Hybrid and multi-field variational principles for geometrically exact three-dimensional beams

Abstract This paper addresses the development of several alternative novel hybrid/multi-field variational formulations of the geometrically exact three-dimensional elastostatic beam boundary-value problem. In the framework of the complementary energy-based formulations, a Legendre transformation is used to introduce the complementary energy density in the variational statements as a function of stresses only. The corresponding variational principles are shown to feature stationarity within the framework of the boundary-value problem. Both weak and linearized weak forms of the principles are presented. The main features of the principles are highlighted, giving special emphasis to their relationships from both theoretical and computational standpoints.

[1]  D. White,et al.  A mixed finite element for three-dimensional nonlinear analysis of steel frames , 2004 .

[2]  Miran Saje,et al.  Finite-element formulation of geometrically exact three-dimensional beam theories based on interpolation of strain measures , 2003 .

[3]  R. Ogden,et al.  Inequalities associated with the inversion of elastic stress-deformation relations and their implications , 1977, Mathematical Proceedings of the Cambridge Philosophical Society.

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

[5]  M. Iura,et al.  A consistent theory of finite stretches and finite rotations, in space-curved beams of arbitrary cross-section , 2001 .

[6]  Ernst Hellinger,et al.  Die allgemeinen Ansätze der Mechanik der Kontinua , 1913 .

[7]  J. Brilla.  Trends in applications of pure mathematics to mechanics , 1983 .

[8]  Dinar Camotim,et al.  Work-conjugacy between rotation-dependent moments and finite rotations , 2003 .

[9]  Debasish Roy,et al.  Consistent quaternion interpolation for objective finite element approximation of geometrically exact beam , 2008 .

[10]  J. C. Simo,et al.  A finite strain beam formulation. The three-dimensional dynamic problem. Part I , 1985 .

[11]  Rakesh K. Kapania,et al.  A formulation and implementation of geometrically exact curved beam elements incorporating finite strains and finite rotations , 2003 .

[12]  Alberto Cardona,et al.  A nonlinear beam element formulation in the framework of an energy preserving time integration scheme for constrained multibody systems dynamics , 2008 .

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

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

[15]  A. Ibrahimbegovic,et al.  Shape Optimization of Elastic Structural Systems Undergoing Large Rotations: Simultaneous Solution Procedure , 2003 .

[16]  Ignacio Romero,et al.  The interpolation of rotations and its application to finite element models of geometrically exact rods , 2004 .

[17]  D. Zupan,et al.  Rotational invariants in finite element formulation of three-dimensional beam theories , 2002 .

[18]  P. Betsch,et al.  Frame‐indifferent beam finite elements based upon the geometrically exact beam theory , 2002 .

[19]  S. Atluri,et al.  Dynamic analysis of finitely stretched and rotated three-dimensional space-curved beams , 1988 .

[20]  E. Reissner.  On a Variational Theorem in Elasticity , 1950 .

[21]  A. Ibrahimbegovic.  On the choice of finite rotation parameters , 1997 .

[22]  A. J. Chen,et al.  Finite Element Method in Dynamics of Flexible Multibody Systems: Modeling of Holonomic Constraints and Energy Conserving Integration Schemes , 2000 .

[23]  E. Reissner.  On finite deformations of space-curved beams , 1981 .

[24]  Gordan Jelenić,et al.  A kinematically exact space finite strain beam model - finite element formulation by generalized virtual work principle , 1995 .

[25]  S. Atluri,et al.  On a consistent theory, and variational formulation of finitely stretched and rotated 3-D space-curved beams , 1988 .

[26]  J. C. Simo,et al.  Non-linear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithms , 1995 .

[27]  H. Kardestuncer,et al.  Finite element handbook , 1987 .

[28]  Gordan Jelenić,et al.  Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics , 1999 .

[29]  Miran Saje,et al.  The three-dimensional beam theory: Finite element formulation based on curvature , 2001 .

[30]  R. J. Knops,et al.  Trends in applications of pure mathematics to mechanics : a collection of invited papers presented at a symposium at Heriot-Watt University in September 1979 , 1978 .

[31]  K. Washizu.  Variational Methods in Elasticity and Plasticity , 1982 .

[32]  S. Atluri,et al.  Primal and mixed variational principles for dynamics of spatial beams , 1996 .

[33]  P. Wriggers,et al.  An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 2: shells , 2011 .

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

[35]  Miran Saje,et al.  A quadratically convergent algorithm for the computation of stability points: The application of the determinant of the tangent stiffness matrix , 1999 .

[36]  J. Mäkinen.  Total Lagrangian Reissner's geometrically exact beam element without singularities , 2007 .

[37]  A. Ibrahimbegovic.  On finite element implementation of geometrically nonlinear Reissner's beam theory: three-dimensional curved beam elements , 1995 .

[38]  J. C. Simo,et al.  A Geometrically-exact rod model incorporating shear and torsion-warping deformation , 1991 .

[39]  Dinar Camotim,et al.  On the differentiation of the Rodrigues formula and its significance for the vector‐like parameterization of Reissner–Simo beam theory , 2002 .

[40]  P. Wriggers,et al.  An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 1: Rods , 2008 .

[41]  Alex H. Barbat,et al.  Static analysis of beam structures under nonlinear geometric and constitutive behavior , 2007 .

[42]  José Paulo Moitinho de Almeida,et al.  A hybrid-mixed finite element formulation for the geometrically exact analysis of three-dimensional framed structures , 2011 .

[43]  Adnan Ibrahimbegovic,et al.  Quadratically convergent direct calculation of critical points for 3d structures undergoing finite rotations , 2000 .

[44]  M. Crisfield,et al.  Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[45]  S. Atluri,et al.  Analysis of flexible multibody systems with spatial beams using mixed variational principles , 1998 .

[46]  A. Ibrahimbegovic,et al.  Computational aspects of vector-like parametrization of three-dimensional finite rotations , 1995 .

[47]  Goran Turk,et al.  A kinematically exact finite element formulation of elastic–plastic curved beams , 1998 .

[48]  E. Reissner,et al.  On One‐Dimensional Large‐Displacement Finite‐Strain Beam Theory , 1973 .

[49]  Rakesh K. Kapania,et al.  On a geometrically exact curved/twisted beam theory under rigid cross-section assumption , 2003 .