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.
Highlights
The classical theory of continuum is not suitable for the mechanical analysis of micro- and nanostructures, since it is scaleless and does not have parameters for the scale of structural elements
The red line denotes the contour plot of the temperature field when solving the problem of classical parabolic heat conduction, without taking into account the contribution of nonlocality
We see that with an increase in the parameter of the non-local material a, the solution to the non-local problem becomes similar to the classical one. This is due to the fact that a larger number of material structures fall into the zone of influence of nonlocality
Summary
The classical theory of continuum is not suitable for the mechanical analysis of micro- and nanostructures, since it is scaleless and does not have parameters for the scale of structural elements. Ehrengen's theory uses a nonlocality influence function (decaying with distance), which includes a length scale parameter. This approach allows one to obtain intego-differential nonlocal constitutive equations that take into account interatomic interactions. The non-local approach is based on the idea that the heat flux at a control point is a function of the macroscopic field of the temperature gradient at all points of the body. The law of heat propagation in a medium, taking into account the effect of spatial nonlocality: c p1 This mathematical model allows one to take into account two existing opposing concepts of describing the structure of any rigid body - the concept of continuity and discreteness. Equations (3) – (4) with boundary conditions define an integro-differential formulation of nonlocal heat conduction in a solid
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