Hypertemperature effects in heterogeneous media and thermal flux at small-length scales

Grigor Nika; Adrian Muntean; Grigor Nika; Adrian Muntean

doi:10.3934/nhm.2023052

Networks and Heterogeneous Media

2023, Volume 18, Issue 3: 1207-1225. doi: 10.3934/nhm.2023052

Previous Article Next Article

Research article

Hypertemperature effects in heterogeneous media and thermal flux at small-length scales

Grigor Nika ^,,
Adrian Muntean

Mathematics & Computer Science, Karlstad University, Universitetsgatan 2, Karlstad, Sweden

Received: 26 November 2022 Revised: 02 July 2022 Accepted: 07 March 2023 Published: 12 April 2023

We propose an enriched microscopic heat conduction model that can account for size effects in heterogeneous media. Benefiting from physically relevant scaling arguments, we improve the regularity of the corrector in the classical problem of periodic homogenization of linear elliptic equations in the three-dimensional setting and, while doing so, we clarify the intimate role that correctors play in measuring the difference between the heterogeneous solution (microscopic) and the homogenized solution (macroscopic). Moreover, if the data are of form $ f = {\rm div}\; {\boldsymbol{F}} $ with $ {\boldsymbol{F}} \in {\rm L}^{3}(\Omega, {\mathbb R}^3) $, then we recover the classical corrector convergence theorem.
- correctors,
- scale-size thermal effects,
- generalized Fourier's law,
- microstructure
Citation: Grigor Nika, Adrian Muntean. Hypertemperature effects in heterogeneous media and thermal flux at small-length scales[J]. Networks and Heterogeneous Media, 2023, 18(3): 1207-1225. doi: 10.3934/nhm.2023052

Related Papers:

Abstract

We propose an enriched microscopic heat conduction model that can account for size effects in heterogeneous media. Benefiting from physically relevant scaling arguments, we improve the regularity of the corrector in the classical problem of periodic homogenization of linear elliptic equations in the three-dimensional setting and, while doing so, we clarify the intimate role that correctors play in measuring the difference between the heterogeneous solution (microscopic) and the homogenized solution (macroscopic). Moreover, if the data are of form $ f = {\rm div}\; {\boldsymbol{F}} $ with $ {\boldsymbol{F}} \in {\rm L}^{3}(\Omega, {\mathbb R}^3) $, then we recover the classical corrector convergence theorem.

References

[1]	E. C. Aifantis, Internal length gradient (ILG) material mechanics across scales and disciplines, Adv. Appl. Mech., 49 (2016), 1–110. https://doi.org/10.1016/bs.aams.2016.08.001 doi: 10.1016/bs.aams.2016.08.001
[2]	G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482–1518. https://doi.org/10.1137/0523084 doi: 10.1137/0523084
[3]	G. Allaire, M. Amar, Boundary layer tails in periodic homogenization, ESAIM Contr Optim CA, 4 (1999), 209–243. https://doi.org/10.1051/cocv:1999110 doi: 10.1051/cocv:1999110
[4]	L. Ambrosio, A. Carlotto, A. Massaccesi, Lectures on Elliptic Partial Differential Equations, Berlin: Springer, 2019.
[5]	A. Bensoussan, J. L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures. Netherlands: Elsevier, 1978.
[6]	A. Berezovski, Internal variables representation of generalized heat equations, Contin. Mech. Thermodyn., 31 (2019), 1733–1741. https://doi.org/10.1007/s00161-018-0729-4 doi: 10.1007/s00161-018-0729-4
[7]	F. Bonetto, J. Lebowitz, L. Rey-Bellet, Fourier's law: a challenge for theorists, Mathematical Physics 2000, (2000), 128–150.
[8]	M. Cherdantsev, K. D. Cherednichenko, Two-Scale $\Gamma$–Convergence of Integral Functionals and its Application to Homogenisation of Nonlinear High-Contrast Periodic composites, Arch. Rational Mech. Anal., 204 (2012), 445–478. https://doi.org/10.1007/s00205-011-0481-4 doi: 10.1007/s00205-011-0481-4
[9]	C. I. Christov, On a higher-gradient generalization of Fourier's law of heat conduction, AIP Conference Proceedings, (2007), 11–22.
[10]	C. Ciorănescu, P. Donato, An Introduction to Homogenization, Oxford: Oxford University Press, 2000.
[11]	D. Ciorănescu, A. Damlamian, P. Donato, G. Griso, R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math. Anal., 44 (2012), 718–760. https://doi.org/10.1137/100817942 doi: 10.1137/100817942
[12]	D. Ciorănescu, A. Damlamian, G. Griso, Éclatement périodique et homogénéisation, Comptes Rendus Math., 335 (2022), 99–104. https://doi.org/10.1016/S1631-073X(02)02429-9 doi: 10.1016/S1631-073X(02)02429-9
[13]	D. Ciorănescu, A. Damlamian, G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585–1620.
[14]	D. Ciorănescu, A. Damlamian, G. Griso, The periodic unfolding method, Theory and Applications to Partial Differential Problems, Singapore: Springer Singapore, 2018.
[15]	B. D. Coleman, V. J. Mizel, Thermodynamics and departures from Fourier's law of heat conduction, Arch. Ration. Mech. Anal., 13 (1963), 245–261. https://doi.org/10.1007/BF01262695 doi: 10.1007/BF01262695
[16]	A. Damlamian, An elementary introduction to periodic unfolding, Gakuto Int. Series, Math. Sci. Appl., 24 (2005), 1651–1684. https://doi.org/10.1377/hlthaff.24.6.1684 doi: 10.1377/hlthaff.24.6.1684
[17]	F. Demengel, G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, London: Springer-Verlag London, 2012.
[18]	A. Fehér, R. Kovács, On the evaluation of non-Fourier effects in heat pulse experiments, Int. J. Eng. Sci., 169 (2021), 103577. https://doi.org/10.1016/j.ijengsci.2021.103577 doi: 10.1016/j.ijengsci.2021.103577
[19]	A. Fehér, N. Lukács, L. Somlai, T. Fodor, M. Szücs, T. Fülöp, et al., Size effects and beyond-Fourier heat conduction in room-temperature experiments, J. Non-Equil. Thermody., 46 (2021), 403–411.
[20]	S. Forest, Milieux continus généralisés et matériaux hétérogènes, Paris: Presses des MINES, 2006.
[21]	S. Forest, E. C. Aifantis, Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua, Int. J. Solids Structures, 47 (2010), 3367–3376.
[22]	S. Forest, M. Amestoy, Hypertemperature in thermoelastic solids, CR MECANIQUE, 336 (2008), 347–353.
[23]	S. Forest, F. Pradel, K. Sab, Asymptotic analysis of heterogeneous Cosserat media, Int J Solids Struct, 38 (2001), 4585–4608.
[24]	P. Germain, La méthode des puissances virtuelles en mécanique des milieux continus, I: Théorie du second gradient, J. Mécanique, 12 (1973), 235–274.
[25]	P. Germain, The method of virtual power in continuum mechanics. Part 2: Microstructure, SIAM J. Appl. Math., 25 (1973), 556–575. https://doi.org/10.1016/0026-2714(73)90243-6 doi: 10.1016/0026-2714(73)90243-6
[26]	M. Giaquinta, L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, Berlin: Springer Science & Business Media, 2013.
[27]	G. Griso, Error estimates and unfolding for periodic homogenization, Asym. Anal., 40 (2004), 269–286.
[28]	D. Lukkassen, G. Nguetseng, P. Wall, Two-scale convergence, Int. J. Pure Appl. Math., 2 (2002), 35–86.
[29]	A. Mielke, On evolutionary $\Gamma$–convergence, In A. Muntean, J. Rademacher, A. Zagaris, editors, Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Cham: Springer, 2016,187–249.
[30]	R. D. Mindlin, N. N. Eshel, On first strain-gradient theories in linear elasticity, Int J Solids Struct, 4 (1968), 109–124. https://doi.org/10.1177/002188636800400105 doi: 10.1177/002188636800400105
[31]	S. Moskow, M. Vogelius, First-order corrections to the homogenised eigenvalues of a periodic composite medium. a convergence proof, P ROY SOC EDINB A, 127 (1997), 1263–1299. https://doi.org/10.1017/S0308210500027050 doi: 10.1017/S0308210500027050
[32]	G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608–623. https://doi.org/10.1137/0520043 doi: 10.1137/0520043
[33]	G. Nika, A gradient system for a higher-gradient generalization of Fourier's law of heat conduction, Mod. Phys. Lett. B, (2023), 1–11. https://doi.org/10.1142/S0217984923500112 doi: 10.1142/S0217984923500112
[34]	G. Nika, A. Muntean, Effective medium theory for second-gradient nonlinear elasticity with chirality, arXiv: 2202.00644, [Preprint], (2022) [cited 2023 April 12]. Available from: https://doi.org/10.48550/arXiv.2202.00644
[35]	G. Nika, B. Vernescu, Rate of convergence for a multiscale model of dilute emulsions with non-uniform surface tension, Discrete Continuous Dyn Syst Ser B, 9 (2016), 1553–1564. https://doi.org/10.3934/dcdss.2016062 doi: 10.3934/dcdss.2016062
[36]	D. Onofrei, B. Vernescu, Error estimates for periodic homogenization with non-smooth coefficients, Asym. Anal., 54 (2007), 103–123. https://doi.org/10.1353/tcj.2007.0008 doi: 10.1353/tcj.2007.0008
[37]	L. S. Pan, D. Xu, J. Lou, Q. Yao, A generalized heat conduction model in rarefied gas, EPL, 73 (2006), 846–850. https://doi.org/10.1209/epl/i2005-10473-7 doi: 10.1209/epl/i2005-10473-7
[38]	D. Ruelle, A mechanical model for Fourier's law of heat conduction, Commun. Math. Phys, 311 (2012), 755–768.
[39]	E. Sanchez-Palencia, Non-homogeneous media and vibration theory, Lect. Notes Phys., 320 (1980), 57–65.
[40]	G. Stampacchia, The spaces $\mathcal{L}^{(p, \lambda)}$, ${N}^{(p, \lambda)}$ and interpolation, Ann. Scuola Norm-Sci., 19 (1965), 443–462.
[41]	A. S. J. Suiker, C. S. Chang, Application of higher-order tensor theory for formulating enhanced continuum models, Acta Mech., 142 (2000), 223–234. https://doi.org/10.1007/BF01190020 doi: 10.1007/BF01190020

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)