Nonlinear finite elements/Balance of linear momentum

From Wikiversity
Jump to: navigation, search


Statement of the balance of linear momentum[edit]

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.

Proof[edit]

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,