The method of virtual power in continuum mechanics application to media presenting singular surfaces and interfaces

Summary

The work develops the principle of virtual power for finite velocity fields for so-called simple materials (or first-gradient theory) without further constitutive assumptions when the body is swept out by a singular surface which is either afree singular surface (such as usual strong discontinuities of continuum mechanics) or athermodynamical singular surface (a so-called interface between phases). The formulation given on exemplary cases first shows how to systematically construct the new “internal” contact forces which exist at the discontinuity, as well as the new inertial contributions which arise from mass transfer across the singular surface and the acceleration of particles attached to it. Then it is shown how various virtual velocity fields generate all the dynamical field equations as well as transversality conditions when the description of external forces allows for them. The principle of virtual power here is so formulated that, when combined, forreal velocity fields, with the first principle of thermodynamics in global form, it yields directly the socalled energy theorem both in the bulk and at the singular surface. Then the corresponding rates of entropy production are deduced after introduction of the second principle of thermodynamics. While one does not claim to obtain here essentially new equations, the present formulation of the principle of virtual power paves the way for useful complex extensions which are difficult to deal with through other avenues (e.g., electromagnetic continua with “junctions” such as piezoelectric semiconductors).