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.
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,