Interactions due to a moving heat source in a thin slim rod under memory-dependent dual-phase lag magneto-thermo-visco-e
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Interactions due to a moving heat source in a thin slim rod under memory-dependent dual-phase lag magneto-thermo-visco-elasticity Sudip Mondal1
Received: 3 February 2019 / Accepted: 20 May 2019 © Springer Nature B.V. 2019
Abstract Enlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of magnetic field and moving heat source in a rod in the context of dual-phase lag (DPL) theory of thermo-visco-elasticity. Both ends of the rod are fixed and heat insulated. Employing Laplace transform as a tool, the problem has been transformed into the space-domain and solved analytically. Finally, solutions in the real-time domain are obtained by applying the inverse Laplace transform. Numerical calculation for stress, displacement, and temperature within the rod is carried out and displayed graphically. The effect of moving heat source speed on temperature, stress, and displacement is studied. It is found from the distributions that the temperature, thermally induced displacement, and stress of the rod decrease at large source speed. Keywords Memory-dependent derivative · Magneto-thermoelasticity · Viscosity · Moving heat source · Laplace transform and numerical inversion of Laplace transform · Dual-phase lag model
1 Introduction Traditional coupled dynamic thermoelasticity theories predict the infinite speed of thermal waves, which depend on the mixed parabolic–hyperbolic governing equations of Biot (1956), but the predicted infinite speed of the thermal wave is not an acceptable phenomenon. The infinite speed of thermal waves results in an instantaneous effect on the body, no matter how far from the point of heat source it is applied. The classical theory of thermoelasticity has been generalized and changed into different thermoelastic models that keep Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11043-019-09418-z) contains supplementary material, which is available to authorized users.
B S. Mondal
[email protected]
1
Department of Mathematics, Basirhat College, North 24 Parganas 743412, West Bengal, India
Mech Time-Depend Mater
running under the mark of “hyperbolic thermoelasticity”. At present, there are a few theories of the hyperbolic thermoelasticity. It ensures the finite speed of wave propagation. The first is due to Lord and Shulman (LS) (Lord and Shulman 1967) who acquired a wave-type heat equation by hypothesizing another law of heat equation in the form q(x, t + τq ) = −K∇θ (x, t).
(1)
In addition, the time delay constants τq and τθ , wherein in the present note τq > τθ ≥ 0 will be assumed, are associated with the microstructure of the material under consideration. The dual-phase-lag model, which reduces to Fourier law in the limit (τq , τθ ) → 0, describes a process in which a temperature gradient that is established across a material volume at time t + τθ will not give rise to a thermal flux at a point x wit
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