Nonlinear finite elements/Balance of angular momentum

Statement of the balance of angular momentum edit

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


Proof edit

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




Expressed in direct tensor notation,


where   is the third-order permutation tensor. Therefore,




The balance of angular momentum can then be written as


Since   is an arbitrary volume, we have




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






The material time derivative of   is defined as




Also, from the conservation of linear momentum




The material time derivative of   is defined as




From the balance of mass




In index notation,


Expanding out, we get