Page 457 - Introduction to Continuum Mechanics

Principle of Linear Momentum 441

But

and

Therefore, Eq. (i) becomes

Since

therefore, we have

from which the following field equation of motion is obtained:

This is the same equation as Eq. (4.7.2).
We can also obtain the equation of motion in the reference state as follows:
Let p 0, dS 0, and dV 0 denote the density, surface area and volume respectively at the
reference time t 0 for the differential material having p, dS and dVat timef, then the conser-
vation of mass principle gives

and the definition of the stress vector !<, associated with the first Piola-Kirchhoff stress tensor
gives [see Section 4.10]

Now, using Eqs. (7.6.6) and (7.6.7), Equation (7.6.3) can be transformed to the reference
configuration. That is
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