Mathematical model of a nonlocal medium with internal state parameters

Abstract

Modern structural and functional materials consisting of a combination of micro- and nanostructural elements are often called structure-sensitive materials. The methodology of continuum in its pure form is inapplicable to such materials. Nevertheless, as is often the case, the methods and mathematical models of the classical mechanics of a continuous medium can be applied at the nano- and microlevel. Such a technique of extending the concepts of the classical mechanics of a continuous medium to a medium with micro- and nanostructure is called the method of continuous approximation [1]. The fi eld of research in which the behavior of materials with micro- and nanostructure is studied with the use of the method of continuous approximation is sometimes referred to as the generalized mechanics of a continuous medium. The method of continuous approximation employs such a notion as nonlocality [1]. An important stage in the creation and use of the considered class of materials is the construction of mathematical models allowing one to describe the behavior of these materials in a wide range of variation of external effects. However, the general methodology of constructing such models is still far from complete. In the present work we suggest a thermomechanical model of materials with a fi ne structure that accounts for the temporal effects in accumulation and propagation of heat, as well as the effects of spatial nonlocality, and propose the generalization and application of the earlier obtained results on construction of mathematical models of a continuous medium with internal state parameters [2–8] to a nonlocal medium. The governing equations of a nonlocal medium with internal state parameters can be obtained using the energy conservation law [5–7]: