Advertisement

Γ-convergene e for a geometrically exact Cosserat shell-model of defective elastic crystals

  • Patrizio Neff
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 516)

Abstract

I consider the Γ-limit to a three-dimensional Cosserat model as the aspect ratio h > 0 of a flat domain tends to zero. The bulk model involves already exact rotations as a second independent field intended to describe the rotations of the lattice in defective elastic crystals. The Γ-limit based on the natural scaling consists of a membrane like energy and a. transverse shear energy both scaling with h, augmented by a curvature energy due to the Cosserat bulk, also scaling with h. A technical difficulty is to establish equi-coercivity of the sequence of functional as the aspect ratio h tends to zero. Usually, equi-coercivity follows from a local coerciveness assumption. While the three-dimensional problem is well-posed for the Cosserat couple modulus μc ≥0, equi-coercivity needs a. strictly positive μc > 0. Then the Γ-limit model determines the midsorfaee deformation mH 1,2 (ω, ℝ3). For the true defective crystal case, however, μc=0 is appropriate. Without equi-coercivity, we obtain first an estimate of the Γ-lim in and Γ-lim sup which can be strengthened to the Γ-convergence result. The Reissner-Mindlin model is “almost” the linearization of the Γ-limit for μc=0.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. I. Aganovic, J. Tambaca, and Z. Tutek. Derivation and justification of the models of rods and plates from linearized three-dimensional micropolar elasticity. J. Elasticity, 84:131–152, 2007a.CrossRefMathSciNetGoogle Scholar
  2. I. Aganovic, J. Tambaca, and Z. Tutek. Derivation and justification of the model of micropolar elastic shells from three-dimensional linearized micropolar elasticity. Asympt. Anal., 51:335–361, 2007b.zbMATHMathSciNetGoogle Scholar
  3. H. Altenbach and P.A. Zhilin. The theory of simple elastic shells. In R. Kienzler, H. Altenbach, and I. Ott, editors, Theories of Plates and Shells. Critical Review and New Applications, Euromech Colloquium 444, pages 1–12. Springer, Heidelberg, 2004.Google Scholar
  4. S. Antman. Nonlinear Problems of Elasticity., volume 107 of Applied Mathematical Sciences. Springer, Berlin, 1995.zbMATHGoogle Scholar
  5. G. Anzellotti, S. Baldo, and D. Percivale. Dimension reduction in variational problems, asymptotic development in Γ-convergence and thin structures in elasticity. Asymptotic Anal., 9:61–100, 1994.zbMATHMathSciNetGoogle Scholar
  6. I. Babuska and L. Li. The problem of plate modelling: theoretical and computational results. Comp. Meth. Appl. Mech. Engrg., 100:249–273, 1992.zbMATHCrossRefGoogle Scholar
  7. T. Belytscbko, W.K. Liu, and B. Moran. Nonlinear Finite Elements for Continua and Structures. Wiley, Chichester, 2000.Google Scholar
  8. P. Betsch, P. Gruttmann, and E. Stein. A 4-node finite shell element for the implementation of general hyperelastic 3d-elasticity at finite strains. Comp. Meth. Appl Mech. Engrg., 130:57–79, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  9. K. Bhattacharya and R.D. James. A theory of thin films of martensitic materials with applications to microacLualors. J. Mech. Phys. Solids, 47:531–576, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  10. F. Bourquin, P.G. Ciarlet, G. Geymonat, and A. Raoult. Γ-convergence et analyse asymptotique cles plaques minces. C. R. Acad. Sci. Paris, Ser. I, 315:1017–1024, 1992.zbMATHMathSciNetGoogle Scholar
  11. A. Braides. Γ-Convergence for Beginners. Oxford University Press, Oxford, 2002.zbMATHCrossRefGoogle Scholar
  12. N. Büchter and E. Ramm. Shell theory versus degeneration-a comparison in large rotation finite element analysis, Int. J. Num. Meth. Engrg., 34: 39–59, 1992.zbMATHCrossRefGoogle Scholar
  13. G. C apriz. Continua with Microstructure. Springer, Heidelberg, 1989.Google Scholar
  14. P.G. Ciarlet. Mathematical Elasticity, Vol II: Theory of Plates. North-Holland, Amsterdam, first edition, 1997.Google Scholar
  15. P.G. Ciarlet. Introduction to Linear Shell Theory. Series in Applied Mathematics. Gauthier-Villars, Paris, first edition, 1998.Google Scholar
  16. P.G. Ciarlet. Mathematical Elasticity, Vol III: Theory of Shells. North-Holland, Amsterdam, first edition, 1999.Google Scholar
  17. H. Cohen and C.N. DeSilva. Nonlinear theory of elastic surfaces. J. Mathematical Phys., 7:246–253, 1966a.zbMATHCrossRefMathSciNetGoogle Scholar
  18. H. Cohen and C.N. DeSilva. Nonlinear theory of elastic directed surfaces. J. Mathematical Phys., 7:960–966, 1966b.zbMATHCrossRefMathSciNetGoogle Scholar
  19. H. Cohen and C.C. Wang. A mathematical analysis of the simplest direct models for rods and shells. Arch. Rat. Mech. Anal, 108:35–81, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  20. E. Cosserat and F. Cosserat. Théorie des corps déformables. Librairie Scientifique A. Hermann et Pils (engl, translation by D. Delpb ersieh 2007, pdf available at http://www.matheraatik.tudarmstadt.de/fbereiche/anarysis/pde/staff/neff/patrizio/CosseraL html), Paris, 1909.Google Scholar
  21. P. Destuyrider and M. Saiauri. Mathematical Analysis of Thin Plate Models. Springer, Berlin, 1996.Google Scholar
  22. M. Dikmen. Theory of Thin Elastic Shells. Pitman, London, 1982.zbMATHGoogle Scholar
  23. H. Le Dret and A. Raoult. The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Science, 6:59–84, 1996.zbMATHCrossRefGoogle Scholar
  24. H. Le Dret and A. Raoult. The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl., 74: 549–578, 1995.zbMATHMathSciNetGoogle Scholar
  25. J.L. Ericksen and C. Truesdell. Exact theory of stress and strain in rods and shells. Arch. Rat Mech. Anal, 1:295–323, 1958.zbMATHCrossRefMathSciNetGoogle Scholar
  26. I. Ponseca and G. Francfort. On the inadequacy of the sealing of linear elasticity for 3d-2d asymptotics in a nonlinear setting. J. Math. Pures Appl., 80(5):547–562, 2001.CrossRefMathSciNetGoogle Scholar
  27. D.D. Fox and J.C. Simo. A drill rotation formulation for geometrically exact shells. Comp. Meth. Appl. Mech. Eng., 98:329–343, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  28. D.D. Fox, A. Raoult, and J.C. Simo. A justification of nonlinear properly invariant plate theories. Arch. Rat. Mech. Anal., 124:157–199, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  29. L. Freddi and R. Paroni. The energy density of martensitic thin films via dimension reduction. Interfaces Free Boundaries, 6:439–459, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  30. G. Friesecke, R.D. James, and S. Müller. A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math., 55(11):1461–1506, 2002a.zbMATHCrossRefMathSciNetGoogle Scholar
  31. G. Friesecke, R.D. James, and S. Müller. The Föppl-von Kármán plate theory as a low energy Γ-limit of nonlinear elasticity. C. R. Math. Acad. Sci. Paris, 335(2):201–206, 2002b.zbMATHMathSciNetGoogle Scholar
  32. G. Friesecke, R.D. James, M.G. Mora, and S. Müller. Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Γ-convergence. C. R. Math. Acad. Sci. Paris. 336(8):697–702, 2003.zbMATHMathSciNetGoogle Scholar
  33. G. Friesecke, R.D. James, and S. Müller. A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence. Arch. Rat. Mech. Anal., 180:183–236, 2006.zbMATHCrossRefGoogle Scholar
  34. K. Genevey. Asymptotic analysis of shells via Γ-convergence. J. Comput. Math., 18:337–352, 2000.zbMATHMathSciNetGoogle Scholar
  35. G. Geymoriat, S. Müller, and N. Triantafyllidis. Homogenization of nonlin-early elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity. Arch. Rat. Mech. Anal., 122:231–290, 1993.CrossRefGoogle Scholar
  36. E. De Giorgi. Sulla convergenza. di alcune successioni di integral del tipo dell’ area. Rend. Mat. Roma, 8:227–294, 1975.Google Scholar
  37. E. De Giorgi. Γ-convergenza e G-convergenza. Boll. Un. Mat. Ital., 5: 213–220, 1977.MathSciNetGoogle Scholar
  38. A.E. Green, P.M. Naghdi, and W.L. Wainwright. A generai theory of a Cosserat surface. Arch. Rat. Mech. Anal., 20:287–308, 1965.CrossRefMathSciNetGoogle Scholar
  39. F. Gruttmann and R.L. Taylor. Theory and finite element formulation of rubberlike membrane shells using principle stretches. Int. J. Num. Meth. Engrg., 35:1111–1126, 1992.zbMATHCrossRefGoogle Scholar
  40. P. Gruttmann, E. Stein, and P. Wriggers. Theory and numerics of thin elastic shells with finite rotations. Ing. Arch., 59:54–67, 1989.CrossRefGoogle Scholar
  41. T.J.R. Hughes and F. Brezzi. On drilling degrees of freedom. Comp. Meth. Appl. Mech. Engrg., 72:105–121, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  42. V. Lods and B. Miara. Nonlinearly elastic shell models: a formal asymptotic approach. II. The flexural model. Arch. Rat. Mech. Anal., 142:355–374, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  43. G. Dal Maso. Introduction to Γ-Convergence. Birkhäuser, Boston, 1992.Google Scholar
  44. B. Miara. Nonlinearly elastic shell models: a formal asymptotic approach. I. The membrane model. Arch. Rat Mech. Anal, 142:331–353, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  45. R.D. Mindlin. Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. Trans. ASME, J. Appl. Mech. 18:31–38, 1951.zbMATHGoogle Scholar
  46. P.M. Naghdi. The theory of shells and plates. In Handbuch der Physik, Mechanics of Solids, volume VI a/2. Springer, 1972.Google Scholar
  47. P. Neff. On Korn’s first inequality with nonconstant coefficients. Proc. Roy. Soc. Edinb. A, 132:221–243, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  48. P. Neff. Finite multiplicative elastic-viscoplastic Cosserat micropola theory for poly crystals with grain rotations. Modelling and mathematical analysis. Preprint 2297, http://www3. mathemaiik. tudarmstadt. de/fb/MOJJIE/bibliothek/preprinis.htrrd. appeared partly in Int. J. Eng. SET., 9/2003.Google Scholar
  49. P. Neff. A finites-train elastic-plastic Cosserat theory for poly crystals with grain rotations. Int. J. Eng. Sci., DOI 10.1016/j.ijengsci.2006.04.002, 44:574–594. 2006a.CrossRefMathSciNetGoogle Scholar
  50. P. Neff. The Γ-limit of a finite strain Cosserat model for asymptotically thin domains versus a formal dimensional reduction. In W. Pietraszkiewiecz and C. Szymczak, editors, Shell-Structures: Theory and. Applications., pages 149–152. Taylor and Francis Group, London, 2006b.Google Scholar
  51. P. Neff. Existence of minimizers for a geometrically exact Cosserat solid. Proc. Appl Math. Mech., 4(1):548–549, 2004aCrossRefGoogle Scholar
  52. P. Neff. The Γ-limit of a finite strain Cosserat model for asymptotically thin domains and a consequence for the Cosserat couple modulus. Proc. Appl. Math. Mech., 5(1):629–630, 2005.CrossRefGoogle Scholar
  53. P. Neff. Existence of minimizers for a, finite-strain micromorphic elastic solid. Preprint 2318, http://www3. mathemaiik. tudarrasiadi.de/fb/rrmthe/bibliMhek/preprints.himl, Proc. Roy. Soc. Edinb. A, 136:997–1012, 2006c.zbMATHMathSciNetGoogle Scholar
  54. P. Neff. A geometrically exact Cosserat-sheil model including size effects, avoiding degeneracy in the thin shell limit. Part I: Formal dimensional reduction for elastic plates and existence of minimizers for positive Cosserat couple modulus. Cord. Mech. Thermodynamics, 16(6 (DOI 10.1007/s00161-004-0182-4)):577–628, 2004b.zbMATHCrossRefMathSciNetGoogle Scholar
  55. P. Neff. A geometrically exact planar Cosserat shell-model with microstructure. Existence of minimizers for zero Cosserat couple modulus. Preprint 2357, http://www3.mathematik.tu-darmstadt.de/fb/mathe/bibliothek/preprints.html, Math. Mod. Meth. Appl. Sci.(M3AS), 17(3):363–392, 2007.zbMATHCrossRefMathSciNetGoogle Scholar
  56. P. Neff and K. Chelmiński. A geometrically exact Cosserat shell-model including size effects, avoiding degeneracy in the thin shell limit. Rigourous justification via Γ-convergence for the elastic plate. Preprint 2365, http://www3.mathematik.tu-darmstadt.de/fb/mathe/bibliothek/preprints.html, 10/2004.Google Scholar
  57. P. Neff and S. Forest. A geometrically exact micromorphic model for elastic metallic foams accounting for affine raicrostructure. Modelling, existence of minimizers, identification of moduli and computational results. J. Elasticity, 87:239–276, 2007.zbMATHCrossRefMathSciNetGoogle Scholar
  58. P. Neff and I Münch. Curl bounds Grad on SO(3). Preprint 2455, http://www3.mathematik.tu-darmistadt.de/fb/mathe/bibliothek/preprints.html, ESAIM: Control., Optimisation and Calculus of Variations, published online, DOI: 10.1051/cocv:2007050, 14(1):148–159, 2008.zbMATHCrossRefGoogle Scholar
  59. W. Pompe. Korn’s first inequality with variable coefficients and its generalizations. Comment. Math. Univ. Carolinae, 44,1:57–70, 2003.zbMATHMathSciNetGoogle Scholar
  60. E. Reissner. The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech., 12:A69–A77, 1945.MathSciNetGoogle Scholar
  61. E. Reissner. Reflections on the theory of elastic plates. Appl. Mech. Rev., 38(2):1453–1464, 1985.CrossRefGoogle Scholar
  62. A. Rössle. On the derivation of an asymptotically correct shear correction factor for the Reissner-Mindlin plate model. C. R. Acad. Sei. Paris, Ser. I, Math., 328(3):269–274, 1999.zbMATHGoogle Scholar
  63. M.B. Rubin. Cosserat Theories: Shells, Rods and Points. Kluwer Academic Publishers, Dordrecht, 2000.zbMATHGoogle Scholar
  64. C. Sansour. A theory and finite element formulation of shells at finite deformations including thickness change: circumventing the use of a rotation tensor. Arch. Appl. Mech., 10:194–216, 1995.CrossRefGoogle Scholar
  65. C. Sansour and H. Bednarczyk. The Cosserat surface as a shell model, theory and finite element formulation. Comp. Meth. Appl. Mech. Eng., 120:1–32, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  66. C. Sansour and J. Bocko. On hybrid stress, hybrid strain and enhanced strain finite element formulations for a geometrically exact shell theory with drilling degrees of freedom, Int. J. Num. Meth. Engrg., 43:175–192, 1998.zbMATHCrossRefGoogle Scholar
  67. C. Sansour and H. Buffer. An exact finite rotation shell theory, its mixed variational formulation and its finite element implementation. Int. J. Num. Meth. Engrg., 34:73–115, 1992.zbMATHCrossRefGoogle Scholar
  68. Y.C. Shuh. Heterogeneous thin films of martensitic materials. Arch. Mat. Meek. Anal., 153:39–90, 2000.CrossRefGoogle Scholar
  69. J.C. Simo and D.D. Fox. On a stress resultant geometrically exact shell model. Part I: Formulation and optima,! parametrization. Comp. Meth. Appl Mech. Eng., 72:267–304, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  70. J.C Simo and D.D. Fox. On a stress resultant geometrically exact shell model. Part VI: Conserving algorithms for non-linear dynamics. Comp. Meth. Appl. Mech. Eng., 34:117–164, 1992.zbMATHMathSciNetGoogle Scholar
  71. J.C Simo and J.G. Kennedy. On a stress resultant geometrically exact shell model. Part V: Nonlinear plasticity: formulation and integration algorithms. Comp. Meth. Appl. Mech. Eng., 96:133–171, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  72. J.C. Simo, D.D. Fox, and M.S. Rifai. On a stress resultant geometrically exact shell model. Part II: The linear theory; computational aspects. Comp. Meth. Appl. Mech. Eng., 73:53–92, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  73. J.C. Simo, D.D. Fox, and M.S. Rifai. On a stress resultant geometrically exact shell model. Part III: Computational aspects of the nonlinear theory. Comp. Meth. Appl. Meek. Eng., 79:21–70, 1990a.zbMATHCrossRefMathSciNetGoogle Scholar
  74. J.C Simo, M.S. Rifai, and D.D. Fox. On a stress resultant geometrically exact shell model. Part IV: Variable thickness shells with through the thickness stretching. Comp. Meth. Appl. Meek. Eng., 81:91–126, 1990b.zbMATHCrossRefMathSciNetGoogle Scholar
  75. D.J. Steigmann. Tension-field theory. Proc. R. Soc. London A, 429:141–173, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  76. R. Temam. Mathematical Problems in Plasticity. Gauthier-Villars, New-York, 1985.Google Scholar
  77. K. Wisriiewski and E. Turska. Warping arid in-plane twist parameters in kinematics of finite rotation shells. Comp. Meth. Appl. Meek. Engng., 190:5739–5758, 2001.CrossRefGoogle Scholar
  78. K. Wisniewski and E. Turska. Second-order shell kinematics implied by rotation constraint-equation. J. Elasticity, 67:229–246, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  79. P. Wriggers and F. Gruttmann. Thin shells with finite rotations formulated in Biot stresses: Theory and finite element formulation. Int. J. Num. Meth. Engrg., 36:2049–2071, 1993.zbMATHCrossRefGoogle Scholar
  80. N. Zaafarani, D. Raabe, R.N. Singh, F. Roters, and S. Zaefferer. Three-dimensional investigation of the texture and microstructure below a nanoinclent in a Cu single crystal using 3D EBSD and crystal plasticity finite element simulations. Acta Materialia, 54:1863–1876, 2006.CrossRefGoogle Scholar

Copyright information

© CISM, Udine 2010

Authors and Affiliations

  • Patrizio Neff
    • 1
  1. 1.University Duisburg-EssenEssenGermany

Personalised recommendations