This paper presents time domain FUNDAMENTAL SOLUTIONs for the extended Biot's dynamic formulations of two-dimensional (2D) unsaturated poroelasticity. Unsaturated porous media is considered as a porous media in which the voids are saturated with two immiscible fluids, i.e. liquid and gas. At first, the corresponding explicit Laplace transform domain FUNDAMENTAL SOLUTION is obtained in terms of skeleton displacements, as well as liquid and gas pressures. Subsequently, the closed-form time domain FUNDAMENTAL SOLUTIONs are derived by analytical inversion of the Laplace transform domain SOLUTIONs. Finally, a set of numerical results are presented which verifies the accuracy of the analytically inversed transient FUNDAMENTAL SOLUTION and demonstrates some salient features of the elastic waves in unsaturated media.

Response of a transversely isotropic 3-D half-space subjected to a surface time-‎harmonic excitation is presented in analytical form. The derivation of the ‎FUNDAMENTAL SOLUTIONs expressed in terms of displacements is based on the prefect ‎series of displacement potential functions that have been obtained in the companion ‎by the authors. First the governing equations are uncoupled in the cylindrical ‎coordinates. Then , the uncoupled equations are analytically solved to obtain Green ‎functions that are expressed in terms of Fourier series in the tangential direction of the ‎coordinates and in the terms of Hankel functions in its radial direction. The analytical ‎Green functions of this paper are exactly same as the results of lamb (1904) in the ‎case of isotropic material. The green functions can be used as the kernel functions of ‎the boundary integral equation that is used to solve elastodynamic boundary value ‎problems‏.‏

The main purpose of present article is to find the FUNDAMENTAL SOLUTION of partial differential equations in the generalized theory of thermoelastic diffusion materials with double porosity in case of steady oscillations in terms of elementary functions.

The FUNDAMENTAL SOLUTION of the system of differential equations in bio-thermoelasticity with dual phase lag (DPL) in case of steady oscillations in terms of elementary function is constructed and basic property is established. The tissue is considered as an isotropic medium and the propagation of plane harmonic waves is studied. The Christoffel equations are obtained and modified with the thermal as well as bio thermoelastic coupling parameters. These equations explain the existence and propagation of three waves in the medium. Two of the waves are attenuating longitudinal waves and one is nonattenuating transverse wave. The thermal property has no effect on the transverse wave. The velocities and attenuating factors of longitudinal waves are computed for a numerical bioheat transfer model with phase lag. The variation with frequency, thermal parameters, blood perfusion parameter and phase lag parameter are presented graphically. Also the reflection of plane wave from a stress free isothermal boundary of isotropic bio-thermoelastic half space in the context of DPL theory of thermoelasticity is studied. The amplitude ratios of various reflected waves are obtained and these amplitude ratios are further used to obtain the energy ratios of various reflected waves. These energy ratios are function of the angle of incidence and bio-thermoelastic properties of the medium. The expressions of energy ratios have been computed numerically for a particular model to show the effect of Poisson ratio, blood perfusion rate and phase lag parameters.

The present article deals with the study of propagation of plane waves in isotropic generalized thermo elastic diffusion with voids under initial stress. It is found that, for two dimensional model of isotropic generalized thermo elastic diffusion with voids under initial stress, there exists four coupled waves namely, P wave, Mass Diffusion (MD) wave, thermal (T) wave and Volume Fraction (VF) wave. The phase propagation velocities and attenuation quality factor of these plane waves are also computed and depicted graphically. In addition, the FUNDAMENTAL SOLUTION of system of differential equations in the theory of initially stressed thermo elastic diffusion with voids in case of steady oscillations in terms of elementary functions has been constructed. Some basic properties of the FUNDAMENTAL SOLUTION are established and some particular cases are also discussed.