Continual-discrete modelling of a multicomponent laminar body by using a system of two-dimensional continua

Abstract

A method is considered for constructing continual-discrete models of multicomponent layered bodies by using a system consisting of an arbitrary number of two-dimensional continua with finite intervals between them. Consistency relationships are presented for the fundamental kinematic, deformation, and dynamic parameters which enable rheological relationships to be obtained for the body as a whole taking the properties and nature of the interaction of the individual components into account. An example of the modelling of a thin laminar elastic body is examined. Methods for modelling a biological membrane are discussed. Physical objects exist for which a direct description is impossible by methods of the mechanics of three-dimensional continuous media, or is insufficiently effective because the physical properties of the object are discrete in one of the directions, i.e., the requirements for the continuity hypothesis /1/ are not satisfied in this direction. The object here posssesses fairly continuous properties in the other two directions and allows of a continual description. Among tho discrete objects in the transverse direction is the shell of a live cell, a biological membrane, say, consisting of several layers of macromolecules where the individual layers include molecules of different species. Moreover, a broad class of laminar and stratified bodies exists, whose properties in the transverse direction can possibly be described by a discrete set of parameters. In a number of papers (/2–7/, for example) the concept has been introduced of a two-dimensional continuum (a material surface possessing mass) that is characterized by appropriate kinematic, dynamic, and energy parameters. The ideal of modelling multicomponent laminar bodies by using systems of two-dimensional continua /8/ is natural ∗ ∗ See also: Pribyleva, T.A. Investigations in Biomechanics. Model of a multisheet continuum: kinematics and mass balance. Report 2555, Moscow Univ. Mechanics.Inst., 1981. . Certain elements of this approach are elucidated in /9, 10/.