Page 212 - Introduction to Continuum Mechanics

Piola Kirchhoff Stress Tensors 197

We note that when Cartesian coordinates are used for both the reference and the current

We also note that the first Piola-Kirchhoff stress tensor is in general not symmetric.
(B) The Second Piola-Kirchhoff Stress Tensor
Let

where

In Eq. (4.10.8b), df is the (pseudo) differential force which transforms, under the deformation
gradient F into the (actual) differencial force df at the deformed position (one njay compare
the transformation equation df = Fdf with d\ - ¥dX); thus, the pseudo vector t is in general
in a different direction than that of the Cauchy stress vector t
The second Piola-Kirchhoff stress tensor is a linear transformation T such that

where we recall n 0 is the normal to the undeformed area. From Eqs. (4.10.8a) (4.10.8b) and
(4.10.9). we have

We also have (see Eqs. (4.10.3) and (4.10.4)

Comparing Eqs. (i) and (ii), we have

Equation (4.10.10) gives the relationship between the first Piola-Kirchhoff stress tensor TQ and
the second Piola-Kirchhoff stress tensor T. Now, from Eqs. (4.10.6a) and (4.10.10), one easily
obtain the relationship between the second Piola-Kirchhoff stress tensor and the Cauchy stress
tensor T as

We note that the second Piola-Kirchhoff stress tensor is always a symmetric tensor if the
Cauchy stress tensor is a symmetric one.