An 8-node brick element with mixed formulation for large deformation analyses

Abstract An 8-node brick element with a mixed formulation has been developed in this paper. This element uses a reduced integration scheme (RI) and the hourglass control technique with the aid of mixed formulations. It has the correct rank of the element stiffness matrix so that the hourglass-modes can be avoided completely. It passes large strain patch tests and guarantees stability and convergence. With the aid of Hu-Washizu principle and “assumed strain” method, it gives no volumetric locking for (nearly) incompressible materials and by setting some shear parameters β ij to about zero, “shear locking” can also be reduced or even eliminated for thin bodies in pure bending. It is economical since only ore-point quadrature is used to evaluate the element stiffness matrix. It is robust, accurate and efficient in numerical simulations for almost all kind of problems. It can provide good results not only for non-linear solids but also for thin bodies (plates and shells). At the end of this paper, several examples concerning the patch tests, large deflection of beams and plates as well as some metal forming processes are used to show the performance of this element.

[1]  John Argyris,et al.  Large natural strains and some special difficulties due to non-linearity and incompressibility in finite elements , 1974 .

[2]  H. Bruneel,et al.  Simulation of square-cup deep-drawing with different finite elements , 1995 .

[3]  T. Hughes.  Equivalence of Finite Elements for Nearly Incompressible Elasticity , 1977 .

[4]  F. Barlat,et al.  Prediction of tricomponent plane stress yield surfaces and associated flow and failure behavior of strongly textured f.c.c. polycrystalline sheets , 1987 .

[5]  Robert Charlier,et al.  Some recent developments in computer simulation of metal forming processes , 1989 .

[6]  Frédéric Barlat,et al.  Crystallographic texture, anisotropic yield surfaces and forming limits of sheet metals , 1987 .

[7]  O. C. Zienkiewicz,et al.  Basic formulation and linear problems , 1989 .

[8]  R. Hill.  Theoretical plasticity of textured aggregates , 1979, Mathematical Proceedings of the Cambridge Philosophical Society.

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

[10]  G. Ferron,et al.  A parametric description of orthotropic plasticity in metal sheets , 1994 .

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

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

[13]  Ted Belytschko,et al.  Nonlinear versions of flexurally superconvergent elements , 1988 .

[14]  Frédéric Barlat,et al.  Plastic behavior and stretchability of sheet metals. Part I: A yield function for orthotropic sheets under plane stress conditions , 1989 .

[15]  J. Dienes.  On the analysis of rotation and stress rate in deforming bodies , 1979 .

[16]  Robert Charlier,et al.  FRICTIONAL CONTACT FINITE-ELEMENTS BASED ON MIXED VARIATIONAL-PRINCIPLES , 1993 .

[17]  S. Cescotto.  Finite Deformation of Solids , 1992 .

[18]  Ted Belytschko,et al.  Elements with embedded localization zones for large deformation problems , 1988 .

[19]  S. Timoshenko,et al.  THEORY OF PLATES AND SHELLS , 1959 .

[20]  R. Hill.  A theory of the yielding and plastic flow of anisotropic metals , 1948, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[21]  B. C. Noh,et al.  New improved hourglass control for bilinear and trilinear elements in anisotropic linear elasticity , 1987 .

[22]  J. Z. Zhu,et al.  The finite element method , 1977 .

[23]  Ted Belytschko,et al.  Efficient linear and nonlinear heat conduction with a quadrilateral element , 1983 .

[24]  R. A. Uras,et al.  Finite element stabilization matrices-a unification approach , 1985 .

[25]  P. Jetteur,et al.  A mixed finite element for the analysis of large inelastic strains , 1991 .

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

[27]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[28]  R. N. White,et al.  COLD-BENDING OF THICK HIGH-STRENGTH STEEL PLATES , 1990 .

[29]  Serge Cescotto,et al.  A note on the response to simple shear of elasto-plastic materials with isotropic hardening , 1991 .

[30]  William F. Hosford,et al.  Upper-bound anisotropic yield locus calculations assuming 〈111〉-pencil glide , 1980 .

[31]  T. Belytschko,et al.  A fractal patch test , 1988 .

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

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

[34]  W. Hosford.  A Generalized Isotropic Yield Criterion , 1972 .

[35]  Ted Belytschko,et al.  On flexurally superconvergent four-node quadrilaterals , 1987 .

[36]  R. N. White,et al.  Residual Stresses in Cold‐Bent Thick Steel Plates , 1990 .

[37]  John L. Bassani,et al.  Yield characterization of metals with transversely isotropic plastic properties , 1977 .

[38]  Michael Ortiz,et al.  A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations , 1985 .

[40]  L. Herrmann.  Elasticity Equations for Incompressible and Nearly Incompressible Materials by a Variational Theorem , 1965 .

[41]  D. Malkus,et al.  Mixed finite element methods—reduced and selective integration techniques: a unification of concepts , 1990 .

[42]  Robert Charlier,et al.  Numerical approach of contact with friction between two bodies in large deformations , 1992 .