Development of a one point quadrature shell element for nonlinear applications with contact and anisotropy

A general purpose shell element for nonlinear applications including sheet metal forming simulation is developed based on reduced integration with one point quadrature. The developed shell element has five degrees of freedom and four nodes. It covers flexible warping behavior without artificial warping correction. A physical stabilization scheme with the assumed natural strain method is employed to derive a strain field that can be decomposed into the sum of a constant and a linear term with respect to the natural coordinates. The rigid body projection is introduced to treat rigid body rotations effectively. The shell element incorporates elasto-plastic planar anisotropic material models based on the incremental deformation theory. Linear and nonlinear patch tests are performed and the results are compared with analytical or previously reported results. Simulations that include impact and deformable body contact are performed to show the robustness of the contact algorithm. Finally, to demonstrate the capability of handling anisotropic materials, the developed shell element is used for the circular cup drawing process simulation in order to predict the earing profile of Al 2008-T4 alloy sheet.

[1]  J. C. Simo,et al.  An augmented lagrangian treatment of contact problems involving friction , 1992 .

[2]  Mark O. Neal,et al.  Contact‐impact by the pinball algorithm with penalty and Lagrangian methods , 1991 .

[3]  T. Belytschko,et al.  A uniform strain hexahedron and quadrilateral with orthogonal hourglass control , 1981 .

[4]  Ekkehard Ramm,et al.  EAS‐elements for two‐dimensional, three‐dimensional, plate and shell structures and their equivalence to HR‐elements , 1993 .

[5]  K. Bathe,et al.  A continuum mechanics based four‐node shell element for general non‐linear analysis , 1984 .

[6]  Ted Belytschko,et al.  Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems , 1991 .

[7]  Dong-Yol Yang,et al.  A rigid-plastic finite-element formulation for the analysis of general deformation of planar anisotropic sheet metals and its applications , 1986 .

[8]  T. Pian,et al.  Rational approach for assumed stress finite elements , 1984 .

[9]  H. Saunders,et al.  Finite element procedures in engineering analysis , 1982 .

[10]  Dong-Yol Yang,et al.  Finite element method for sheet forming based on an anisotropic strain-rate potential and the convected coordinate system , 1995 .

[11]  Robert L. Taylor,et al.  Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems☆ , 1993 .

[12]  F. Barlat,et al.  Strain rate potential for metals and its application to minimum plastic work path calculations , 1993 .

[13]  T. Hughes.  Generalization of selective integration procedures to anisotropic and nonlinear media , 1980 .

[14]  P. Wriggers,et al.  Application of augmented Lagrangian techniques for non‐linear constitutive laws in contact interfaces , 1993 .

[15]  T. Hughes,et al.  Finite rotation effects in numerical integration of rate constitutive equations arising in large‐deformation analysis , 1980 .

[16]  T. Hughes,et al.  Finite Elements Based Upon Mindlin Plate Theory With Particular Reference to the Four-Node Bilinear Isoparametric Element , 1981 .

[17]  Renato Natal Jorge,et al.  Development of shear locking‐free shell elements using an enhanced assumed strain formulation , 2002 .

[18]  Dong-Yol Yang,et al.  Elasto-plastic finite element method based on incremental deformation theory and continuum based shell elements for planar anisotropic sheet materials , 1999 .

[19]  K. Bathe,et al.  Finite Element Methods for Nonlinear Problems , 1986 .

[20]  E. Hinton,et al.  A nine node Lagrangian Mindlin plate element with enhanced shear interpolation , 1984 .

[21]  T. Belytschko,et al.  Physical stabilization of the 4-node shell element with one point quadrature , 1994 .

[22]  Peter Betsch,et al.  Numerical implementation of multiplicative elasto-plasticity into assumed strain elements with application to shells at large strains , 1999 .

[23]  K. Park,et al.  A Curved C0 Shell Element Based on Assumed Natural-Coordinate Strains , 1986 .

[24]  Thomas J. R. Hughes,et al.  Nonlinear finite element analysis of shells: Part I. three-dimensional shells , 1981 .

[25]  J. C. Simo,et al.  A perturbed Lagrangian formulation for the finite element solution of contact problems , 1985 .

[26]  T. Belytschko,et al.  Efficient implementation of quadrilaterals with high coarse-mesh accuracy , 1986 .

[27]  H. Parisch,et al.  A critical survey of the 9-node degenerated shell element with special emphasis on thin shell application and reduced integration , 1979 .

[28]  T. Belytschko,et al.  A stabilization procedure for the quadrilateral plate element with one-point quadrature , 1983 .

[29]  Medhat A. Haroun,et al.  Reduced and selective integration techniques in the finite element analysis of plates , 1978 .

[30]  E. Ramm,et al.  Shear deformable shell elements for large strains and rotations , 1997 .

[31]  Jerry I. Lin,et al.  Explicit algorithms for the nonlinear dynamics of shells , 1984 .

[32]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model. Part III: computational aspects of the nonlinear theory , 1990 .

[33]  J. M. Kennedy,et al.  Hourglass control in linear and nonlinear problems , 1983 .

[34]  F. Barlat,et al.  A six-component yield function for anisotropic materials , 1991 .

[35]  D. Owen,et al.  Computational model for 3‐D contact problems with friction based on the penalty method , 1992 .

[36]  Dong-Yol Yang,et al.  Earing predictions based on asymmetric nonquadratic yield function , 2000 .

[37]  T. Belytschko,et al.  Projection schemes for one-point quadrature shell elements , 1994 .

[38]  Manabu Gotoh,et al.  A finite element analysis of rigid-plastic deformation of the flange in a deep-drawing process based on a fourth-degree yield function , 1978 .

[39]  Lawrence H.N. Lee,et al.  Postbifurcation behavior of wrinkles in square metal sheets under Yoshida Test , 1993 .

[40]  David J. Benson,et al.  Implementation of a modified Hughes-Liu shell into a fully vectorized explicit finite element code , 1985 .

[41]  O. C. Zienkiewicz,et al.  Reduced integration technique in general analysis of plates and shells , 1971 .

[42]  E. Ramm,et al.  Large elasto-plastic finite element analysis of solids and shells with the enhanced assumed strain concept , 1996 .

[43]  Frédéric Barlat,et al.  A general elasto-plastic finite element formulation based on incremental deformation theory for planar anisotropy and its application to sheet metal forming , 1999 .

[44]  F. Barlat,et al.  Yield function development for aluminum alloy sheets , 1997 .

[45]  T. Hughes,et al.  Nonlinear finite element shell formulation accounting for large membrane strains , 1983 .

[46]  K. Chung,et al.  Finite element simulation of sheet metal forming for planar anisotropic metals , 1992 .

[47]  Ted Belytschko,et al.  Resultant-stress degenerated-shell element , 1986 .

[48]  Edward L. Wilson,et al.  Incompatible Displacement Models , 1973 .

[49]  O. C. Zienkiewicz,et al.  Analysis of thick and thin shell structures by curved finite elements , 1970 .

[50]  Z. Zhong.  Finite Element Procedures for Contact-Impact Problems , 1993 .