Intrinsic methods in elasticity: a mathematical survey

In the classical approach to elasticity problems, the components of the displacement field are the primary unknowns. In an "intrinsic'' approach, new unknowns with more physical or geometrical meanings, such as a strain tensor field or a rotation field for instance, are instead taken as the primary unknowns. We survey here recent progress about the mathematical analysis of such methods applied to linear and nonlinear three-dimensional elasticity and shell problems.

[1]  D. A. Danielson,et al.  Nonlinear Shell Theory With Finite Rotation and Stress-Function Vectors , 1972 .

[2]  B. D. Veubeke,et al.  A new variational principle for finite elastic displacements , 1972 .

[3]  P. G. Ciarlet,et al.  Injectivity and self-contact in nonlinear elasticity , 1987 .

[4]  Existence of a solution for an unsteady elasticity problem in large displacement and small perturbation , 2002 .

[5]  J. J. Telega,et al.  The complementary energy principle as a dual problem for a specific model of geometrically non-linear elastic shells with an independent rotation vector : general results , 1992 .

[6]  Philippe G. Ciarlet,et al.  An estimate of the H1-norm of deformations in terms of the L1-norm of their Cauchy–Green tensors , 2004 .

[7]  Philippe G. Ciarlet,et al.  A New Approach to the Fundamental Theorem of Surface Theory , 2008 .

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

[9]  P. G. Ciarlet,et al.  An Introduction to Differential Geometry with Applications to Elasticity , 2006 .

[10]  On the characterizations of matrix fields as linearized strain tensor fields , 2006 .

[11]  Robert V. Kohn,et al.  New integral estimates for deformations in terms of their nonlinear strains , 1982 .

[12]  Philippe G. Ciarlet,et al.  On rigid and infinitesimal rigid displacements in three-dimensional elasticity , 2003 .

[13]  V. Girault,et al.  Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension , 1994 .

[14]  Compatibility equations for large deformations , 1992 .

[15]  P. Hartman,et al.  On the Embedding Problem in Differential Geometry , 1950 .

[16]  Sorin Mardare.  ON ISOMETRIC IMMERSIONS OF A RIEMANNIAN SPACE WITH LITTLE REGULARITY , 2004 .

[17]  Stuart S. Antman,et al.  Ordinary differential equations of non-linear elasticity I: Foundations of the theories of non-linearly elastic rods and shells , 1976 .

[18]  P. G. Ciarlet,et al.  Another approach to the fundamental theorem of riemannian geometry in R3, by way of rotation fields , 2007 .

[19]  Philippe G. Ciarlet,et al.  Continuity of a Deformation as a Function of its Cauchy-Green Tensor , 2003 .

[20]  Elie Cartan.  La géométrie des espaces de Riemann , 1925 .

[21]  P. Ciarlet,et al.  A nonlinear Korn inequality on a surface , 2006 .

[22]  Adel Blouza,et al.  Existence and uniqueness for the linear Koiter model for shells with little regularity , 1999 .

[23]  Abdelhafid Hamdouni.  Interprétation géométrique de la décomposition des équations de compatibilité en grandes déformations , 2000 .

[24]  B. Dacorogna.  Direct methods in the calculus of variations , 1989 .

[25]  John Fritz,et al.  Bounds for Deformations in Terms of Average Strains , 1985 .

[26]  Philippe G. Ciarlet,et al.  JUSTIFICATION OF A TWO-DIMENSIONAL NONLINEAR SHELL MODEL OF KOITER＇S TYPE , 2001 .

[27]  P. Ciarlet,et al.  ANOTHER APPROACH TO LINEARIZED ELASTICITY AND A NEW PROOF OF KORN'S INEQUALITY , 2005 .

[28]  P. G. Ciarlet.  Un modèle bi-dimensionnel non linéaire de coque analogue à celui de W.T. Koiter , 2000 .

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

[30]  R. Shield.  The Rotation Associated with Large Strains , 1973 .

[31]  Sorin Mardare.  On Pfaff systems with Lp coefficients and their applications in differential geometry , 2005 .

[32]  On Poincaré's and J.L. Lions' lemmas , 2005 .

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

[34]  Yu. G. Reshetnyak.  Mappings of Domains in Rn and Their Metric Tensors , 2003 .

[35]  C∞-REGULARITY OF A MANIFOLD AS A FUNCTION OF ITS METRIC TENSOR , 2006 .

[36]  Yavuz Başar.  A consistent theory of geometrically non-linear shells with an independent rotation vector , 1987 .

[37]  A NEW APPROACH TO LINEAR SHELL THEORY , 2005 .

[38]  J. Lions,et al.  Les inéquations en mécanique et en physique , 1973 .

[39]  SAINT VENANT COMPATIBILITY EQUATIONS ON A SURFACE APPLICATION TO INTRINSIC SHELL THEORY , 2008 .

[40]  T. Valent,et al.  Boundary Value Problems of Finite Elasticity , 1988 .

[41]  RECOVERY OF A SURFACE WITH BOUNDARY AND ITS CONTINUITY AS A FUNCTION OF ITS TWO FUNDAMENTAL FORMS , 2005 .

[42]  Philippe G. Ciarlet,et al.  An Existence Theorem for Nonlinearly Elastic ‘Flexural’ Shells , 1998 .

[43]  F. John.  Rotation and strain , 1961 .

[44]  W. Pietraszkiewicz.  On Using Rotations as Primary Variables in The Non-linear Theory of Thin Irregular Shells , 2001 .

[45]  A new look at the compatability problem of elasticity theory , 1990 .

[46]  Philippe G. Ciarlet,et al.  On the recovery of a surface with prescribed first and second fundamental forms , 2002 .

[47]  Bernadette Miara.  Nonlinearly Elastic Shell Models: A Formal Asymptotic Approach I. The Membrane Model , 1998 .

[48]  J. Moreau.  Duality characterization of strain tensor distributions in an arbitrary open set , 1979 .

[49]  Jerrold E. Marsden,et al.  On the rotated stress tensor and the material version of the Doyle-Ericksen formula , 1984 .

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

[51]  Sorin Mardare.  On systems of first order linear partial differential equations with $L^p$ coefficients , 2007, Advances in Differential Equations.

[52]  SAINT VENANT COMPATIBILITY EQUATIONS IN CURVILINEAR COORDINATES , 2007 .

[53]  R. Weiner.  Lecture Notes in Economics and Mathematical Systems , 1985 .

[54]  Philippe G. Ciarlet,et al.  The Continuity of a surface as a function of its two fundamental forms , 2003 .

[55]  Philippe G. Ciarlet,et al.  Recovery of a Manifold with Boundary and its Continuity as a Function of its Metric Tensor , 2004 .

[56]  Janusz Badur,et al.  Finite rotations in the description of continuum deformation , 1983 .

[57]  G. Friesecke,et al.  Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence , 2003 .

[58]  Sur la possibilité de plonger un espace riemannien donné dans un espace euclidien , 1927 .

[59]  On rigid and infinitesimal rigid displacements in shell theory , 2004 .

[60]  Charles B. Morrey,et al.  QUASI-CONVEXITY AND THE LOWER SEMICONTINUITY OF MULTIPLE INTEGRALS , 1952 .

[61]  H. Dret.  Structure of the set of equilibrated loads in nonlinear elasticity and applications to existence and nonexistence , 1987 .

[62]  Sorin Mardare.  The Fundamental Theorem of Surface Theory for Surfaces with Little Regularity , 2003 .

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

[64]  Wojciech Pietraszkiewicz,et al.  Intrinsic equations for non-linear deformation and stability of thin elastic shells ? , 2004 .

[65]  A. Raoult,et al.  The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity , 1995 .

[66]  P. G. Ciarlet,et al.  Three-dimensional elasticity , 1988 .