Advertisement

An introduction to Λ-convergence methods for thin structures

  • Danilo Percivale
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 503)

Abstract

In these lectures an essential introduction to lower semicontinuity and Λ-convergence basic facts is given together with simple applications to thin structures in elasticity.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [1]
    L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press Oxford, (2000).zbMATHGoogle Scholar
  2. [2]
    L. Ambrosio, A. Coscia & G. Dal Maso, Fine properties of functions with bounded deformation, Arch.Rat. Mech.Anal., 139,3(1997), 201–238.zbMATHCrossRefGoogle Scholar
  3. [3]
    L. Ambrosio & A. Braides, Energies in SBV and variational models in fracture mechanics, Homogeneization and Appl. to Material sciences, (Nice, 1995), 9GAKUTO Int.Ser.Math.Sci Appl.Google Scholar
  4. [4]
    G. Anzellotti, S. Baldo & D. Percivale, Dimension reduction in variational problems, asymptotic development in Λ-convergence and thin structures in elasticity, Asymptotic Anal., 9,(1994) 61–100.zbMATHMathSciNetGoogle Scholar
  5. [5]
    C. Baiocchi, G. Buttazzo, F. Gastaldi & F. Tomarelli, General existence theorems for unilateral problems in continuum mechanics, Arch.Rat.Mech. Anal., 100, 2(1988), 149–189.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    G.I. Barenblatt The formation of equilibrium cracks during brittle fracture, general ideas and hypotheses. Axially symmetric cracks, Appl.Math. Mech. (PMM) 23,(1959), 622–636.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    A. Braides, G. Dal Maso & A. Garroni, Variational formulation of soft-ening phenomena in fracture mechanics: the one-dimensional case, Arch.Rat. Mech. Anal., 146,(1999) 23–58.zbMATHCrossRefGoogle Scholar
  8. [8]
    A. Braides & I. Fonseca, Brittle thin films, Appl.Math.Opt. 44(2001) 3, 299–323.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    A. Braides, I. Fonseca & G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films, Indiana Univ.Math.J 49(2000) 4, 1367–1404.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    M. Carriero, A. Leaci & F. Tomarelli, Plastic free discontinuities and special bounded hessian, C. R. Acad. Sci. Paris, 314(1992), 595–600.zbMATHMathSciNetGoogle Scholar
  11. [11]
    M. Carriero, A. Leaci & F. Tomarelli, Special Bounded Hessian and elastic-plastic plate, Rend. Accad. Naz. delle Scienze (dei XL), (109)XV (1992), 223–258.MathSciNetGoogle Scholar
  12. [12]
    M. Carriero, A. Leaci & F. Tomarelli, Strong solution for an Elastic Plastic Plate, Calc. Var., 2(1994), 219–240.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    M. Carriero, A. Leaci, F. Tomarelli, Second Order Variational Problems with Free Discontinuity and Free Gradient Discontinuity, in: Calculus of Variations: Topics from the Mathematical Heritage of Ennio De Giorgi, D. Pallara (Ed.), Quaderni di Matematica, 14, Series edited by Dipartimento di Matematica, Seconda Università di Napoli, (2004), 135–186.Google Scholar
  14. [14]
    V. Casarino, D. Percivale, A variational model for non linear elastic plates, J. Convex Anal., 3(1996) 221–243.zbMATHMathSciNetGoogle Scholar
  15. [15]
    P.G. Ciarlet, Mathematical Elasticity, vol II: Theory of Plates, Studies in Math. and its Appl., North-Holland, (1997).Google Scholar
  16. [16]
    F. Colombo, F. Tomarelli, Boundary value problems and obstacle problem for elastic bodies with free cracks, in: Calculus of Variations: Topics from the Mathematical Heritage of Ennio De Giorgi, D. Pallara (ed.), Quaderni di Matematica, Series edited by Dipartimento di Matematica, Seconda Universita’ di Napoli, (2004), 221–243.Google Scholar
  17. [17]
    G. Dal Maso An Introduction to Λ — convergence. Birkhauser (1993)Google Scholar
  18. [18]
    E. De Giorgi Free discontinuity problems in calculus of variations, Frontiers in Pure & Applied Mathematics, R. Dautray ed., North-Holland, Amsterdam 1991, 55–61.Google Scholar
  19. [19]
    E. De Giorgi & L. Ambrosio, Un nuovo tipo di funzionale del Calcolo delle Variazioni, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Natur., 82(1988), 199–210.zbMATHMathSciNetGoogle Scholar
  20. [20]
    G. Del Piero, One-dimensional ductile-brittle transition, yielding, and structured deformations Variations of domain and free-boundary problems in solid mechanics (Paris 1997), 203–210, Solid Mech. Appl., 66, Kluwer Acad. Publ., Dordrecht (1999)Google Scholar
  21. [21]
    G.Del Piero & D.R. Owen, Structured deformations of continua, Arch.Rat.Mech. Anal., 1241993, 99–155.Google Scholar
  22. [22]
    G. Del Piero & L. Truskinovsky, A one-dimensional model for localized and distributed failure, J.Phys., IV France, pr. 8, (1998), 95–102.Google Scholar
  23. [23]
    G. Francfort & J.J. Marigo, Revisiting brittle fracture as an energy minimization problem, J.Mech.Phys. Solids, 46(1998), 1319–1342.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    A.A. Griffith, The phenomenon of rupture and flow in solids, Phyl.Trans. Roy.Soc. A,221, (1920), 163–198.CrossRefGoogle Scholar
  25. [25]
    G. Kirchhoff, Über das Gleichgewicht und die Bewegung einer elastischen Scheibe, J.Reine Angew.Math40(1850), 51–88zbMATHGoogle Scholar
  26. [26]
    D. Percivale, Perfectly Plastic Plates: a variational definition J.Reine Angew.Math.411(1990), 39–50.MathSciNetGoogle Scholar
  27. [27]
    D. Percivale, Thin elastic beams: St.Venant’s problem, Asympt. Anal.,20,(1999),39–60.zbMATHMathSciNetGoogle Scholar
  28. [28]
    D. Percivale & F. Tomarelli Scaled Korn-Poincaré inequality in BD and a model of elastic plastic cantilever, Asymptotic Analysis,23, (2000) 291–311.zbMATHMathSciNetGoogle Scholar
  29. [29]
    D. Percivale & F. Tomarelli From SBD to SBH: the elastic plastic plate, Interfaces and Free Boundaries 4(2002), 137–165.zbMATHMathSciNetGoogle Scholar
  30. [30]
    D. Percivale & F. Tomarelli From Special Bonded Deformation to Special Bounded Hessian: the elastic plastic beam, Quad.551/P, Dip.Mat. Politecnico Milano (2003), 1–41. to appear on M 3 AS Google Scholar
  31. [31]
    G. Savarè & F. Tomarelli, Superposition and Chain Rule for Bounded Hessian Functions, Advances in Math.140(1998), 237–281.zbMATHCrossRefGoogle Scholar
  32. [32]
    M.A. Save & C.E. Massonet, Plastic analysis and design of plates, shells and disks, North-Holland Ser. Appl.Math Mech.(1972).Google Scholar
  33. [33]
    R. Temam, Problèmes Mathematiques en Plasticité, Gauthier-Vllars, (1983), Paris.Google Scholar
  34. [34]
    R. Temam & G. Strang Functions of bounded deformation, Arch.Rat.Mech. Anal.,75 (1980), 7–21.zbMATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    L. Truskinowsky Fracture as a phase transition, in R.C. Batra, M.F. Beatty, (eds), Contemporary Research in the Mechanics and Mathematics of Materials (ded.to J.L.Ericksen), C.I.M.N.E., Barcelona, (1996), 322–332Google Scholar
  36. [36]
    P. Villaggio: Qualitative Methods in Elasticity, Nordhoff (1977).Google Scholar

Copyright information

© CISM, Udine 2008

Authors and Affiliations

  • Danilo Percivale

There are no affiliations available

Personalised recommendations