Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 283

The tensor T is called the Cauchy stress tensor .
The key assertion of this famous theorem is the representability of the stress
vector t in terms of the tensor T . Using the Gauss integral theorem, it follows
from (1.8) that

f(x)dx + T (x)nds = [ f(x) + div T(x) ] dx = 0 .

V ∂V V

This relation also implies the differential equation (1.11). The equilibrium equa-
tions (1.9) for the moments imply the symmetry (1.12).

The Piola Transform

We have formulated the equilibrium equations in terms of the coordinates of the
deformed body B (as did Euler). Since these coordinates have to be computed in the
first place, it is useful to transform the variables to the reference configuration. To
distinguish the expressions, in the following we add a subscript R when referring
to the reference configuration. In particular, x = φ(x R ).
The transformation of the body forces follows directly from the well-known
transformation theorem for integrals, where the volume element is given by dx =
det ( φ)dx R . The forces are proportional to density. Densities are transformed ac-

cording to conservation of mass: ρ(x)dx = ρ R (x R )dx R which implies ρ(φ(x R )) =
det ( ∇ φ − 1 )ρ R (x R ) . Consequently,

∇

det ( ∇ φ − 1 )f R (x R ). f(x) = ( 1 . 13 )

The equation (1.13) makes implicit use of the assumption that under the deforma-
tion, point masses do not move to positions where we have a different force field.
In this case we speak of a dead load .
The transformation of stress tensors is more complicated, but can be computed
by elementary methods; cf. Ciarlet [1988]. In terms of the reference configuration,
we have

div R T R + f R = 0 ( 1 . 14 )

with

det ( ∇ φ) T ( ∇ φ) − T . T R : = ( 1 . 15 )

Equation (1.14) is the analog of (1.11). However, in contrast to T , the so-called first
Piola-Kirchhoff stress tensor T R in (1.15) is not symmetric. To achieve symmetry,
we introduce the second Piola-Kirchhoff stress tensor

det ( ∇ φ) ( ∇ φ) − 1 T( ∇ φ) − T .

R : = ( 1 . 16 )

Clearly, R = ( ∇ φ) − 1 T R .
The differences between the three stress tensors can be neglected for small
deformation gradients.