Using finite strain 3D‐material models in beam and shell elements

In this paper an interface is derived between arbitrary three‐dimensional material laws and finite elements which include special stress conditions. The mechanical models of beams and shells are usually based upon zero‐stress conditions. This requires a material law respecting the stress condition for each finite element formulation. Complicated materials, e.g. finite strain models are often described in the 3D‐continuum. Considering the zero‐stress condition requires a reformulation of these material laws, which is often complicated. The subject of this paper is to incorporate physically non‐linear 3D‐material laws in beam and shell elements. To this effect a local algorithm will be developed to condense an arbitrary 3D‐material law with respect to the zero‐stress condition. The algorithm satisfies the stress condition at each integration point on the element level.

[1]  J. C. Simo,et al.  Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory , 1992 .

[2]  K. Bathe,et al.  Large displacement analysis of three‐dimensional beam structures , 1979 .

[3]  Peter Betsch,et al.  FE plane stress analysis incorporating arbitrary 3D large strain constitutive models , 1997 .

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

[5]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model , 1990 .

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

[7]  Werner Wagner,et al.  THEORY AND NUMERICS OF THREE-DIMENSIONAL BEAMS WITH ELASTOPLASTIC MATERIAL BEHAVIOUR ∗ , 2000 .

[8]  S. Klinkel,et al.  A finite rotation shell theory with application to composite structures , 1995 .

[9]  En-Jui Lee.  Elastic-Plastic Deformation at Finite Strains , 1969 .

[10]  F. Gruttmann,et al.  A geometrical nonlinear eccentric 3D-beam element with arbitrary cross-sections , 1998 .

[11]  D. W. Scharpf,et al.  On large displacement-small strain analysis of structures with rotational degrees of freedom , 1978 .

[12]  Sven Klinkel,et al.  A continuum based three-dimensional shell element for laminated structures , 1999 .

[13]  M. Géradin,et al.  A beam finite element non‐linear theory with finite rotations , 1988 .

[14]  Adnan Ibrahimbegovic,et al.  Finite elastoplastic deformations of space-curved membranes , 1994 .

[15]  J. C. Simo,et al.  Geometrically non‐linear enhanced strain mixed methods and the method of incompatible modes , 1992 .

[16]  J. C. Simo,et al.  On stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization , 1989 .

[17]  Richard Von Mises,et al.  Mechanik der plastischen Formänderung von Kristallen , 1928 .

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

[19]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model. Part V: Nonlinear plasticity: formulation and integration algorithms , 1992 .

[20]  R. Borst.  The zero-normal-stress condition in plane-stress and shell elastoplasticity , 1991 .

[21]  R. Ogden.  Large deformation isotropic elasticity – on the correlation of theory and experiment for incompressible rubberlike solids , 1972, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[22]  David W. Murray,et al.  Nonlinear Finite Element Analysis of Steel Frames , 1983 .

[23]  Eduardo N. Dvorkin,et al.  A formulation of the MITC4 shell element for finite strain elasto-plastic analysis , 1995 .

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

[25]  A. Ibrahimbegovic,et al.  Computational aspects of vector-like parametrization of three-dimensional finite rotations , 1995 .

[26]  E. Stein,et al.  A 4-node finite shell element for the implementation of general hyperelastic 3D-elasticity at finite strains , 1996 .

[27]  E. Stein,et al.  An assumed strain approach avoiding artificial thickness straining for a non‐linear 4‐node shell element , 1995 .

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

[29]  E. Ramm,et al.  Three‐dimensional extension of non‐linear shell formulation based on the enhanced assumed strain concept , 1994 .

[30]  C. Miehe,et al.  Aspects of the formulation and finite element implementation of large strain isotropic elasticity , 1994 .

[31]  J. P. Boehler,et al.  Introduction to the Invariant Formulation of Anisotropic Constitutive Equations , 1987 .