Skip to main content
SpringerLink
Log in
SpringerLink
Log in

An introduction to Λ-convergence methods for thin structures

  • Chapter
Classical and Advanced Theories of Thin Structures

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

  • 743 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press Oxford, (2000).

    MATH  Google Scholar 

  2. L. Ambrosio, A. Coscia & G. Dal Maso, Fine properties of functions with bounded deformation, Arch.Rat. Mech.Anal., 139,3(1997), 201–238.

    Article  MATH  Google Scholar 

  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. 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.

    MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  8. A. Braides & I. Fonseca, Brittle thin films, Appl.Math.Opt. 44(2001) 3, 299–323.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Braides, I. Fonseca & G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films, Indiana Univ.Math.J 49(2000) 4, 1367–1404.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Carriero, A. Leaci & F. Tomarelli, Plastic free discontinuities and special bounded hessian, C. R. Acad. Sci. Paris, 314(1992), 595–600.

    MATH  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  12. M. Carriero, A. Leaci & F. Tomarelli, Strong solution for an Elastic Plastic Plate, Calc. Var., 2(1994), 219–240.

    Article  MATH  MathSciNet  Google Scholar 

  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. V. Casarino, D. Percivale, A variational model for non linear elastic plates, J. Convex Anal., 3(1996) 221–243.

    MATH  MathSciNet  Google Scholar 

  15. P.G. Ciarlet, Mathematical Elasticity, vol II: Theory of Plates, Studies in Math. and its Appl., North-Holland, (1997).

    Google Scholar 

  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. G. Dal Maso An Introduction to Λ — convergence. Birkhauser (1993)

    Google Scholar 

  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. 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.

    MATH  MathSciNet  Google Scholar 

  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. G.Del Piero & D.R. Owen, Structured deformations of continua, Arch.Rat.Mech. Anal., 1241993, 99–155.

    Google Scholar 

  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. G. Francfort & J.J. Marigo, Revisiting brittle fracture as an energy minimization problem, J.Mech.Phys. Solids, 46(1998), 1319–1342.

    Article  MATH  MathSciNet  Google Scholar 

  24. A.A. Griffith, The phenomenon of rupture and flow in solids, Phyl.Trans. Roy.Soc. A,221, (1920), 163–198.

    Article  Google Scholar 

  25. G. Kirchhoff, Über das Gleichgewicht und die Bewegung einer elastischen Scheibe, J.Reine Angew.Math40(1850), 51–88

    MATH  Google Scholar 

  26. D. Percivale, Perfectly Plastic Plates: a variational definition J.Reine Angew.Math.411(1990), 39–50.

    MathSciNet  Google Scholar 

  27. D. Percivale, Thin elastic beams: St.Venant’s problem, Asympt. Anal.,20,(1999),39–60.

    MATH  MathSciNet  Google Scholar 

  28. D. Percivale & F. Tomarelli Scaled Korn-Poincaré inequality in BD and a model of elastic plastic cantilever, Asymptotic Analysis,23, (2000) 291–311.

    MATH  MathSciNet  Google Scholar 

  29. D. Percivale & F. Tomarelli From SBD to SBH: the elastic plastic plate, Interfaces and Free Boundaries 4(2002), 137–165.

    MATH  MathSciNet  Google Scholar 

  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. G. Savarè & F. Tomarelli, Superposition and Chain Rule for Bounded Hessian Functions, Advances in Math.140(1998), 237–281.

    Article  MATH  Google Scholar 

  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. R. Temam, Problèmes Mathematiques en Plasticité, Gauthier-Vllars, (1983), Paris.

    Google Scholar 

  34. R. Temam & G. Strang Functions of bounded deformation, Arch.Rat.Mech. Anal.,75 (1980), 7–21.

    Article  MATH  MathSciNet  Google Scholar 

  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), 322–332

    Google Scholar 

  36. P. Villaggio: Qualitative Methods in Elasticity, Nordhoff (1977).

    Google Scholar 

Download references

Author information

Authors and Affiliations

    Authors
    1. Danilo Percivale
      View author publications

      You can also search for this author in PubMed Google Scholar

    Editor information

    Editors and Affiliations

    1. University of Udine, Italy

      Antonino Morassi

    2. University of Sassari, Italy

      Roberto Paroni

    Rights and permissions

    Reprints and permissions

    Copyright information

    © 2008 CISM, Udine

    About this chapter

    Cite this chapter

    Percivale, D. (2008). An introduction to Λ-convergence methods for thin structures. In: Morassi, A., Paroni, R. (eds) Classical and Advanced Theories of Thin Structures. CISM International Centre for Mechanical Sciences, vol 503. Springer, Vienna. https://doi.org/10.1007/978-3-211-85430-3_3

    Download citation

    • DOI: https://doi.org/10.1007/978-3-211-85430-3_3

    • Publisher Name: Springer, Vienna

    • Print ISBN: 978-3-211-85429-7

    • Online ISBN: 978-3-211-85430-3

    • eBook Packages: EngineeringEngineering (R0)

    Publish with us

    Policies and ethics

    Search

    Navigation