Tensor-based finite element formulation for geometrically nonlinear analysis of shell structures

In the present study, a finite element computational model for the nonlinear analysis of shell structures is presented. A tensor-based finite element formulation is presented to describe the mathematical model of a shell in a natural and simple way by using curvilinear coordinates. In addition, a family of high-order elements with Lagrangian interpolations is used to avoid membrane and shear locking, and no mixed interpolations are employed. A first-order shell theory with seven parameters is derived with exact nonlinear deformations and under the framework of the Lagrangian description. This approach takes into account thickness stretching and, therefore, 3D constitutive equations are utilized. Numerical simulations and comparisons of the present results with those found in the literature for typical benchmark problems involving isotropic and laminated composites, as well as functionally graded shells, are found to be excellent and show the validity of the developed finite element model. Moreover, the simplicity of this approach makes it attractive for applications in contact mechanics and damage propagation of shells.

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

[2]  J. Reddy.  An introduction to nonlinear finite element analysis , 2004 .

[3]  E. Stein,et al.  On the parametrization of finite rotations in computational mechanics: A classification of concepts with application to smooth shells , 1998 .

[4]  Patrick Massin,et al.  Nine node and seven node thick shell elements with large displacements and rotations , 2002 .

[5]  E. Hinton,et al.  A family of quadrilateral Mindlin plate elements with substitute shear strain fields , 1986 .

[6]  T. Charlton.  Progress in Solid Mechanics , 1962, Nature.

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

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

[9]  Otso Ovaskainen,et al.  Shell deformation states and the finite element method : a benchmark study of cylindrical shells , 1995 .

[10]  Harri Hakula,et al.  Scale resolution, locking, and high-order finite element modelling of shells , 1996 .

[11]  Y. Başar,et al.  Finite-rotation theories for composite laminates , 1993 .

[12]  E. Ramm,et al.  Shell theory versus degeneration—a comparison in large rotation finite element analysis , 1992 .

[13]  J. Reddy,et al.  Consistent Third-Order Shell Theory with Application to Composite Cylindrical Cylinders , 2005 .

[14]  Carlo Sansour,et al.  Families of 4-node and 9-node finite elements for a finite deformation shell theory. An assesment of hybrid stress, hybrid strain and enhanced strain elements , 2000 .

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

[16]  W. Pietraszkiewicz.  Finite rotations and Lagrangean description in the non-linear theory of shells , 1979 .

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

[18]  Carlo Sansour,et al.  An exact finite rotation shell theory, its mixed variational formulation and its finite element implementation , 1992 .

[19]  J. P. Pontaza,et al.  Least-squares finite element formulation for shear-deformable shells , 2005 .

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

[21]  J. N. Reddy,et al.  Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates , 1998 .

[22]  J. Chróścielewski,et al.  Genuinely resultant shell finite elements accounting for geometric and material non-linearity , 1992 .

[23]  J. Reddy.  An introduction to the finite element method , 1989 .

[24]  Phill-Seung Lee,et al.  Insight into finite element shell discretizations by use of the basic shell mathematical model , 2005 .

[25]  J. N. Reddy,et al.  THERMOMECHANICAL ANALYSIS OF FUNCTIONALLY GRADED CYLINDERS AND PLATES , 1998 .

[26]  Carlo Sansour,et al.  A theory and finite element formulation of shells at finite deformations involving thickness change: Circumventing the use of a rotation tensor , 1995, Archive of Applied Mechanics.

[27]  Hamdan N. Al-Ghamedy,et al.  Finite element formulation of a third order laminated finite rotation shell element , 2002 .

[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]  B. Brank,et al.  On implementation of a nonlinear four node shell finite element for thin multilayered elastic shells , 1995 .

[31]  D. Chapelle,et al.  MITC elements for a classical shell model , 2003 .

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

[33]  M. De.  Handbuch der Physik , 1957 .

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

[35]  Juhani Pitkäranta,et al.  On the membrane locking of h-p finite elements in a cylindrical shell problem , 1994 .

[36]  Maenghyo Cho,et al.  Development of geometrically exact new shell elements based on general curvilinear co‐ordinates , 2003 .

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

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

[39]  Thomas J. R. Hughes,et al.  Consistent linearization in mechanics of solids and structures , 1978 .

[40]  J. N. Reddy,et al.  Energy principles and variational methods in applied mechanics , 2002 .

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

[42]  Dr.-Ing. C. Sansour.  A Theory and finite element formulation of shells at finite deformations involving thickness change , 1995 .

[43]  Ekkehard Ramm,et al.  An assessment of assumed strain methods in finite rotation shell analysis , 1989 .

[44]  J. Meixner,et al.  S. Flügge, Herausgeber: Handbuch der Physik, Band III/3: Die nicht‐linearen Feldtheorien der Mechanik. Von C. Truesdell und W. Noll. Springer‐Verlag, Berlin/Heidelberg/New York 1965. VIII/602 Seiten. Preis: 198,‐ DM , 1967, Berichte der Bunsengesellschaft für physikalische Chemie.

[45]  P. Chadwick.  Continuum Mechanics: Concise Theory and Problems , 1976 .

[46]  P. M. Naghdi,et al.  The Theory of Shells and Plates , 1973 .

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

[48]  L. Della Croce,et al.  Hierarchic finite elements for thin Naghdi shell model , 1998 .

[49]  J. Reddy.  Analysis of functionally graded plates , 2000 .

[50]  Arciniega Aleman,et al.  On a tensor-based finite element model for the analysis of shell structures , 2006 .

[51]  Wilfried B. Krätzig,et al.  ‘Best’ transverse shearing and stretching shell theory for nonlinear finite element simulations , 1993 .

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

[53]  Dieter Weichert,et al.  Nonlinear Continuum Mechanics of Solids , 2000 .

[54]  O. C. Zienkiewicz,et al.  Analysis of thick and thin shell structures by curved finite elements , 1970 .

[55]  Boštjan Brank,et al.  Nonlinear shell problem formulation accounting for through-the-thickness stretching and its finite element implementation , 2002 .

[56]  P. M. Naghdi,et al.  FOUNDATIONS OF ELASTIC SHELL THEORY , 1962 .

[57]  Liviu Librescu,et al.  Refined geometrically nonlinear theories of anisotropic laminated shells , 1987 .

[58]  R. D. Wood,et al.  Nonlinear Continuum Mechanics for Finite Element Analysis , 1997 .

[59]  Chahngmin Cho,et al.  An efficient assumed strain element model with six DOF per node for geometrically non‐linear shells , 1995 .

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

[61]  J. N. Reddy,et al.  Shear Deformation Plate and Shell Theories: From Stavsky to Present , 2004 .

[62]  Werner Wagner,et al.  A simple finite rotation formulation for composite shell elements , 1994 .