Thermo-elastodynamic response of a spherical cavity in saturated poroelastic medium

Ganbin Liu, Kanghe Xie, Rongyue Zheng
2010 Applied Mathematical Modelling  
In this paper, an attempt has been made to investigate the thermo-hydro-elastodynamic response of a spherical cavity in isotropic saturated poroelastic medium when subjected to a time dependent thermal/mechanical source. The fully coupling thermo-hydro-elastodynamic model is presented on the basis of equations of motion, fluid flow, feat flow and constitutive equation with effective stress and temperature change. Solutions of displacement, temperature and stresses are obtained by using a
more » ... alytical approach in the domain of Laplace transform. Numerical results are also performed for portraying the nature of variations of the field variables, i.e. the coefficient of thermo-osmosis, the permeability. In addition, comparisons are presented with the corresponding partially thermo-hydroelastodynamic model and thermo-elastodynamic model to ascertain the validity and the difference between these models. Crown proposed by Ignaczak [5]. Recently, Bagri et al. [6] suggested a new system of coupled equations that contains the LS, GL and GN models in a unified form. They employed the suggested formulation and then analytically solved the coupled system of equations for a layer using the Laplace transform. Based on the Lord Sulman theory, Rehbinder [7], Chatterjee and Roychoudhuri [8], Erbay et al. [9] , considered an infinite isotropic elastic medium with a spherical cavity when the surface of the cavity subjected to different boundary conditions. The one-dimensional problem of distribution of thermal stresses and temperature in a generalized thermoelastic infinite medium with a spherical cavity subjected to a sudden change in temperature of its internal boundary was studied by Sherief and Saleh [10] . Mukhopadhyay [11] concerned with the thermally induced vibration in a homogeneous and isotropic unbounded body with a spherical cavity subjected to harmonically varying temperature. The distribution of stresses due to step input of temperature at the boundary of a spherical hole in a homogeneous isotropic unbounded body in the context of generalized theories of thermoelasticity was investigated by Kar and Kanoria [12] . In addition, the boundary integral equation formulation was done to solve the thermoelastic problem by Anwar and Sherief [13] . Several investigators employed the GN models to solve a variety of thermoelastic problems. The uniqueness of solution of the governing equations for the GN theory formulated in terms of stress and energy-flux is established by in Chandrasekharaiah [14] . Furthermore, Chandrasekharaiah [15] studied the one-dimensional thermal wave propagation in a half-space based on the GN model due to a sudden exposure of temperature to the boundary, using the Laplace transform method. In above literatures, however, the fluid flux in the thermoelasticity is neglected. In fact, porous materials make their appearance in a wide variety of settings, natural and artificial, and in diverse technological applications. As a consequence, a number of problems arises dealing with, among other issues, statics and strength, fluid flow and heat conduction, and dynamics. During last decades, much attention has been drawn to find reliable and efficient formulations and solution procedures. The theoretical developments in this area have matured from a simple extension of Biot's isotropic poroelastic theory to a more general approach that can handle the coupling along with the material anisotropy [16] . The extension of Biot's theory of poroelasticity to incorporate thermal effects has been used in various forms [17] . Booker and Savvidou [18] presented an exact solution for the consolidation of soil around a point heat source and proposed an approximate solution for a cylindrical source. Zimmerman [19] gave brief derivation of the equations of linearized poroelasticity and thermoelasticity. Giraud et al. [20] analysed the case of a heat source that decreases exponentially with time by considering a low-permeability clay for nuclear waste disposal. Wang and Papamichos [21] discussed solutions for a cylindrical wellbore and a spherical cavity subjected to a constant temperature change and heat flow rate. A thermodynamics theory for elastic saturated porous solids has been presented by Biot [22] . A coupled finite element model was presented by Lewis et al. [23] , in which a partitioned solution procedure is carried out after the time domain interaction to restore the symmetry of the coefficient. Coussy [24] gave a general theory of saturated porous solids under finite deformation employing the total Lagrangian formulation for the solid skeleton. Masters et al. [25] proposed a model for deforming fissured porous media, in which two different porosities are defined for the porous medium and the fissured network, respectively. Bai and Abousleiman [26] discussed the necessity for a full coupling and the possibility of decoupling and tried to provide practical framework for the thermoporo-elasto-plastic coupling analyses. A formulation for the thermoporo-elasto-plastic coupling analysis is presented by Wang and Dong [27], in which the energy balance equation is rederived based on the concept of free enthalpy. The corresponding finite element procedures are developed and implemented into the commercial software ADINA. Most of the above literatures are based on classical Darcy's law and Fourier's law for fluid flow and heat flow, respectively. Due to the complexity of the governing equations, most of the above-mentioned analytical solutions are obtained assuming a semi-coupled thermoporoelastic theory. Specifically, the equation describing the temperature variation is usually uncoupled from other field quantities. In fact, since detailed and deep knowledge of hydrothermal, hydromechanical and thermalmechanical processes and their interdependence is necessary for describing the fully coupled behaviour of fluid-saturated media, the thermo-mechanical coupling in the poroelastic medium turns out to be of much greater complexity than in the classical case of an impermeable elastic solid. In addition to thermal and mechanical interaction within each phase, thermal and mechanical coupling occurs between the phases. Therefore, a mechanical or thermal change in one phase results mechanical and thermal change throughout the aggregate. The generation and dissipation of thermally induced pore pressure with time is a complex phenomenon, involving gradients of pore pressure and temperature, hydraulic and thermal flows within the mass of soil, and changes in the mechanical properties of the soil with temperature [28] . In other words, the influences of temperature gradient on fluid flow (i.e., thermo-osmosis) and pore pressure gradient on heat flow (i.e., thermal-filtration) are neglected. Previous studies [29, 30] showed that, for semi-impermeable porous materials (e.g., very low-permeability clays), the thermo-osmosis and thermal-filtration effects play an important role in the thermal responses of the media. Based on the thermodynamics of irreversible processes, the mass conservation equation and heat energy balance equation are established and the governing equations of thermal consolidation for homogeneous isotropic materials are presented by Bai and Li [31], accounting for the coupling effects of the temperature, stress and displacement fields. Using the Biot's wave equation and the theory of thermodynamic, Liu et al. [32] investigated the dynamic response of saturated porous elastic medium. Furthermore, on the basis of the theory of thermodynamic of Biot's, Darcy law of fluid and the modified Fourier law of heat conduction, the non-linear model fully coupled thermo-hydro-elastodynamic response (THED) for a saturated poroelastic medium was derived by Liu et al. [33] . The compressibility of the medium, the influence of fluid flux on the heat flux and the influence of change of temperature on the fluid flux were considered in this model.
doi:10.1016/j.apm.2009.10.031 fatcat:7z6y4negrfdodcqkhsa6xqp4ty