Theory of elasticity with spatial dispersion one-dimensional complex structure

Abstract

An elastic medium of simple structure with spatial dispersion was considered in [1]. In the present paper we construct a more general model of a macroscopically homogeneous medium of complex structure which cannot be described adequately in terms of a single kinematic variable. As our initial micromodel we have chosen the familiar model of a complex chain each of whose unit cells has N degrees of freedom [2 and 3]. In sec. 1 this model is generalized for the case of a continuous mass distribution. New kinematic variables, namely the displacement of the centers of mass of the cells and microdeformations of various orders, are introduced. Appropriately, force micromoments are introduced as the force variables. The algorithm introduced in [1] is used to effect a transition to equations of an elastic medium with spatial dispersion. The corresponding operations are expressed explicitly in terms of the initial micro parameters. With the phenomenological approach, the resulting equations describe the most general one-dimensional model of a homogeneous linearly elastic medium of complex structure with spatial dispersion. We also consider the general equations of a nonhomogeneous medium of periodic structure with a single kinematic variable. It is established that under certain conditions both of these models can be considered as representations of one and the same physical model, but with different areas of applicability. Formulas for converting from one representation to the other are presented. Section 2 contains a discussion of certain specific models and a derivation of the sufficient conditions under which the equations of a medium of complex structure admit of exact transformation into the equations of a medium of simple structure. It is shown that in the case of weak dispersion, the equations of a medium of complex structure with certain additional limitations coincide with the one-dimensional equations of the couple stress theory of elasticity [4 to 6]. It is difficult to justify these limitations physically, however. The acoustic frequency range is of primary interest in elasticity theory. In sec. 3 it is shown that in the acoustic range it is always possible to transform the system of equations describing a complex structure into an equation with a single kinematic variable (the displacement of the centers of mass of the cells) and into equations explicitly solved for the remaining variables. This involves both spatial and temporal dispersion, although the latter is not related to energy dissipation. It is important to note that the operators which appear in the equation involving the centers of mass are directly related to macroexperiment. Specifically, in the zeroth approximation there occurs a transition to the ordinary equation of elasticity theory with an elasticity constant which is determined by experiment. Because of this, it is our view that this representation is more adequate to the macroscopic description of a medium of complex structure than the representations considered in sec. 1.