6.5 Residual Stress
277
During closure (B to B 0 ), the length of the circumference at a given radius changes
from 2φ 0 R to 2πρ, where the angle φ 0 = π − φ/2 is defined in Fig. 6.7. The ratio
of these lengths gives the circumferential stretch ratio, with the other stretch ratios
having the same basic forms as those listed above. Relative to B, therefore, the
stretch ratios in B 0 are
λ R =
∂ρ
∂R
,
λ =
πρ
φ 0 R
,
λ Z = ,
(6.39)
where is the axial stretch ratio that occurs during the closure phase. In general,
taking ≈ 1 is a reasonable approximation.
As in the analysis of growth, the total stretch ratios from B to B L are obtained
by multiplying the stretch ratios between the successive configurations. With
Eqs. (6.38) and (6.39), this process yields (Fig. 6.7)
λ r = λ R λ ρ =
∂r
∂R
λ θ = λ λ ϑ =
πr
φ 0 R
λ z = λ Z λ ζ = λλ.
(6.40)
For φ 0 = π and = 1 (no residual stress), these relations reduce to Eqs. (4.52).
To derive the stretch ratios using tensor analysis, we determine the component
deformation gradient tensors F 1 and F 2 indicated in Fig. 6.7, and then the total
deformation gradient tensor is F = F 2 · F 1 .
The position vectors to a point in B, B 0 , and B L are
B :
R = R e R + Z e Z
B 0 :
ρ = ρ e ρ + ζ e ζ
B L :
r = r e r + z e z ,
(6.41)
and the mappings
B → B 0 :
ρ = ρ(R),
ϑ = ππ/φ 0 ,
ζ = Z
B 0 → B L :
r = r(ρ),
θ = ϑ,
z = λζ
(6.42)
define the deformations. The expression relating ϑ and satisfies the requirement
that points originally along the radial line = 0 transform into points along ϑ =
0, whereas points originally along = φ 0 transform into points along ϑ = π
(Fig. 6.7). The other relations follow from Eqs. (4.48).
Computing F 1 and F 2 requires the gradient operators
In B :
∇ = e R
∂
∂R
+
e
R
∂
∂∂
+ e Z
∂
∂Z
277
During closure (B to B 0 ), the length of the circumference at a given radius changes
from 2φ 0 R to 2πρ, where the angle φ 0 = π − φ/2 is defined in Fig. 6.7. The ratio
of these lengths gives the circumferential stretch ratio, with the other stretch ratios
having the same basic forms as those listed above. Relative to B, therefore, the
stretch ratios in B 0 are
λ R =
∂ρ
∂R
,
λ =
πρ
φ 0 R
,
λ Z = ,
(6.39)
where is the axial stretch ratio that occurs during the closure phase. In general,
taking ≈ 1 is a reasonable approximation.
As in the analysis of growth, the total stretch ratios from B to B L are obtained
by multiplying the stretch ratios between the successive configurations. With
Eqs. (6.38) and (6.39), this process yields (Fig. 6.7)
λ r = λ R λ ρ =
∂r
∂R
λ θ = λ λ ϑ =
πr
φ 0 R
λ z = λ Z λ ζ = λλ.
(6.40)
For φ 0 = π and = 1 (no residual stress), these relations reduce to Eqs. (4.52).
To derive the stretch ratios using tensor analysis, we determine the component
deformation gradient tensors F 1 and F 2 indicated in Fig. 6.7, and then the total
deformation gradient tensor is F = F 2 · F 1 .
The position vectors to a point in B, B 0 , and B L are
B :
R = R e R + Z e Z
B 0 :
ρ = ρ e ρ + ζ e ζ
B L :
r = r e r + z e z ,
(6.41)
and the mappings
B → B 0 :
ρ = ρ(R),
ϑ = ππ/φ 0 ,
ζ = Z
B 0 → B L :
r = r(ρ),
θ = ϑ,
z = λζ
(6.42)
define the deformations. The expression relating ϑ and satisfies the requirement
that points originally along the radial line = 0 transform into points along ϑ =
0, whereas points originally along = φ 0 transform into points along ϑ = π
(Fig. 6.7). The other relations follow from Eqs. (4.48).
Computing F 1 and F 2 requires the gradient operators
In B :
∇ = e R
∂
∂R
+
e
R
∂
∂∂
+ e Z
∂
∂Z