Двойственная вариационная модель стационарного процесса теплопроводности, учитывающая пространственную нелокальность

Abstract

Microcontinuum theories boast a great potential for simulating structurally sensitive materials. There exists a sufficiently large number of works delineating the basics of non-local mechanics using the theory of elasticity as an example. Estimating the investigative capacity of non-local mechanics is at present particularly relevant to simulating nanodevices, nanoelectromechanical systems (NEMS), and media featuring complex internal micro- and nanostructures. Typically, analysing these simulations involves overcoming certain difficulties caused by the necessity to solve integro-differential equations numerically. Variational methods may be successfully applied to analysing mathematical models of continuous media as an additional tool. The paper describes plotting an alternative functional for the problem of steady-state thermal conductivity in a homogeneous body, taking into account non-locality effects and featuring a temperature-independent thermal conductivity coefficient. We show that the stationary conditions for this functional do not differ from those in the absence of non-locality. The alternative functional combined with the fundamental functional presented previously constitute a dual variational model. We quantitatively analyse the problem of an infinite planar plate featuring constantly active internal heat sources. The dual variational formulation of the problem allows us not only to obtain an approximate solution to the problem under consideration, but also to estimate its error, as well as to reduce this error by selecting other approximating functions if necessary