On rational treatments of the general laws of balance and jump, with emphasis on configurational formulations

Abstract

The present contribution intends to provide a rational framework for manipulating the general differential laws of balance and jump of continuum mechanics. At first, multiplications of the latter laws by proper physical fields are introduced, together with identical expansions. We then point out that the results of these mathematical manipulations, while being of the form of the general differential balance law again, are characterized by a possible lack of integrability with respect to volume production terms in the presence of a singular surface, the production terms being necessary for consistency. Lack of integrability is not to be expected when the general law is specified to the fundamental laws of balance of mass, momentum, total energy and entropy. When it must be supposed that integrability could be missing, however, the integral form of the general law of balance cannot be written down for a region containing a singular surface, and the so-called pillbox procedure fails for deriving the general relation of jump at the singular surface. The specific relations of balance and jump then have to be derived from proper relations that follow from consistent mathematical manipulations of the general differential law of balance and the general law of jump. In the present paper, these general relations are listed explicitly for balance of scalar and vectorial entities, multiplied and expanded by scalar, vector or tensor fields, respectively. Based on this rationale, various important formulations on configurational forces known from the literature on the mechanics in material space are recalled: The specific relation of balance of pseudomomentum, the configurational forces and the driving force at an interface are derived from the above general relations by involving the fundamental laws of balance and jump of linear momentum. The driving traction at an interface is then set into connection with the derived relations of balance and jump of kinetic, internal and free energy, respectively. The latter relations follow from the general relations by involving the fundamental laws of balance of linear momentum, total energy and entropy. As a result, we present relations between the driving traction at an interface and the surface production terms that appear in the various relations of jump, which should allow new interpretations of configurational formulations.