Elastic surface—substrate interactions

A theory for three–dimensional finite deformations of elastic solids with conforming elastic films attached to their bounding surfaces is described. The Gurtin–Murdoch theory incorporating elastic resistance of the film to strain is generalized to account for the effects of intrinsic flexural resistance. This modification yields a model that can be used to describe equilibrium deformations in the presence of compressive–surface stress fields. An associated variational theory is given and material symmetry considerations are discussed. The theory is illustrated by examples.

[1]  H. Bufler,et al.  Large strain analysis of rubber-like membranes under dead weight, gas pressure, and hydrostatic loading , 1994 .

[2]  Giorgio Vergara Caffarelli,et al.  Surface interaction potentials in elasticity , 1990 .

[3]  R. Ogden,et al.  The Effect of Elastic Surface Coating on the Finite Deformation and Bifurcation of a Pressurized Circular Annulus , 1997 .

[4]  W. Pietraszkiewicz,et al.  Work-Conjugate Boundary Conditions in the Nonlinear Theory of Thin Shells , 1989 .

[5]  A. Pipkin,et al.  Plane deformations of nets with bending stiffness , 1987 .

[6]  H. Cohen,et al.  On a nonlinear theory of elastic shells. , 1968 .

[7]  J. L. Sanders,et al.  NONLINEAR THEORIES FOR THIN SHELLS , 1963 .

[8]  Plane Infinitesimal Waves in Elastic Sheets with Bending Stiffness , 1997 .

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

[10]  J. Ericksen.  Wave propagation in thin elastic shells , 1971 .

[11]  H. Ziegler,et al.  On the Concept of Elastic Stability , 1956 .

[12]  M. Gurtin,et al.  On the inclusion of the complete symmetry group in the unimodular group , 1966 .

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

[14]  M. Epstein,et al.  The symmetry group of second-grade materials , 1992 .

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

[16]  P. M. Naghdi,et al.  The influence of the reference geometry on the response of elastic shells , 1972 .

[17]  H. Cohen,et al.  On the response and symmetry of elastic and hyperelastic membrane points , 1984 .

[18]  M. Gurtin,et al.  Addenda to our paper A continuum theory of elastic material surfaces , 1975 .

[19]  Configuration dependent pressure potentials , 1988 .

[20]  G. Kirchhoff,et al.  Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. , 1850 .

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

[22]  H. Whitney.  Geometric Integration Theory , 1957 .

[23]  A note on pressure potentials , 1991 .

[24]  A. Pipkin,et al.  Bending energy of highly elastic membranes II , 1996 .

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

[26]  N. Meyers.  QUASI-CONVEX1TY AND LOWER SEMI-CONTINUITY OF MULTIPLE VARIATIONAL INTEGRALS OF ANY ORDER , 2010 .

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

[28]  E. D. Giorgi,et al.  Γ — Convergence and calculus of variations , 1983 .

[29]  Walter Noll,et al.  A mathematical theory of the mechanical behavior of continuous media , 1958 .

[30]  J. G. Simmonds,et al.  The Nonlinear Theory of Elastic Shells , 1998 .

[31]  R. Hill.  Uniqueness criteria and extremum principles in self-adjoint problems of continuum mechanics , 1962 .

[32]  A. Pipkin,et al.  ELASTIC SHEETS WITH BENDING STIFFNESS , 1992 .

[33]  M. G. Faulkner,et al.  Variational theory for spatial rods , 1993 .

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

[35]  Maïté Carrive-Bedouani,et al.  Approximations par éléments finis d'un modèle de coques minces géométriquement exact , 1995 .

[36]  A. Murdoch.  Symmetry considerations for materials of second grade , 1979 .

[37]  Q.-s. Zheng,et al.  Two-dimensional tensor function representation for all kinds of material symmetry , 1993, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[38]  P. M. Naghdi,et al.  Simple Force Multipoles in the Theory of Deformable Surfaces , 1967 .

[39]  R. Ogden.  Non-Linear Elastic Deformations , 1984 .

[40]  J. G. Simmonds,et al.  The strain energy density of rubber-like shells , 1985 .

[41]  C. Bert,et al.  Buckling of a coating bonded to a round bar subjected to axial extension , 1993 .

[42]  M. Gurtin,et al.  On the wrinkling of a free surface , 1977 .

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

[44]  D. Steigmann,et al.  Equilibrium of prestressed networks , 1992 .

[45]  A. Pipkin,et al.  Energy-minimizing deformations of elastic sheets with bending stiffness , 1993 .

[46]  R. Howe,et al.  Stress and Microstructure in Lpcvd Polycrystalline Silicon Films: Experimental Results and Closed Form Modeling of Stresses , 1991 .

[47]  Proof of a Conjecture in Elastic Membrane Theory , 1986 .

[48]  A. Pipkin,et al.  Bending energy of highly elastic membranes , 1992 .

[49]  S. Brendle,et al.  Calculus of Variations , 1927, Nature.

[50]  A. Murdoch,et al.  Symmetry considerations for material surfaces , 1979 .

[51]  Jerrold E. Marsden,et al.  Quasiconvexity at the boundary, positivity of the second variation and elastic stability , 1984 .

[52]  L. Graves.  The Weierstrass condition for multiple integral variation problems , 1939 .

[53]  R. Ogden,et al.  Plane deformations of elastic solids with intrinsic boundary elasticity , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[54]  C. Pearson.  General theory of elastic stability , 1956 .

[55]  David J. Steigmann,et al.  Equilibrium of elastic nets , 1991, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.