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This work is devoted to a study of the induced temperature and stress fields in an elastic half space in context of clas-sical coupled thermoelasticity and generalized thermoelasticity in a unified system of equations. The half space is con-sidered to be made of an isotropic homogeneous thermoelastic material. The bounding plane surface is heated by a non-Gaussian laser beam with pulse duration of 2 ps. An exact solution of the problem is first obtained in Laplace transform space. Since the response is of more interest in the transient state, the inversion of Laplace transforms have been carried numerically. The derived expressions are computed numerically for copper and the results are presented in graphical form.

Although thermomechanical phenomena in the majority of practical engineering applications are adequately simulated with the classical Fourier heat conduction equation, there is an important body of problems that require due consideration of thermomechanical coupling: it is appropriate in these cases to apply the generalized theory of thermoelasticity. Serious attention has been paid to the generalized thermoelasticity theories in solving thermoelastic problems in place of the classical uncoupled/coupled theory of thermoelasticity.

The absence of any elasticity term in the heat conduction equation for uncoupled thermoelasticity appears to be unrealistic, since due to the mechanical loading of an elastic body, the strain so produced causes variation in the temperature field. Moreover, the parabolic type of the heat conduction equation results in an infinite velocity of thermal wave propagation, which also contradicts the actual physical phenomena. Introducing the strain-rate term in the uncoupled heat conduction equation, Biot extended the analysis to incorporate coupled thermoelasticity [

The so-called ultra-short lasers are those with pulse duration ranging from nanoseconds to femtoseconds in general. In the case of ultra-short-pulsed laser heating, the high-intensity energy flux and ultra-short durationlaser beam, have introduced situations where very large thermal gradients or an ultra-high heating speed mayexist on the boundaries [

The present investigation is devoted to a study of the induced temperature and stress fields in an elastic half space under the purview of classical coupled thermoelasticity and generalized thermoelasticity in a unified system of field equations. The half space continuum is considered to be made of an isotropic homogeneous thermoelastic material, the bounding plane surface being subjected to a Non-Gaussian laser pulse. An exact solution of the problem is first obtained in Laplace transform space. Since the response is of more interest in the transient state, the inversion of Laplace transforms have been carried numerically. The derived expressions are computed numerically for copper and the results are presented in graphical form.

All the field equations represented by (CTE), (L-S) and (G-L) can be formulated in the following unified system [

which constitute equation of motion where are Lame’s constants, is the displacement component, is the body force component, and is the thermal expansion, is relaxation time, T is the temperature of the body and is the density.

which constitute equation of heat conduction where K is the thermal conductivity, C_{E} is the specific heat at constant strain, is relaxation time, is the reference temperature, n is a parameter and Q is the heat source.

which is called constitutive equation where is the stress tensor and is the Kronecker function.

Equations (1)-(3) reduce to coupled thermoelasticity (CTE) when. Putting, and, the equations reduce to Lord-Shulman (L-S) model, while when, and, the equations reduce to Green-Lindsay (G-L) model [13,14].

We will consider the medium is heated uniformly by a laser pulse with non-Gaussian form temporal profile [

where is a characteristic time of the laserpulse (the time duration of a laser pulse), L_{0} is the laser intensity which is defined as the total energy carried by a laser pulse per unit area of the laser beam, see

The conduction heat transfer in the medium can be modeled as a one-dimensional problem with an energy source near the surface, i.e.

where is the absorption depth of heating energy and R_{a} is the surface reflectivity [

When we consider the laser pulse lie on the surface of the mediumwhen (see

We consider half-space () with the x-axis pointing into the medium with initial temperature distribution T_{o}. This half-space is irradiated uniformly the bounding plane (x = 0) by a laser pulse with non-Gaussian temporal profile as in (6). We assume that there is no body forces affecting the medium and all the state functions initially are equal to zero.

The displacement vector has the components:

Hence, the governing equations (1)-(3) in one-dimensional will take the following forms:

The equation of motion

where is the temperature increment.

The heat equation:

where

The constitute equation:

For simplicity, we will use the following non-dimensional variables Youssef (2006):

where is the longitudinal wave speed and is the thermal viscosity.

Hence, we have the following system of equations (we have dropped the prime for convenient)

where is the dimensionless thermoelastic coupling constant, and.

Applying the Laplace transform for Equations (13)-(15) defined by the formula

.

Hence, we obtain the following system of differential equations

where all the state functions initially are equal to zero,

and.

Eliminating between the equations (17) and (18), we get

where and .

The solution of equation (20) takes the following form:

where are the roots of the characteristic equation

and

To get the value of the parameters and we have to apply the boundary conditions on the bounding plane of the assumed half space as follows:

which gives after applying Laplace transform

After applying the above boundary conditions, we get

and

Finally, we can write the solution in the Laplace transform domain as follows:

and

By using equations (19), (26) and (27), we get

We get the displacement form equations (10) and (27) in the form

In order to get the inversion of the Laplace transform, the Riemann-sum approximation method is used. In this method, any function in Laplace domain can be inverted to the time domain as

where Re is the real part and is imaginary number unit. For faster convergence, numerous numerical experiments have shown that the value of satisfies the relation [

With a view to illustrating the analytical procedure presented earlier, we now consider a numerical example for which computational results are given. For this purpose, copper is taken as the thermoelastic material, [

The computations were carried out for t = 0.2 and the temperature, the stress, the strain and the displacement distributions are represented graphically at different positions of x.

The figures 2-5 show that, the laser pulse makes the difference between the results in the context of the three studied models CTE, L-S and G-L is very clear and we can differentiate between them, while it was very difficult previously when we used thermal loading by using thermal shock or ramp-type heating as in [13,14].

The authors are grateful for the supports for this work

provided by Institute of Scientific Research and Revival of Islamic Heritage, Umm Al-Qura University by grant number 42905012.