Kinematics and Balance Relations for Bidimensional Continua

Abstract

This work is concerned with the formulation of the kinematics and balance relations for a so-called bidimensional continuum, which can be used in modeling thin layers and interface regions such as phase boundaries. Such a continuum represents a thin, shell-like 3-dimensional region in which the upper and lower surfaces move relative to each other as well as relative to the dividing, non-material interface between them. As such, it is more general than standard interfaces or shells. The standard balance relations of three-dimensional continua are adapted to this dynamic bidimensional geometry using the differential geometric notion of a flow. On this basis, the adapted balance relations are averaged over the dynamic thickness of the bidimensional continuum to obtain reduced 2-dimensional, surface forms of these on the dividing interface. In addition to the usual influence of the surface geometry on their form, the resulting adapted and averaged surface balance relations contain flux terms accounting for the effect of relative motion, i.e., diffusion, on the balances. In the limit that the thickness of the bidimensional continuum goes to zero, the generalized surface balance relations reduce to the classical jump balance relations across an interface.