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Memory response of hyperbolic two-temperature thermoelastic diffusive half-space with variable material properties

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Abstract

In the manufacturing of micro-electromechanical, geotectonic, nuclear devices, etc., mathematical modeling of thermoelastic diffusion phenomenon has significant importance. In this light, the present problem studies two-dimensional thermodiffusive elastic half-space with variable thermal conductivity and diffusivity. Adopting hyperbolic two-temperature dual-phase-lag thermoelastic diffusion model based on memory-dependent derivatives augments the novelty of the present work. Initially, the medium is kept quiescent and the bounding surface of half-space is subjected to thermal, mechanical, and concentration loadings. Linearizing heat conduction and mass diffusion equations using Kirchhoff’s transformation method, the problem is solved using eigenvalue approach in Laplace–Fourier transformed domain with initial and boundary restrictions. To obtain the results in space-time domain, numerical inversion technique is applied. The impact of various parameters on dimensionless field variables involved in the study is analyzed graphically. The results reveal that nonlinear kernel function and greater values of time delay contribute to smoother flow of temperature change and concentration in the medium. With classical two-temperature theory, physical quantities attain comparatively lower magnitudes than with one-temperature or hyperbolic two-temperature theories. The variable thermal conductivity and diffusivity have an increasing impact on temperature change and concentration, respectively.

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Abbreviations

\(\lambda , ~ \mu \) :

Lamè’s elastic constants

\(T_0\) :

Reference temperature (constant temperature of the medium at an unstrained state)

\(T_d\) :

Absolute thermodynamic temperature

\(T_\textrm{c}\) :

Absolute conductive temperature

\(T=T_d-T_0\) :

Change in thermodynamic temperature

\(\phi =T_\textrm{c}-T_0\) :

Change in conductive temperature

\(K(\phi )\) :

Variable thermal conductivity

\(K_0\) :

Temperature-independent thermal conductivity

\(u_i\) :

Components of displacement vector

\(e_{ij} \) :

Components of strain tensor

\(\sigma _{ij}\) :

Components of stress tensor

\(\rho \) :

Mass density

\(C_e\) :

Specific heat at constant strain

a :

Measurement of thermodiffusive effect

b :

Measure of diffusive effect

P :

Chemical potential per unit mass

C :

Concentration of diffusive material

D(C):

Variable diffusivity

\(D_0\) :

Concentration-independent diffusivity

S :

Entropy per unit mass

\(\delta ^*\) :

(Hyperbolic) two-temperature parameter

\(\tau _q, \tau _\phi \) :

Thermal relaxation times

\(\tau _\eta , \tau _P\) :

Diffusion relaxation times

\(\delta _{ij} \) :

Kronecker’s delta

\((\beta _1,\beta _2)=(3\lambda +2\mu )(\alpha _t,\alpha _c)\); \(\alpha _t\), \(\alpha _c\) :

\(\alpha _t\), \(\alpha _c\) are the coefficients of linear thermal and diffusion expansion, respectively.

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  1. Department of Mathematics and Scientific Computing, National Institute of Technology, Hamirpur, Himachal Pradesh, 177005, India

    G. Geetanjali, Ankit Bajpai & P. K. Sharma

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  1. G. Geetanjali
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Geetanjali, G., Bajpai, A. & Sharma, P.K. Memory response of hyperbolic two-temperature thermoelastic diffusive half-space with variable material properties. Arch Appl Mech 93, 467–485 (2023). https://doi.org/10.1007/s00419-022-02276-1

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