Variational Formulation of the Mathematical Model of Stationary Heat Conduction with Account for Spatial Nonlocality

Abstract

In creating and using new structurally sensitive materials, it is essential to construct mathematical models which make it possible to describe the materials’ behavior in widely changing external influences. Modeling of non-local media is a class of methods of generalized continuum mechanics. The capability of analyzing mathematical models of a continuous medium can be extended through variational methods. The paper describes the construction of a functional for the problem of stationary heat conduction in a homogeneous body, taking into account the effects of nonlocality and with a temperature-independent heat conduction coefficient. A variational formulation used in the case of a linear problem in combination with acceptable admissible functions allowed us to quantify this effect. The study gives an example of using the variational formulation, which takes into account the influence of the spatial nonlocality of the stationary heat conduction process on the parameters that determine the well-known phenomenon of thermal explosion with an exponential temperature dependence of the volumetric heat release rate in the plate. The quantitative analysis was carried out for two versions of the function that is admissible for the constructed integral functional and describes the possible temperature distribution over the thickness of the plate under consideration. Having compared the obtained results, we opted for the admissible function, as its use leads to a smaller difference between the maximum value of the parameter characterizing the intensity of heat release in the plate and the known result found with no account for the influence of spatial nonlocality