New insights on generalized heat conduction and thermoelastic coupling models

doi:10.1007/s10483-025-3280-7

Abstract

Abstract:

With the miniaturization of devices and the development of modern heating technologies, the generalization of heat conduction and thermoelastic coupling has become crucial, effectively emulating the thermodynamic behavior of materials in ultrashort time scales. Theoretically, generalized heat conductive models are considered in this work. By analogy with mechanical viscoelastic models, this paper further enriches the heat conduction models and gives their one-dimensional physical expression. Numerically, the transient thermoelastic response of the slim strip material under thermal shock is investigated by applying the proposed models. First, the analytical solution in the Laplace domain is obtained by the Laplace transform. Then, the numerical results of the transient responses are obtained by the numerical inverse Laplace transform. Finally, the transient responses of different models are analyzed and compared, and the effects of material parameters are discussed. This work not only opens up new research perspectives on generalized heat conductive and thermoelastic coupling theories, but also is expected to be beneficial for the deeper understanding of the heat wave theory.

Key words: generalized heat conduction, thermoelastic coupling, transient response, generalized Cattaneo-Vernotte (CV) model, generalized Green-Naghdi (GN) model

2010 MSC Number:

O343.6

Yue HUANG, Lei YAN, Hua WU, Yajun YU. New insights on generalized heat conduction and thermoelastic coupling models. Applied Mathematics and Mechanics (English Edition), 2025, 46(8): 1533-1550.

TrendMD

Figures/Tables 10

Fig. 1

Fig. 2

Table 1

Equations of modified generalized heat conduction models and Laplace transform"

Model	Model equation	$M$	$N$	$M ¯$	$N ¯$
CV	$q + τ q ˙ = − k ∇ T$	$1 + τ ∂ ∂ t$	1	$1 + τ s$	1
MCV1	$q + 2 τ q ˙ = − k ∇ T − k τ ∇ T ˙$	$1 + 2 τ ∂ ∂ t$	$1 + τ ∂ ∂ t$	$1 + 2 τ s$	$1 + τ s$
MCV2	$q + τ q ˙ = − 2 k ∇ T − k τ ∇ T ˙$	$1 + τ ∂ ∂ t$	$2 + τ ∂ ∂ t$	$1 + τ s$	$2 + τ s$
MCV3	$q + 3 τ q ˙ + τ 2 q ¨ = − k ∇ T − k τ ∇ T ˙$	$1 + 3 τ ∂ ∂ t + τ 2 ∂ 2 ∂ t 2$	$1 + τ ∂ ∂ t$	$1 + 3 τ s + τ 2 s 2$	$1 + τ s$
MCV4	$q + 3 τ q ˙ + τ 2 q ¨ = − 2 k ∇ T − k τ ∇ T ˙$	$1 + 3 τ ∂ ∂ t + τ 2 ∂ 2 ∂ t 2$	$2 + τ ∂ ∂ t$	$1 + 3 τ s + τ 2 s 2$	$2 + τ s$
MCV5	$q + 2 τ q ˙ + τ 2 q ¨ = − 2 k ∇ T − 2 k τ ∇ T ˙$	$1 + 2 τ ∂ ∂ t + τ 2 ∂ 2 ∂ t 2$	$2 + 2 τ ∂ ∂ t$	$1 + 2 τ s + τ 2 s 2$	$2 + 2 τ s$
MCV6	$q + 3 τ q ˙ + τ 2 q ¨ = − k ∇ T − 2 k τ ∇ T ˙$	$1 + 3 τ ∂ ∂ t + τ 2 ∂ 2 ∂ t 2$	$1 + 2 τ ∂ ∂ t$	$1 + 3 τ s + τ 2 s 2$	$1 + 2 τ s$
GN	$τ q ˙ = − k ∇ T − k τ ∇ T ˙$	$τ ∂ ∂ t$	$1 + τ ∂ ∂ t$	$τ s$	$1 + τ s$
MGN1	$2 τ q ˙ + τ 2 q ¨ = − k ∇ T − k τ ∇ T ˙$	$2 τ ∂ ∂ t + τ 2 ∂ 2 ∂ t 2$	$1 + τ ∂ ∂ t$	$2 τ s + τ 2 s 2$	$1 + τ s$
MGN2	$τ q ˙ + τ 2 q ¨ = − k ∇ T − 2 k τ ∇ T ˙$	$τ ∂ ∂ t + τ 2 ∂ 2 ∂ t 2$	$1 + 2 τ ∂ ∂ t$	$τ s + τ 2 s 2$	$1 + 2 τ s$
MGN3	$2 τ q ˙ + 2 τ 2 q ¨ = − k ∇ T − 2 k τ ∇ T ˙ − k τ 2 ∇ T ¨$	$2 τ ∂ ∂ t + 2 τ 2 ∂ 2 ∂ t 2$	$1 + 2 τ ∂ ∂ t + τ 2 ∂ 2 ∂ t 2$	$2 τ s + 2 τ 2 s 2$	$1 + 2 τ s + τ 2 s 2$
MGN4	$τ q ˙ + τ 2 q ¨ = − k ∇ T − 3 k τ ∇ T ˙ − k τ 2 ∇ T ¨$	$τ ∂ ∂ t + τ 2 ∂ 2 ∂ t 2$	$1 + 3 τ ∂ ∂ t + τ 2 ∂ 2 ∂ t 2$	$τ s + τ 2 s 2$	$1 + 3 τ s + τ 2 s 2$

Table 1

Fig. 3

Fig. 4

Table 2

Material properties of the copper"

$λ$ /GPa	$μ$ /GPa	$α θ / (m ⋅ K − 1)$	$ρ / (kg ⋅ m − 3)$	$c E / (J ⋅ kg − 1 ⋅ K − 1)$	$T 0$ /K
77.6	38.6	$1.78 × 10 − 5$	8 945	381	293

Table 2

Fig. 5

Fig. 6

Fig. 7

Fig. 8

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