288
6 Growth
6.7 Growth-Induced Residual Stress in Tubes
In Example 6.6, we showed how the presence of residual stress in an unloaded
artery can decrease transmural stress gradients in the loaded vessel. However, we did
not consider mechanisms that generate residual stress. Since experimental studies
suggest that growth plays a major role in this process, this section examines the
distributions of residual stress caused by some relatively simple growth patterns.
Because the results are not always intuitively obvious, the main objective is to build
a qualitative understanding of how growth can produce residual stress in tubular
structures.
Model Consider a tube consisting of a single layer of incompressible, orthotropic
material. In the ZSS, the tube has an inner radius a 0 = 2 mm and outer radius b 0 =
3 mm. The strain-energy density function is given by Eq. (6.37) with the material
constants listed in (6.47). For axisymmetric growth defined by
G(R) = G r (R) e r e r + G θ (R) e θ e θ + e z e z ,
we herein compute wall stress in the unloaded, unconstrained tube for specified
patterns of radial and circumferential growth.
Analysis By symmetry, the specified growth causes axisymmetric deformation
without shear (in cylindrical coordinates), and so the analysis is based on the
equations in Sect. 6.6.4. For an incompressible material (J ∗ = 1), Eqs. (6.59)
and (6.61) give
J = λ r λ θ λ z =
∂r
∂R
r
R
λ = J G ,
where J G = det G = G r G θ . The last two terms yield
λr dr = J G R dR,
which can be integrated to obtain
r
2
=
2
λ
R
a 0
J G R dR + C,
where the limits have been selected for convenience while maintaining the integral
as a function of R. Enforcing the boundary condition r(a 0 ) = a gives C = a 2 , and
thus
r(R) =
a
2
+
2
λ
R
a 0
J G R dR
1
2
.
(6.72)
6 Growth
6.7 Growth-Induced Residual Stress in Tubes
In Example 6.6, we showed how the presence of residual stress in an unloaded
artery can decrease transmural stress gradients in the loaded vessel. However, we did
not consider mechanisms that generate residual stress. Since experimental studies
suggest that growth plays a major role in this process, this section examines the
distributions of residual stress caused by some relatively simple growth patterns.
Because the results are not always intuitively obvious, the main objective is to build
a qualitative understanding of how growth can produce residual stress in tubular
structures.
Model Consider a tube consisting of a single layer of incompressible, orthotropic
material. In the ZSS, the tube has an inner radius a 0 = 2 mm and outer radius b 0 =
3 mm. The strain-energy density function is given by Eq. (6.37) with the material
constants listed in (6.47). For axisymmetric growth defined by
G(R) = G r (R) e r e r + G θ (R) e θ e θ + e z e z ,
we herein compute wall stress in the unloaded, unconstrained tube for specified
patterns of radial and circumferential growth.
Analysis By symmetry, the specified growth causes axisymmetric deformation
without shear (in cylindrical coordinates), and so the analysis is based on the
equations in Sect. 6.6.4. For an incompressible material (J ∗ = 1), Eqs. (6.59)
and (6.61) give
J = λ r λ θ λ z =
∂r
∂R
r
R
λ = J G ,
where J G = det G = G r G θ . The last two terms yield
λr dr = J G R dR,
which can be integrated to obtain
r
2
=
2
λ
R
a 0
J G R dR + C,
where the limits have been selected for convenience while maintaining the integral
as a function of R. Enforcing the boundary condition r(a 0 ) = a gives C = a 2 , and
thus
r(R) =
a
2
+
2
λ
R
a 0
J G R dR
1
2
.
(6.72)