A GEOMETRICALLY EXACT PLANAR COSSERAT SHELL-MODEL WITH MICROSTRUCTURE: EXISTENCE OF MINIMIZERS FOR ZERO COSSERAT COUPLE MODULUS

The existence of minimizers to a geometrically exact Cosserat planar shell model with microstructure is proven. The membrane energy is a quadratic, uniformly Legendre–Hadamard elliptic energy in contrast to traditional membrane energies. The bending contribution is augmented by a curvature term representing the interaction of the rotational microstructure in the Cosserat theory. The model includes non-classical size effects, transverse shear resistance, drilling degrees of freedom and accounts implicitly for thickness extension and asymmetric shift of the midsurface. Upon linearization with zero Cosserat couple modulus μc = 0, one recovers the infinitesimal-displacement Reissner–Mindlin model. It is shown that the Cosserat shell formulation admits minimizers even for μc = 0, in which case the drill-energy is absent. The midsurface deformation m is found in H1(ω, ℝ3). Since the existence of energy minimizers rather than equilibrium solutions is established, the proposed analysis includes the large deformation/large rotation buckling behaviour of thin shells.

[1]  M. Ortiz,et al.  The morphology and folding patterns of buckling-driven thin-film blisters , 1994 .

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

[3]  P. Podio-Guidugli,et al.  Extreme elastic deformations , 1991 .

[4]  Clifford Ambrose Truesdell,et al.  Exact theory of stress and strain in rods and shells , 1957 .

[5]  Équations aux dérivées partielles et applications , 2008 .

[6]  J. C. Simo,et al.  A justification of nonlinear properly invariant plate theories , 1993 .

[7]  P. Podio-Guidugli,et al.  An exact derivation of the thin plate equation , 1989 .

[8]  Rolf Stenberg,et al.  A new finite element formulation for the plate bending problem , 1993 .

[9]  G. Friesecke,et al.  A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity , 2002 .

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

[11]  Alexander Mielke,et al.  On the justification of plate theories in linear elasticity theory using exponential decay estimates , 1995 .

[12]  Patrizio Neff,et al.  Local existence and uniqueness for a geometrically exact membrane‐plate with viscoelastic transverse shear resistance , 2005 .

[13]  H. Cohen,et al.  A mathematical analysis of the simplest direct models for rods and shells , 1989 .

[14]  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.

[15]  Antonio DeSimone,et al.  Folding energetics in thin-film diaphragms , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[16]  M. Ortiz,et al.  Delamination of Compressed Thin Films , 1997 .

[17]  P. Neff.  Local existence and uniqueness for quasistatic finite plasticity with grain boundary relaxation , 2005 .

[18]  Oscillatory Thermomechanical Instability of an Ultrathin Catalyst , 2003, Science.

[19]  Bernadette Miara,et al.  Nonlinearly Elastic Shell Models: A Formal Asymptotic Approach II. The Flexural Model , 1998 .

[20]  P. Neff.  On Korn's first inequality with non-constant coefficients , 2002, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[21]  P. M. Naghdi,et al.  A general theory of a Cosserat surface , 1965 .

[22]  P. Neff.  A finite-strain elastic–plastic Cosserat theory for polycrystals with grain rotations , 2006 .

[23]  Carlo Sansour,et al.  The Cosserat surface as a shell model, theory and finite-element formulation , 1995 .

[24]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model. Part VI: Conserving algorithms for non‐linear dynamics , 1992 .

[25]  J. C. Simo,et al.  A drill rotation formulation for geometrically exact shells , 1992 .

[26]  A. Raoult,et al.  The membrane shell model in nonlinear elasticity: A variational asymptotic derivation , 1996 .

[27]  S. Antman.  Nonlinear problems of elasticity , 1994 .

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

[29]  W. Pompe.  Korn's First Inequality with variable coefficients and its generalization , 2003 .

[30]  I. Babuska,et al.  Finite Element Analysis , 2021 .

[31]  Philippe G. Ciarlet,et al.  Ellipticity of bending and membrane shell equations , 1995 .

[32]  R. Monneau.  Justification of the Nonlinear Kirchhoff-Love Theory of Plates as the Application of a New Singular Inverse Method , 2003 .

[33]  A. Eringen.  Theory of micropolar plates , 1967 .

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

[35]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model. Part V: Nonlinear plasticity: formulation and integration algorithms , 1992 .

[36]  Patrizio Neff,et al.  Existence of minimizers for a finite-strain micromorphic elastic solid , 2006, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[37]  Douglas N. Arnold,et al.  On the Range of Applicability of the Reissner–Mindlin and Kirchhoff–Love Plate Bending Models , 2002 .

[38]  R. James,et al.  A theory of thin films of martensitic materials with applications to microactuators , 1999 .

[39]  H. Cohen,et al.  Nonlinear theory of elastic surfaces. , 1966 .

[40]  L. D. Marini,et al.  A nonconforming element for the Reissner–Mindlin plate , 2003 .

[41]  Philippe G. Ciarlet,et al.  Asymptotic methods for elastic structures : proceedings of the international conference, Lisbon, Portugal, October 4-8, 1993 , 1995 .

[42]  M. Rubin.  Cosserat Theories: Shells, Rods and Points , 2000 .

[43]  M. Gurtin,et al.  On the characterization of geometrically necessary dislocations in finite plasticity , 2001 .

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

[45]  P. Neff,et al.  The Cosserat couple modulus for continuous solids is zero viz the linearized Cauchy‐stress tensor is symmetric , 2006 .

[46]  R. Taylor,et al.  Theory and finite element formulation of rubberlike membrane shells using principal stretches , 1992 .

[47]  Philippe Destuynder,et al.  Mathematical Analysis of Thin Plate Models , 1996 .

[48]  J. Ball.  Convexity conditions and existence theorems in nonlinear elasticity , 1976 .

[49]  Carlo Sansour,et al.  On hybrid stress, hybrid strain and enhanced strain finite element formulations for a geometrically exact shell theory with drilling degrees of freedom , 1998 .

[50]  Phill-Seung Lee,et al.  On the asymptotic behavior of shell structures and the evaluation in finite element solutions , 2002 .

[51]  Patrizio Neff,et al.  A geometrically exact viscoplastic membrane-shell with viscoelastic transverse shear resistance avoiding degeneracy in the thin-shell limit. , 2005 .

[52]  D. Steigmann,et al.  Tension-field theory , 1990, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[53]  H. Cohen,et al.  Nonlinear theory of elastic directed surfaces. , 1966 .

[54]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model. Part II: the linear theory; computational aspects , 1989 .

[55]  Peter Wriggers,et al.  Theory and numerics of thin elastic shells with finite rotations , 1989 .

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

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

[58]  J. L. Sanders,et al.  Theory of thin elastic shells , 1982 .

[59]  F. Brezzi,et al.  On drilling degrees of freedom , 1989 .

[60]  Peter Wriggers,et al.  Thin shells with finite rotations formulated in biot stresses : theory and finite element formulation , 1993 .

[61]  D. Chapelle,et al.  The Finite Element Analysis of Shells - Fundamentals , 2003 .