Page 210 - Introduction to Continuum Mechanics

Piola Kirchhoff Stress Tensors 195

(c)The boundary conditions are

Thus,

From Eqs. (viii) and (ix), we obtain

Thus,

4.10 Piola Kirchhoff Stress Tensors
Let dA 0 be the differential material area with unit normal n 0 at the reference time t 0 and
dA that at the current time t of the same material area with unit normal n. We may refer to
dA 0 as the undeformed area and dA as the deformed area. Let df be the force acting on the
deformed area dAn. In Section 4.1, we defined the Cauchy stress vector t and the associated
Cauchy stress tensor T based on the deformed area dAn, that is

and

In this section, we define two other pairs of (pseudo) stress vectors and tensors, based on the
undeformed area.
(A) The First Piola-Kirchhoff Stress Tensor
Let