Optimal solid shells for non-linear analyses of multilayer composites. II. Dynamics

Abstract We are presenting a simple low-order solid-shell element formulation––having only displacement degrees of freedom (dofs), i.e., without rotational dofs––that has an optimal number of parameters to pass the patch tests, and thus allows for efficient and accurate analyses of large deformable multilayer shell structures using elements at extremely high aspect ratio. With the dynamics referred to a fixed inertial frame, the elements can be used to analyze multilayer shell structures undergoing large overall motion. The formulation of this element is based on the mixed Hu-Washizu variational principle leading to a novel enhancing strain tensor (enhanced assumed strain (EAS) method) that renders the computation particularly efficient, with improved in-plane and out-of-plane bending behavior (Poisson thickness locking), especially in refined analyses of composite structures involving a large number of high aspect-ratio layers. The energy–momentum conserving algorithm in the context of current solid shell element is presented. We discuss the EAS formulation based on the displacement gradient and its complexity compared to formulation on the Green–Lagrange strain. Shear locking and curvature thickness locking are treated using the assumed natural strain (ANS) method. The element has an optimal combination of the ANS method and the minimal number of EAS parameters required to pass the plate bending patch test. Numerical examples involving dynamic analyses (with conservation of energy and momentum) of multilayer shell structures having a large range of element aspect ratios are presented. Several implicit direct integration methods with/without numerical dissipation are used and compared in terms of the accuracy, stability and cost in multilayer shell structure. Finally, we note that the topic in this paper is a fitting dedication to Professor Ekkehard Ramm, who has made important pioneering contributions in this research direction.

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

[2]  Multilayer shells: Geometrically-exact formulation of equations of motion , 2000 .

[3]  Peter Betsch,et al.  A nonlinear extensible 4-node shell element based on continuum theory and assumed strain interpolations , 1996 .

[4]  J. Marsden,et al.  Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems , 2000 .

[5]  Bruce M. Irons,et al.  An engineers' defence of the patch test , 1983 .

[6]  Vijay K. Varadan,et al.  FINITE ELEMENT MODELLING OF STRUCTURES INCLUDING PIEZOELECTRIC ACTIVE DEVICES , 1997 .

[7]  Christian Miehe,et al.  A theoretical and computational model for isotropic elastoplastic stress analysis in shells at large strains , 1998 .

[8]  O. C. Zienkiewicz,et al.  The finite element method, fourth edition; volume 2: solid and fluid mechanics, dynamics and non-linearity , 1991 .

[9]  L. Vu-Quoc,et al.  General Multilayer Geometrically‐Exact Beams/1‐D Plates with Deformable Layer Thickness: Equations of Motion , 2000 .

[10]  M. Crisfield,et al.  Energy‐conserving and decaying Algorithms in non‐linear structural dynamics , 1999 .

[11]  J. C. Simo,et al.  Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics , 1992 .

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

[13]  J. Reddy.  Mechanics of laminated composite plates : theory and analysis , 1997 .

[14]  E. Ramm,et al.  On the physical significance of higher order kinematic and static variables in a three-dimensional shell formulation , 2000 .

[15]  J. C. Simo,et al.  A CLASS OF MIXED ASSUMED STRAIN METHODS AND THE METHOD OF INCOMPATIBLE MODES , 1990 .

[16]  J. C. Simo,et al.  On the Variational Foundations of Assumed Strain Methods , 1986 .

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

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

[19]  Carlos A. Felippa,et al.  On the Original Publication of the General Canonical Functional of Linear Elasticity , 2000 .

[20]  Whirley.  DYNA3D: A nonlinear, explicit, three-dimensional finite element code for solid and structural mechanics , 1993 .

[21]  O. Zienkiewicz,et al.  The finite element patch test revisited a computer test for convergence, validation and error estimates , 1997 .

[22]  O. C. Zienkiewicz,et al.  The patch test—a condition for assessing FEM convergence , 1986 .

[23]  Jintai Chung,et al.  A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method , 1993 .

[24]  R. L. Harder,et al.  A proposed standard set of problems to test finite element accuracy , 1985 .

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

[26]  T. R. Hughes,et al.  Mathematical foundations of elasticity , 1982 .

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

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

[29]  L. Vu-Quoc,et al.  Geometrically exact sandwich shells: The dynamic case , 2000 .

[30]  J. Reddy,et al.  THEORIES AND COMPUTATIONAL MODELS FOR COMPOSITE LAMINATES , 1994 .

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

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

[33]  H. Saunders.  Book Reviews : NUMERICAL METHODS IN FINITE ELEMENT ANALYSIS K.-J. Bathe and E.L. Wilson Prentice-Hall, Inc, Englewood Cliffs, NJ , 1978 .

[34]  E. Ramm.  From Reissner Plate Theory to Three Dimensions in Large Deformation Shell Analysis , 2000 .

[35]  F. Armero,et al.  Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems , 1998 .

[36]  L. E. Malvern.  Introduction to the mechanics of a continuous medium , 1969 .

[37]  John Argyris,et al.  Recent Advances in Computational Thermostructural Analysis of Composite Plates and Shells With Strong Nonlinearities , 1997 .

[38]  L. Brillouin.  Les tenseurs en mécanique et en élasticité , 1987 .

[39]  Yavuz Başar,et al.  Refined shear-deformation models for composite laminates with finite rotations , 1993 .

[40]  L. Vu-Quoc,et al.  Galerkin Projection for Geometrically Exact Sandwich Beams Allowing for Ply Drop-off , 1995 .

[41]  J. Whitney,et al.  Bending-Extensional Coupling in Laminated Plates Under Transverse Loading , 1969 .

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

[43]  K. Bathe.  Finite Element Procedures , 1995 .

[44]  M Puso,et al.  NIKE3D a nonlinear, implicit, three-dimensional finite element code for solid and structural mechanics user's manual update summary , 1991 .

[45]  Ekkehard Ramm,et al.  Nonlinear shell formulations for complete three-dimensional constitutive laws including composites and laminates , 1994 .

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

[47]  R. Spilker,et al.  Hybrid stress reduced‐Mindlin elements for thin multilayer plates , 1986 .

[48]  Ekkehard Ramm,et al.  Generalized Energy–Momentum Method for non-linear adaptive shell dynamics , 1999 .

[49]  K. Y. Sze,et al.  A hybrid stress ANS solid‐shell element and its generalization for smart structure modelling. Part I—solid‐shell element formulation , 2000 .

[50]  Robert L. Taylor,et al.  Complementary mixed finite element formulations for elastoplasticity , 1989 .

[51]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[52]  R. Hauptmann,et al.  A SYSTEMATIC DEVELOPMENT OF 'SOLID-SHELL' ELEMENT FORMULATIONS FOR LINEAR AND NON-LINEAR ANALYSES EMPLOYING ONLY DISPLACEMENT DEGREES OF FREEDOM , 1998 .

[53]  Richard H. Macneal,et al.  A simple quadrilateral shell element , 1978 .

[54]  E. Wilson,et al.  A non-conforming element for stress analysis , 1976 .

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

[56]  Sven Klinkel,et al.  A geometrical non‐linear brick element based on the EAS‐method , 1997 .

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

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

[59]  O. C. Zienkiewicz,et al.  An alpha modification of Newmark's method , 1980 .

[60]  H. Parisch.  A continuum‐based shell theory for non‐linear applications , 1995 .

[61]  I. Ebcioǧlu,et al.  General multilayer geometrically-exact beams and 1-D plates with piecewise linear section deformation , 1996 .

[62]  Christian Miehe,et al.  Energy and momentum conserving elastodynamics of a non‐linear brick‐type mixed finite shell element , 2001 .