Page 217 - Introduction to Continuum Mechanics

202 Stress

By using the following identity [see Prob. 4.40] for any tensor A^^^s)

we obtain,

so that

Thus,

Substituting Eq. (vii) in the Cauchy's Equation of motion [ Eq.(4.7.2b)], we get

Since dV = (detF>/K 0 [See Eq. (3.29.3)], therefore,

where p 0is the initial density. Thus, we have, in terms of the first Piola-Kirchhoff stress tensor
and with respect to the material coordinates, the equations of motion take the following form

whereas in terms of the Cauchy stress tensor and with respect to the spatial coordinates, the
equations of motion take the form

In invariant notation, Eq. (4.11.4) reads