Page 211 - Introduction to Continuum Mechanics

196 Stress

The stress vector ^ , defined by the above equation is a pseudo-stress vector in that, being
based on the undeformed area, it does no describe the actual intensity of the force. We note
however, that t 0 has the same direction as the Cauchy stress vector L
The first Piola-Kirchhoff stress tensor (also known as the Lagrangian Stress tensor) is a
linear transformation T 0 such that

The relation between the first Piola-Kirchhoff stress tensor and the Cauchy stress tensor
can be obtained as follows:
Since

therefore

Using Eqs. (4.10.2) and (4.10.4), Eq. (ii) becomes

Using Eq. (3.28.6), i.e.,

we have,

The above equation is to be true for all n 0, therefore,

This is the desired relationship.
In Cartesian components, Eq. (4.10.6a) reads

From Eq. (4.10.6a), we obtain

which in Cartesian components, reads