3.4 Analysis of Stress
101
base vectors, i.e., across any three mutually orthogonal planes passing through
the point, then the stress vector and associated stress components acting on any
other plane (with unit normal n) passing through the point can be computed using
Eq. (3.112).
Similarly, we define the first Piola-Kirchhoff stress tensor 6 P and the second
Piola-Kirchhoff stress tensor S using relations analogous to Eq. (3.112), i.e.,
T
0
= N · P
˜
T = N · S,
(3.115)
where N is the unit normal to a plane in the undeformed body, and the pseudo-stress
vectors T 0 and ˜
T are defined by Eqs. (3.101) and (3.103).
Please Note In this book, T without a subscript is the Cauchy stress vector acting
on a plane with arbitrary orientation defined by n in the deformed body, whereas
T 0 and ˜
T without subscripts are Piola-Kirchhoff stress vectors on an arbitrary plane
with unit normal N in the undeformed configuration.
All three stress vectors, and thus all three stress tensors, are related to the
same force vector df. Since Eq. (3.102) gives df = F · d ˜ f = d ˜ f · F T , combining
Eqs. (3.100)–(3.103) yields
df = T dA = T
0 dA
0
= ˜
T · F
T dA
0 ,
(3.116)
and substituting Eqs. (3.112) and (3.115) produces
df = n · σ dA = N · P dA
0
= N · S · F
T dA
0 .
(3.117)
With the differential area vectors given by dA 0 = N dA 0 and dA = n dA, this
equation becomes
df = dA · σ = dA
0
· P = dA
0
· S · F
T .
(3.118)
Substituting dA = J dA 0 · F −1 from (3.87) and effectively canceling out dA 0 in
Eq. (3.118) give
J F
−1
· σ = P = S · F
T ,
which yields
σ = J −1 F · P = J −1 F · S · F T .
(3.119)
6 Some authors define P T as the first Piola-Kirchhoff stress tensor.