Incomplete thoughts on mass flux and superposed RBM

Attached are some (hand-written) observations on wanting to do continuum mechanics when mass is not conserved for fixed sets of particles of the body (so, situations transcending the rocket-losing-mass type). I feel (un)comfortable with these observations, depending upon the day I think about such things.

The primary reason to even consider this is I would like to explore if any physically realistic model can be made in dealing with damage and fracture without assuming that the topology of the mathematical body in 3-d Euclidean space is changed during a motion representative of damage evolution and fracture - of course, in 'reality' the topology changes (or so it seems), but then there are serious difficulties in setting up a *dynamical* model. I am interested in understanding the kinematical apsects of this question clearly as well as the dynamical ones (in this regard, I should mention that I have a reasonable understanding of what gets done in peridynamics and would still like to consider alternatives). 

Perhaps someone on imechanica will straighten me out on these questions.