Skip to main content
SpringerLink
Log in

On boundary regularity for the stress in problems of linearized elasto-plasticity

  • Original Paper
  • Published:
International Journal of Advances in Engineering Sciences and Applied Mathematics Aims and scope Submit manuscript

Abstract

We investigate regularity properties of the stress tensor near the boundary for models of elasto-plasticity in arbitrary dimension. Focusing on special geometries, namely on balls and infinite strips, we obtain L 2-estimates for the tangential derivatives of the stress tensor near the boundary. We indicate why these estimates may fail for more general domains. In addition, we establish L 2-estimates for (the trace of) the stress tensor on the boundary.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anzellotti G. and Giaquinta M., Existence of the displacements field for an elastoplastic body subject to Hencky’s law and von Mises yield condition, Manuscripta Math., 32, 101–136 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  2. Anzellotti G. and Giaquinta M., On the existence of the fields of stresses and displacements for an elasto-perfectly plastic body in static equilibrium, J. Math. Pures et Appl., 61, 219–244 (1982)

    MATH  MathSciNet  Google Scholar 

  3. Bensoussan A. and Frehse J., Asymptotic behaviour of Norton-Hoff’s law in plasticity theory and H 1 regularity. In: Boundary value problems for partial differential equations and applications, volume 29 of RMA Res. Notes Appl. Math., pages 3–25. Masson, Paris (1993)

    Google Scholar 

  4. Bensoussan A. and Frehse J., Asymptotic behaviour of the time dependent Norton-Hoff law in plasticity theory and H 1 regularity, Comment. Math. Univ. Carolin., 37, 285–304 (1996)

    MATH  MathSciNet  Google Scholar 

  5. Bildhauer M. and Fuchs M., Smoothness of weak solutions of the Ramberg/Osgood equations on plane domains, Z. Angew. Math. Mech., 87, 70–76 (2009)

    Article  MathSciNet  Google Scholar 

  6. Blum H. and Frehse J., Boundary differentiability for the solution to Hencky’s law of elastic plastic plane stress. Preprint 435 SFB611, University of Bonn (2008)

  7. Bulíček M., Gwiazda P., Málek J. and Świerzcewska-Gwiazda A., On flows of an incompressible fluid with a discontinuous power-law like rheology, Adv. Calc. Var. (to appear) (2009)

  8. Demyanov A., Regularity of stresses in Prandtl-Reuss perfect plasticity, Calc. Var. Partial Differential Equations, 34, 23–72 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Duvaut G. and Lions J.L., Inequalities in Mechanics and Physics, Springer-Verlag, Paris (1976)

    MATH  Google Scholar 

  10. Frehse J. and Málek J., Boundary regularity results for models of elasto-perfect plasticity, Math. Models Methods Appl. Sci., 9(9), 1307–1321 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hardt R.M. and Kinderlehrer D., Some regularity results in plasticity. In: Geometric measure theory and the calculus of variations (Arcata, Calif., 1984), volume 44 of Proc. Sympos. Pure Math., pages 239–244. Amer. Math. Soc., Providence, RI (1986)

    Google Scholar 

  12. Hencky H., Zur Theorie Plastischer Deformationen, Z. Angew. Math. Mech., 4, 323–334 (1924)

    Article  Google Scholar 

  13. Hlaváček I., Haslinger J., Nečas J. and Lovíšek J., Solution of Variational Inequalities in Mechanics, Springer, New York (1988)

    MATH  Google Scholar 

  14. Hoff H.J., Aproximate analysis of structures in the presence of moderately large creep deformations, Quart. Appl. Math., 12, 49 (1954)

    MATH  MathSciNet  Google Scholar 

  15. Hooke R., Lectures de potentia restitutiva or of spring explaining the power of springing bodies, Early Science in Oxford, 8, 331–356 (1678)

    Google Scholar 

  16. Johnson C., Existence theorems for plasticity problems, J. Math. Pures Appl. (9), 55(4), 431–444 (1976)

    MATH  MathSciNet  Google Scholar 

  17. Johnson C., On plasticity with hardening, J.Math. Anal. Appl., 62(2), 325–336 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  18. Knees D., Global regularity of the elastic fields of a power-law model on Lipschitz domains, Math. Methods Appl. Sci., 29(12), 1363–1391 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kohn R. and Temam R., Dual Spaces of Stresses and Strains, with Applications to Hencky Plasticity, Applied Mathematics and Optimization, 10, 1–35 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  20. Korn A., Sur leséquations d’élasticité, Ann. École Norm., 24, 9–75 (1907)

    MathSciNet  Google Scholar 

  21. Kratochvíl J., Málek J., Rajagopal K.R. and Srinivasa A.R., Modeling of response of elastic plastic materials treated as a mixture of hard and sift regions, Z. angew. Math. Phys., 55, 500–518 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lichtenstein L., Über die erste Randwertaufgaber der Elastizitätstheorie, Math. Z., 20, 21–28 (1921)

    Article  MathSciNet  Google Scholar 

  23. Löbach D., Interior Stress Regularity for the Prandtl-Reuss and Hencky model of perfect plasticity using the Perzyna approximation, Bonner Math. Schriften, 386, 1–56 (2007)

    Google Scholar 

  24. Lubliner J., Plasticity Theory, Macmilan Publishing Company, New York (1990)

    MATH  Google Scholar 

  25. von Mises R., Methodik der festen Körper im plastisch-deformablen Zustand, Nachr. Akad. Wiss. Göttingen, Math.-phys. Kl., 1913, 582–592 (1913)

    Google Scholar 

  26. Málek J., Mathematical properties of flows of incompressible power-law fluids that are described by implicit constitutive relations, Electronic Trans. Numerical Anal., 31, 110–125 (2008)

    MATH  Google Scholar 

  27. Nečas J. and Hlaváček I., Mathematical theory of elastic and elasto-plastic bodies: an introduction, volume 3 of Studies in Applied Mechanics, Elsevier Scientific Publishing Co., Amsterdam (1980)

    Google Scholar 

  28. Norton F.H., The creep of steel at high temperature, McGraw-Hill, New York (1929)

    Google Scholar 

  29. Prager W. and Hodge P.G., Theory of perfectly plastic solids, John Wiley & Sons Inc., New York, N.Y. (1951)

    MATH  Google Scholar 

  30. Prandtl L., Spannungsverteilung in plastischen Körpern. In: Proceeding of the First Int. Congr. Appl. Mech., pages 43–54, Delft (1924)

  31. Rajagopal K.R., Multiple configurations in continuum mechanics. Report 6, Institute for computational and applied mechanics, University of Pittsburgh (1995)

  32. Rajagopal K.R., On implicit constitutive theories, Appl. Math., 48(4), 279–319 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  33. Rajagopal K.R., On implicit constitutive theories for fluids, J. Fluid Mech., 550(4), 243–249 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Rajagopal K.R. and Srinivasa A.R., On the thermomechanics of materials that have multiple natural configurations. Part I: Viscoelasticity and classical plasticity, Z. angew. Math. Phys., 55, 861–893 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  35. Rajagopal K.R. and Srinivasa A.R., On the response of non-dissipative solids, Proc. R. Soc. A, 463, 357–367 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  36. Rajagopal K.R. and Srinivasa A.R., On the thermodynamics of fluids defined by implicit constitutive relations, Z. angew. Math. Phys., 59, 715–729 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  37. Ramberg W. and Osgood W.R., Description of stress-strain curves by three parameters, Tech. Notes Nat. Adv. Comm. Aeronaut., 1943(902), 13pp. (15 plates) (1943)

  38. Reuss A., Berücksichtigung der elastischen Formänderung in der Plastizitätstheorie, Z. Angenew. Math. Mech., 10, 266–271 (1930)

    Article  MATH  Google Scholar 

  39. Seregin G.A., Regularity of weak solutions of variational problems in plasticity theory, Dokl. Akad. Nauk SSSR, 314(6), 1344–1349 (1990)

    MathSciNet  Google Scholar 

  40. Seregin G.A., On the regularity of minimizers of some variational problems in the theory of plasticity, Algebra i Analiz, 4(5), 181–218 (1992)

    MATH  MathSciNet  Google Scholar 

  41. Seregin G.A., Two-dimensional variational problems in plasticity theory, Izv. Ross. Akad. Nauk Ser. Mat., 60(1), 175–210 (1996a)

    MathSciNet  Google Scholar 

  42. Seregin G.A., Remarks on regularity up to the boundary for solutions to variational problems in plasticity theory, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 233 (Kraev. ZadachiMat. Fiz. i Smezh. Vopr. Teor. Funkts. 27), (1996b)

  43. Steinhauer M., On Analysis of Some Nonlinear Systems of Partial Differential Equations of Continuum Mechanics. Bonner Mathematische Schriften [Bonn Mathematical Publications], 359. Universität Bonn Mathematisches Institut, Bonn (2003). Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn (2002)

    MATH  Google Scholar 

  44. Temam R., Mathematical Problems in Plasticity, Gauthier-Villars/Bordas, Paris (1985)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute of Charles University, Sokolovská 83, 186 75, Prague, Czech Republic

    M. Bulíček & J. Málek

  2. Institute of Applied Mathematics, Endenicher Allee 60, D-53121, Bonn, Germany

    J. Frehse

Authors
  1. M. Bulíček
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Frehse
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Málek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Bulíček.

Additional information

Miroslav Bulíček thanks Collaborative Research Center (SFB) 611 and Jindřich Nečas Center for Mathematical Modeling, for their support extended under project LC06052, financed by MŠMT.

Jens Frehse acknowledges Jindřich Nečas Center for Mathematical Modeling for its support under the project LC06052.

Josef Málek’s contribution is a part of the research project MSM 0021620839, financed by MŠMT; the support of GAČR 201/09/0917 is also acknowledged.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bulíček, M., Frehse, J. & Málek, J. On boundary regularity for the stress in problems of linearized elasto-plasticity. Int J Adv Eng Sci Appl Math 1, 141–156 (2009). https://doi.org/10.1007/s12572-010-0001-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12572-010-0001-z

Keywords

Search

Navigation