CONTINUUM MECHANICS for ENGINEERS

Note that if principal axes of B ij are chosen the invariants of Eq 8.2-2 are
written in terms of the stretch ratios λ 1, λ 2, and λ 3 as follows:
(8.2-3)
Many rubber or elastomeric materials have a mechanical response that is
often nearly incompressible. Even though a perfectly incompressible material
is not possible, a variety of problems can be solved by assuming incompressibility.
The incompressible response of the material can be thought of as
a constraint on the deformation gradient. That is, the incompressible nature
of the material is modeled by an addition functional dependence between
the deformation gradient components. For an incompressible material the
density remains constant. This was expressed in Section 5.3 by v i,i = 0
(Eq 5.3-8) or ρJ = ρ o (Eq 5.3-10a). With the density remaining constant and
Eq 5.3-10a as the expression of the continuity equation it is clear that
(8.2-4)
For the time, regress from specific incompressibility conditions to a more
general case of a continuum with an internal constraint. Assume a general
constraint of the form
or, since C AB = F iAF iB,
I
2
1 1
I
2 2
2 1 2
I
= λ + λ + λ
2
2
2 2 2
3 1 2 3
(8.2-5)
(8.2-6)
This second form of the internal constraint has the advantage that it is invariant
under superposed rigid body motions. Differentiation of φ results in
(8.2-7)
where it is understood partial differentiation with respect to a symmetric
tensor results in a symmetric tensor. That is,
2
3
= λλ + λλ + λλ
= λλλ
2 2
2 3
J = det{ FiA}= 1
φ F ( iA)=
0
˜ φ( CAB)= 0
2 2
1 3
( iA iB iA iB)
∂
=
∂
∂ φ φ
+
C ∂
C˙
F˙ AB F F F˙
C
AB
AB
∂ ∂
≡ +
∂ ∂
∂
φ 1 ⎛ φ φ ⎞
C 2
⎜
⎝ C ∂C
⎟
⎠
AB AB BA
(8.2-8)