THERMAL SHOCK PROBLEM FOR ONE DIMENSIONAL GENERALIZED THERMOELASTIC LAYERED COMPOSITE MATERIAL

The dynamic treatment of one-dimensional generalized thermoelastic problem of heat conduction is made for a layered thin plate, which is exposed, to a uniform thermal shock. The basic equations are transformed by Laplace transform and solved by a direct method. The solution was applied for a plate of sandwich structure. The inverses of Laplace transforms are obtained numerically. The temperature, the stress and the displacement distributions are represented in graphs, which show the coupled and the generalized cases. Keywords– Thermoelasticity, Laplace Transforms, Layered Composite


INTRODUCTION
Lord and Shulman [1] obtained the governing equations of generalized thermoelasticity involving one relaxation time for isotropic homogeneous media.These equations predict finite speeds of propagation of heat and displacement distribution, the corresponding equations for an isotropic case were obtained by Dhaliwal and Sherief [2].Due to the complexity of the governing equations and the mathematical difficulties associated with their solution several simplifications have been used.For example some authors [3,4] use the framework of coupled thermoelastic\city where the relaxation time is taken as zero resulting in a parabolic system of partial differential equations.The solution of this system exhibits infinite speed of propagation of heat signals contradictory to physical observation.Some other authors use still further simplifications by ignoring the inertia effects in a coupled theory [5] or by neglecting the coupling effect.
This work deals with a plate consisting of layers of unidentical substances, each of which is homogeneous and isotropic.When this plate, which is initially at rest and having a uniform temperature, is suddenly heated at the free surfaces, a heat flow occurs in the plate and change in thermal and the mechanical field is brought about.

THE BASIC EQUATIONS
The coordinate system is so chosen that the x-axis is taken perpendicularly to the layer, and the y-and z-axes in parallel.We are dealing with one-dimensional generalized thermoelasticity with one relaxation time.The equation of motion The constitutive equation The heat equation where and.
The above equations can be put into a more convenient form by using the following non-dimensional variables where After dropping the primes for convenience, we obtain [ ] where Taking Laplace transform as define Then, equations ( 4), ( 5) and ( 6) will take the forms where x D ∂ ∂ = .
By eliminating e , we get Using the above two equations, we obtain

APPLICATION
Considering a layered plat of sand-witch structure such as shown in Figure 1, where layers I , III made from the same metal, and the layer II is a deferent metal.Layer II is put in the middle of the plate, and its thickness is a half of that of the plate.
The solution of the equations ( 12) and (13) take the form x The solution of the equations ( 12) and (13) take the form ( ) ( ) The solution of the equations ( 12) and (13) take the form The Boundary Conditions: (1)-The thermal boundary conditions (2)-The mechanical boundary conditions 0 , and (ii)- , and (23) Applying the pervious conditions into equations ( 14)-( 19), we obtain ( We can get the displacement by using equation ( 4), such that x p sinh k sinh

THE SOLUTION IN THE PHYSICAL DOMAIN
In order to invert the Laplace transform in equations ( 24)-( 29), we adopt a numerical inversion method based on a Fourier series expansion [6].By this method the inverse ) t ( f of the Laplace transform ( ) where N is a sufficiently large integer representing the number of terms in the truncated Fourier series, chosen such that ( ) where ε 1 is a prescribed small positive number that corresponds to the degree of accuracy required.The parameter c is a positive free parameter that must be greater than the real part of all the singularities of ( ) s f .The optimal choice of c was obtained according to the criteria described in [7].The copper material and the type 316 stainless steel are chosen for purposes of numerical evaluations [6], [8].The computations were carried out for value of time, namely t = 0.2 and for length 1 = l (unit length) The numerical values of the temperature, displacement component and stress component for the two cases, coupled and generalized are obtained and represented graphically.