Technical information

5-2
In terms of a cylindrical coordinate system (
zr ,,
θ
), with z as the axis of
material symmetry, and assuming an axially symmetric deformation field (i.e.,
0
==
θθ
ε
ε
rz
), the constitutive law becomes:
zrr
aaa
σ
σ
σ
ε
θ
⋅
+
⋅+⋅=
131211
................................................................... (5.1.2a)
zr
aaa
σ
σ
σ
ε
θθ
⋅
+
⋅+⋅=
131112
................................................................... (5.1.2b)
zrz
aaa
σ
σ
σ
ε
θ
⋅
+
⋅+⋅=
331313
................................................................... (5.1.2c)
rzrz
a
τ
ε
⋅= )2/(
44
.......................................................................................... (5.1.2d)
in which
x
Ea /1
11
=
,
z
Ea /1
33
=
,
xxy
Ea /
12
ν
−
=
,
zzx
Ea /
13
ν
−
=
and
xz
Ga /1
44
=
.
Hence, five elastic constants are included, namely: two Young’s moduli
)(
yx
EE =
and
z
E
; two Poisson’s ratios
)(
yxxy
ν
ν
=
and
)/(
xzxzzx
EE
⋅
=
ν
ν
; and one shear modulus
)(
yzxz
GG =
. The condition that the strain energy must be positive imposes the following
property restrictions (PRs) on the values of the elastic constants (e.g., Poulus and Davis,
1974): (PR1)
0,, >
xzzx
GEE
; (PR2)
021 >
⋅
⋅
−
−
zxxzxy
ν
ν
ν
; and (PR3)
01 >−
xy
ν
.
Following Lekhnitskii (1963) and Singh (1986), the stresses (
rzzr
τ
σ
σ
σ
θ
,,, ) and
displacements (
wu, in the z
r
, directions respectively) can be derived from a stress
function
),( zr
φ
as follows:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
⋅+
∂
∂
⋅+
∂
∂
∂
∂
−=
2
2
2
2
z
a
rr
b
r
z
r
φφφ
σ
............................................................. (5.1.3a)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
⋅+
∂
∂
⋅+
∂
∂
⋅
∂
∂
−=
2
2
2
2
1
z
a
rr
r
b
z
φφφ
σ
θ
......................................................... (5.1.3b)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
⋅+
∂
∂
⋅+
∂
∂
⋅
∂
∂
=
2
2
2
2
z
d
rr
c
r
c
z
z
φφφ
σ
............................................................ (5.1.3c)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
⋅+
∂
∂
⋅+
∂
∂
∂
∂
=
2
2
2
2
1
z
a
rr
r
r
rz
φφφ
τ
................................................................ (5.1.3d)