By B. Das
This e-book provides difficulties and strategies of the mathematical theories of thermoelasticity and magnetothermoelasticity. The classical, coupled and generalized theories are solved utilizing the eigenvalue method. diverse equipment of numerical inversion of the Laplace rework are provided and their direct functions are illustrated. The booklet is especially necessary to these drawn to continuum mechanics.
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Sample text
3 (in Eq. 32), we now take λ and ω(R, α) are both scalar and scalar function respectively. 16) which leads the equation L ω = 0 e−α R . 16), as discussed in Chap. 17) The roots of the characteristic equation corresponding to the matrix A are the eigenvalues and the corresponding eigenvectors are X . 20) i=1 Where Z (R, p) is calculated from the expression of of V . dZ dR which is the component Using these values (Eq. 20), Eqs. 3 Solution Procedure 37 where Bi ’s are the arbitrary constants which are to be determined from the boundary conditions.
Thermal relaxation time parameter. 9 Basic Equations and Formulation of the Problem In the absence of external heat source and body force and taking the displacement component (u) in r -direction, the basic equations are written for a homogeneous and isotropic infinite medium with circular cylindrical cavity with radius “a” with a constant reference temperature θ0 . 9 Basic Equations and Formulation of the Problem 45 the radial displacement (u) exists and other displacement components are zero and taking u = u(r, t), and r is the radial distance.
45) i=1 where X i ’s are coefficients, and X i ’s are the corresponding roots of the Legendre equation Pn (u) = 0. Putting the values p = 1, 2, 3, . . , N in Eq. 45), we get X 1 h(u 1 ) + X 2 h(u 2 ) + · · · + X n h(u n ) = f (1) X 1 u 1 h(u 1 ) + X 2 u 2 h(u 2 ) + · · · + X n u n h(u n ) = f (2) .................. 46) From Eq. 46), we get the values of h(u i ’s, i = 1(1)n, where f (−logu 1 ) = h(u 1 ), f (−logu 2 ) = h(u 2 ), . . , f (−logu n ) = h(u n ). 46), we get the numerical inversion of Laplace transform according to the numeric values of p.