Modeling of Variable Lamé ’ s Modulii for a FGM Generalized Thermoelastic Half Space

In this work we consider a problem in the context of the generalized theory of thermoelasticity for a half space. The material of the half space is functionally graded in which Lamé’s modulii are functions of the vertical distance from the surface of the medium. The surface is traction free and subjected to a time dependent thermal shock. The problem was solved by using the Laplace transform method together with the perturbation technique. The obtained results are discussed and compared with the solution when Lamé’s modulii are constants. Numerical results are computed and represented graphically for the temperature, displacement and stress distributions.


INTRODUCTION
The theory of generalized thermoelasticity with one relaxation time was introduced by Lord and Shulman (1967).In this theory Cattaneo-Maxwell law of heat conduction replaces the conventional Fourier's law.The heat equation associated with this theory is a hyperbolic one and hence automatically eliminates the paradox of infinite speeds of propagation inherent in both the uncoupled and the coupled theories of thermoelasticity.For many problems involving steep heat gradients and when short time effects are sought this theory is indispensable.Sherief andEl-Maghraby (2003, 2005) solved some crack problems for this theory.Sherief and Hamza has obtained the solution of axisymmetric problems in cylindrical regions in (1994) and in spherical regions in (1996).Sherief and Ezzat (1994) have obtained the solution in the form of series.Sherief and Dhaliwal (1981) used asymptotic expansions to obtain the solution of a 1D problem and to find the locations of the wave fronts and the speed of propagation of thermoelastic waves.This theory was extended to deal with thermoelastic diffusion in (2004), micropolarity of the medium in (2005), viscoelastic effects in Latin American Journal of Solids and Structures 13 (2016) 715-730 (2011).Sherief et.al (2010) extended this theory using fractional derivatives.Under this theory, Sherief and Abd-Ellatief (2013, 2014a, 2014b) have solved some problems.
Functionally graded materials (FGMs) are a new class of materials with the material properties varying continuously along specified directions.Due to their desirable properties, FGMs have been increasing used in modern engineering applications (Wang(2013)).For example, the functionally graded metal-ceramic composites have been used as thermal barriers or thermal shields in various applications (Lee et.al(1996), Tsukamoto(2010)).Especially, Praveen et.al. (1999) in severe temperature environments, such as extremely high temperature and thermal shock, widely potential applications are opening for FGMs.
In recent years, Asgari and Akhlaghi (2009) composition of several different materials is often used in structural components in order to optimize the thermal resistance and temperature distribution of structures subjected to thermal loading.Other works that consider FGM can be found in (Nahvi et.al. (2008), Zafarmand et. al. (2015), Dey et.al. (2015)).

FORMULATION OF THE PROBLEM
In this work, we consider a homogeneous isotropic thermoelastic solid occupying the region 0 x  composed of a FGM material whose Lamé's parameters depend on the vertical distance '' x '' from the surface.The surface of the half-space is taken to be traction free and is subjected to a thermal shock that is a function of time.We assume also that there are no body forces or heat sources inside the medium.The initial conditions are assumed to be homogenous.
From the physics of the problem, it is clear that all the variables depend on x and t only.The displacement vector will thus have the form.

 
( , ),0,0 u u x t  (1) The equation of motion is given by where  is the density.The equation of the heat conduction has the form where E C is the specific heat per unit mass in the absence of deformation, k is the thermal conductivity and 0 is the relaxation time.From now on, we shall take ,   in the form where  0 ,  0 and "a" are constants.Thus equations (3-5) take the form We shall use the following non-dimensional variables Using the above non-dimensional variables, equations ( 7)-( 9) take the form Latin American Journal of Solids and Structures 13 (2016) 715-730 The remaining components of the stress tensor are given by: where We assume that the boundary conditions have the form where f(t) is a known function of t

SOLUTION IN THE LAPLACE TRANSFORM DOMAIN
Applying the Laplace transform with parameter s defined by the relation 0 ( , ) ( , ) to both sides of equations ( 10)-( 13), we get the following equations In order to solve the above equations, we shall use the perturbation method.We expand the temperature, displacement and stress functions as follows: Latin American Journal of Solids and Structures 13 (2016) 715-730  gives, upon equating the coefficients of "a " in both sides up to order 1 Using expansion of equation ( 17), we obtain  between equations ( 21) and ( 23), we get where D x

  
The general solution of equation ( 27) which is bounded at infinity can be written as where From equations ( 21) and ( 28), we get The boundary conditions (15-16) give In order to determine 1 2 , A A we shall use the boundary conditions (30a), (30c) to obtain Equations ( 28), (29) become Eliminating (1) u between Eq. ( 22) and Eq. ( 24), we get Substituting from equations (31-32) into the right hand side of equation ( 33) and factorizing the left hand side as before, we obtain where The solution of the above non homogenous differential equation takes the form where B1 and B2 are unknown parameters depending on s while the remaining parameters, due to the particular solution, are given by By the same manner the displacement differential equation takes the form where The solution of Eq. ( 36) is given by where L1 and L2 are unknown parameters depending on s while the remaining parameters, due to the particular solution, are given by Substituting form Eqs. ( 28), ( 35) and (37) into Eq.( 24), comparing the coefficient of the exponentials in the resulting equation, we get Latin American Journal of Solids and Structures 13 (2016) 715-730 Substituting from Eqs. ( 35), (37) into the boundary conditions (30b), (30d), we obtain Solving the above system, we obtain The other constants can be easily obtained from equation ( 38a, b,c).Thus we have the solution in the Laplace transform domain for the temperature and displacement functions.The stress  can be determined from Eqs. (25)(26).The solution in the physical domain can be obtained by using numerical inversion method outlined in (Honig and Hirdes (1984)).This method has been successfully used in solving many problems in the theory of thermoelasticity (see e.g.(Sherief and Megahed (1999)) , (Sherief and Anwar (1988))

NUMERICAL RESULTS AND DISCUSSION
The copper material was chosen for purposes of numerical evaluations.The constants of the problem are shown in table  The computations were carried out for different functions f(t).We have chosen the following two cases: Case 1 Solids and Structures 13 (2016) 715-730 Case 2 The temperature, displacement and stress components were calculated by using the numerical method for the inversion of Laplace transform illustrated in Honig and Hirdes (1984).The FORTRAN programming language was used on a personal computer.The accuracy maintained was 5 digits for the numerical program.

FIGURES and Discussion
Figures (1) and ( 2) show the temperature distribution for t = 0.1 for cases 1 and 2, respectively.In these figures and subsequent ones the dotted lines denote the case of constant Lamé's modulii (CL) while solid lines represent the case of variable Lamé's modulii (VL).We note that changing of Lamé's modulii has very small effect on the temperature profile.4) show the displacement distribution for t = 0.1 for cases 1 and 2, respectively.We note that an increase in the values of Lamé's modulii results in an increase in the value of the displacement.Figures (7)(8)(9)(10)(11)(12) show the distributions of θ, u and σ for different times.In these figures the solid lines and the dotted lines represent t = 0.08 and t = 0.12, respectively.We note that the increase of the time results in an increase in the thermal wave penetration into the medium.The same effect is noticed for the displacement and the stress distributions.Comparison between the values of all the functions are shown in tables 2-4.As usual in dealing with problems of the theory of generalized thermoelasticity, the finite speed of wave propagation is apparent.For t = 0.1 the solution is identically zero for x > 0.6341, approximately, for both cases.
We note that for case 2, the temperature and stress distributions have two discontinuities.The first discontinuity is at x = 0.973 approximately while the second discontinuity is at x = 0.6341 approximately.The first discontinuity in the temperature is very small in value and does not show in the figure.The displacement distribution is continuous but has discontinuous first derivatives at the above locations.These locations are the locations of the wave fronts.These discontinuities are due to the fact that the input thermal shock f 2 (t) is a discontinuous function.
For case 1, where the thermal shock is represented by a continuous function f 1 (t), all the considered functions are continuous.The temperature and stress have discontinuous first derivatives at the wave fronts.

Figures ( 5 Figure 5 :
Figures (5) and (6)  show the stress distribution for t = 0.1 for cases 1 and 2, respectively.We note that an increase in the values of Lamé's modulii results in a decrease in the value of the stress component xx  .

Table 1 :
The material constants.

Table 3 :
Displacement Values for t = 0.1.with VL  with CL  with VL  with CL