State‐Space Approach to Characterize Rayleigh Waves in Nonlocal Piezothermoelastic Orthotropic Medium With Phase‐Lags

Rayleigh wave propagation with two temperature and diffusion in context of three phase lag thermoelasticity

This article delves into the intricate dynamics of Rayleigh wave propagation within a nonlocal orthotropic medium, where the presence of void and diffusion adds an intriguing layer to the analysis. Grounded in Eringen’s nonlocal elasticity theory and embracing the three-phase-lag model of hyperbolic thermoelasticity, the study focuses on the interplay between the mass diffusion principles of Fick’s and the Fourier’s law under the framework of hyperbolic thermoelasticity. The investigation employs a methodological approach centered around normal mode analysis to navigate the complexities of the problem at hand. The derived frequency equation governing Rayleigh waves undergoes meticulous scrutiny through the exploration of specific cases. The elliptical trajectory of surface particles and its eccentricity during Rayleigh wave propagation are identified and calculated. Graphical representations cover propagation speed, attenuation coefficient, penetration depth, and specific loss of Rayleigh waves relative to wave number for isothermal surfaces, providing insights into their nuanced behavior in orthotropic mediums in the inclusion and exclusion of the nonlocal variable, diffusion and void.

Effect of phase-lags on Rayleigh wave propagation in initially stressed magneto-thermoelastic orthotropic medium

Purpose This paper is concerned with the study of the propagation of Rayleigh waves in a homogeneous isotropic, generalized thermoelastic medium with mass diffusion and double porosity structure using the theoretical framework of three-phase-lag model of thermoelasticity. Design/methodology/approach Using Eringen’s nonlocal elasticity theory and normal mode analysis technique, this paper solves the problem. The medium is subjected to isothermal, thermally insulated stress-free, and chemical potential boundary conditions. Findings The frequency equation of Rayleigh waves for isothermal and thermally insulated surfaces is derived. Propagation speed, attenuation coefficient, penetration depth and specific loss of the Rayleigh waves are computed numerically. The impact of nonlocal, void and diffusion parameters on different physical characteristics of Rayleigh waves like propagation speed, attenuation coefficient, penetration depth and specific loss with respect to wave number for isothermal and thermally insulated surfaces is depicted graphically. Originality/value Some limiting and particular cases are also deduced from the present investigation and compared with the existing literature. During Rayleigh wave propagation, the path of the surface particle is found to be elliptical. This study can be extended to fields like earthquake engineering, geophysics and the degradation of old building materials.

Propagation of Rayleigh waves on curved surfaces

This paper aims to investigate the influence of rotation, initial stress and gravity field on the propagation of Rayleigh waves in a homogeneous orthotropic elastic medium. The government equations and Lame’s potentials are used to obtain the frequency equation which determines the velocity of Rayleigh waves, including rotation, initial stress and gravity field, in a homogeneous, orthotropic elastic medium has been investigated. The numerical results analyzing the frequency equation are discussed and presented graphically. It is important to note that the Rayleigh wave velocity in an orthotropic elastic medium increases a considerable amount in comparison to the Rayleigh wave velocity in an isotropic material. The results indicate that the effects of rotation, initial stress and gravity field on Rayleigh wave velocity are very pronounced.

The main aim of present study is to show the effect of gravity, initial stress and magnetic field on the Rayleigh waves propagation in homogeneous orthotropic magneto-thermoelastic medium in the context of Three Phase Lag (TPL) model at two temperature. The governing equations of thermoelasticity have been solved by normal mode technique to deduce the frequency equation for Rayleigh wave with relevant boundary conditions. Special cases have been derived for isothermal and thermally insulated surfaces. Computer simulation is used for numerical discussion to show the effects of various parameters on phase velocity of Rayleigh waves. The variation in phase velocity corresponding to wave number has been demonstrated graphically in the presence of gravity, initial stress and magnetic field.

The main aim of present study is to show the effect of gravity, initial stress and magnetic field on the Rayleigh waves propagation in homogeneous orthotropic magneto-thermoelastic medium in the context of Three Phase Lag (TPL) model at two temperature. The governing equations of thermoelasticity have been solved by normal mode technique to deduce the frequency equation for Rayleigh wave with relevant boundary conditions. Special cases have been derived for isothermal and thermally insulated surfaces. Computer simulation is used for numerical discussion to show the effects of various parameters on phase velocity of Rayleigh waves. The variation in phase velocity corresponding to wave number has been demonstrated graphically in the presence of gravity, initial stress and magnetic field.

In this paper the effects of heat conduction upon the propagation of Rayleigh surface waves in a semi-infinite elastic solid are studied theoretically in two special cases: (i) when the surface of the solid is maintained at constant temperature (case 1); and (ii) when the surface is thermally insulated (case 2). The investigation is carried out within the framework of the linear theory of thermoelasticity, a principal objective being the clarification of the relation between so-called thermoelastic Rayleigh waves and the Rayleigh waves of classical elastokinetics. The secular equation for thermoelastic Rayleigh waves is shown to define a many-valued algebraic function μ(X ) (X being a dimensionless frequency) the branches of which represent possible modes of surface wave propagation. Two different types of surface mode are recognized: E-modes, which resemble classical Rayleigh waves but are subject to damping and dispersion; and T-modes, which are essentially diffusive in character. Necessary and sufficient conditions for the existence of a surface wave are formulated, and questions of the existence and multiplicity of Rayleigh E- and T-modes in particular situations are resolved by submitting the branches of μ(X) to these requirements. The algebraic function μ(X ) has singular points at X = 0 and X = ∞, and approximations, valid at sufficiently low or at sufficiently high frequencies, to the speed of propagation v and the attenuation coefficient q of a given surface mode are obtainable from series representations of the appropriate branch of μ(X) in neighbourhoods of these singularities. The singular point X = 0 is associated with adiabatic deformations of the solid, and hence with classical Rayleigh waves, and the singularity at X = ∞ with isothermal deformations. Particular attention is devoted to the Rayleigh E-modes and the main conclusions reached are as follows. In case 1 there exists a single E-mode (mode 2) at low frequencies and two distinct E-modes (modes 1 and 2) at high frequencies. For mode 2, v/vR = 1 + O(X½), q = O(X3/2) Xdistinct 0 (vR being the speed of propagation of classical Rayleigh waves), and for both modes v and q approach finite limits as X → ∞. In case 2 the converse situation applies, there being two distinct E-modes (modes 1 and 2) at low frequencies and only one (mode 1) at high frequencies. For both modes, v/vR = 1 + O(X3/2), q = O(X2) as X → 0, and for mode 1, v and q approach finite limits as X → ∞. Detailed numerical results referring to a medium of worked pure copper at a reference temperature of 20 °C are given. In particular the frequency dependence of the speeds of propagation and attenuation coefficients of the various E-modes are exhibited, and the frequencies at which mode 1 appears in case 1 and at which mode 2 disappears in case 2 are determined.

Propagation of harmonic plane waves is considered in an orthotropic elastic medium in the presence of initial stress and gravity. Roots of a quadratic equation define the propagation of one quasi-longitudinal wave and one quasi-transverse wave in a symmetry plane in this medium. These two waves are coupled in the identical phase to define the propagation of Rayleigh waves at the boundary of the medium. Two conditions at the stress-free boundary translate into a complex frequency equation, which explains the dispersive behavior of this Rayleigh wave. For the presence of radical terms, this complex equation is rationalized into a real algebraic equation. Only one root of this algebraic equation satisfies the mother frequency equation and hence represents the propagation of dispersive Rayleigh waves at the boundary of the orthotropic solid. The influence of initial stress and gravity on velocity and polarization of Rayleigh waves is observed through a numerical example.

Nonlocal scale effect on Rayleigh wave propagation in porous fluid-saturated materials

The present article deals with the propagation of Rayleigh surface waves in homogeneous transversely isotropic medium. Effect of rotation on Rayleigh waves in thermoelastic half-space is studied under the purview of the three-phase-lag model in presence of magnetic field. The normal mode analysis is used to obtain the expressions for the displacement components, stresses, and temperature distribution. The frequency equations in the closed form are derived and the path of particles during Rayleigh wave propagation is found elliptical. In order to illustrate the analytical developments, the numerical solution is carried out and the computer-simulated results in respect of Rayleigh wave velocity, attenuation coefficient, and specific loss are presented graphically.

This study examines Rayleigh wave propagation dynamics in a nonlocal orthotropic medium with thermoelastic diffusion, utilizing Eringen's nonlocal elasticity theory and the Lord–Shulman hyperbolic thermoelasticity model. Normal mode analysis is used to solve the problem, deriving the frequency equation for Rayleigh waves and analyzing specific cases. The elliptical path of surface particles and its eccentricity are calculated. Graphs illustrate the relationships of propagation speed, attenuation coefficient, penetration depth, and specific loss of Rayleigh waves concerning wave number for both thermally insulated and isothermal surfaces. The results reveal that the presence of nonlocal parameters and diffusion significantly increases the values of physical variables, especially with higher wave numbers, highlighting their considerable impact on the system's dynamics.

Transference of Rayleigh Waves in Corrugated Orthotropic Layer over a Pre-stressed Orthotropic Half-Space with Self Weight

Under the action of Rayleigh waves, pile head is easy to rotate with a concrete pile cap, and pure fixed‐head condition is rarely achieved, which is a common phenomenon for it usually occurs on the precast piles with insufficient anchorage. In addition, the propagation characteristics of Rayleigh wave have been changed significantly due to the existence of capillary pressure and the coupling between phases in unsaturated soil, which significantly affects the pile‐soil interaction. In order to study the above problems, a coupled vibration model of unsaturated soil–pile system subjected to Rayleigh waves is established on the basis that the pile cap is equivalent to a rigid mass block. Meanwhile, the soil constitution is simplified to linear‐elastic and small deformations are assumed to occur during the vibration phase of soil–pile system. Then, the horizontal dynamic response of a homogeneous free‐field unsaturated soil caused by propagating Rayleigh waves is obtained by using operator decomposition theory and variable separation method. The dynamic equilibrium equation of a pile is established by using the dynamic Winkler model and the Timoshenko beam theory, and the analytical solutions of the horizontal displacement, rotation angle, bending moment and shear force of pile body are derived according to the boundary conditions of flexible constraint of pile top. Based on the present solutions, the rationality of the proposed model is verified by comparing with the previous research results. Through parametric study, the influence of rotational stiffness and yield bending moment of pile top on the horizontal dynamic characteristics of Rayleigh waves induced pile is investigated in detailed. The analysis results can be utilized for the seismic design of pile foundation under Rayleigh waves.