Page 220 - Introduction to Continuum Mechanics

Stress Power 205

where £> {y are the components of the rate of deformation tensor defined in Section 3.13.
Equation (4.12.8a) can be written in the invariant form

Show that the stress power can be expressed in terms of the first Piola-Kirchhoff stress
tensor T 0 and the deformation gradient F as the following

Solution. In Sect. 3.12, we obtained [see Eq. (3.12.4)],

Since dx - F dX [ see Eq. (3.7.2)], therefore

Equation (i) is to be true for all dX, thus

or,

Using Eqs. (4.12.7) and (4.12.10b), the stress power can be written

Since [see Eq. (4.10.7)]

therefore,