Nonlinear finite elements/Balance of angular momentum

Statement of the balance of angular momentumEdit

The balance of angular momentum in an inertial frame can be expressed as:

 

ProofEdit

We assume that there are no surface couples on   or body couples in  . Recall the general balance equation

 

In this case, the physical quantity to be conserved the angular momentum density, i.e.,  . The angular momentum source at the surface is then   and the angular momentum source inside the body is  . The angular momentum and moments are calculated with respect to a fixed origin. Hence we have

 

Assuming that   is a control volume, we have

 

Using the definition of a tensor product we can write

 

Also,  . Therefore we have

 

Using the divergence theorem, we get

 

To convert the surface integral in the above equation into a volume integral, it is convenient to use index notation. Thus,

 

where   represents the  -th component of the vector. Using the divergence theorem

 

Differentiating,

 

Expressed in direct tensor notation,

 

where   is the third-order permutation tensor. Therefore,

 

or,

 

The balance of angular momentum can then be written as

 

Since   is an arbitrary volume, we have

 

or,

 

Using the identity,

 

we get

 

The second term on the right can be further simplified using index notation as follows.

 

Therefore we can write

 

The balance of angular momentum then takes the form

 

or,

 

or,

 

The material time derivative of   is defined as

 

Therefore,

 

Also, from the conservation of linear momentum

 

Hence,

 

The material time derivative of   is defined as

 

Hence,

 

From the balance of mass

 

Therefore,

 

In index notation,

 

Expanding out, we get

 

Hence,