Two-dimensional generalized thermo-elastic problem for anisotropic half-space

This paper concerns with the study of wave propagation in fibre reinforced anisotropic half space under the influence of temperature and hydrostatic initial stress. Lord-Shulman theory is applied to the heat conduction equation. The resulting equations are written in the form of vector matrix differential equation by using Normal Mode technique, finally which is solved by Eigen value approach.

natural variables constitute convergent solution.So, the alternative method of potential function approach is eigenvalue approach.In this method, we obtain a vector-matrix differential equation from the basic equations which reduces finally to an algebraic eigenvalue problem and the solutions for the field variables are obtained by determining the eigenvalues and eigenvectors from the corresponding coefficient matrix.In this theory, body forces and/or heat sources are also accommodated as in Das and Lahiri [7], Bachher et al. [8].Now, two different models of generalized thermoelastisities are extensively used.One is Lord and Shulman (L-S) [9] theory and the other is Green and Lindsay (G-L) [10] theory.Introducing one relaxtion time parameter in L-S theory the heat conduction equation becomes hyperbolic type without violating conventional Fourier's law.Whereas the G-L theory modified the heat conduction equation as well as the equation of motion in coupled thermoelasticitywo relaxation time parameters.There are other three models (Model I, II and III by Green and Nagdhi [11][12][13]) for generalized thermoelasticity concerned to the theory of with or without energy dissipation.

Development of governing equations
The stress-strain relation and the governing equations of motion without body forces and heat sources are written as follow: We consider the problem of a elastic half-space ( ≥ 0) in fibre-reinforced anisotropic material with ≡ ( , , ) where + + = 1 as in I. A. Abbas [14], where the displacements are given: We consider the direction of fibre as ≡ (1,0,0) with -axis as prefered direction, and Eqs.(1)(2)(3)(4)(5), reduces as given: with: where , are linear thermal expansion coeeficients.To transform the above governing equations in non-dimensional forms, we introduce the nondimensional variables as follows: ( Using non-dimensional Eq. ( 13), the governing equations reduces to (eleminating primes for convenience): ( , , ) = 3. Solution procedure

Solution of the vector-matrix differential equation
To solve the vector-matrix differential Eq. ( 27), we apply the method of eigenvalue approach, The characteristic equation of the matrix is given by: The roots of the characteristic Eq. ( 28) are = , = 1, 2, 3 which are of the form = ± , = ± and = ± and they are also eigenvalues of the matrix.The eigenvector, corresponding to the eigenvalue can obtained as: where = − , = − , = − .As in Lahiri et al. [7], the general solution of Eq. ( 27) which is regular as can be written as: Hence the field variables can be written as the following: The simplified form of Eqs.(21-23) can be written as: where:

Boundary conditions
Considering the problem of a half-space , defined as follows: In order to determine the arbitrary constants , = 1, 2, 3, we consider the boundary conditions as follows.

Case 1 a) Mechanical Boundary condition:
For stress-free surface =0, = 0, = 0. b) Thermal Boundary condition: where is Biot's number.For fixed , nominal decreasing of has been seen as t increases.For fixed , numerical values of decreases as increases.Also, significant changes occur in the region 0.6 ≤ ≤ 1.0 and 0 ≤ ≤ 1.0.For fixed , the numerical value of gradually decreases as increases.For fixed , the numerical value of gradually increases as increases.For fixed numerical value of gradually increases in the region 0 ≤ ≤ 0.3 (approx.)For fixed numerical value of nominally increases as increases.

4. 2 . 1 .
Fig. 1 represents distribution of normal stress for = 0.3.For fixed time , gradually increases as increases.For fixed numerical values of gradually decreases as t increases.Fig. 2 represents distribution of normal stress for = 0.2.For fixed time , gradually decreases as increases.For fixed x numerical values of gradually increases as increases.Fig. 3 represents distribution of normal stress for = 0.5.For fixed time , gradually decreases as increases.For fixed numerical values of gradually increases as increases.

For
fixed numerical value of gradually decreases as increases.For fixed the numerical value of gradually decreases as increases.Significant changes occur in the region 0.2 ≤ ≤ 0.6 and 0.6 ≤ ≤ 1.0.Fig. 6 represent distribution of stress component at for different values of and for fixed = 0.1 and 0.1.

Fig. 9 .
Fig. 9.The variation of at = 0.3 and = 2 verses and For fixed numerical value of nominally increases as increases.For fixed numerical value of gradually increases as increases.Fig. 11 represent distribution of temperature, for different values of and for fixed = 0.5 and 3.For fixed numerical value of nominally increases as increases.For fixed numerical values of decreases as increases.