Continuum mechanics/Balance of angular momentum

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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

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,