A nonlocal Fourier's law and its application to the heat conduction of one-dimensional and two-dimensional thermal lattices

Noël Challamel; Cécile Grazide; Vincent Picandet; Arnaud Perrot; Yingyan Zhang

doi:10.1016/j.crme.2016.01.001

A nonlocal Fourier's law and its application to the heat conduction of one-dimensional and two-dimensional thermal lattices

Noël Challamel ¹ ; Cécile Grazide ¹ ; Vincent Picandet ¹ ; Arnaud Perrot ¹ ; Yingyan Zhang ²

¹ Université de Bretagne Sud, UBS – Institut Dupuy de Lôme, Centre de Recherche, Rue de Saint Maudé, BP92116, 56321 Lorient Cedex, France
² School of Computing, Engineering & Mathematics, Western Sydney University, Locked Bag 1797, Penrith NSW 2751, Australia

Comptes Rendus. Mécanique, Volume 344 (2016) no. 6, pp. 388-401.

Résumé

This study focuses on heat conduction in unidimensional lattices also known as microstructured rods. The lattice thermal properties can be representative of concentrated thermal interface phases in one-dimensional segmented rods. The exact solution of the linear time-dependent spatial difference equation associated with the lattice problem is presented for some given initial and boundary conditions. This exact solution is compared to the quasicontinuum approximation built by continualization of the lattice equations. A rational-based asymptotic expansion of the pseudo-differential problem leads to an equivalent nonlocal-type Fourier's law. The differential nonlocal Fourier's law is analysed with respect to thermodynamic models available in the literature, such as the Guyer–Krumhansl-type equation. The length scale of the nonlocal heat law is calibrated with respect to the lattice spacing. An error analysis is conducted for quantifying the efficiency of the nonlocal model to capture the lattice evolution problem, as compared to the local model. The propagation of error with the nonlocal model is much slower than that in its local counterpart. A two-dimensional thermal lattice is also considered and approximated by a two-dimensional nonlocal heat problem. It is shown that nonlocal and continualized heat equations both approximate efficiently the two-dimensional thermal lattice response. These extended continuous heat models are shown to be good candidates for approximating the heat transfer behaviour of microstructured rods or membranes.

Reçu le : 2015-10-23
Accepté le : 2016-01-02
Publié le : 2016-03-17

DOI : 10.1016/j.crme.2016.01.001

Mots-clés : Heat equation, Lattice, Fourier's law, Nonlocal thermodynamics, Gradient Fourier's law, Nonlocality, Diffusion equation

Affiliations des auteurs :

Noël Challamel ¹ ; Cécile Grazide ¹ ; Vincent Picandet ¹ ; Arnaud Perrot ¹ ; Yingyan Zhang ²

¹ Université de Bretagne Sud, UBS – Institut Dupuy de Lôme, Centre de Recherche, Rue de Saint Maudé, BP92116, 56321 Lorient Cedex, France
² School of Computing, Engineering & Mathematics, Western Sydney University, Locked Bag 1797, Penrith NSW 2751, Australia

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     author = {No\"el Challamel and C\'ecile Grazide and Vincent Picandet and Arnaud Perrot and Yingyan Zhang},
     title = {A nonlocal {Fourier's} law and its application to the heat conduction of one-dimensional and two-dimensional thermal lattices},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {388--401},
     publisher = {Elsevier},
     volume = {344},
     number = {6},
     year = {2016},
     doi = {10.1016/j.crme.2016.01.001},
     language = {en},
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Noël Challamel; Cécile Grazide; Vincent Picandet; Arnaud Perrot; Yingyan Zhang. A nonlocal Fourier's law and its application to the heat conduction of one-dimensional and two-dimensional thermal lattices. Comptes Rendus. Mécanique, Volume 344 (2016) no. 6, pp. 388-401. doi : 10.1016/j.crme.2016.01.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2016.01.001/

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Rajesh Kumar; Seema Thakran; Ankush Gunghas; Kapil Kumar Kalkal Transient disturbances in a nonlocal functionally graded thermoelastic solid under Green–Lindsay model, International Journal of Numerical Methods for Heat Fluid Flow, Volume 31 (2021) no. 7, p. 2288 | DOI:10.1108/hff-08-2020-0514
Dinesh Kumar Sharma; Mitali Bachher; Mahesh Kumar Sharma; Nantu Sarkar On the Analysis of Free Vibrations of Nonlocal Elastic Sphere of FGM Type in Generalized Thermoelasticity, Journal of Vibration Engineering Technologies, Volume 9 (2021) no. 1, p. 149 | DOI:10.1007/s42417-020-00217-2
Sergey A. Lurie; Dmitrii B. Volkov-Bogorodskii; Petr A. Belov Analytical Solution of Stationary Coupled Thermoelasticity Problem for Inhomogeneous Structures, Mathematics, Volume 10 (2021) no. 1, p. 90 | DOI:10.3390/math10010090
J. Sladek; V. Sladek; M. Repka The Heat Conduction in Nanosized Structures, Physical Mesomechanics, Volume 24 (2021) no. 5, p. 611 | DOI:10.1134/s102995992105012x
Nantu Sarkar Thermoelastic responses of a finite rod due to nonlocal heat conduction, Acta Mechanica, Volume 231 (2020) no. 3, p. 947 | DOI:10.1007/s00707-019-02583-9
Kapil Kumar Kalkal; Devender Sheoran; Sunita Deswal Reflection of plane waves in a nonlocal micropolar thermoelastic medium under the effect of rotation, Acta Mechanica, Volume 231 (2020) no. 7, p. 2849 | DOI:10.1007/s00707-020-02676-w
K. Zhukovsky; D. Oskolkov Modeling of heat transport and exact analytical solutions in thin films with account for constant non-relativistic motion, International Journal of Heat and Mass Transfer, Volume 150 (2020), p. 119085 | DOI:10.1016/j.ijheatmasstransfer.2019.119085
Jan Sladek; Vladimir Sladek; Miroslav Repka; Ernian Pan A novel gradient theory for thermoelectric material structures, International Journal of Solids and Structures, Volume 206 (2020), p. 292 | DOI:10.1016/j.ijsolstr.2020.09.023
Ali Hashmat; Jahangir Adnan; Khan Aftab Reflection of thermo-elastic wave in semiconductor nanostructures nonlocal porous medium, Journal of Central South University, Volume 27 (2020) no. 11, p. 3188 | DOI:10.1007/s11771-020-4472-1
Dinesh Kumar Sharma; Dinesh Thakur; Vishal Walia; Nantu Sarkar Free vibration analysis of a nonlocal thermoelastic hollow cylinder with diffusion, Journal of Thermal Stresses, Volume 43 (2020) no. 8, p. 981 | DOI:10.1080/01495739.2020.1764425
Nantu Sarkar Thermoelastic responses of a nonlocal elastic rod due to nonlocal heat conduction, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Volume 100 (2020) no. 4 | DOI:10.1002/zamm.201900252
Sergey Lurie; Petr Belov From Generalized Theories of Media with Fields of Defects to Closed Variational Models of the Coupled Gradient Thermoelasticity and Thermal Conductivity, Higher Gradient Materials and Related Generalized Continua, Volume 120 (2019), p. 135 | DOI:10.1007/978-3-030-30406-5_8
Nantu Sarkar; S. K. Tomar Plane waves in nonlocal thermoelastic solid with voids, Journal of Thermal Stresses, Volume 42 (2019) no. 5, p. 580 | DOI:10.1080/01495739.2018.1554395
Sudip Mondal; Nihar Sarkar; Nantu Sarkar Waves in dual-phase-lag thermoelastic materials with voids based on Eringen’s nonlocal elasticity, Journal of Thermal Stresses, Volume 42 (2019) no. 8, p. 1035 | DOI:10.1080/01495739.2019.1591249
Mitali Bachher; Nantu Sarkar Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer, Waves in Random and Complex Media, Volume 29 (2019) no. 4, p. 595 | DOI:10.1080/17455030.2018.1457230
Konstantin Zhukovsky; Dmitrii Oskolkov; Nadezhda Gubina Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative, Axioms, Volume 7 (2018) no. 3, p. 48 | DOI:10.3390/axioms7030048
K. Zhukovsky; D. Oskolkov Exact harmonic solutions to Guyer–Krumhansl-type equation and application to heat transport in thin films, Continuum Mechanics and Thermodynamics, Volume 30 (2018) no. 6, p. 1207 | DOI:10.1007/s00161-018-0648-4
K. Zhukovsky Exact harmonic solution to ballistic type heat propagation in thin films and wires, International Journal of Heat and Mass Transfer, Volume 120 (2018), p. 944 | DOI:10.1016/j.ijheatmasstransfer.2017.12.091
B. Hérisson; N. Challamel; V. Picandet; A. Perrot; C. M. Wang Static and Dynamic Behaviors of Microstructured Membranes within Nonlocal Mechanics, Journal of Engineering Mechanics, Volume 144 (2018) no. 1 | DOI:10.1061/(asce)em.1943-7889.0001379
K. V. Zhukovsky A Harmonic Solution for the Hyperbolic Heat Conduction Equation and Its Relationship to the Guyer–Krumhansl Equation, Moscow University Physics Bulletin, Volume 73 (2018) no. 1, p. 45 | DOI:10.3103/s0027134918010186

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