184
4 Problems in Soft Tissue Biomechanics
Stress and Equilibrium Because the only nonzero shear strain is E θz = E zθ , the
Cauchy stress tensor is expected to have the form
σ = σ rr e r e r + σ θθ e θ e θ + σ zz e z e z + σ θz e θ e z + σ zθ e z e θ .
(4.77)
By symmetry, all stress components depend only on r, and expanding the equilibrium equation ∇ · σ = 0 involves the shear terms
e r
∂
∂r
+ e θ
1
r
∂
∂θ
+ e z
∂
∂z
· [σ θz (r) e θ (θ )e z + σ zθ (r) e z e θ (θ )]
in addition to those in Eq. (4.57). Fortunately, these terms turn out to be zero. Thus,
with or without torsion, equilibrium is governed by Eq. (4.58), i.e.,
∂σ rr
∂r
+
σ rr − σ θθ
r
= 0.
(4.78)
Constitutive Relations For an incompressible material, Eq. (3.239) 1 gives the
constitutive relation
σ = ¯
σ − p I =
⎡
⎣
¯
σ rr − p
0
0
0
¯
σ θθ − p ¯
σ θz
0
¯
σ zθ
¯
σ zz − p
⎤
⎦
(e i e j )
,
(4.79)
where (i, j ) = (r, θ, z) and
¯
σ = F ·
∂W
∂E
· F
T .
(4.80)
To obtain specific expressions for the stress components, we substitute
∂W
∂E
= e i e j W ij ,
(4.81)
in which
W ij ≡
∂W
∂E ij
.
For the present problem, Eq. (4.69) yields
W rr = W θθ = 2c 1
W zz = 2
c 1 + 2c 3 E zz e
4c 4 E 2
zz
(4.82)