Nonlinear finite elements/Balance of linear momentum

Statement of the balance of linear momentumEdit

The balance of linear momentum can be expressed as:

  

where   is the mass density,   is the velocity,   is the Cauchy stress, and   is the body force density.

ProofEdit

Recall the general equation for the balance of a physical quantity

 

In this case the physical quantity of interest is the momentum density, i.e.,  . The source of momentum flux at the surface is the surface traction, i.e.,  . The source of momentum inside the body is the body force, i.e.,  . Therefore, we have

 

The surface tractions are related to the Cauchy stress by

 

Therefore,

 

Let us assume that   is an arbitrary fixed control volume. Then,

 

Now, from the definition of the tensor product we have (for all vectors  )

 

Therefore,

 

Using the divergence theorem

 

we have

 

or,

 

Since   is arbitrary, we have

 

Using the identity

 

we get

 

or,

 

Using the identity

 

we get

 

From the definition

 

we have

 

Hence,

 

or,

 

The material time derivative of   is defined as

 

Therefore,

 

From the balance of mass, we have

 

Therefore,

 

The material time derivative of   is defined as

 

Hence,