226
5 Contraction
where
I 1 = λ
2
r + λ
2
θ + λ
2
z
in terms of the total stretch ratios λ i relative to cylindrical coordinates (r, θ, z). The
active strain-energy function is given by Eq. (5.21), with λ ∗ = λ ∗
z and c a given
by (5.22).
Ignoring velocity effects, determine the total Cauchy stress σ z as a function of
λ z and the contraction ratio K z = K. Plot σ z versus λ z for selected values of K
between K min = 0.5 and 1.
Solution
Combined extension and torsion of a passive, incompressible bar was discussed in
Sect. 4.5. Much of that analysis remains valid for the present problem. For extension
only (ψ = 0), Eqs. (4.72) and (4.75) give
F = λ r e r e r + λ θ e θ e θ + λ z e z e z ,
(5.29)
where
λ r = λ θ = λ
−1/2
z
.
(5.30)
The total Cauchy stress tensor is
σ = σ r e r e r + σ θ e θ e θ + σ z e z e z ,
(5.31)
and the equilibrium equation (4.78) and boundary condition (4.84) are the same.
Now, we bring contraction into the analysis. Since contraction occurs only in the
z-direction, extending Eq. (5.3) to 3D gives
λ r = K r λ
∗
r = λ
∗
r
λ θ = K θ λ
∗
θ = λ
∗
θ
λ z = K z λ
∗
z = K(t)λ
∗
z ,
(5.32)
since K r = K θ = 1 in the passive directions (r and θ ). Next, Eqs. (5.11), (5.21),
and (5.28) yield
( ¯
σ r ) p = λ r
∂W p
∂λ r
= 2c p λ
2
r e
β
λ 2
r +λ 2
θ +λ 2
z −3
( ¯
σ θ ) p = λ θ
∂W p
∂λ θ
= 2c p λ
2
θ e
β
λ 2
r +λ 2
θ +λ 2
z −3
( ¯
σ z ) p = λ z
∂W p
∂λ z
= 2c p λ
2
z e
β
λ 2
r +λ 2
θ +λ 2
z −3