {l_{2} }}} \right. \kern-\nulldelimiterspace} {l_{2} }}} < 1\). In such a special case with a restrictive condition of \(s_{0} = s_{1}\), a non-dispersive Rayleigh wave exists with a classical Rayleigh wave velocity \(c = s_{0} c_{{\text{s}}}\).

If \(l_{1} < l_{2}\) and \(s_{0} < s_{1}\), the relation between the normalized Rayleigh wave speed \({c \mathord{\left/ {\vphantom {c {c_{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {c_{{\text{s}}} }}\) and the wave number \(k\) is given in Fig. 3 with a fixed \(\nu = 0.25\). One can easily see that with an increasing wave number (corresponding to a decreasing wavelength or an increasing wave frequency), Rayleigh waves will transform from non-dispersive to dispersive with a higher wave speed than the classical one. Furthermore, when the wave number is sufficiently large, two wave speeds emerge. Such novel phenomenon can be obviously attributed to the surface effect. In contrast, only one Rayleigh wave with a constant normalized wave speed \(s_{0} = 0.9194\) is predicted by the classical model without the surface effect. Interestingly, such a peculiar phenomenon is not sustainable. When the wave number continues to increase, the wave speed becomes unique again with a value larger than the classical one. Similar to case 1, the surface energy effect will enhance the wave speed, while the surface inertial effect will reduce it. Moreover, it is worth noting that the second Rayleigh wave has a cut-off wavelength (or cut-off wave number or cut-off frequency) as shown in Fig. 3, which means that the second Rayleigh wave cannot exist when its wavelength is larger than the cut-off one. It is also found that the cut-off wavelength decreases with an increasing intrinsic length scale \(l_{1}\) and increases with an increasing intrinsic length scale \(l_{2}\). Such a result indicates that the surface energy effect will reduce the cut-off wavelength, while surface inertial effect will enlarge it.

Fig. 3
figure3

Relation between the normalized Rayleigh wave speed \({c \mathord{\left/ {\vphantom {c {c_{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {c_{{\text{s}}} }}\) and the wave number \(k\) in the case of \(l_{1} < l_{2}\) and \(s_{0} < s_{1}\)

In the case of \(l_{1} < l_{2}\) and \(s_{0} > s_{1}\), many face-centered-cubic (fcc) metal materials abide by such conditions as shown in Table 1 [3, 25, 37, 39]. We take Au half space as an example. The relation between the normalized Rayleigh wave speed \({c \mathord{\left/ {\vphantom {c {c_{\text{s}} }}} \right. \kern-\nulldelimiterspace} {c_{\text{s}} }}\) and the wave number \(k\) is shown in Fig. 4. It is interesting to find that with an increasing wave number, the Rayleigh wave becomes dispersive but with a lower speed than the classical one (\(s < s_{0} = 0.9480\)). When the wave number continues to increase (\(9.04 \times 10^{9} < k < 1 \times 10^{12}\)), the Rayleigh wave may have two wave speeds. Both the wave speeds are less than the classical counterpart (\(s < s_{0}\)), except for one wave speed in the region of wave number \(k \in \left[ {9.04 \times 10^{9} ,\;1.063 \times 10^{10} } \right]\). When the wave number is larger than \(10^{12}\), the two wave speeds become a unique constant again, as shown in Fig. 4.

Table 1 Material parameters of several fcc metals [3, 25, 37, 39]
Fig. 4
figure4

Relation between the normalized Rayleigh wave speed \({c \mathord{\left/ {\vphantom {c {c_{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {c_{{\text{s}}} }}\) and the wave number \(k\) in the case of \(l_{1} < l_{2}\) and \(s_{0} > s_{1}\)

From above, one can see that the propagation behavior of Rayleigh waves is sensitive to the two surface material parameters, i.e., the bulk surface energy density and surface mass density, especially for a Rayleigh wave of high frequency. The dispersive behavior of Rayleigh waves can be modulated by two intrinsic length scales \(l_{1}\), \(l_{2}\), which should be useful for the design of surface acoustic wave devices.

Conclusion

Based on the new theory of surface elastodynamics, the behavior of a Rayleigh wave propagating on an elastic half space is analyzed, in which the surface effect is considered, including the surface energy effect and the surface initial effect. It is found that the propagating speed of Rayleigh waves is controlled by two intrinsic length scales, both of which have clearly physical meanings. One is the ratio of the bulk surface energy density to the bulk shear modulus and the other is the ratio of the surface mass density to the bulk mass density. For Rayleigh waves with a relatively large wavelength, the surface effect is very weak and the Rayleigh wave will propagate with a classical wave speed. When the wavelength shrinks to nanometers, the surface effect would induce a dispersive Rayleigh wave, in which the surface energy effect enhances the wave speed, while the surface inertial effect has an opposite tendency. An interesting phenomenon is that the surface effect may even lead to two Rayleigh waves existing with different wave speeds. The results of this paper are not only helpful to understand the dispersive mechanism of elastic waves, but also helpful for the fine design and measurement of nanowave devices.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants 11532013, 11872114, 11772333, and 12002033) and the Project of State Key Laboratory of Explosion Science and Technology (Grant ZDKT17-02).

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Jia, N., Peng, Z., Li, J. et al. Dispersive behavior of high frequency Rayleigh waves propagating on an elastic half space. Acta Mech. Sin. (2021). https://doi.org/10.1007/s10409-020-01009-3

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Keywords

  • Rayleigh wave
  • High frequency
  • Dispersion
  • Elastodynamics
  • Surface effect