_{1}

This work is dealing with two-temperature generalized thermoelasticity without energy dissipation infinite medium with spherical cavity when the surface of this cavity is subjected to laser heating pulse. The closed form solutions for the two types of temperature, strain, and the stress distribution due to time exponentially decaying laser pulse are constructed. The Laplace transformation method is employed when deriving the governing equations. The inversion of Laplace transform will be obtained numerically by using the Riemann-sum approximation method. The results have been presented in figures to show the effect of the time exponentially decaying laser pulse and the two temperature parameter on all the studied fields.

The two temperatures theory of thermoelasticity was introduced by Gurtin and Williams [

Among the authors who contribute to developing this theory, Quintanilla studied existence, structural stability, convergence and spatial behavior for this theory [

The present paper is devoted to a study of the induced temperature and stress fields in aninfinite elastic medium with aspherical cavity under the purview of two-temperature thermoelasticity without energy dissipation. The medium is considered to be an isotropic homogeneous thermoelastic material. The bounding plane surface of the cavity is thermally loaded bytime exponentially decaying laser pulse. An exact solutions of the problem is obtained in Laplace transformdomain, and the inversions of the Laplace transforms have been culculated numerically. The derived formulations are computed numerically for copper, and the results are presented in graphical form.

We will consider perfectly conducting, elastic, isotropic, and homogeneous medium and the governing equations will be taken in the context of two-temperature generalized thermoelasticity without energy dissipation.

According to Youssef model, the heat conduction equation takes the form [

, (1)

The conduction-dynamical heat equation takes the form [

The equations of motion take the form

The constitutive equations take the form

and

where

temperature,

We will consider perfectly conducting elastic infinite bodies with spherical cavity occupy the region

Thus, the field equations in spherical one-dimensional case can be put as:

and

The non-Fourier heat transfer equation due to a laser heating pulse decaying exponentially in time can be written as [

where _{0} is laser peak power intensity, _{ }

And

where

The constitutive equations will take the following forms

and

where

We shall use the following non-dimensional variablesfor convenience [

where

Equation (1) and Equations (4)-(9) assume the form (where the primes are suppressed for simplicity)

By using Equation (13) into Equation (15), we get

also, we have

and

where

We use the Laplace transform of both sides of the last equations defined as:

Hence, we obtain

and

Eliminating

where

From Equations (28) and (24), we obtain

From Equations (22) and (28), we have

where

By eliminating

where

By eliminating

where

The bounded solutions of the Equations (31) and (32) take the forms

and

where

By using Equations (33) and (34) into Equation (30), we obtain

Hence, we get

To get the constants

Thus, the system of the equations on (34) and (37) gives the following linear equations

and

By solving the above system, we get

Those complete the solutions as following

Substituting from Equations (41) and (42) in (28), (27) and (25) we get

To determine the solutions in the time domain, the Riemann-sum approximation method is used to obtain the numerical results. In this method, any function in Laplace domain can be inverted to the time domain as:

where Re is the real part and

We now consider a numerical example for which computational results are given. For this purpose, copper is taken as the thermoelastic material for which we take the following values of the different physical constants [

From the above values, we get the non-dimensional values of the problem as:

Figures 1-5 represent the conductive temperature distribution, the thermodynamic temperature distribution, the strain distribution, the displacement distribution, and the stress distribution respectively, in the context of one-temperature type (solid lines) and two-temperature type (dashed lines). We can notice that the two-temper- ature parameter has significant effects on all distribution. The material reaches the steady state through the two- temperature type before the one-temperature type. The peak points decrease when we use the two-temperature type.

I want thank Prof. Hamdy M. Youssef (Mechanics Department, Faculty of Engineering, Umm Al-Qura University, Makkah KSA) for his help and advises to me to complete this work and to choose this respected journal.

Eman A. N.Al-Lehaibi, (2015) Two-Temperature Generalized Thermoelasticity without Energy Dissipation of Infinite Medium with Spherical Cavity Thermally Excited by Time Exponentially Decaying Laser Pulse. Modeling and Numerical Simulation of Material Science,05,55-62. doi: 10.4236/mnsms.2015.54006