Variational Convergence for Nonlinear Shell Models with Directors and Related Semicontinuity and Relaxation Results

Abstract: We use a variational convergence method to study the consistency of various Cosserat hypotheses in shell theory with the limit nonlinear membrane model derived from three-dimensional elasticity. In the course of the analysis, we introduce a generalization of quasiconvexity that is suitable for problems of the calculus of variations with two vectorial unknowns, one of which appears through its gradient, the other one through its value, in a weak W1,p×Lp framework.

[1]  Stuart S. Antman,et al.  The Theory of Rods , 1973 .

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

[3]  Paolo Marcellini,et al.  Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals , 1985 .

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

[5]  J. Paumier.  Asymptotic Consistency Of The Polynomial Approximation In The Linearized Plate Theory. Application T , 1997 .

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

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

[8]  Bernard Dacorogna,et al.  Quasiconvexity and relaxation of nonconvex problems in the calculus of variations , 1982 .

[9]  J. Marsden,et al.  The limits of hamiltonian structures in three-dimensional elasticity, shells, and rods , 1996 .

[10]  I. Fonseca,et al.  3D-2D asymptotic analysis of an optimal design problem for thin films , 1998 .

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

[12]  Irene Fonseca,et al.  A -Quasiconvexity. lower semicontinuity, and young measures , 1999 .

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

[14]  J. Ball,et al.  W1,p-quasiconvexity and variational problems for multiple integrals , 1984 .

[15]  Paolo Marcellini,et al.  Semicontinuity problems in the calculus of variations , 1980 .

[16]  Philippe G. Ciarlet,et al.  Mathematical elasticity. volume II, Theory of plates , 1997 .

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

[18]  H. Attouch.  Variational convergence for functions and operators , 1984 .

[19]  C. C. Wang,et al.  Introduction to Rational Elasticity , 1973 .

[20]  P. Ciarlet,et al.  Mathematical elasticity, volume I: Three-dimensional elasticity , 1989 .

[21]  A. Pipkin.  Relaxed energy densities for large deformations of membranes , 1993 .

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

[23]  Jerrold E. Marsden,et al.  The Limits of Hamiltonian Structures in Three Dimensional Elasticity , Shells , and Rods , 1996 .

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

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

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

[27]  Hervé Le Dret,et al.  The quasiconvex envelope of the Saint Venant–Kirchhoff stored energy function , 1995, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[28]  G. Buttazzo,et al.  A variational definition of the strain energy for an elastic string , 1991 .

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

[30]  Philippe G. Ciarlet,et al.  A Justi cation of a Nolinear Model in Plate Theory , 1979 .