Spatial Nonlocality Effect Influence on the Plate Temperature State
Authors: Savelyeva I.Yu. | Published: 21.02.2024 |
Published in issue: #1(112)/2024 | |
DOI: 10.18698/1812-3368-2024-1-28-40 | |
Category: Mathematics and Mechanics | Chapter: Mathematical Simulation, Numerical Methods and Software Packages | |
Keywords: mathematical model, nonlocality, heat transfer, multiscale approach, nonequilibrium molecular dynamics |
Abstract
Models used in the multiscale simulation are ranging from the quantum mechanics to the continuum mechanics models. At the microscale, the molecular dynamics models are computationally intensive. Thus, the macroscale models imply the modified continuum models, make it possible to take into account relationship between the material characteristics at the macro and micro levels, and are remaining relevant. One of the well-known approaches to constructing such models involves considering the influence of the continuum spatial and temporal nonlocality. All the nonlocality medium models include certain nonlocality parameters, which variation and selection make it possible to account for connection between medium characteristics at the macro and micro levels. Currently, establishing connections between these macroscale models parameters and parameters of the atomistic and molecular dynamics models becomes an urgent task. The paper uses the example of a problem in the stationary temperature state of a homogeneous plate taking into account the spatial nonlocality, and demonstrates a possibility to establish the indicated relationships between parameters of the different-scale models. An analytical solution was obtained for the one-dimensional problem when considering a simple influence function. Influence of the nonlocality coefficient on the degree of deviation of the temperature distribution over the plate thickness from the classical solution was analyzed. The plate temperature state obtained within the framework of the macroscale approach was compared with the mathematical simulation results at the temperature distribution nanolevel using the nonequilibrium molecular dynamics method
The work was supported by the Ministry of Science and Higher Education of the Russian Federation (project no. 0705-2023-0012)
Please cite this article in English as:
Savelyeva I.Yu. Spatial nonlocality effect influence on the plate temperature state. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2024, no. 1 (112), pp. 28--40 (in Russ.). EDN: EZWDQS
References
[1] Nieminen R.M. From atomistic simulation towards multiscale modelling of materials. J. Phys.: Condens. Matter., 2002, vol. 14, no. 11, pp. 2859--2876. DOI: https://doi.org/10.1088/0953-8984/14/11/306
[2] Elliott J.A. Novel approaches to multiscale modelling in materials science. Int. Mater. Rev., 2011, vol. 56, iss. 4, pp. 207--225. DOI: https://doi.org/10.1179/1743280410Y.0000000002
[3] Shaat M. A reduced micromorphic model for multiscale materials and its applications in wave propagation. Compos. Struct., 2018, vol. 201, pp. 446--454. DOI: https://doi.org/10.1016/j.compstruct.2018.06.057
[4] Bouvard J.L., Ward D.K., Hossain D., et al. Review of hierarchical multiscale modeling to describe the mechanical behavior of amorphous polymers. J. Eng. Mater. Technol., 2009, vol. 131, iss. 4, art. 41206. DOI: https://doi.org/10.1115/1.3183779
[5] Ogata S., Lidorikis E., Shimojo F., et al. Hybrid finite-element/molecular-dynamics/electronic-density-functional approach to materials simulations on parallel computers. Comput. Phys. Commun., 2001, vol. 138, iss. 2, pp. 143--154. DOI: https://doi.org/10.1016/S0010-4655(01)00203-X
[6] Miller R.E., Tadmor E.B. Hybrid continuum mechanics and atomistic methods for simulating materials deformation and failure. Mater. Res. Soc. Bull., 2007, vol. 32, no. 11, pp. 920--926. DOI: https://doi.org/10.1557/mrs2007.189
[7] Волегов П.С., Герасимов Р.М., Давлятшин Р.П. Модели молекулярной динамики: обзор EAM-потенциалов. Часть 2. Потенциалы для многокомпонентных систем. Вестник ПНИПУ. Механика, 2018, № 2, с. 114--132. DOI: https://doi.org/10.15593/perm.mech/2018.2.11
[8] Frenkel D., Smit B. Understanding molecular simulation. Academic Press, 2001.
[9] Ibragimov I.M., Kovshov A.N., Nazarov Yu.F. Osnovy kompyuternogo modelirovaniya nanosistem [Fundamentals of computer modeling of nanosystems]. St. Petersburg, Lan Publ., 2010.
[10] Eringen A.C. Nonlocal continuum field teories. Springer, 2002. 393 p.
[11] Zarubin V.S., Kuvyrkin G.N., Savel’eva I.Yu. Mathematical model of a nonlocal medium with internal state parameters. J. Eng. Phys. Thermophy., 2013, vol. 86, no. 4, pp. 820--826. DOI: https://doi.org/10.1007/s10891-013-0900-5
[12] Shaat M., Ghavanloob E., Fazelzadeh S.A. Review on nonlocal continuum mechanics: physics, material applicability, and mathematics. Mech. Mater., 2020, vol. 150, art. 103587. DOI: https://doi.org/10.1016/j.mechmat.2020.103587
[13] Polizzotto C. Nonlocal elasticity and related variational principles. Int. J. Solids Struct., 2001, vol. 38, iss. 42-43, pp. 7359--7380. DOI: https://doi.org/10.1016/S0020-7683(01)00039-7
[14] Kuvyrkin G.N., Savelieva I.Yu. Thermomechanical model of nonlocal deformation of a solid. Mech. Solids, 2016, vol. 51, no. 3, pp. 256--262. DOI: https://doi.org/10.3103/S002565441603002X
[15] Kuvyrkin G., Savelyeva I., Kuvshinnikova D. Temperature distribution in a composite rod, taking into account nonlocal spatial effects. E3S Web Conf., 2019, vol. 128, art. 09006. DOI: https://doi.org/10.1051/e3sconf/201912809006
[16] Kuvyrkin G.N., Savelyeva I.Yu. Numerical solution of integrodifferential heat conduction equation for a nonlocal medium. Math. Models Comput. Simul., 2014, vol. 6, no. 1, pp. 1--8. DOI: https://doi.org/10.1134/S2070048214010104
[17] Savelyeva I.Yu. Variational formulation of the mathematical model of stationary heat conduction with account for spatial nonlocality. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2022, no. 2 (101), pp. 68--86 (in Russ.). DOI: http://dx.doi.org/10.18698/1812-3368-2022-2-68-86
[18] Savelyeva I.Yu. Dual variational model of a steady-state thermal conductivity process taking into account spatial non-locality. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2022, no. 5 (104), pp. 45--61 (in Russ.). DOI: http://dx.doi.org/10.18698/1812-3368-2022-5-45-61
[19] Jolley K., Gill S.P.A. Modelling transient heat conduction in solids at multiple length and time scales: a coupled non-equilibrium molecular dynamics/continuum approach. J. Comput. Phys. Sci., 2009, vol. 228, iss. 19, pp. 7412--7425. DOI: https://doi.org/10.1016/j.jcp.2009.06.035
[20] Zarubin V.S., Kuvyrkin G.N., Savelyeva I.Yu. Fizicheskie i matematicheskie modeli mikromekhaniki [Physical and mathematical models of micromechanics]. Moscow, BMSTU Publ., 2021.