Two temperature fractional order thermoelasticity theory in a spherical domain

This article is an application of fractional thermoelasticity in association with two-temperature theory. The fractional heat conduction model has been proposed to investigate the thermal variations within the bounded spherical region. The corresponding heat conduction equation has been derived in the context of the generalized two-temperature theory of fractional thermoelasticity. The analytical solutions of thermal variations have been obtained in the Laplace domain, which are inverted using the Gaver–Stehfest algorithm in the time domain. Kuznetsov convergence criterion has been discussed for the bounded variations and stability of the problem. The delay time translations used in the heat flux vector and the temperature gradient result in the finite speed of thermal wave propagation. As a special case of time fractional derivative, the classical and generalized thermoelasticity theories have been recovered.     

Transient response analysis of a spherical shell embedded in an infinite thermoelastic medium based on a memory-dependent generalized thermoelasticity

In the context of a memory-dependent generalized thermoelasticity, the thermal-induced transient response in an infinite elastic body containing a spherical shell is investigated. A thermal shock is applied on the inner surface of the spherical shell. The infinite body and the spherical shell are assumed to be isotropic but two dissimilar materials. By using an analytical technique based on the Laplace transform along with its numerical inversion, the governing equations of the problem are solved and the non-dimensional physical quantities in the two materials, i.e., temperature, displacement and stress, are obtained and illustrated graphically respectively. In simulation, the accuracy of memory-dependent derivative (MDD) is verified by degrading the present model into L-S model to compare the results obtained from the cases with interfacial thermal resistance and without interfacial thermal resistance. In addition, the effects of the different kernel functions as well as the ratios of the two materials, including the ratios of the density, the thermal-conductivity and the time-delay, on the distributions of the considered variables are obtained and demonstrated graphically respectively.     

The effect of fractional thermoelasticity on two-dimensional problems in spherical regions under axisymmetric distributions

Abstract

Two-dimensional axisymmetric problems are considered within the context of the fractional order thermoelasticity theory. The general solution is obtained in the Laplace transform domain by using a direct approach without the use of potential functions. The resulting formulation is used to solve two problems of a solid sphere and of an infinite space with a spherical cavity. The surface in each case is taken to be traction free and subjected to a given axisymmetric temperature distribution. The inversion of the Laplace transforms is carried out using the inversion formula of the transform together with Fourier expansion techniques. Numerical methods are used to accelerate the convergence of the resulting series to obtain the temperature, displacement, and stress distributions in the physical domain. Numerical results are represented graphically and discussed. Some comparisons are shown in figures to estimate the effect of the fractional order parameter on all studied fields.     

A study on generalized thermoelasticity theory based on non-local heat conduction model with dual-phase-lag

Non-local continuum theory helps to analyze the influence of all the points of the body at a material point. Involvement of non-local factor, i.e., size effect in heat conduction theory enhances the microscopic effects at a macroscopic level. The present work is concerned with the generalized thermoelasticity theory based on the recently introduced non-local heat conduction model with dual-phase-lag effects by Tzou and Guo. We formulate the generalized governing equations for this non-local heat conduction model and investigate a one-dimensional elastic half-space problem. Danilovskaya’s problem is taken, i.e., we assume that thermal shock is applied at the traction free boundary of the half-space. Laplace transformation is used to solve the problem and numerical method is applied to solve the problem by finding Laplace inversion through the Stehfest method. Various graphs are plotted to analyze the effects of different parameters and to mark the variation of this non-local model with previously established models.     

Fundamental solution for a line source of heat in the fractional order theory of thermoelasticity using the new Caputo definition

Abstract

The new Caputo Fabrizio fractional differential operator is used to investigate a problem in the fractional order theory of thermoelasticity. The problem concerns an infinite elastic space under the effect of a continuous line source of heat. The problem is solved using asymptotic expansions valid for short times. Laplace and Hankel transforms are used to solve the problem. A brief study to the nature of propagation of waves is introduced. Graphical results are presented and discussed.     

An investigation on strain and temperature rate-dependent thermoelasticity and its infinite speed behavior

Abstract

The present work is aimed at a mathematical analysis of the newly proposed strain and temperature rate-dependent thermoelasticity theory, also called a modified Green–Lindsay model (MGL) theory, given by Yu et al. (2018). This model is also an attempt to remove the discontinuity in the displacement field observed under temperature rate-dependent thermoelasticity theory proposed by Green and Lindsay. We study thermoelastic interactions in an infinite homogeneous, isotropic elastic medium with a cylindrical cavity based on this model when the surface of the cavity is subjected to thermal shock. The solutions for the distribution of displacement, temperature, and stress components are obtained by using the Laplace transform technique. The inversion of the Laplace transform is carried out by short-time approximation. A detailed comparison of the analytical results predicted by the MGL model with the corresponding predictions by the Lord–Shulman model and the Green–Lindsay model is performed. It is observed that strain rate terms in the constitutive equation avoid the prediction of discontinuity in the displacement field and other significant effects are noted. However, the new theory predicts the infinite speed of disturbance like the classical theory. Variations of field variables at different time are graphically displayed for different models and compared by using a numerical method.     

Eigenvalue approach to fractional order generalized thermoelasticity with line heat source in an infinite medium

Applying the eigenvalue approach method along with Laplace transform, a general solution scheme for the thermoelastic deformation of an unbounded transversely isotropic medium has been developed on fractional order generalized thermoelasticity with an instantaneous heat source. Solution has been achieved in Laplace transform domain for the perturbed temperature field and other field variables. Several graphs have been presented, and analyses of the results have been made.     

An asymptotic approach to one-dimensional model of nonlinear thermoelasticity at low temperatures and small strains*

A one-dimensional nonlinear homogeneous isotropic thermoelastic model with an elastic heat flow at low temperatures and small strains is analyzed using the method of weakly nonlinear asymptotics. For such a model, both the free energy and the heat flux vector depend not only on the absolute temperature and strain tensor but also on an elastic heat flow that satisfies an evolution equation. The governing equations are reduced to a matrix partial differential equations, and the associated Cauchy problem with a weakly perturbed initial condition is solved. The solution is given in the form of a power series with respect to a small parameter, the coe?cients of which are functions of a slow variable that satisfy a system of nonlinear second-order ordinary differential transport equations. A family of closed-form solutions to the transport equations is obtained. For a particular Cauchy problem in which the initial data are generated by a closed-form solution to the transport equations, the asymptotic solution in the form of a sum of four traveling thermoelastic waves admitting blow-up amplitudes is presented.     

Analytical solution of generalized coupled thermoelasticity problem in a rotating disk subjected to thermal and mechanical shock loads

In this article, a fully analytical solution of the generalized coupled thermoelasticity problem in a rotating disk subjected to thermal and mechanical shock loads, based on Lord–Shulman model, is presented. The general forms of axisymmetric thermal and mechanical boundary conditions as arbitrary time-dependent heat transfer and traction, respectively, are considered at the inner and outer radii of the disk. The governing equations are solved analytically using the principle of superposition and the Fourier–Bessel transform. The general closed form solutions are presented for temperature and displacement fields. To validate the solutions, the results of this study are compared with the numerical results available in the literature, which show good agreement. For the temperature, displacement and stresses, radial distributions, and time histories are plotted and discussed. The propagation of thermoelastic waves and their reflection from the boundary of the disk are clearly shown. Moreover, effects of relaxation time and angular velocity on temperature, displacement, and stress fields are investigated.     

Analysis of a problem arising in porous thermoelasticity of type II

In this article, we consider, from the numerical point of view, a linear thermo-porous-elastic model. The heat conduction is assumed to be of type II. The mechanical problem is written as a coupled system of three hyperbolic partial differential equations for the displacements, the porosity and the thermal displacements. Then, its variational formulation is derived, which is written as a coupled system of three hyperbolic linear variational equations in terms of the velocity, the porous speed and the temperature. An existence and uniqueness result, as well as an energy decay property, is recalled. The fully discrete approximation of the aforementioned problem is introduced by using the finite element method for the spatial approximation and the implicit Euler scheme to discretize the time derivatives. A stability property is proved, from which the energy decay of the discrete energy is deduced. Then, a priori error estimates are obtained, from which, under suitable regularity conditions, the linear convergence of the algorithm is derived. Finally, some numerical simulations are presented to show the accuracy of the approximation and the behavior of the solution.     

The effect of diffusion on two-dimensional problem of generalized thermoelasticity with Green–Naghdi theory

The present paper is concerned with the investigation of disturbances in a homogeneous, isotropic elastic medium with generalized thermoelastic diffusion. The formulation is applied to the generalized thermoelasticity based on the Green and Naghdi (GN) theory under the effect of diffusion. The analytical expressions for displacement components, stresses, temperature field, concentration and chemical potential are obtained in the physical domain by using the normal mode analysis. These expressions are calculated numerically for a copper-like material and depicted graphically. Effect of presence of diffusion is analyzed theoretically and numerically. Comparisons are made with the results predicted by the type II and type III in the presence and absence of diffusion.     

Structure and intrinsic properties of dispersion relation in hyperbolic thermoelasticity

For a system of field equations of hyperbolic thermoelasticity, we derive a propagation condition for a thermoelastic disturbance in a form of homogeneous plane wave in deformation and temperature. The corresponding dispersion relation is given in an explicit form, together with the dependence of characteristic coefficients on the principal invariants of the tensor of isothermal elasticity, heat conductivity, and thermoelasticity. Discussion of different types of homogeneous thermoelastic plane waves is given as well. Derived methodology is applied in the analysis of Green–Naghdi model.     

Thermoelastic interaction in the semi-infinite solid medium due to three-phase-lag effect involving memory-dependent derivative

The present study deals with the thermoelastic interaction in a semi-infinite elastic solid with a heat source in the context of three-phase-lag model with memory-dependent derivative. The governing coupled equations, involving time delay and kernel functions are expressed in the vector matrix differential equation form in the Laplace transform domain. The analytical formulations of the problem have been solved by eigenvalue technique. The Honig–Hirdes numerical method is used for the inversion of Laplace transformation. Numerical results are obtained by choosing various types of time delay parameters and kernel functions and graphical representations have been performed accordingly. An extrapolative capability is established by considering the memory-dependent derivative into a three-phase-lag model.     

Effect of stochastic thermal input on elastic and thermal properties of an infinitely long annular cylinder

Random elastic and thermal properties for an infinitely long solid conducting circular cylinder are investigated under the effect of random thermal input. The problem is considered in the context of a generalized thermoelasticity theory with one relaxation time. The lateral surface of the solid is traction free and subjected to known stochastic temperature, driven by an additive Gaussian white noise. Laplace transform technique is used to obtain the solution in the transformed domain. Statistically, we derive and analyze the mean and variance for temperature, displacement and stress. Numerical inversion of the transformed solution is carried out, represented graphically and discussed.     

Fractional order simulation for unsteady MHD nanofluid flow in porous medium with Soret and heat generation effects

In this paper, unsteady magnetohydrodynamics nanofluid flow with thermo-diffusion and heat generation effects is studied. The fluid flow at the plate is considered exponentially accelerated through a porous medium. The governing system of equations is made dimensionless with the help of similarity transformation. A Caputo–Fabrizio fractional-order derivative is employed to generalize the momentum, energy, and concentration equations, and the exact expression is obtained using Laplace transformation techniques. To realize the physics of the problem, numerical results of velocity, temperature, and concentration profiles are obtained and presented through graphs. Also, the numerical values of the Nusselt number and Sherwood number are obtained and compared which strongly agree with the previous studies. From the results, it is concluded that velocity distribution decline by improving the value of the chemical reaction and magnetic field while the reverse trend is observed for volume fraction and micropolar parameter. It is also seen that the heat transfer process improves with heat generation and thermal radiation whereas, mass transfer declines with the chemical reaction parameter.     

Spatial and temporal behavior in the theory of thermoelasticity for solids with double porosity

In this article, we study the spatial and the temporal behavior of solutions to the initial boundary value problem associated with the linear theory of thermoelastic materials with a double porosity structure. We consider two appropriate time-weighted integral measures and we deduce some exponential estimates that describe the spatial behavior of solutions. For bounded bodies, we obtain estimates of Saint-Venant type, while for unbounded bodies we deduce some alternatives of Phragmén–Lindelöf type. The temporal behavior of solutions is described using the Cesáro means of various parts of the total energy.     

Thermoelastic Problems of Thin Circular and Rectangular Plates

In this paper an attempt is made to determine the temperature, displacement and stress functions of a thin circular plate by applying finite Hankel transform and Laplace transform techniques. This plate that is assumed to be in the plane state of stress is subjected to axisymmetric boundary conditions. As a further simplification, special cases of the third kind of boundary condition are used on the two plane surfaces, while zero temperature is maintained on the outer curved surface of the thin circular plate. A particular case of the boundary conditions is discussed in detail, and numerical results are presented graphically. A mathematically similar problem is that of determining temperature distribution, displacement and stress functions on an edge of a thin rectangular plate with the stated boundary conditions. The results are obtained by applying finite Marchi–Fasulo transform and Laplace transform techniques.     

Fractional heat conduction in a thin hollow circular disk and associated thermal deflection

The time nonlocal generalization of the classical Fourier law with the “Long-tail” power kernel can be interpreted in terms of fractional calculus and leads to the time fractional heat conduction equation. The solution to the fractional heat conduction equation under a Dirichlet boundary condition with zero temperature and the physical Neumann boundary condition with zero heat flux are obtained by integral transform. Thermal deflection has been investigated in the context of fractional-order heat conduction by quasi-static approach for a thin hollow circular disk. The numerical results for temperature distribution and thermal deflection using thermal moment are computed and represented graphically for copper material.     

Integral transform methods for inverse problem of heat conduction with known boundary of a thin rectangular object and its stresses

The three-dimensional inverse transient thermoelastic problem for a thin rectangular object is considered within the context of the theory of generalized thermoelasticity. The upper surface of the rectangular object occupying the space D: a≤x≤a; b≤y≤b; 0≤z≤h; with the known boundary conditions. Laplace and Finite Marchi-Fasulo transform techniques are used to determine the unknown temperature, temperature distribution, displacement and thermal stresses on upper plane surface of a thin rectangular object. The distributions of the considered physical variables are obtained and represented graphically.     

Role of heat source/sink in transient free convective flow through a vertical cylinder filled with a permeable medium: An analytical approach

In this study, an investigation is performed to analyze the impact of the heat source/sink parameter on the laminar transient free convective flow through a vertical cylinder filled with a permeable medium. The governing nondimensional PDEs of the mathematical model along with their appropriate initial and boundary conditions are solved analytically by incorporating the Laplace transform scheme. Moreover, we explored the impact of emerging physical parameters of the considering model in the presence of the source/sink on the velocity profiles by graphs and tables. It is found that the velocity profile has a increasing tendency with enhancement in the numerical values of the time, which finally attains its steady-state solution in the presence of heat source/sink. Moreover, the Prandtl number, sink parameter, and viscosity ratio parameter lead to a decrease in the velocity profiles, whereas the reverse phenomenon occurs with the Darcy number and source parameter. Finally, the numerical values of the Nusselt number, skin friction, and mass flux are given in the tabular forms. The main result obtained in this paper is that the velocity is higher in the case of the source parameter, whereas an opposite behavior is observed in the case of the sink parameter.