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The purpose of this paper is to deal with the three-dimensional analysis of free vibrations in a stress-free and rigidly fixed homogeneous transversely isotropic hollow cylinder in the context of three-phase-lag (TPL) model of hyperbolic thermoelasticity.

The matrix Frobenius method of extended power series is employed to obtain the solution of coupled ordinary differential equations along the radial coordinate.

The natural frequency, dissipation factor and inverse quality factor in the stress-free and rigidly fixed hollow cylinder get significantly affected due to thermal vibrations and thermo-mechanical coupling.

The modified Bessel functions and matrix Frobenius method have been directly used to study the vibration model of a homogeneous, transversely isotropic hollow cylinder in the context of TPL model based on three-dimensional thermoelasticity.

The purpose of this paper is to study the propagation of inhomogeneous waves in a partially saturated poro-thermoelastic media through the examples of the free surface of such media..

The mathematical model evolved by Zhou et al. (2019) is solved through the Helmholtz decomposition theorem. The propagation velocities of bulk waves in partially saturated poro-thermoelastic media are derived by using the potential functions. The phase velocities and attenuation coefficients are expressed in terms of inhomogeneity angle. Reflection characteristics (phase shift, loci of vertical slowness, amplitude, energy) of elastic waves are investigated at the stress-free thermally insulated boundary of a considered medium. The boundary can be permeable or impermeable. The incident wave is portrayed with both attenuation and propagation directions (i.e. inhomogeneous wave). Numerical computations are executed by using MATLAB.

In this medium, the permanence of five inhomogeneous waves is found. Incidence of the inhomogeneous wave at the thermally insulated stress-free surface results in five reflected inhomogeneous waves in a partially saturated poro-thermoelastic media. The reflection coefficients and splitting of incident energy are obtained as a function of propagation direction, inhomogeneity angle, wave frequency and numerous thermophysical features of the partially saturated poro-thermoelastic media. The energy of distinct waves (incident wave, reflected waves) accompanying interference energies between distinct pairs of waves have been exhibited in the form of an energy matrix.

The sensitivity of propagation characteristics (velocity, attenuation, phase shift, loci of vertical slowness, energy) to numerous aspects of the physical model is analyzed graphically through a particular numerical example. The balance of energy is substantiated by virtue of the interaction energies at the thermally insulated stress-free surface (opened/sealed pores) of unsaturated poro-thermoelastic media through the bulk waves energy shares and interaction energy.

The purpose of this paper is to present a model to analyze free vibrations in a transradially isotropic, thermoelastic hollow sphere subjected to stress free, thermally insulated or stress free, isothermal and rigidly fixed, thermally insulated or rigidly fixed, isothermal boundary conditions.

The potential functions along with spherical wave solution have been used to reduce the system of governing partial differential equations to a coupled system of ordinary differential equations in radial coordinates after employing non-dimensional quantities. Matrix Frobenius method of extended power series has been employed to obtain accurate solution of coupled differential equations in terms of radial coordinates. The mathematical model of the considered problem has been solved analytically to obtain the characteristics equations after imposing the appropriate boundary conditions at the outer and inner surfaces of the hollow sphere. The characteristic equations which govern various types of vibration modes expected to exist have been derived in the compact form. The special cases of spheroidal and toroidal modes of vibrations have been deduced from the characteristic equations and discussed.

The toroidal mode has been found to be independent of temperature change. The magnitude of lowest frequency and damping factor are significantly affected in the presence of thermal field and increase with an increase in the spherical harmonics in addition to geometry of the structure.

The matrix Frobenius method has been used to develop analytical solutions and functional iteration technique to carry out numerical simulations of such structures for the first time. The simulated results are presented graphically and compared with the available literature.

The purpose of this manuscript is to study the vibration characteristics of the spherically symmetric solid and hollow spheres poised of a homogeneous thermoelastic material, based on the three dimensional coupled thermoelasticity.

In this paper, matrix Fröbenius series solution is used to derive the frequency equations, for the field functions. Results have been applied on rigidly fixed boundary conditions.

The main finding of this paper is that the frequency of vibration of spherically symmetric sphere (structure is independent of theta and phi) increases with the increase of radius, for solid spheres and for hollow spheres with thickness to mean radius ratio. Deformation in the given materials increases with thickness to mean radius ratio of the hollow sphere.

A numerical simulation has been done with the help of functional iteration method for solid and hollow thermoelastic spheres made of zinc and poly methyl meth acrylate materials for different boundary conditions. The computer simulated results in contempt of frequency, damping of vibration modes and displacement have been obtained graphically and compared with the existed results.

The purpose of this paper is to analyze the free vibrations in a stress free and thermally insulated (or isothermal), homogeneous, transversely isotropic, solid cylinder based on three‐dimensional coupled thermoelasticity, which is initially undeformed and kept at uniform temperature.

The displacement potential functions have been introduced in the equations of motion and heat conduction in order to decouple the purely shear and longitudinal motions. The system of governing partial differential equations is reduced to four second‐order coupled ordinary differential equations in radial coordinate by using the method of separation of variables. The matrix Frobenius method of extended power series is employed to obtain the solution of coupled ordinary differential equations along the radial coordinate. The convergence analysis of matrix Frobenius method has been successfully carried out.

The purely transverse mode is found to be independent of the rest of the motion and temperature change. The natural frequency, dissipation factor, inverse quality factor and frequency shift of vibrations in a stress free solid cylinder get significantly affected due to thermal variations and thermo‐mechanical coupling.

A new procedure is used and compared to other methods available in the literature.

The purpose of this paper is to study the analytical solutions of transversely isotropic thermo-piezoelectric interactions in a polygonal cross-sectional fiber immersed in fluid using the Fourier expansion collocation method.

A mathematical model is developed for the analytical study on a transversely isotropic thermo-piezoelectric polygonal cross-sectional fiber immersed in fluid using a linear form of three-dimensional piezothermoelasticity theories. After developing the formal solution of the mathematical model consisting of partial differential equations, the frequency equations have been analyzed numerically by using the Fourier expansion collocation method (FECM) at the irregular boundary surfaces of the polygonal cross-sectional fiber. The roots of the frequency equation are obtained by using the secant method, applicable for complex roots.

From the literature survey, it is evident that the analytical formulation of thermo-piezoelectric interactions in a polygonal cross-sectional fiber contact with fluid is not discussed by any researchers. Also, in this study, a polygonal cross-section is used instead of the traditional circular cross-sections. So, the analytical solutions of transversely isotropic thermo-piezoelectric interactions in a polygonal cross-sectional fiber immersed in fluid are studied using the FECM. The dispersion curves for non-dimensional frequency, phase velocity and attenuation coefficient are presented graphically for lead zirconate titanate (PZT-5A) material. The present analytical method obtained by the FECM is compared with the finite element method which shows a good agreement with present study.

This paper contributes the analytical model to find the solution of transversely isotropic thermo-piezoelectric interactions in a polygonal cross-sectional fiber immersed in fluid. The dispersion curves of the non-dimensional frequency, phase velocity and attenuation coefficient are more prominent in flexural modes. Also, the surrounding fluid on the various considered wave characteristics is more significant and dispersive in the hexagonal cross-sections. The aspect ratio (a/b) of polygonal cross-sections is critical to industry or other fields which require more flexibility in design of materials with arbitrary cross-sections.

This paper addresses a significant research gap in the study of Rayleigh surface wave propagation within a piezoelectric medium characterized by piezoelectric properties, thermal effects and voids. Previous research has often overlooked the crucial aspects related to voids. This study aims to provide analytical solutions for Rayleigh waves propagating through a medium consisting of a nonlocal piezo-thermo-elastic material with voids under the Moore–Gibson–Thompson thermo-elasticity theory with memory dependencies.

The analytical solutions are derived using a wave-mode method, and roots are computed from the characteristic equation using the Durand–Kerner method. These roots are then filtered based on the decay condition of surface waves. The analysis pertains to a medium subjected to stress-free and isothermal boundary conditions.

Computational simulations are performed to determine the attenuation coefficient and phase velocity of Rayleigh waves. This investigation goes beyond mere calculations and examines particle motion to gain deeper insights into Rayleigh wave propagation. Furthermore, this investigates how kernel function and nonlocal parameters influence these wave phenomena.

The results of this study reveal several unique cases that significantly contribute to the understanding of Rayleigh wave propagation within this intricate material system, particularly in the presence of voids.

This investigation provides valuable insights into the synergistic dynamics among piezoelectric constituents, void structures and Rayleigh wave propagation, enabling advancements in sensor technology, augmented energy harvesting methodologies and pioneering seismic monitoring approaches.

This study formulates a novel governing equation for a nonlocal piezo-thermo-elastic medium with voids, highlighting the significance of Rayleigh waves and investigating the impact of memory.

This chapter contributes to the ongoing debate about how digitalisation affects the internationalisation of small- and medium-sized firms (SMEs). By applying the Uppsala Internationalisation Process model, this chapter examines the impact of e-commerce on the internationalisation of SMEs. The study uses a unique dataset, which includes 14,513 SMEs across several sectors in 34 countries. The results show that firms using the Internet as a means to provide information about the firm exhibit a higher degree of internationalisation, while using the Internet to facilitate transactions was found to have a positive impact on the ratio of foreign sales to the total sales; however, these foreign sales are likely to be concentrated in less regions/markets. Furthermore, perceived export barriers were found to be a significant moderator of the effects of e-commerce usage on international intensity and international diversification. This suggests that e-commerce does not automatically facilitate the internationalisation of SMEs.

The purpose of this paper is to study the effect of rotation and gravity on a homogeneous, isotropic, and generalized thermo-viscoelastic material with voids. The problem is studied in the context of the coupled theory, Lord-Shulman theory with one relaxation time, and Green-Lindsay theory with two relaxation times.

The analytical method used was the normal mode analysis technique.

Numerical results for the physical quantities were analyzed and presented graphically. The graphical results indicated that the effects of rotation and gravity were observable physical effects on the thermo-viscoelastic material with voids. Comparisons were made between the results obtained in the absence and presence of rotation and gravity.

In the present work, the authors investigated the effect of rotation and gravity on thermo-viscoelastic medium with voids. Comparisons were also made between the three theories in the absence and the presence of rotation and gravity. Such problems are very important in many dynamical systems.