Nonlinear finite elements/Balance of mass

Statement of the balance of mass edit

The balance of mass can be expressed as:


where   is the current mass density,   is the material time derivative of  , and   is the velocity of physical particles in the body   bounded by the surface  .

Proof edit

We can show how this relation is derived by recalling that the general equation for the balance of a physical quantity   is given by


To derive the equation for the balance of mass, we assume that the physical quantity of interest is the mass density  . Since mass is neither created or destroyed, the surface and interior sources are zero, i.e.,  . Therefore, we have


Let us assume that the volume   is a control volume (i.e., it does not change with time). Then the surface   has a zero velocity ( ) and we get


Using the divergence theorem


we get




Since   is arbitrary, we must have


Using the identity


we have


Now, the material time derivative of   is defined as