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Elasto-thermodiffusive nonlocal responses for a spherical cavity due to memory effect

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Abstract

The present work is devoted to the derivation of fundamental equations in generalized thermoelastic diffusion theory. The main aim is to establish a size-dependent model with the consideration of spatial nonlocal effects of concentration and strain fields. The heat transport equation for the present problem is considered in the context of Moore–Gibson–Thompson (MGT) generalized thermoelasticity theory involving linear and nonlinear kernel functions in a delayed interval in terms of the memory-dependent derivative. The medium is considered to be one-dimensional having a spherical cavity where the boundary of the cavity is traction-free and is subjected to prescribed thermal and chemical shocks. The Laplace transform technique is incorporated for the solution of the basic equations. For numerical evaluation, the analytical expressions have been inverted in the space-time domain using the method of Zakian. From numerical results, the effects of the nonlocality parameters in the heat transport law and the nonlocality of mass-flux have been discussed. The effect of different kernel functions, the delay time, and the effect of thermodiffusion are also reported. A comparative study between the MGT theory and the hyperbolic Lord–Shulman theory is also explained.

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Abbreviations

\(\lambda \), \(\mu\) :

Lame’s constants

\(\alpha _{t}\) :

coefficient of linear thermal expansion

\(\alpha _{c}\) :

coefficient of linear diffusion

\(\beta _{1} = (3\lambda +2\mu )\alpha _{t}\) :

thermal coupling parameter

\(\beta _{2} = (3\lambda +2 \mu ) \alpha _{c}\) :

diffusion coupling parameter

\(k\) :

thermal conductivity

\(k^{\star }\) :

conductivity rate parameter

\(\rho \) :

density

\(c_{E} \) :

specific heat

\(T \) :

absolute temperature

\(T_{0} \) :

reference temperature

\(\sigma _{ij} \) :

stress tensor

\(e_{ij} \) :

strain tensor

\(\textbf{u} \) :

displacement vector

\(\textbf{q }\) :

heat flux vector

\(\eta \) :

flow of diffusing mass vector

\(\delta _{ij} \) :

Kronecker delta

\(\tau \) :

relaxation time for MGT theory

\(\tau _{1} \) :

relaxation time for mass flux

\(\omega \) :

delay time

\(D \) :

diffusion coefficient

\(P \) :

chemical potential

\(C \) :

mass concentration of diffusive material

\(U \) :

internal energy per unit mass

\(F_{i} \) :

body force per unit mass

\(S \) :

entropy per unit mass

\(F \) :

Helmhotz free energy per unit volume

\(\chi _{D} \) :

Phonon mean-free path

\(e_{0}a_{0} \) :

nonlocal parameter

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Acknowledgements

The author would like to thank the editor and the anonymous referees for their comments and suggestions on this paper.

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  1. Department of Mathematics, Sister Nivedita University, West Bengal, India

    Abhik Sur

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Sur, A. Elasto-thermodiffusive nonlocal responses for a spherical cavity due to memory effect. Mech Time-Depend Mater (2023). https://doi.org/10.1007/s11043-023-09626-8

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