The proper formulation of constitutive laws involves more physical considerations such as frame indifference
in which it is required that the response of the system cannot depend on the manner in which the
Cartesian coordinate system for the spacial coordinates was chosen.

For Q

(t)

an orthogonal transformation, (see Problem 21 on Page 1082) and

′ ′
y = q (t)+ Q (t)y,n = Qn,

the new spacial coordinates are denoted by y^{′}. Recall an orthogonal transformation is just one which
satisfies

T T
Q (t) Q (t) = Q (t)Q(t) = I.

The stress has to do with the traction force area density produced by internal changes in the body and has
nothing to do with the way the body is observed. Therefore, it is required that

′ ′
T n = QT n

Thus

′
T Qn = QTn

Since this is true for any n normal to the boundary of any piece of the material considered, it must be the
case that

′
T Q = QT

and so

′ T
T = QT Q .

This is called frame indifference.

By (17.14), the Piola Kirchhoff stress S is related to T by

−T
S = det(F )TF ,F ≡ Dxy.

This stress involves the use of the material coordinates and a normal N to a piece of the body in reference
configuration. Thus SN gives the force on a part of ∂V_{t} per unit area on ∂V_{0}. Then for a different choice
of spacial coordinates, y^{′} = q

(t)

+ Q

(t)

y,

′ ′ ′ ′−T
S = det(F )T (F )

but

′ ′
F = Dxy = Q (t)Dxy = QF

and so frame indifference in terms of S is

′ T −T T −T
S = det(F)QT Q (QF ) = det(F )QT Q QF = QS

This principle of frame indifference is sometimes ignored and there are certainly interesting
mathematical models which have resulted from doing this, but such things cannot be considered physically
acceptable.

There are also many other physical properties which can be included, which require a certain form for
the constitutive equations. These considerations are outside the scope of this book and require a
considerable amount of linear algebra.

There are also balance laws for energy which you may study later but these are more problematic than
the balance laws for mass and momentum. However, the divergence theorem is used in these
also.