Nonlinear finite elements/Balance of energy

Statement of the balance of energyEdit

The balance of energy can be expressed as:

 

where   is the mass density,   is the internal energy per unit mass,   is the Cauchy stress,   is the particle velocity,   is the heat flux vector, and   is the rate at which energy is generated by sources inside the volume (per unit mass).

ProofEdit

Recall the general balance equation

 

In this case, the physical quantity to be conserved the total energy density which is the sum of the internal energy density and the kinetic energy density, i.e.,  . The energy source at the surface is a sum of the rate of work done by the applied tractions and the rate of heat leaving the volume (per unit area), i.e,   where   is the outward unit normal to the surface. The energy source inside the body is the sum of the rate of work done by the body forces and the rate of energy generated by internal sources, i.e.,  .

Hence we have

 

Let   be a control volume that does not change with time. Then we get

 

Using the relation  , the identity  , and invoking the symmetry of the stress tensor, we get

 

We now apply the divergence theorem to the surface integrals to get

 

Since   is arbitrary, we have

 

Expanding out the left hand side, we have

 

For the first term on the right hand side, we use the identity   to get

 

For the second term on the right we use the identity   and the symmetry of the Cauchy stress tensor to get

 

After collecting terms and rearranging, we get

 

Applying the balance of mass to the first term and the balance of linear momentum to the second term, and using the material time derivative of the internal energy

 

we get the final form of the balance of energy: