A New Locking-Free Thick/Thin Shell Element
101
The strain-displacement relation of the linear elastic problem in the curvilinear
coordinate system is expressed as
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ε 11 = ∂u 1 /(h 1 ∂α 1 ) + u 2 ∂h 1 /(h 1 h 2 ∂α 2 ) + u 3 ∂h 1 /(h 1 ∂α 3 )
ε 22 = u 1 ∂h 2 /(h 1 h 2 ∂α 1 ) + ∂u 2 /(h 2 ∂α 2 ) + u 3 ∂h 2 /(h 2 ∂α 3 )
ε 33 ≈ 0
γ 12 = ∂u 1 /(h 2 ∂α 2 ) − u 1 ∂h 1 /(h 1 h 2 ∂α 2 ) + ∂u 2 /(h 1 ∂α 1 ) − u 2 ∂h 2 /(h 1 h 2 ∂α 1 )
γ 23 = ∂u 2 /∂α 3 − ∂u 2 h 2 /(h 2 ∂α 3 ) + ∂u 3 /(h 2 ∂α 2 )
γ 31 = ∂u 1 /∂α 3 − u 1 ∂h 1 /(h 1 ∂α 3 ) + ∂u 3 /(h 1 ∂α 2 )
,
(15)
where h 1 , h 2 and h 3 = 1 are the Lame coefficients along axes α 1 , α 2 and α 3 , respectively.
The substitution of Eq. (11) into Eq. (15) results in
ε = Ba
e
,
(16)
where ε =
ε 11 ε 22 γ 12 γ 23 γ 31
T is the strain vector and B is the strain matrix.
For an isotropic linear elastic material, the stress-strain relation in element e i is
expressed as
σ = DBa
e
,
(17)
where the elasticity matrix
D = D 0
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 ν
0
0
0
ν 1
0
0
0
0 0 (1 − ν)/2
0
0
0 0
0
(1 − ν)/(2k)
0
0 0
0
0
(1 − ν)/(2k)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,
(18)
where D 0 = E/
1 − ν 2
; E is the elastic modulus; ν is the Poisson ratio. Because
σ 33 << σ 11 and σ 33 << σ 22 in the Mindlin/Reissner assumption, σ 33 and its influence
on the deformation have been neglected in Eqs. (16) and (17).
k in Eq. (18) is the shear correction factor of the cross section. The Reissner/Mindlin
assumption implies that the shear strain on the cross section is evenly distributed, but
this may not be the case. If this strain is distributed in a quadratic parabola with the
maximum value on the mid-surface and the minimum value (equals to 0) on the shell
surfaces, k = 1.2 can be derived for a rectangular cross section according to the same
shear strain energy of the two distributions [5].
Thus, the strain energy of element e i can be expressed as
e
=
a
e
T
H / 2
−H / 2
¨
e i
B
T DBh 1 h 2 dα 1 dα 2
dα 3 a
e
/2.
(19)
101
The strain-displacement relation of the linear elastic problem in the curvilinear
coordinate system is expressed as
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ε 11 = ∂u 1 /(h 1 ∂α 1 ) + u 2 ∂h 1 /(h 1 h 2 ∂α 2 ) + u 3 ∂h 1 /(h 1 ∂α 3 )
ε 22 = u 1 ∂h 2 /(h 1 h 2 ∂α 1 ) + ∂u 2 /(h 2 ∂α 2 ) + u 3 ∂h 2 /(h 2 ∂α 3 )
ε 33 ≈ 0
γ 12 = ∂u 1 /(h 2 ∂α 2 ) − u 1 ∂h 1 /(h 1 h 2 ∂α 2 ) + ∂u 2 /(h 1 ∂α 1 ) − u 2 ∂h 2 /(h 1 h 2 ∂α 1 )
γ 23 = ∂u 2 /∂α 3 − ∂u 2 h 2 /(h 2 ∂α 3 ) + ∂u 3 /(h 2 ∂α 2 )
γ 31 = ∂u 1 /∂α 3 − u 1 ∂h 1 /(h 1 ∂α 3 ) + ∂u 3 /(h 1 ∂α 2 )
,
(15)
where h 1 , h 2 and h 3 = 1 are the Lame coefficients along axes α 1 , α 2 and α 3 , respectively.
The substitution of Eq. (11) into Eq. (15) results in
ε = Ba
e
,
(16)
where ε =
ε 11 ε 22 γ 12 γ 23 γ 31
T is the strain vector and B is the strain matrix.
For an isotropic linear elastic material, the stress-strain relation in element e i is
expressed as
σ = DBa
e
,
(17)
where the elasticity matrix
D = D 0
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 ν
0
0
0
ν 1
0
0
0
0 0 (1 − ν)/2
0
0
0 0
0
(1 − ν)/(2k)
0
0 0
0
0
(1 − ν)/(2k)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,
(18)
where D 0 = E/
1 − ν 2
; E is the elastic modulus; ν is the Poisson ratio. Because
σ 33 << σ 11 and σ 33 << σ 22 in the Mindlin/Reissner assumption, σ 33 and its influence
on the deformation have been neglected in Eqs. (16) and (17).
k in Eq. (18) is the shear correction factor of the cross section. The Reissner/Mindlin
assumption implies that the shear strain on the cross section is evenly distributed, but
this may not be the case. If this strain is distributed in a quadratic parabola with the
maximum value on the mid-surface and the minimum value (equals to 0) on the shell
surfaces, k = 1.2 can be derived for a rectangular cross section according to the same
shear strain energy of the two distributions [5].
Thus, the strain energy of element e i can be expressed as
e
=
a
e
T
H / 2
−H / 2
¨
e i
B
T DBh 1 h 2 dα 1 dα 2
dα 3 a
e
/2.
(19)