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State-space approach to two-temperature generalized thermoelasticity without energy dissipation of medium subjected to moving heat source

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Abstract

In this work, a model of two-temperature generalized thermoelasticity without energy dissipation for an elastic half-space with constant elastic parameters is constructed. The Laplace transform and state-space techniques are used to obtain the general solution for any set of boundary conditions. The general solutions are applied to a specific problem of a half-space subjected to a moving heat source with a constant velocity. The inverse Laplace transforms are computed numerically, and the comparisons are shown in figures to estimate the effects of the heat source velocity and the two-temperature parameter.

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Abbreviations

λ, μ :

Lamé’s constants

ρ :

density

C E :

specific heat at constant strain

t :

time

T :

dynamical temperature

T 0 :

reference temperature

θ :

dynamical temperature increment (T−T 0)

φ :

conductive temperature

Q :

heat source

α T :

coefficient of linear thermal expansion

Γ :

stress-temperature coefficient (3λ+2μ)αT

σ ij :

components of stress tensor

e ij :

components of strain tensor

u i :

components of displacement vector

K*:

characteristic of theorem

c 0 :

longitudinal wave speed \(\sqrt {\frac{{\lambda + 2\mu }} {\rho }}\)

η :

thermal viscosity \(\frac{{\rho C_E }} {{K^* }}\)

ɛ T :

dimensionless thermoelastic coupling constant \(\frac{{\gamma c_0^2 }} {{K^* }}\)

C T :

dimensionless conductive-dynamical heat coupling constant ηc 20

a :

two-temperature parameter

β :

dimensionless two-temperature parameter ac 20 η 2

b :

dimensionless mechanical coupling constant \(\frac{{\gamma T_0 }} {{\lambda + 2\mu }}\)

v :

heat source velocity

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Authors and Affiliations

  1. Department of Mathematical, Faculty of Education, Alexandria University, Alexandria, 21527, Egypt

    H. M. Youssef

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  1. H. M. Youssef
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Correspondence to H. M. Youssef.

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Youssef, H.M. State-space approach to two-temperature generalized thermoelasticity without energy dissipation of medium subjected to moving heat source. Appl. Math. Mech.-Engl. Ed. 34, 63–74 (2013). https://doi.org/10.1007/s10483-013-1653-7

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  • DOI: https://doi.org/10.1007/s10483-013-1653-7

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