178
4 Problems in Soft Tissue Biomechanics
is not satisfied identically.
Constitutive Relations For the present problem, it is convenient to write the
constitutive equations in the form
σ r = ¯
σ r − p,
σ θ = ¯
σ θ − p,
σ z = ¯
σ z − p,
(4.59)
where the response functions are
¯
σ r = λ r
∂W
∂λ r
,
¯
σ θ = λ θ
∂W
∂λ θ
,
¯
σ z = λ z
∂W
∂λ z
.
(4.60)
These relations are provided by Eq. (3.249) 1 with J = 1.
Boundary Conditions The inner surface of the tube is subjected to the pressure
p i , while the outer surface is traction free. Since a positive pressure acts normal
to a surface, pushes against it, and is defined as force per unit deformed area, the
Cauchy (true) stress vector on the inner surface of the wall is T = p i e r . With the
outward-directed normal to the deformed inner surface being n = −e r , Eqs. (3.112)
and (4.56) yield
T = n · σ
= −e r · (σ r e r e r + σ θ e θ e θ + σ z e z e z )
= −σ r e r = p i e r
at the inner surface. Hence, the boundary conditions are
r = a :
σ r = −p i
r = b :
σ r = 0.
(4.61)
In other words, a positive pressure exerts a compressive radial stress on the inside
surface of the tube.
Solution With W (λ r , λ θ , λ z ) given by Eq. (4.46), the stresses can be computed
using Eqs. (4.59) and (4.60) once the Lagrange multiplier p is determined. To find
p, we substitute (4.59) into (4.58) to obtain
∂
∂r
( ¯
σ r − p) +
1
r
( ¯
σ r − ¯
σ θ ) = 0,
(4.62)
which is integrated to get
p(r) = ¯
σ r (r) +
r
b
( ¯
σ r − ¯
σ θ )
dr
r