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3.2 Interpolation of the Green–Lagrangean strains and associated variations The element has to fulfil membrane and bending patch test, see e.g. [27]. As has been shown in appendix B the bending patch test – when using the present mixed interpolation for the stress resultants and shell strains – can be fulfilled with substitute shear strains defined in [23] but not with the bilinear displacement interpolation inserted in the transverse shear strains (5) 3 . Thus the finite element interpolation of the Green–Lagrangean strains reads ⎡ ε h ⎤ ⎡ 1 11 2 (xh , 1 ·x h , 1 −X h , 1 ·X h ⎤ , 1 ) ε h 1 22 2 (xh , 2 ·x h , 2 −X h , 2 ·X h , 2 ) 2ε h 12 x h , 1 ·x h , 2 −X h , 1 ·X h , 2 ε h κ h 11 x h , 1 ·d h , 1 −X h , 1 ·D h , 1 G = κ h = 22 x h , 2 ·d h , 2 −X h , 2 ·D h (19) , 2 2κ h 12 x h , 1 ·d h , 2 +x h , 2 ·d h , 1 −X h , 1 ·D h , 2 −X h , 2 ·D h , 1 ⎧ ⎢ γ1 h ⎥ ⎢ ⎨ 1 ⎣ ⎦ ⎣ J [(1 − η) ⎫ −1 2 γB ξ +(1+η) γξ D ⎬ ⎥ ⎦ γ2 h ⎩ 1 [(1 − ξ) 2 γA η +(1+ξ) γη C ] ⎭ . The strains at the midside nodes A, B, C, D of the element are specified as follows γξ M = [x, ξ ·d − X, ξ ·D] M M = B,D γη L = [x, η ·d − X, η ·D] L (20) L = A, C , where the following quantities are given with the bilinear interpolation (12) and(17) d A = 1 (d 2 4 + d 1 ) D A = 1 (D 2 4 + D 1 ) d B = 1 (d 2 1 + d 2 ) D B = 1 (D 2 1 + D 2 ) d C = 1 (d 2 2 + d 3 ) D C = 1 (D 2 2 + D 3 ) d D = 1 (d 2 3 + d 4 ) D D = 1 (D 2 3 + D 4 ) x A , η = 1 (x 2 4 − x 1 ) X A , η = 1 (X 2 4 − X 1 ) x B , ξ = 1 (x 2 2 − x 1 ) X B , ξ = 1 (X 2 2 − X 1 ) x C , η = 1 (x 2 3 − x 2 ) X C , η = 1 (X 2 3 − X 2 ) x D , ξ = 1 (x 2 3 − x 4 ) X D , ξ = 1 (X 2 3 − X 4 ) . Accordingly the interpolated virtual strains read ⎡ δε h ⎤ ⎡ 11 δx h , 1 ·x h ⎤ , 1 δε h 22 δx h , 2 ·x h , 2 2δε h 12 δx h , 1 ·x h , 2 +δx h , 2 ·x h , 1 δε h δκ h 11 δx h , 1 ·d h , 1 +δd h , 1 ·x h , 1 G = δκ h = 22 δx h , 2 ·d h , 2 +δd h , 2 ·x h , 2 2δκ h 12 δx h , 1 ·d h , 2 +δx h , 2 ·d h , 1 +δd h , 1 ·x h , 2 +δd h , 2 ·x h , 1 ⎧ ⎢ δγ1 h ⎥ ⎢ ⎨ 1 ⎣ ⎦ ⎣ J [(1 − η) ⎫ −1 2 δγB ξ +(1+η) δγξ D ⎬ ⎥ ⎦ δγ2 h ⎩ 1 [(1 − ξ) 2 δγA η +(1+ξ) δγη C ] ⎭ 8 (21) (22)
with δγξ M = [δx, ξ ·d + x, ξ ·δd] M M = B,D δγη L = [δx, η ·d + x, η ·δd] L L = A, C , where δx, ξ ,δx, η ,δd are evaluated at the midside nodes considering (21). The virtual vectors δx h , α and δd h , α using (17) are determined (23) 4∑ δx h , α = N I , α δu I δd h , α = I=1 with the virtual nodal displacements δu I and 4∑ N I , α δd I , (24) I=1 where according to [30] δw I = H I δω I , δd I = δw I × d I = W T I δw I W I =skewd I (25) H I = 1 + 1 − cos ω I ω 2 I Ω I + ω I − sin ω I ω 3 I Ω 2 I . (26) ThecoefficientsofΩ I and Ω 2 I possess the limit values 1/2 and1/6 for ω I → 0. At nodes which are not positioned on intersections a drilling stiffness is not available and a transformation of the virtual rotation vector to the local coordinate system is necessary: { 13 for nodes on shell intersections δω I = T 3I δβ I T 3I = [a 1I , a 2I ] (3×2) for all other nodes { (27) [δβxI ,δβ yI ,δβ zI ] T for nodes on shell intersections δβ I = [δβ 1I ,δβ 2I ] T for all other nodes where δβ αI denote local rotations. Furthermore the drilling degree of freedom is fixed, thus δβ 3I = 0. The element possesses six degrees of freedom at all nodes on intersections and five at all other nodes. In this context we also refer to [28, 29]. Next combining (25) –(27) weobtain δd I = T I δβ I T I = W T I H I T 3I (28) Thus we are able to summarize the finite element approximation of the virtual shell strains (22) considering (23) -(28) ⎡ δε h ⎤ ⎡ N 11 I , 1 x T ⎤ , 1 0 δε h 22 N I , 2 x T , 2 0 2δε h 12 N I , 1 x T , 2 +N I , 2 x T , 1 0 δκ h 11 4∑ N I , 1 d T , 1 N I , 1 b T [ ] I1 δuI δκ h = 22 N I=1 I , 2 d T , 2 N I , 2 b T I2 δβ I 2δκ h 12 N ⎢ ⎣ δγ1 h I , 1 d T , 2 +N I , 2 d T , 1 N I , 1 b T I2 + N I , 2 b T I1 (29) ⎥ ⎧ ⎫ ⎧ ⎫ ⎦ ⎢ ⎨ N ⎣ I , ξ d T ⎬ J −1 M ⎨ N I , ξ ξ I b T ⎬ J −1 M ⎥ ⎦ δγ2 h ⎩ N I , η d T ⎭ ⎩ L N I , η η I b T ⎭ L 4∑ δε h G = B I δv I I=1 9
Page 1 and 2: Universität Karlsruhe (TH) Institu
Page 3 and 4: A robust nonlinear mixed hybrid qua
Page 5 and 6: the strain field and the non-consta
Page 7 and 8: with membrane strains ε αβ , cur
Page 9: as close as possible to the coordin
Page 13 and 14: To avoid numerical difficulties the
Page 15 and 16: where numel denotes the total numbe
Page 17 and 18: 4 Examples The derived element form
Page 19 and 20: 4.3 Hemispherical shell with a 18
Page 21 and 22: Load P [N] 3,5 3,0 2,5 2,0 1,5 1,0
Page 23 and 24: Load P in kN 20,0 18,0 16,0 14,0 12
Page 25 and 26: 200 175 150 Load P [kN] 125 100 u_z
Page 27 and 28: B Remarks on patch test and stabili
Page 29 and 30: The plane stress condition S 33 (E
Page 31 and 32: E J 2 -plasticity model for small s
Page 33 and 34: [16] Betsch P, Gruttmann F, Stein,

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