Dual Variational Model of the Nonlinear Heat Conduction Problem with Consideration of Spatial Nonlocality

The microcontinuum theories have great potential for modeling structure-sensitive materials. Research into the possibilities of using nonlocal thermomechanics in modeling nanodevices and nanoelectromechanical systems, media with a complex internal micro- and nanostructure is of great interest today. Analysis of such models is associated, as a rule, with overcoming certain difficulties caused by the necessity of numerical solution of integro-differential equations. The possibilities of analyzing mathematical models of a continuous medium can be expanded through the use of variational methods. This paper describes the construction of relations for the dual variational model of a stationary nonlinear heat conduction problem with consideration for the nonlocality effects. It is shown that the conditions of stationarity of the alternative functional coincide with known similar conditions in the absence of nonlocality. A quantitative analysis was carried out using the example of a problem about a plate that is infinite in its plane with active internal heat-release sources. Such problems are typical, for example, of the phenomena of a thermal explosion when exothermic chemical reactions occur in the body material or of thermal breakdown of a dielectric layer under the influence of electrical potentials. The dual variational formulation of the problem allows one not only to obtain an approximate solution to the problem under consideration, but also to estimate the error of this solution, and, if necessary, to reduce this error by selecting approximating functions.