By B. Das
This e-book offers difficulties and options 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 purposes are illustrated. The publication is particularly worthwhile to these attracted to continuum mechanics.
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Additional resources for Problems and Solutions in Thermoelasticity and Magneto-thermoelasticity
Initially, the body is at rest and the state of undeformed and unstressed. So, any type of displacement components and their time derivatives are zero. The body is also maintaining constant reference temperature T0 . 16) 36 4 Generalized Thermoelasticity where, L = p ε ; b12 cT2 4 = d2 d R2 + 2 d − R22 R dR 2 1; b22 = p (1+ε) cT2 dZ T dR ; V = U and A = [bi j ]i, j=1,2 b11 = p 2 ; b21 = Solution of the Vector–Matrix Differential Equation As the solution discussed in Chap. 3 (in Eq. 32), we now take λ and ω(R, α) are both scalar and scalar function respectively.
40 4 Generalized Thermoelasticity Distribution of Temperature(Z) verses time for different values of R Fig. 5 t Fig. 5. 3. 2. (ii) finally tangential stress vanishes for all times. 7 Concluding Remarks 41 Distribution of σφφ verses R for different time Fig. 5 Variation of Stress case I. 2 Fig. 6 Variation of Displacement 14 Case II. 5 t Case - II 4. 3, and we conclude that The variation of displacement, stresses, and temperature almost depicts same behavior. The absolute value of displacement, stresses and temperature is increased as the increment of the value of η0 .
For fixed time t 5 Case III. 5 R Nomenclature Jn (α) = Bessel function of order n. α0 = Coefficient of volume expansion. k ∗ = Parameter of Green and Naghdi’s theory parameter. λ, μ = Lame` constants. 2 . Denoting, u¨ = ∂∂t u2 and θ˙ = ∂θ ∂t ρ = Mass density of the medium. c = Specific heat at constant deformation. θ = Temperature distribution of the medium above the reference temperature θ0 . k = Thermal conductivity of the material. τ = 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 .