Transactions of JSCES, Paper No Development of Shell Element with Thickness Stretch Takeki YAMAMOTO, Takahiro YAMADA, and Kazumi MATSUI

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1 Transactions of JSCES, Paper No Development of Shell Element with Thickness Stretch 1 Takeki YAMAMOTO, Takahiro YAMADA, and Kazumi MATSUI 1 ( ) ( ) The finite element method is commonly used to simulate the behavior of sheet forming processes, in order to realize high precision machining. In the conventional shell elements, the plane stress condition which ignores the transverse normal stress is assumed. Thus, the conventional shell elements are not sufficient to simulate the complex behaviors, such as, the deformation of the sheet and the contact force at the sheet-die interface. In this paper, we present a formulation for considering the thickness change and the stress distribution by the surface traction in the shell element. We introduce a displacement variation along the transverse direction to MITC shell element, which is widely used for avoiding the transverse shear locking. Then, we can evaluate the equilibrium equation for the transverse direction by using the introduced variation. Further, we verify the proposed approach to compare the results of the proposed shell with that of the continuum elements. Key Words: Finite Element Method, Shell Element, Thick Sheet, Total Lagrangian Method , , , c 015. Manuscript received, January 09, 015; final revision, March 6, 015; published, April 16, 015. Copyright c 015 by the Japan Society for Computational Engineering and Science (1) () thickness locking (3) (4) thickness locking Braun et al. (1) (1) () Andelfinger and Ramm (5) EAS

2 thickness locking EAS EAS 3 (6) 3D (7) (9) thickness locking EAS (6) (7) (8) Sussman and Bathe (9) thickness locking Braun et al. (1) multidirector (10) multidirector C 0 multidirector layer-wise multilayer (11) (13) Carrera (14) multilayer (15) Pseudo (16) (17) EAS (18) Fig. 1 4-node shell element with 5 transverse nodes (16) (17) 4 MITC4 (19) MITC4. Total Lagrange Green-Lagrange.1 4 Fig. 1 Fig. 1 4 Fig. 1 5

3 Fig. 1 (ξ, η, ζ) = (0, 0, 0) M M t (0,0) a e f 3(a) M N a N a(0, 0)e f 3(a) t (0,0) = (1) N a (0, 0)e f 3(a) t (0,0). X X(ξ, η, ζ) = X(ξ, η) + ˆX(ξ, η, ζ) () X ˆX x x(ξ, η, ζ) = x(ξ, η) + ˆx(ξ, η, ζ) (3) x ζ = 0 ˆx X x N a a = 1 4 X(ξ, η, ζ) = N a(ξ, η)x a+z(ζ) N a(ξ, η)e f 3(a) (4) x(ξ, η, ζ) = N a (ξ, η)x a + z(ζ) N a (ξ, η)e f 3(a) (5) X a x a E f 3(a) ef 3(a) Z z Z h e Z(ζ) = ζ h e (6) Z = z z z(ζ) = ζ h e + w(ζ) = Z(ζ) + w(ζ) (7) w (7) ζ = 0 w 1 ζ i ζ ζ i+1 1 w(ζ) = ζ i+1 ζ ζ i+1 ζ i w(ζ i ) + ζ ζ i w(ζ i+1 ) ζ i+1 ζ i = ζi+1 ζ w i + ζ ζi w i+1 (8) ζ i+1 ζ i ζ i+1 ζ i w i ζ i u u = x X = N a(ξ, η)(x a X a) = + z(ζ) N a(ξ, η)e f 3(a) Z(ζ) N a(ξ, η)e f 3(a) N a(ξ, η)u a + Z(ζ) N a(ξ, η)(e f 3(a) Ef 3(a) ) + w(ζ) N a(ξ, η)e f 3(a) (9) u a w t (0,0) (9) 3 M (9) 3 t (0,0) u u = N a(ξ, η)u a + Z(ζ) N a(ξ, η)(e f 3(a) Ef 3(a) ) + w(ζ)t (0,0) (10) (10) (10)

4 Dvorkin and Bathe (19) Green-Lagrange MITC4 Fig. 1 A B C D Green- Lagrange ẼB 3 ẼD 3 ẼC 31 ẼA 31 Ẽ 3 = 1 (1 + ξ) ẼB (1 ξ) ẼD 3 Ẽ 31 = 1 (1 + η) ẼC (1 η) ẼA 31 (11) ξ η ζ Ẽ3 Ẽ31 Green-Lagrange (11) MITC4 (10) (11) Reissner- Mindlin (0) MITC4 MITC4.4 u TD ŪTD ū TD u TD = ŪTD + ūtd (1) (1) (10) u t (0,0) (1) ŪTD ū TD u TD ŪTD Ū TD = = he h e he h e = ŪTD + u TD dz (ŪTD + ū TD ) dz he h e ū TD dz (13) ū TD he h e ū TD dz = 0 (14) (14).5 n e f etd n A top δw ext δw ext = f etd δu (n) etd da A top (15) δu (n) etd e t e(0,0) δu (n) etd = δūetd + δū (n) etd = N a (ξ, η)δūatd + δū (n) etd t e(0,0) (16) δūatd a δū (n) etd n (15) δw ext ( ) δw ext = f etd N a (ξ, η)δūatd da A top + f (n) etd A topδū (n) etd (17) f (n) etd w n f etd f (n) etd f etd da = f (n) etd t e(0,0)da = f (n) etd A topt e(0,0) A top A top (18) (17) f etd.6 u IP u TD [ ] { } { } k 11 k 1 u IP f IP = k t 1 k u TD f TD (19) k ij (i, j = 1, )

5 Table 1 Boundary condition of each vertex Fig. Patch test of element Point x y z θ 1 θ 1 fixed fixed fixed free free free fixed fixed free free 3 free free fixed free free 4 fixed free fixed free free Table Model parameters of pure-bending Young s modulus E [GPa] 100 Poisson s ratio ν 0.30 Thickness h [m] 0.10 Transverse nodes 3, 5, 7, 9, 11, 13, 15 Fig. 3 Simulation model of pure-bending k 1 = k t 1 t (19) f IP f TD (19) ku IP = f (0) (0) k f k = k 11 k 1 k 1 k t 1 (1) f = f IP k 1 k 1 f TD () (0) k MITC (0) Fig. 3. thickness locking thickness locking Fig. 3 Table 1 Table 1 Fig. 4 Fig. 4 MITC4 MITC4 3 Fig. 3 Fig. 4 MITC4 Fig. 5 Fig. 5 thickness locking (3) (4) (8) Fig. 5 Fig. 5(a) (b)

6 Fig. 4 Relative error versus the number of transverse nodes Fig. 6 Simulation model with surface tractions Fig. 5 Transverse normal strain distributions thickness locking Fig [%] thickness locking Fig. 5 9 thickness locking Fig. Fig ADINA MITC4 Fig. 6 0 x y 0.00 E = [N/m ] ν = 0.3 Fig.6 MITC4 z [N/m ] MITC MITC , 00 Fig. 6 Fig. 7 Fig. 7(a) Fig. 7(b) Reissner-Mindlin Fig. 7(c) von-mises Reissner-Mindlin

7 Fig. 8 Displacement variation in transverse direction with surface tractions Fig. 9 Plane strain bending: geometry and loading MITC ,147 59, Fig. 7 Stress distributions in the simulation model under surface tractions Fig. 6 Fig. 8 Fig. 6 A-B z z 0 w i w mid w i = w i w mid (3) Fig Rolling up Fig. 9 π rolling up (9) St. Venant-Kirchhoff E = 100 [N/mm ] ν = 0.4 y x z ADINA MITC4 ADINA Total Lagrange 3 y MITC4

8 Fig. 11 Cauchy stress distributions through the thickness at fixed end for θ = π Fig. 10 bending Displacement variation for plane strain z x y z x z , ,000 Fig. 10 z z 0 Fig. 1 Green-Lagrange strain distributions through the thickness at fixed end for θ = π

9 Fig. 14 Plane strain bending with surface tractions: geometry and loading Fig.13 Comparison of mid surface with neutral surface in shell element (3) Cauchy x σ xx z σ zz Fig. 11 Cauchy Cauchy x σ xx z Cauchy Green-Lagrange x E xx z E zz Fig. 1 Green-Lagrange Green-Lagrange x E xx z E zz MITC4 Green-Lagrange x E xx z E zz (10) u w ζ (10) u ξ η Green-Lagrange w MITC4 Green-Lagrange Fig. 1 Fig. 1 Fig. 13(a) S in S out Fig. 13(b) s in s out s in s out Fig. 1 Fig Fig. 14 Fig. 14 z z 1 [m] y St. Venant-Kirchhoff E = [N/m ] ν = 0.3 Fig. 14 MITC4 z [N/m ] x z ADINA MITC4 ADINA Total Lagrange 3 y MITC4

10 Fig. 15 Displacement variation for plane strain bending with surface tractions Fig. 16 Cauchy stress distributions (f top : N/m / f btm : N/m ) z x y z x z , Fig. 15 z z 0 (3) Fig. 15(a) (b) Fig. 15 Fig. 16 Fig Cauchy Green-Lagrange x z Fig. 18 Fig Cauchy Green-Lagrange x z Fig. 16 Fig. 18 Cauchy Fig. 17 Green-Lagrange strain distributions (f top : N/m / f btm : N/m )

11 Fig. 17 Fig. 19 Green-Lagrange Fig. 16 Cauchy x σ xx z σ zz Fig. 17 Green-Lagrange x E xx z E zz Fig. 15(a) Fig. 18(a) Cauchy x σ xx Fig. 19(a) Green-Lagrange x E xx Fig. 19(b) Green-Lagrange z E zz Green-Lagrange z E zz Fig. 18(b) Cauchy z σ zz Fig. 18 Cauchy stress distributions (f top : N/m / f btm : N/m ) 5. Fig. 19 Green-Lagrange strain distributions (f top : N/m / f btm : N/m )

12 (1) M. Braun, M. Bischoff, and E. Ramm, Nonlinear shell formulations for complete three-dimensional constitutive laws including composites and laminates, Computational Mechanics, 15, 1994, pp () N. Büchter, E. Ramm, and D. Roehl, Threedimensional extension of non-linear shell formulation based on the enhanced assumed strain concept, Int. J. Numer. Meth. Engng., 37, 1994, pp (3) E. Carrera and S. Brischetto, Analysis of thickness locking in classical, refined and mixed multilayered plate theories, Composite Structures, 8, 008, pp (4) E. Carrera and S. Brischetto, Analysis of thickness locking in classical, refined and mixed theories for layered shells, Composite Structures, 85, 008, pp (5) U. Andelfinger and E. Ramm, EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements, Int. J. Numer. Meth. Engng., 36, 1993, pp (6) R. Hauptmann and K. Schweizerhof, A systematic development of solid-shell element formulations for linear and non-linear analyses employing only displacement degrees of freedom, Int. J. Numer. Meth. Engng., 4, 1998, pp (7) D. Chapelle, A. Ferent, and K. J. Bathe, 3D-shell elements and their underlying mathematical model, Math. Models Methods Appl. Sci., 14, 004, pp (8) D. N. Kim and K. J. Bathe, A 4-node 3D-shell element to model shell surface tractions and incompressible behavior, Computers and Structures, 86, 008, pp (9) T. Sussman and K. J. Bathe, 3D-shell elements for structures in large strains, Computers and Structures, 1, 013, pp. 1. (10) P. M. Pinsky and K. O. Kim, A multi-director formulation for elastic-viscoelastic layered shells, Int. J. Numer. Meth. Engng., 4, 1986, pp (11) M. Epatein and H. P. Huttelmaier, A finite element formulation for multilayered and thick plates, Computers and Structures, 16, 1983, pp (1) H. P. Huttelmaier and M. Epstein, A finite element formulation for multilayered and thick shells, Computers and Structures, 1, 1985, pp (13) D. R. J. Owen and Z. H. Li, A refined analysis of laminated plates by finite element displacement methods I. Fundamentals and static analysis, Computers and Structures, 6, 1987, pp (14) E. Carrera, Multilayered Shell Theories Accounting for Layerwise Mixed Description, Part 1: Governing Equations, Am Inst. Aeronaut Astronaut J., 37, 1999, pp (15),,,, FEM 1,, 45, 004, pp (16),,,,,,, 1 3,, 55, 004, pp (17) N. Iwata, H. Tsutamori, M. Niihira, H. Ishikura, Y. Umezu, A. Murata, Y. Yogo, Numerical Prediction of Springback Shape of Severely Bent Sheet Metal, NUMIF ORM, 39, 007, pp (18),,,,,,, 54, 013, pp (19) E. N. Dvorkin and K. J. Bathe, A continuum mechanics based four-node shell element for general non-linear analysis, Engineering Computations, 1, 1984, pp (0) K. J. Bathe, F inite Element P rocedure, Prentical- Hall, Inc., 1996

Developement of Plastic Collocation Method Extension of Plastic Node Method by Yukio Ueda, Member Masahiko Fujikubo, Member Masahiro Miura, Member Sum

Developement of Plastic Collocation Method Extension of Plastic Node Method by Yukio Ueda, Member Masahiko Fujikubo, Member Masahiro Miura, Member Summary Previously, the authors developed the plastic