An EFG method for the nonlinear analysis of plates undergoing arbitrarily large deformations

Abstract The applicability of a meshfree approximation method, namely the EFG method, on fully geometrically exact analysis of plates is investigated. Based on a unified nonlinear theory of plates, which allows for arbitrarily large rotations and displacements, a Galerkin approximation via MLS functions is settled. A hybrid method of analysis is proposed, where the solution is obtained by the independent approximation of the generalized internal displacement fields and the generalized boundary tractions. A consistent linearization procedure is performed, resulting in a semi-definite generalized tangent stiffness matrix which, for hyperelastic materials and conservative loadings, is always symmetric (even for configurations far from the generalized equilibrium trajectory). Besides the total Lagrangian formulation, an updated version is also presented, which enables the treatment of rotations beyond the parameterization limit. An extension of the arc-length method that includes the generalized domain displacement fields, the generalized boundary tractions and the load parameter in the constraint equation of the hyper-ellipsis is proposed to solve the resulting nonlinear problem. Extending the hybrid-displacement formulation, a multi-region decomposition is proposed to handle complex geometries. A criterium for the classification of the equilibrium's stability, based on the Bordered–Hessian matrix analysis, is suggested. Several numerical examples are presented, illustrating the effectiveness of the method. Differently from the standard finite element methods (FEM), the resulting solutions are (arbitrary) smooth generalized displacement and stress fields.

[1]  Hyun Gyu Kim,et al.  Analysis of thin beams, using the meshless local Petrov–Galerkin method, with generalized moving least squares interpolations , 1999 .

[2]  M. Crisfield.  A FAST INCREMENTAL/ITERATIVE SOLUTION PROCEDURE THAT HANDLES "SNAP-THROUGH" , 1981 .

[3]  Paul M.E. Shutler,et al.  Constrained critical points , 1995 .

[4]  Wing Kam Liu,et al.  Meshfree and particle methods and their applications , 2002 .

[5]  S. Timoshenko.  Theory of Elastic Stability , 1936 .

[6]  P. Lancaster,et al.  Surfaces generated by moving least squares methods , 1981 .

[7]  P. Pimenta,et al.  Geometrically exact analysis of space frames by a meshless method , 2005 .

[8]  V. Leitão,et al.  Eliminating Shear-Locking in Meshless Methods: A Critical Overview and a New Framework for Structural Theories , 2007 .

[9]  J. Oden,et al.  H‐p clouds—an h‐p meshless method , 1996 .

[10]  J. Argyris.  An excursion into large rotations , 1982 .

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

[12]  Ted Belytschko,et al.  Volumetric locking in the element free Galerkin method , 1999 .

[13]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[14]  P. Wriggers,et al.  A fully nonlinear multi-parameter shell model with thickness variation and a triangular shell finite element , 2004 .

[15]  Mark A Fleming,et al.  Meshless methods: An overview and recent developments , 1996 .

[16]  Zhao Qi,et al.  The adaptive parameter incremental method for the analysis of snapping problems , 1995 .

[17]  H. B. Mann.  Quadratic Forms with Linear Constraints , 1943 .

[18]  A. Ibrahimbegovic.  Stress resultant geometrically nonlinear shell theory with drilling rotations - Part I : A consistent formulation , 1994 .

[19]  Continuity conditions for finite element analysis of solids , 1992 .

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

[21]  Mark A Fleming,et al.  Smoothing and accelerated computations in the element free Galerkin method , 1996 .

[22]  Vitor M. A. Leitão,et al.  Development of a EFG formulation for damage analysis of reinforced concrete beams , 2004 .

[23]  Kjell Mattiasson,et al.  Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals , 1981 .

[24]  Wing Kam Liu,et al.  MESHLESS METHODS FOR SHEAR-DEFORMABLE BEAMS AND PLATES , 1998 .

[25]  Peter Wriggers,et al.  Consistent linearization for path following methods in nonlinear FE analysis , 1986 .

[26]  P. Pimenta.  Geometrically exact analysis of initially curved rods , 1996 .

[27]  J. A. Freitas,et al.  Non-conventional formulations for the finite element method , 1996 .

[28]  Elmer Rees,et al.  The index of a constrained critical point , 1993 .

[29]  T. Belytschko,et al.  A new implementation of the element free Galerkin method , 1994 .

[30]  K. Hoffman,et al.  The Bordered Operator and the Index, of a Constrained Critical Point , 2000 .

[31]  C. Duarte,et al.  hp-Clouds in Mindlin's thick plate model , 2000 .

[32]  K. Y. Sze,et al.  Popular benchmark problems for geometric nonlinear analysis of shells , 2004 .

[33]  Paulo M. Pimenta,et al.  Geometrically Exact Analysis of Spatial Frames , 1993 .

[34]  Peter Wriggers,et al.  A triangular finite shell element based on a fully nonlinear shell formulation , 2003 .

[35]  Sung Bo Kim,et al.  Geometrically Nonlinear Analysis of Thin-Walled Space Frames , 1999 .

[36]  T. Belytschko,et al.  Nodal integration of the element-free Galerkin method , 1996 .

[37]  S. N. Atluri,et al.  Analysis of shear flexible beams, using the meshless local Petrov‐Galerkin method, based on a locking‐free formulation , 2001 .

[38]  Carlos J. S. Alves,et al.  Advances in meshfree techniques , 2007 .

[39]  David Spring.  On the Second Derivative Test for Constrained Local Extrema , 1985 .

[40]  Brian Moran,et al.  Treatment of material discontinuity in the Element-Free Galerkin method , 1996 .