An exact finite rotation shell theory, its mixed variational formulation and its finite element implementation

A non-linear shell theory, including transverse shear strains, with exact description of the kinematical fields is developed. The strain measures are derived via the polar decomposition theorem allowing for an explicit use of a three parametric rotation tensor. Thus in-plane rotations, also called drilling degrees of freedom, are included in a natural way. Various alternatives of the theory are derived. For a special version of the theory, with altogether six kinematical fields, different mixed variational principles are given. A hybrid finite element formulation, which does not exhibit locking phenomena, is developed. Numerical examples of shell deformation at finite rotations, with excellent element performance, are presented. Comparison with results reported in the literature demonstrates the features of the theory as well as the proposed finite element formulation.

[1]  D. A. Danielson,et al.  Nonlinear Shell Theory With Finite Rotation and Stress-Function Vectors , 1972 .

[2]  F. Brezzi,et al.  On drilling degrees of freedom , 1989 .

[3]  W. Wunderlich,et al.  Rotations as primary unknowns in the nonlinear theory of shells and corresponding finite element models , 1986 .

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

[5]  S. Atluri.  ALTERNATE STRESS AND CONJUGATE STRAIN MEASURES, AND MIXED VARIATIONAL FORMULATIONS INVOLVING RIGID ROTATIONS, FOR COMPUTATIONAL ANALYSES OF FINITELY DEFORMED SOLIDS, WITH APPLICATION TO PLATES AND SHELLS-I , 1984 .

[6]  Wojciech Pietraszkiewicz,et al.  Lagrangian description and incremental formulation in the non-linear theory of thin shells , 1984 .

[7]  Clifford Ambrose Truesdell,et al.  Exact theory of stress and strain in rods and shells , 1957 .

[8]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model. Part II: the linear theory; computational aspects , 1989 .

[9]  P. Pinsky,et al.  An assumed covariant strain based 9‐node shell element , 1987 .

[10]  Ekkehard Ramm,et al.  Strategies for Tracing the Nonlinear Response Near Limit Points , 1981 .

[11]  Atef F. Saleeb,et al.  A hybrid/mixed model for non‐linear shell analysis and its applications to large‐rotation problems , 1990 .

[12]  H. Bufler.  The Biot stresses in nonlinear elasticity and the associated generalized variational principles , 1985 .

[13]  Siegfried F. Stiemer,et al.  Improved arc length orthogonality methods for nonlinear finite element analysis , 1987 .

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

[15]  J. G. Simmonds,et al.  Nonlinear Elastic Shell Theory , 1983 .

[16]  W. Pietraszkiewicz.  Finite Rotations in Structural Mechanics , 1986 .

[17]  Bernard Budiansky,et al.  Notes on nonlinear shell theory. , 1968 .

[18]  E. Hinton,et al.  A new nine node degenerated shell element with enhanced membrane and shear interpolation , 1986 .

[19]  Peter Wriggers,et al.  Theory and numerics of thin elastic shells with finite rotations , 1989 .

[20]  H. Bufler.  Finite Rotations and Complementary Extremum Principles , 1986 .

[21]  J. C. Simo,et al.  The (symmetric) Hessian for geometrically nonlinear models in solid mechanics: intrinsic definition and geometric interpretation , 1992 .

[22]  M. Crisfield.  A consistent co-rotational formulation for non-linear, three-dimensional, beam-elements , 1990 .

[23]  K. Bathe,et al.  A four‐node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation , 1985 .

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

[25]  J. L. Sanders,et al.  NONLINEAR THEORIES FOR THIN SHELLS , 1963 .

[26]  H. Bufler.  On the work theorems for finite and incremental elastic deformations with discontinuous fields: A unified treatment of different versions , 1983 .

[27]  J. Makowski,et al.  Buckling equations for elastic shells with rotational degrees of freedom undergoing finite strain deformation , 1990 .

[28]  Janusz Badur,et al.  Finite rotations in the description of continuum deformation , 1983 .

[29]  J. C. Simo,et al.  A three-dimensional finite-strain rod model. Part II: Computational aspects , 1986 .

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

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

[32]  J. Makowski,et al.  Finite axisymmetric deformation of shells of revolution with application to flexural buckling of circular plates , 1989 .

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