A.M. Abd El-Latief

A one-dimensional fractional order thermoelastic problem for a spherical cavity

We apply the fractional order theory of thermoelasticity to a one-dimensional problem for a spherical cavity subjected to a thermal shock. The predictions of the theory are discussed and compared with those for the generalized theory of thermoelasticity.

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.

Application of fractional order theory of thermoelasticity to a 2D problem for a half-space

A 2D half space problem for the new fractional order theory of thermoelasticity.Laplace and exponential Fourier transform techniques are used.The inverse transforms are obtained using a numerical technique.Predictions of the theory are discussed and compared with older theory. In this work, we apply the fractional order theory of thermoelasticity to a 2D problem for a half-space. The surface of the half-space is taken to be traction free and is subject to heating. There are no body forces or heat sources affecting the medium. Laplace and exponential Fourier transform techniques are used to solve the problem. The inverse Laplace transforms are obtained using a numerical technique.The predictions of the theory are discussed and compared with those for the generalized theory of thermoelasticity. We also study the effect of the fractional derivative parameter on the behavior of the solution. Numerical results are computed and represented graphically for the temperature, displacement and stress distributions.

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.

New state-space approach and its application in thermoelasticity

ABSTRACT In this work, a new state-space approach in the context of fractional-order theory of thermoelasticity is introduced. This new method utilizes the diagonalization of a square matrix by the use of its eigenvalues and vectors. This new approach is applied to a 1-D problem for a half-space subjected to a thermal shock. Laplace transform technique is used throughout. The inversion of the transforms is performed using a numerical inversion technique based on the Fourier series expansion technique. The effects of the time and the fractional parameter on variable field distributions are discussed and represented graphically.ABSTRACTIn this work, a new state-space approach in the context of fractional-order theory of thermoelasticity is introduced. This new method utilizes the diagonalization of a square matrix by the use of its eigenvalues and vectors. This new approach is applied to a 1-D problem for a half-space subjected to a thermal shock. Laplace transform technique is used throughout. The inversion of the transforms is performed using a numerical inversion technique based on the Fourier series expansion technique. The effects of the time and the fractional parameter on variable field distributions are discussed and represented graphically.