Farid A. Hamza

GENERALIZED THERMOELASTIC PROBLEM OF A THICK PLATE UNDER AXISYMMETRIC TEMPERATURE DISTRIBUTION

The two-dimensional problem of a thick plate whose lower and upper surfaces are traction free and subjected to a given axisymmetric temperature distribution is considered within the context of the theory of generalized thermoelasticity with one relaxation time. Potential functions together with Laplace and Hankel transform techniques are used to derive the solution in the transformed domain. The Hankel transforms are inverted analytically. The inversion of the Laplace transforms are 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 and to evaluate the improper integrals involved to obtain the temperature and stress distributions in the physical domain. Analysis of wave propagation in the medium is presented. Numerical results are represented graphically and discussed. A comparison is made with the solution of the corresponding coupled problem.

THEORY OF GENERALIZED MICROPOLAR THERMOELASTICITY AND AN AXISYMMETRIC HALF-SPACE PROBLEM

ABSTRACT The general equations of motion and constitutive equations are derived for a general homogeneous anisotropic medium with a microstructure, taking into account the effects of heat and allowing for second sound effects. A uniqueness theorem is also derived. As an illustration of the obtained equations, we solve a problem for a half-space whose boundary is rigidly fixed and subjected to an axisymmetric thermal shock. There are no body forces, body couples, or heat sources affecting the medium. Laplace and Hankel transform techniques are used. Numerical results are obtained and graphically illustrated.

GENERALIZED TWO-DIMENSIONAL THERMOELASTIC PROBLEMS IN SPHERICAL REGIONS UNDER AXISYMMETRIC DISTRIBUTIONS

Two-dimensional thermoelastic problems under axisymmetric temperature distributions are considered within the context of the theory of generalized thermoelasticity with one relaxation time. The general solution is obtained in the Laplace transform domain by using a direct approach without the customary 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 tractionfree and subjected to a given axisymmetric temperature distribution. The inversion of the Laplace transforms are 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. A comparison is made with the solution of the corresponding coupled pr...

Generalized Fractional Thermoelasticity Associated with Two Relaxation Times

In this work, a new theory of thermoelasticity associated with two relaxation times is derived using the methodology of fractional calculus. The theories of coupled thermoelasticity and of generalized thermoelasticity (Green–Lindsay Model) with two relaxation times follow as limiting cases. A uniqueness theorem and a reciprocity theorem for this model are derived. A variational principle theorem is obtained.

Electro-Magneto-Thermoelastic Plane Waves in Micropolar Solid Involving Two Temperatures

The model of equations of micropolar generalized magneto-thermoelasticity is introduced within the context of the theory of two temperatures generalized thermoelasticity and we consider a problem of an isotropic homogeneous micropolar medium taking into account the heat effects and allowing the magnetic field effects. A plane wave analysis is employed to obtain the exact formulas of the two temperatures (conductive and mechanical), displacement components, micro-rotation components, stresses, couple stresses, induced electric current, electric field and magnetic field. Arbitrary application is chosen to enable us to get the complete solution. The considered variables are presented graphically and discussions are made for the results.

1D applications on fractional generalized thermoelasticity associated with two relaxation times

ABSTRACT Here, this work is concerned with some different one-dimensional (1D) problems of distribution of the thermal stresses and temperature in fractional generalized thermoelastic material, as follows: (i) 1D problem for a half-space of elastic material in the presence of heat sources has been solved by using Laplace transform and state space techniques; and (ii) 1D problem of distribution of thermal stresses and temperature in infinite medium with a spherical cavity subjected to a sudden change in the temperature of its internal boundary, which is assumed to be traction free, has been solved by Laplace transform and direct approach techniques. In the preceding problems, the numerical results for dimensionless variable fields are given and illustrated graphically. According to the numerical results, some comparisons have been shown in figures to estimate the effect of the fractional derivative parameter α about the new theory on all variable fields and discussion has been established for a copper-like material.

Memory time effect on electromagnetic-thermoelastic materials

In this work, new mathematical model of Maxwell’s equations in an electromagnetic field is derived using the physical principles of fractional calculus. The advantage of our model appears according to the comparison between our model and a previous fractional electromagnetic model which was introduced by Gomez et al. Our model is applied to fractional thermoelastic material associated with one relaxation time interaction due to periodically varying heat source. The Laplace–Fourier double transform technique has been used to get the solution. The inversion of the Fourier transform has been done using residual calculus, where poles of the integrand are obtained numerically in a complex domain using Leguerre’s method and the inversion of the Laplace transformation is done numerically using a method based on a Fourier series expansion technique. Numerical results of temperature, displacement, stress, strain, and induced electric/magnetic fields are obtained for a hypothetical material. Finally, some comparisons have been shown in figures to estimate the effect of fractional-order parameters on all the studied variable fields.

Thermomechanical Fractional Model of TEMHD Rotational Flow

In this work, the fractional mathematical model of an unsteady rotational flow of Xanthan gum (XG) between two cylinders in the presence of a transverse magnetic field has been studied. This model consists of two fractional parameters α and β representing thermomechanical effects. The Laplace transform is used to obtain the numerical solutions. The fractional parameter influence has been discussed graphically for the functions field distribution (temperature, velocity, stress and electric current distributions). The relationship between the rotation of both cylinders and the fractional parameters has been discussed on the functions field distribution for small and large values of time.

Thermomechanical Fractional Model of Two Immiscible TEMHD

We introduce a mathematical model of unsteady thermoelectric MHD flow and heat transfer of two immiscible fractional second-grade fluids, with thermal fractional parameters and mechanical fractional parameters , . The Laplace transform with respect to time is used to obtain the solution in the transformed domain. The inversion of Laplace transform is obtained by using numerical method based on a Fourier-series expansion. The numerical results for temperature, velocity, and the stress distributions are represented graphically for different values of and . The graphs describe the fractional thermomechanical parameters effect on the case of two immiscible fluids and the case of a single fluid.