Page 214 - Introduction to Continuum Mechanics

Piola Kirchhoff Stress Tensors 199

Example 4.10.2

The equilibrium configuration of a body is described by

If the Cauchy stress tensor for this body is

(a) What is the corresponding first Piola-Kirchhoff stress tensor.
(b) What is the corresponding second Piola-Kirchhoff stress tensor and
(c) calculate the pseudo stress vector associated with the first Piola-Kirchhoff stress tensor on
the 63 - plane in the deformed state.
(d) calculate the pseudo-stress vector associated with the second Piola-Kirchhoff stress tensor
on the 63 - plane in the deformed state.
Solution. From Eqs. (i), we have

Thus, from Eq. (4.10.6), we have, the first Piola-Kirchhoff stress tensor:

The second Piola-Kirchhoff stress tensor is, from Eq. (4.10.11)