An Eulerian approach for large displacements of thin shells including geometrical non-linearities

Abstract This paper is devoted to the problem of large displacement of shells, the only non-linearity being geometrical. The geometry of the mid-surface of the shell is determined thanks to the knowledge of nodes and normal vectors, and is continuously updated. Stresses are also carried from one configuration to another, which enables us to linearize the equilibrium equations. It is then possible to build an Eulerian formulation. Moreover, some control strategies are proposed, in order to find limit points and unstable solutions. Finally, we check in some classical examples, the accuracy of this technique.

[1]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[2]  Ph. Destuynder,et al.  Some numerical aspects of mixed finite elements for bending plates , 1990 .

[3]  Philippe Destuynder,et al.  A mixed finite element for shell model with free edge boundary conditions Part 1. The mixed variational formulation , 1995 .

[4]  M. Bernadou,et al.  Approximation of general shell problems by flat plate elements , 1989 .

[5]  M. Crisfield.  An arc‐length method including line searches and accelerations , 1983 .

[6]  Michel Salaün,et al.  Approximation of shell geometry for non-linear analysis , 1998 .

[7]  Cv Clemens Verhoosel,et al.  Non-Linear Finite Element Analysis of Solids and Structures , 1991 .

[8]  G. Dhatt,et al.  Modélisation des structures par éléments finis , 1990 .

[9]  P. G. Bergan,et al.  Solution algorithms for nonlinear structural problems , 1980 .

[10]  Gouri Dhatt,et al.  Incremental displacement algorithms for nonlinear problems , 1979 .

[11]  Philippe Destuynder,et al.  A mixed finite element for shell model with free edge boundary conditions Part 2. The numerical scheme , 1995 .

[12]  E. Riks.  The Application of Newton's Method to the Problem of Elastic Stability , 1972 .

[13]  P. Boisse,et al.  A C0 three‐node shell element for non‐linear structural analysis , 1994 .

[14]  A new finite element scheme for bending plates , 1988 .

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

[16]  J. F. Jullien,et al.  Buckling of shell structures, on land, in the sea, and in the air , 1991 .

[17]  P. G. Ciarlet,et al.  Sur L’Ellipticite du Modele Lineaire de coques de W.T. Koiter , 1976 .

[18]  Isaac Fried,et al.  Orthogonal trajectory accession to the nonlinear equilibrium curve , 1984 .