4.4 Extension and Inflation of a Circular Tube
179
= ¯
σ r (r) +
b
r
( ¯
σ θ − ¯
σ r )
dr
r
.
(4.63)
To understand the limits on the integral, note that this relation for p(r) provides the
radial stress in the form
σ r = ¯
σ r − p = −
b
r
( ¯
σ θ − ¯
σ r )
dr
r
.
The lower limit maintains the integral as a function of r. The upper limit is chosen
to satisfy the boundary condition σ r (b) = 0. Similarly, applying the boundary
condition at r = a yields the internal pressure
p i = −σ r (a) =
b
a
( ¯
σ θ − ¯
σ r )
dr
r
.
(4.64)
The above equations can be integrated numerically over the spatial coordinate
r. However, since computations can be somewhat more convenient in material
coordinates, we illustrate the method here. With Eqs. (4.52) giving r = λ θ R and
dr = λ r dR, Eqs. (4.63) and (4.64) can be written in terms of the material coordinate
R as
p(R) = ¯
σ r (R) +
b 0
R
( ¯
σ θ − ¯
σ r )
λ r
λ θ
dR
R
p i =
b 0
a 0
( ¯
σ θ − ¯
σ r )
λ r
λ θ
dR
R
.
(4.65)
For numerical calculations, it is helpful to define a grid across the wall from a 0
to b 0 in terms of the material coordinate R. Then, the computational procedure is
the following:
1. Use (4.54) to compute r at each grid point R.
2. Compute λ θ = r/R at each grid point and then λ r = 1/λ θ λ z = 1/λ θ λ from
incompressibility.
3. Compute ¯
σ i (R) at each point using Eqs. (4.46) and (4.60).
4. Integrate Eqs. (4.65) numerically to obtain p at each grid point, as well as the
pressure p i .
5. Compute the Cauchy stresses at each point using (4.59).
4.4.3 Illustrative Results
Results are shown in Fig. 4.10 for a tube with b 0 /a 0 = 1.2. For a neo-Hookean
tube (c 1 = 1, c 3 = 0) held at a fixed length (λ = 1, 2), the pressure-radius