Abstract
The theory of generalized thermoelasticity with fractional order strain is employed to study the problem of one-dimensional disturbances in a viscoelastic solid in the presence of a moving internal heat source and subjected to a mechanical load. The problem is in the context of Green-Naghdi theory of thermoelasticity with energy dissipation. Laplace transform and state space techniques are used to obtain the general solution for a set of boundary conditions. To tackle the expression of heat source, Fourier transform is also employed. The expressions for different field parameters such as displacement, stress, thermodynamical temperature, and conductive temperature in the physical domain are derived by the application of numerical inversion technique. The effects of fractional order strain, two-temperature parameter, viscosity, and velocity of internal heat source on the field variables are depicted graphically for copper material. Some special cases of interest have also been presented.
Highlights
The classical thermoelasticity theory based on Fourier’s law of heat conduction suffers from the deficiency of admitting thermal signals propagating with infinite speed
Using dimensionless variables and Laplace transform in (9), the displacement component may be evaluated as conductive temperature, stress, thermodynamical temperature, and displacement are obtained in a viscothermoelastic medium with two-temperature and fractional order strain as φ = L 2σ∗ (e−√λ2x − e−√λ1x), s (λ1 − λ2)
In the first group (Figures 1–4), we have shown the effects of mechanical relaxation time τ and parameter β on the considered physical variables with location x
Summary
The classical thermoelasticity theory based on Fourier’s law of heat conduction suffers from the deficiency of admitting thermal signals propagating with infinite speed. Wang et al [30] suggested a new theory of generalized thermoelasticity for elastic media with variable properties in the context of fractional order heat conduction equation. They derived the formulations of anisotropic heterogeneous material with temperature dependent material properties by making use of the Clausius inequality and the higher expansions of free energy. Youssef [31] derived a new theory of thermoelasticity by modifying the Duhamel-Neumann stressstrain relation In this theory, this relation depends on the fractional order of strain which adds knowledge about the time history to the deformation of materials after being acted upon by mechanical or thermal loadings. Some comparisons are exhibited in figures to demonstrate the effects of fractional order strain, viscosity, two-temperature parameter, and the presence of internal heat source
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