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The work is dealing with the studying the effect of the moving heat source with constant velocity on the biological skin tissue. The heat conduction equation has been considered to be a two-temperature high-order thermal lagging model. A limited length of skin tissue has been constructed to be initially at rest and the first end subjected to constant heat flux while the second end has zero heat flux. Laplace transform has been applied and the inversion has been calculated numerically. The results show that the two-temperature parameter and the value of the heat source velocity have significant effects on the conductive and dynamical temperatures increment.

In this work, a mathematical model of three layers system represents how the external thermal shock passing through the three layers of the head: skin, bone, and brain. In this model, we will take into account the internal heat source generated within the human head from the chemical reactions of the tissue and the lag-time to study the response time of the tissue takes due to the perturbation. The materials parameters of the head for an adult and a child have been used to studying and comparing the two cases. Laplace transforms have been used and the inversions have been calculated numerically. We studied the thermal distribute through the human head with different values of time, depth, relaxation time, the frequency of power transmission, and power density. It was shown that all that parameters have significant effects on the human head and the thermal distribution is more harmful to the child than the adult.

A unified mathematical model of electro-thermoelasticity has been constructed in the context of a new consideration of heat conduction law with fractional order derivative. Some essential theories follow as limit cases. The governing coupled equations are applied to several concrete problems: (a) time-dependent thermal shock problem; (b) a problem for a half-space subjected to an arbitrary heating and (c) a layer media problem. Laplace transforms are used to derive the solution in the Laplace transform domain. A numerical method is employed for the inversion of the Laplace transforms. According to the numerical results and its graphs, conclusion about the new theory has been constructed. The predictions of the theory are discussed and compared with dynamic classical coupled theory, Lord–Shulman, Green–Naghdi (GN) and fractional coupled thermoelasticity theories. The result provides a motivation to investigate conducting fractional thermoelectric materials as a new class of applicable materials.

Comb drive actuators are very important microelectromechanical systems structures that are widely used in many applications. They are currently used in force balanced accelerometers, resonant accelerometers and resonant gyroscopes, micro-grippers and micro-engines. In addition, they are extensively used in opto-mechanical subsystems such as switches and variable optical attenuators. The goal of this paper is to present a comb drive actuator model design with ultra-low pull-in voltage to achieve large displacement, deriving the exerted forces equations, mechanical analysis and the most important problems, also comparing with the previous work done to treat these problems.

A mathematical model of two-temperature phase-lag Green–Naghdi thermoelasticty theories based on fractional derivative heat transfer is given. The GN theories as well as the theories of coupled and of generalized thermoelasticity with thermal relaxation follow as limit cases. The resulting non dimensional coupled equations together with the Laplace transforms techniques are applied to a specific problem of a half space subjected to arbitrary heating which is taken as a function of time and is traction free. The inverse transforms are obtained by using a numerical method based on Fourier expansion techniques. The predictions of the theory are discussed and compared with those for the generalized theory of thermoelasticity with one relaxation time. The effects of temperature discrepancy and fractional order parameters on copper-like material are discussed in different types of GN theories.

The model of fractional magneto-thermoelasticity is applied to one-dimensional problems of a thermoelectric spherical shell subjected to an arbitrary thermal loading of its external boundary in the presence of a uniform magnetic field. By means of the Laplace transform and numerical Laplace inversion, the problems are solved. The distributions of the considered temperature, stress and displacement, are represented graphically. The theories of coupled and generalized magneto-thermoelasticity follow as limited cases. Some comparisons are shown in the figures to estimate the effects of the fractional order and ramping parameters.

In this work, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the theory generalized thermoelasticity. A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent and subjected to moving heat source with constant velocity in one direction. Laplace and Fourier transform techniques are used to obtaining the general solution for any set of boundary conditions. The inverse Laplace and Fourier transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating and the velocity of the heat source.

In this paper, we constructed a mathematical model for a thermoelastic half-space medium based on two-temperature generalized thermoelasticity without energy dissipation. The Laplace transforms techniques have been used to get the general solution for a particular model when the medium is subjected to thermal shock by using stat-space approach. The inversion of the Laplace transforms has been calculated numerically and after that we presented the results graphically with some comparisons to study the impact of thermal or mechanical load on the speed of progress of mechanical and thermal waves through the medium. Also, to studying the effect of the two-temperature parameter rotation parameter on all the studied field.

The theory of generalized thermoelasticity, based on the theory of Lord–Shulman (L-S) with one relaxation time, is used to solve boundary value problems of a one-dimensional layer of semi-infinite elastic medium. We considered the medium is rotating, and one of its boundaries subjected to harmonic heat and traction free, and the other boundary connected to the rigid foundation and has zero heat flux. The governing partial differential equations are solved in the Laplace transform domain by using the state-space approach of the modern control theory. The inverse Laplace transforms are obtained numerically. The temperature, the displacement, and the stress distributions are represented graphically with some comparisons to stand on the effect of the rotation of the medium and the harmonic heat on all the studied functions.

Present work deals with the problem of an infinite long hollow cylinder with variable thermal conductivity in the context of generalized thermo-viscoelasticity theory with thermal relaxation. A mapping of Kirchhoff's transformation was used to solve a problem with variable thermal conductivity. The Laplace transform is used. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the displacement and the stress distributions are carried out and represented graphically. The results indicate that the thermal conductivity play a major role in all considered distributions.

A fractional model of the equations of generalized magneto-thermoelasticity for a perfect conducting isotropic thermoelastic media is given. This model is applied to solve a problem of an infinite body with a cylindrical cavity in the presence of an axial uniform magnetic field. The boundary of the cavity is subjected to a combination of thermal and mechanical shock acting for a finite period of time. The solution is obtained by a direct approach by using the thermoelastic potential function. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the displacement and the stress distributions as well as for the induced magnetic and electric fields are carried out and represented graphically. Comparisons are made with the results predicted by the generalizations, Lord–Shulman theory, and Green–Lindsay theory as well as to the coupled theory.

The new model of the equations of the linear theory of magneto-thermo-viscoelasticity with two relaxation times and fractional heat transfer involving fractional relaxation operator is given. The resulting formulation is applied to thermal shock problems for a perfect electrically conducting half-space in the presence of a transverse magnetic field. Laplace transform techniques are used. Some essential theorems on the linear coupled and generalized theories of thermo-viscoelasticity with two relaxation times are established. According to the numerical results and its graphs, conclusion about the new theory has been constructed. The effects of the fractional relaxation operator on viscoelastic material like poly (methyl methacrylate) (Perspex) are discussed.