Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 285

1.5 Axiom of Material Frame-Indifference. The Cauchy stress vector t(x,n) =
T(x)n is independent of the choice of coordinates, i.e., Qt (x, n) = t (Qx, Qn)
for all Q ∈ O

3
+

.

A frame-indifferent material is also called objective .

1.6 Theorem. Suppose the axiom of material frame-indifference holds. Then for
every orthogonal transformation Q ∈ O

3
+

,

T (QF ) = Q T(F)Q T .

( 1 . 19 )

Moreover, there exists a mapping ˜

3
>

3

: S −→ S such that

ˆ

(F) = ˜

(F T F),

( 1 . 20 )

i.e., ˆ depends only on F T F .
Proof. Instead of rotating the coordinate system, we rotate the deformed body:

x −→ Qx,
φ −→ Qφ,
∇ φ −→ Q ∇ φ,
n Q − T n

−→ = Qn,

t(x,n) −→ Qt (x, n).

By Axiom 1.5, t (Qx, Qn) = Qt (x, n) , and thus T (QF )Q · n = Q T(F) · n .
Replacing Qn by n and using Q T Q = I , we get (1.19).

It follows from (1.18) and (1.19) after some elementary manipulations that

ˆ

(QF ) = ˆ

3
+

(F) for Q ∈ O . ( 1 . 21 )

3
+

M

To prove (1.20), we consider the two nonsingular matrices F and G in with

F T F = G T G. Set Q : = FG − 1 . Then Q T Q = I and det (Q) > 0. Now (1.21)

implies ˆ = ˆ (G) , and so in fact ˆ

depends only on the product F T F .

(F)

The axiom of frame-indifference holds for all materials. On the other hand,
isotropy is purely a material property, which means that no direction in the material
is preferred. Layered materials such as wood or crystal are not isotropic. Isotropy
implies that the stress vectors do not change if we rotate the nondeformed body,
i.e., before the deformation takes place.