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Applications of a Rotation-Free Triangular Shell Element 73
substituting (3) into (9) and using the usual membrane strain vector [11]
equation (7) gives
εm =
εm =[εm11, εm12, εm12] T
3
i=1
1
2 ¯Ni N
The virtualmembrane strains are expressed by
3
δεm = N¯Ni
i=1
2.2 Computation of Curvatures
i
ϕ′1 · ϕ i ′ 1 − 1
ϕ i ′ 2 · ϕ i ′ 2 − 1
2ϕ i ′ 1 · ϕ i
′ 2
ϕ i ′ 1 · δϕ i ′ 1
ϕ i 2 · δϕ i ′ 2
δϕ i ′ 1 · ϕ i ′ 2 + ϕ i ′ 1 · δϕ i 2
(10)
(11)
. (12)
The curvatures (second fundamental form) of the middle surface are defined by []
καβ = 1
2 (ϕ′ α · t3 ′ β + ϕ′ β · t3 ′ α) = −t3 · ϕ′αβ , α, β = 1, 2 (13)
We will assume the following constant curvature field within each element
καβ =ˆκαβ
where ˆκαβ is the assumed constant curvature field obtained as
ˆκαβ = − 1
A 0 M
t3 · ϕ′ βα dA
A0
M
0
and A 0 M is thearea(in the original configuration) of the central element in the patch.
Substituting (15) into (14) and integrating by parts the area integral gives the
curvature vector [11] within the element in terms of the following closed line integral
κ =
κ11
κ22
2κ12
= 1
A 0 M
Γ 0
M
−n1 0
0 −n2
−n2 −n1
t3 · ϕ′ 1
t3 · ϕ′ 2
where ni are the components (in the local system) of the normals to the element
sides in the initial configuration Γ 0 Γ M.
For the definition of the normal vector t3, the linear interpolation of the position
vector over the central element is used.
ϕ M 3
= L M i ϕi (17)
i=1
where L M i are the standard linear shape functions of the central triangle (area coordinates)
[11]. In this case the tangent plane components are
ϕ M ′ α =
3
i=1
dΓ 0
(14)
(15)
(16)
L M i,αϕi , α = 1, 2 (18)