Nonlocal Thermodynamics: Mathematical Model of Two-Dimensional Thermal Conductivity

Department of Applied Mathematics, Bauman Moscow State Technical University, 105005, 2nd Baumanskaya st 5, Moscow, Russian Federation

Abstract

Nonlocal models of thermodynamics are becoming more and more popular in the modern world. Such models make it possible to describe materials with a complex structure and unique strength and temperature properties. Models of nonlocal thermodynamics of a continuous medium establish a relationship between micro and macro characteristics of materials. A mathematical model of thermal conductivity in nonlocal media is considered. The model is based on the theory of nonlocal continuum by A.K. Eringen. The interaction of material particles is described using local and nonlocal terms in the law of heat conduction. The nonlocal term describes the effect of decreasing the influence of the surrounding elements of the material structure with increasing distance. The effect of nonlocal influence is described using the standard non-locality function based on the Gaussian distribution. The nonlocality function depends on the distance between the elements of the material structure. The mathematical model of heat conduction in a nonlocal medium consists of an integro-differential heat conduction equation with initial and boundary conditions. A numerical solution to the problem of heat conduction in a nonlocal plate is obtained. The numerical solution of a two-dimensional problem based on the finite element method is described. The influence of nonlocal effects and material parameters on the thermal conductivity in a plate under highintensity surface heating is analyzed. The importance of nonlocal characteristics in modelling the thermodynamic behaviour of materials with a complex structure is demonstrated.