6.5 Residual Stress
279
The E i in Eq. (6.37) for W are strains relative to the ZSS given by
E r =
1
2 (λ
2
r − 1),
E θ =
1
2 (λ
2
θ − 1),
E z =
1
2 (λ
2
z − 1).
With W written in terms of these strain components, it is convenient to use the
constitutive relations (3.249) 1 in the form (with J = 1)
σ i = ¯
σ i − p
¯
σ i = λ
2
i
∂W
∂E i
,
(6.46)
in which i = r, θ, z.
The rest of the analysis follows that in Sect. 4.4. The equilibrium equation (4.58)
and boundary conditions (4.61) are the same, and the Lagrange multiplier p and
internal pressure p i are again given by Eqs. (4.63) and (4.64). Given λ and a (with
= 1), Eq. (6.45) provides r(R) and the deformed outer radius b = r(b 0 ), and we
follow the basic procedure on page 179 to compute p i and the stress distributions.
Results
The following results are based on data provided by Chuong and Fung (1986) for
rabbit arteries. For W given by Eq. (6.37), representative material constants are
c = 11.2 kPa
α 1 = 0.0499
α 2 = 1.0672
α 3 = 0.4775
α 4 = 0.0042
α 5 = 0.0903
α 6 = 0.0585,
(6.47)
and geometric measurements for an artery before and after transmural cutting are
(Fig. 6.7)
a = 1.4 mm
b = 2.0 mm (unloaded)
a 0 = 3.9 mm
b 0 = 4.5 mm (cut)
φ = 220
◦ (φ 0 = 70
◦ )
)= 1.
Pressure-radius curves in the normal physiological range differ relatively little
between the cases with (φ = 220 ◦ ) and without (φ = 0 ◦ ) residual stress (Fig. 6.8a).
In both cases, the inner radius is normalized by its value a = 1.4 mm in the unloaded
configuration in Barney.
In the pressurized artery, E θ decreases across the wall from the inner to the outer
surface, but the transmural gradient is reduced by the presence of residual strain
relative to the zero-stress state (Fig. 6.8b). The reason can be seen by considering
the schematic in Fig. 6.9. Residual stress and strain are produced by equal and
opposite moments M, which bend the cut section into a closed tube. As in a beam,
this bending causes circumferential strains that are negative near the inner surface
and positive near the outer surface, while the strain gradient generated by internal
279
The E i in Eq. (6.37) for W are strains relative to the ZSS given by
E r =
1
2 (λ
2
r − 1),
E θ =
1
2 (λ
2
θ − 1),
E z =
1
2 (λ
2
z − 1).
With W written in terms of these strain components, it is convenient to use the
constitutive relations (3.249) 1 in the form (with J = 1)
σ i = ¯
σ i − p
¯
σ i = λ
2
i
∂W
∂E i
,
(6.46)
in which i = r, θ, z.
The rest of the analysis follows that in Sect. 4.4. The equilibrium equation (4.58)
and boundary conditions (4.61) are the same, and the Lagrange multiplier p and
internal pressure p i are again given by Eqs. (4.63) and (4.64). Given λ and a (with
= 1), Eq. (6.45) provides r(R) and the deformed outer radius b = r(b 0 ), and we
follow the basic procedure on page 179 to compute p i and the stress distributions.
Results
The following results are based on data provided by Chuong and Fung (1986) for
rabbit arteries. For W given by Eq. (6.37), representative material constants are
c = 11.2 kPa
α 1 = 0.0499
α 2 = 1.0672
α 3 = 0.4775
α 4 = 0.0042
α 5 = 0.0903
α 6 = 0.0585,
(6.47)
and geometric measurements for an artery before and after transmural cutting are
(Fig. 6.7)
a = 1.4 mm
b = 2.0 mm (unloaded)
a 0 = 3.9 mm
b 0 = 4.5 mm (cut)
φ = 220
◦ (φ 0 = 70
◦ )
)= 1.
Pressure-radius curves in the normal physiological range differ relatively little
between the cases with (φ = 220 ◦ ) and without (φ = 0 ◦ ) residual stress (Fig. 6.8a).
In both cases, the inner radius is normalized by its value a = 1.4 mm in the unloaded
configuration in Barney.
In the pressurized artery, E θ decreases across the wall from the inner to the outer
surface, but the transmural gradient is reduced by the presence of residual strain
relative to the zero-stress state (Fig. 6.8b). The reason can be seen by considering
the schematic in Fig. 6.9. Residual stress and strain are produced by equal and
opposite moments M, which bend the cut section into a closed tube. As in a beam,
this bending causes circumferential strains that are negative near the inner surface
and positive near the outer surface, while the strain gradient generated by internal