Nonlinear finite elements/Balance of mass

Statement of the balance of massEdit

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  .

ProofEdit

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

 

or,

 

Since   is arbitrary, we must have

 

Using the identity

 

we have

 

Now, the material time derivative of   is defined as

 

Therefore,