On non-locally elastic Rayleigh wave

Abstract

The Rayleigh-type wave solution within a widely used differential formulation in non-local elasticity is revisited. It is demonstrated that this wave solution does not satisfy the equations of motion for non-local stresses. A modified differential model taking into account a non-local boundary layer is developed. Correspondence of the latter model to the original integral theory with the kernel in the form of the zero-order modified Bessel function of the second kind is addressed. Asymptotic behaviour of the model is investigated, resulting in a leading-order non-local correction to the classical Rayleigh wave speed due to the effect of the boundary layer. The suitability of a continuous set-up for modelling boundary layers in the framework of non-local elasticity is analysed starting from a toy problem for a semi-infinite chain. This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)’.

Anti-plane shear waves in an elastic strip rigidly attached to an elastic half-space

“…After Korteweg (1901) and Mindlin (1965) it was also established that the surface elasticity is closely related to strain gradient elasticity, see e.g. , and to other nonlocal theories, see recent works by Chebakov et al (2016); Ghayesh & Farajpour (2019); Li et al (2020); Jiang et al (2022b); Kaplunov et al (2022); Yang et al (2023). Similar surfacerelated phenomena could be results of deformations localization in the vicinity of a surface (Kaplunov & Prikazchikov, 2017;Kaplunov et al, 2019).…”

Expressions of nonlocal quantities and application to Stoneley waves in weakly nonlocal orthotropic elastic half-spaces

“…With the help of equations ( 24), (31), and (33), it is not difficult to verify that the square brackets [ . ] are given by (up to a factor θ =…”

“…It is well-known that the differential equation (1) is not equivalent to the integral relation (2) at all if V is a strict subset of R 3 with nonempty boundary S (see Romano et al [29], Romano and Barretta [30], and Kaplunov et al [31, 32]). In fact, for this case, the integral equality (2) is equivalent to the differential equation (1) (for given g ) along with boundary conditions (on S ) which are the same as the boundary conditions for Green’s function α ( x , x ′ ) (see, for example, Melnikov and Melnikov [27] and Duffy [28]).…”

“…In fact, for this case, the integral equality (2) is equivalent to the differential equation (1) (for given g ) along with boundary conditions (on S ) which are the same as the boundary conditions for Green’s function α ( x , x ′ ) (see, for example, Melnikov and Melnikov [27] and Duffy [28]). They are called the constitutive boundary conditions [29] or the extra boundary conditions [31, 32]. We call the nonlocal model of elasticity whose constitutive law consists of equation (2) and the corresponding extra boundary conditions the equivalent differential model .…”

“…The equivalent differential model is unfortunately ill-posed as shown by Romano et al [29] for the beam problem (for structural problems, in general), by Kaplunov et al [31, 32] for the Rayleigh wave problem (for harmonic plane wave problems, in general), and for the antiplane motion of a half-space subjected to a surface loading of a traveling harmonic wave. In particular, these problems have no solution due to the number of required boundary conditions is bigger than the number of (integral) constants to be determined.…”