Magdy A. Ezzat

On fractional thermoelasticity

Two general models of fractional heat conduction for non-homogeneous anisotropic elastic solids are introduced and the constitutive equations for thermoelasticity theory are obtained, uniqueness and reciprocal theorems are proved and the convolutional variational principle is established and used to prove a uniqueness theorem with no restriction on the elasticity or thermal conductivity tensors except for symmetry conditions. The dynamic coupled, Lord—Shulman, Green—Naghdi and fractional coupled thermoelasticity theories result as limit cases. The reciprocity relation in the case of quiescent initial state is found to be independent of the order of differintegration.

Fractional order theory of a perfect conducting thermoelastic medium

In this work, a new mathematical model of magneto-thermoelasticity theory is constructed in the context of a new consideration of heat conduction law with time-fractional order. This model is applied to a one-dimensional application for a perfect conducting half-space of elastic material, which is thermally shocked in the presence of magnetic field. Laplace transforms and state-space techniques (Ezzat. Can. J. Phys. 86, 1242, (2008)) will be used to obtain the general solution for any set of boundary conditions. According to numerical results and graphs, it is found that introducing a fractional derivative of order α has a significant effect on the temperature, stress, and heat flux distributions as well as the induced electric and magnetic fields; the curves are smoother in the case of 0 < α < 1 due to weak thermal conductivity. Some comparisons are made and shown in figures to estimate the effects of the fractional order parameter on all the studied fields.

A thermal-shock problem in magneto-thermoelasticity with thermal relaxation

Abstract The one-dimensional problem of distribution of thermal stresses and temperature is considered in a generalized thermoelastic electrically conducting half-space permeated by a primary uniform magnetic field when the bounding plane is suddenly heated to a constant temperature. The Laplace transform technique is used to solve the problem. Inverse transforms are obtained in an approximate manner using asymptotic expansions valid for small values of time. Nurnerical computations for two particular cases are carried out.

Electromagneto-thermoelastic plane waves with two relaxation times in a medium of perfect conductivity

The model of the two-dimensional equations of generalized magneto-thermoelasticity with two relaxation times in a perfectly conducting medium is established. The normal mode analysis is used to obtain the exact expressions for the temperature distribution, thermal stresses and the displacement components. The resulting formulation is applied to two different concrete problems. The first deals with a thick plate of perfect conductivity subjected to time-dependent heat source on each face, while the second concerns the case of a heated punch moving across the surface of a semi-infinite thermoelastic half-space of perfect conductivity subject to appropriate boundary conditions. Numerical computations for the horizontal component of the displacement are carried out and represented graphically for each problem. A comparison was made with the results obtained in the absence of a magnetic field.

ELECTROMAGNETO-THERMOELASTIC PLANE WAVES WITH THERMAL RELAXATION IN A MEDIUM OF PERFECT CONDUCTIVITY

The model of the two-dimensional equations of generalized magneto-thermoelasticity with one relaxation time in a perfectly conducting medium is established. The normal mode analysis is used to obtain the exact expressions for the temperature distribution, thermal stresses, and the displacement components. The resulting formulation is applied to three different concrete problems. The first deals with a thick plate of perfect conductivity subjected to a time-dependent heat source on each face; the second concerns the case of a heated punch moving across the surface of a semi-infinite thermoelastic half-space of perfect conductivity subject to appropriate boundary conditions; and the third problem deals with a plate with thermo-isolated surfaces subjected to time-dependent compression. Numerical results are given and illustrated graphically for each problem. Comparisons are made with the results predicted by the coupled theory and with the theory of generalized thermoelasticity with one relaxation time.The model of the two-dimensional equations of generalized magneto-thermoelasticity with one relaxation time in a perfectly conducting medium is established. The normal mode analysis is used to obtain the exact expressions for the temperature distribution, thermal stresses, and the displacement components. The resulting formulation is applied to three different concrete problems. The first deals with a thick plate of perfect conductivity subjected to a time-dependent heat source on each face; the second concerns the case of a heated punch moving across the surface of a semi-infinite thermoelastic half-space of perfect conductivity subject to appropriate boundary conditions; and the third problem deals with a plate with thermo-isolated surfaces subjected to time-dependent compression. Numerical results are given and illustrated graphically for each problem. Comparisons are made with the results predicted by the coupled theory and with the theory of generalized thermoelasticity with one relaxation time.

Convolutional Variational Principle, Reciprocal and Uniqueness Theorems in Linear Fractional Two-Temperature Thermoelasticity

Two general models of fractional heat conduction law for non-homogeneous anisotropic elastic solid is introduced and the constitutive equations for the two-temperature fractional thermoelasticity theory are obtained, uniqueness and reciprocal theorems are proved and the convolutional variational principle is established and used to prove a uniqueness theorem with no restrictions imposed on the elasticity or thermal conductivity tensors except symmetry conditions. The two-temperature dynamic coupled, Lord-Shulman and fractional coupled thermoelasticity theories result as limit cases. The reciprocity relation in case of quiescent initial state is found to be independent of the order of differintegration.

Generation of generalized magnetothermoelastic waves by thermal shock in a perfectly conducting half-space

The problem of distribution of thermal stresses and temperature is considered in a perfectly conducting half-space, in contact with a vacuum, permeated by an initial magnetic field when the bounding plane is suddenly heated to a constant temperature. The problem is in the context of generalized magnetothermoelasticity with one relaxation time. The solution is obtained using the method of potentials. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using asymptotic expansions valid for small values of time. Numerical computations for the temperature and stress distributions are carried out and represented graphically. A comparison is made with the results obtained in the absence of a magnetic field.

State space approach to generalized magneto-thermoelasticity with two relaxation times in a medium of perfect conductivity

In this work the state space formulation for one-dimensional problems of generalized magneto-thermoelasticity with two relaxation times in a perfectly conducting medium is introduced. The Laplace transform technique is used. The resulting formulation is applied to a thermal shock problem, a problem of a layer medium and a problem for the infinite space in the presence of heat sources. A numerical method is employed for the inversion of the Laplace transforms. Numerical results are given and illustrated graphically for the problem considered.

State space approach to two-dimensional generalized thermo-viscoelasticity with two relaxation times

The model of the two-dimensional equations of generalized thermo-viscoelasticity with two relaxation times is established. The state space formulation for two-dimensional problems is introduced. Laplace and Fourier integral transforms are used. The resulting formulation is applied to a problem of a thick plate subject to heating on parts of the upper and lower surfaces of the plate that varies exponentially with time. The Fourier transforms are inverted analytically. A numerical method is employed for the inversion of the Laplace transforms. Numerical results are given and illustrated graphically for the problem considered. Comparisons are made with the results predicted by the coupled theory.

Fundamental solution in thermoelasticity with two relaxation times for cylindrical regions

The solution of the problem of determining stress and temperature distributions with a continuous line source of heat in an infinite elastic body governed by the equations of generalized thermoelasticity with two relaxation times are obtained by using the Hankel and Laplace transform techniques. Inverse transforms are obtained in an approximate manner for small values of time. Numerical results for a particular case are given.