CONTINUUM MECHANICS for ENGINEERS

where dependence on I 3 has been included for subsequent algebra. Using
the Cayley-Hamilton theorem in the form
results in
(8.2-23)
(8.2-24)
∂
Incompressibility requires I3 = 1 and
∂
. Furthermore, since the
pressure term of the internal constraint stress is arbitrary, the first term of
Eq 8.2-24 may be combined with the pressure term. The result is the following
expression for the Cauchy stress
=
W
I
0
3
or
ˆσ = γ δ + γ B + γ B B
ij o ij 1 ij 2 ik jk
( ) + ( − )
−1
= γ IB + γ −γ
I δ γ γ I B
2 3 ij o 2 2 ij 1 2 1 ij
ˆσ
W
I I δ
I
W W W
B I
I I I B
⎛ ∂ ∂ ⎞
= ⎜ −
⎝ ∂ ∂
⎟ +
⎠
∂ ∂
3 2
+ 3
∂ ∂
ij ij ij
−1
ij
3
2 1
2
W
σ pδ
I B
W
I B
=− +∂ +
∂
∂
ij ij
1
ij
∂ 2
−1
ij
W
σ pδ
I B
W
I c
=− +∂ +
∂
∂
∂
ij ij ij ij
1 2
(8.2-25)
(8.2.26)
where c ij is defined in Eq 4.6-16.
At this point, the strain energy has been assumed a function of I 1 and I 2,
but the exact functional form has not been specified. Rivlin (1948) postulated
the strain energy should be represented as a general polynomial in I 1 and I 2
∑ αβ
( ) ( − )
W = C I −3
I 3
1 2
α β
(8.2-27)
It is noted that the strain energy is written in terms of I 1 –3 and I 2 – 3 rather
than I 1 and I 2 to ensure zero strain corresponds to zero strain energy.
Depending on the type of material and deformation, that is, experimental
test data, the number of terms used in Eq 8.2-27 is chosen. For instance,
choosing C 10 = G and all other coefficients zero results in a neo-Hookean
response where G is the shear modulus. Stresses are evaluated with Eq 8.2-26
and used in the equations of motion, Eq 5.4-4. This results in a set of differential
equations solved with the use of the problem’s appropriate boundary