Skip to main content
SpringerLink
Log in
Book cover

Non-Equilibrium Thermodynamics with Application to Solids pp 259–329Cite as

Stability and Constitutive Inequalities in Plasticity

  • Chapter

  • 329 Accesses

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 336))

Abstract

A thermodynamic theory of stability in solids with intrinsic dissipation of plastic type is developed, and various related constitutive inequalities are discussed. The thermodynamic formalism for finite strain elasto-plasticity is presented, with rate-dependent plastic behaviour and its rate-independent limit described with the help of internal variables. A general condition sufficient for stability of equilibrium in the sense of Lyapunov is formulated and then successively transformed as additional assumptions are introduced, with special attention focused on isothermal rate-independent plasticity. In the latter case the conditions for stability of a quasi-static process at varying loading are also derived and shown to differ from the respective conditions of stability of equilibrium. The stability conditions are formulated for an arbitrary continuous system with specified boundary conditions as well as for a homogeneous material element embedded in a continuum. Relation between intrinsic instability at the level of a material element and propagation of acceleration waves or strain localization in shear bands is discussed.

This is a preview of subscription content, 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   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Hill, R.: Aspects of invariance in solids mechanics, Advances in Applied Mechanics, vol. 18, Acad. Press, New York 1978, 1–75.

    Article  MATH  Google Scholar 

  2. Mandel, J.: Plasticité Classique et Viscoplasticité, CISM Course, Udine, Springer 1971.

    Google Scholar 

  3. Rice, J.R.: Inelastic constitutive relations for solids: an internai- variable theory and its application to metal plasticity, J. Mech. Phys. Solids, 19 (1971) 433–455.

    Article  MATH  Google Scholar 

  4. de Groot, S.R. and Mazur, P.: Non-equilibrium Thermodynamics, North-Holland, Amsterdam 1962.

    Google Scholar 

  5. Muschik, W.: Internal variables in non-equilibrium thermodynamics, J. Non-Equilib. Thermodyn., 15 (1990) 127–137.

    Google Scholar 

  6. Kestin, J.: A note on the relation between the hypothesis of local equilibrium and the Clausius-Duhem inequality, J. Non-Equilib. Thermodyn., 15 (1990) 193–212.

    MATH  Google Scholar 

  7. Rice, J.R.: Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms, in: Constitutive Equations in Plasticity (Ed. A. S. Argon), MIT Press, Cambridge Mass. 1975, 23–79.

    Google Scholar 

  8. Lehmann, T. (Ed.): The Constitutive Law in Thermoplast ici ty, CISM Course, Udine, Springer 1984.

    Google Scholar 

  9. Kleiber, M. and Raniecki, B.: Elastic-plastic materials at finite strains, in: Plasticity Today: Modelling, Methods and Applications (Ed. A. Sawczuk and G. Bianchi), Elsevier, London 1985, 3–46.

    Google Scholar 

  10. Neuhäuser, H.: Physical manifestation of instabilities in plastic flow, in: Mechanical Properties of Solids: Plastic instabilities (Ed. V. Balakrishnan and E.C. Bottani), World Scientific, Singapore 1986, 209–252.

    Google Scholar 

  11. Hill, R. and Rice, J.R.: Elastic potentials and the structure of inelastic constitutive laws, SIAMJ. Appl. Math., 25 (1973) 448–461.

    Article  MATH  MathSciNet  Google Scholar 

  12. Kestin, J. and Rice, J.R.: Paradoxes in the application of thermodynamics to strained solids, in: A Critical Review of Thermodynamics (Ed. E.B. Stuart et. al.), Mono Book Corp., Baltimore 1970, 275–298.

    Google Scholar 

  13. Morreau, J. J.: Sur les lois de frottement, de plasticité et de viscosité, C. R. Acad. Sci. Paris, t. 271, Serie A (1970) 608–611.

    Google Scholar 

  14. Morreau, J.J.: On Unilateral Constraints, Friction and Plasticity, CIME Course, Bressansone, in: New Variational Techniques in Mathematical Physics, Edizioni Cremonese, 1974, 175–322.

    Google Scholar 

  15. Nguyen, Q.S.: Stabilité et bifurcation des systèmes dissipatifs standards à comportement indépendant du temps physique, C. R. Acad. Sci. Paris, t. 310, Serie II (1990) 1375–1380.

    MATH  Google Scholar 

  16. Nguyen, Q.S.: Bifurcation and stability of time-independent standard dissipative systems, CISM Course, Udine 1991.

    Google Scholar 

  17. M.J. Sewell: A survey of plastic buckling, in: Stability (H.H.E. Leipholz, ed.), Univ. of Waterloo Press, Ontario 1972, 85–197.

    Google Scholar 

  18. Hill, R. and Rice, J.R.: Constitutive analysis of elastic-plastic crystals at arbitrary strain, J. Mech. Phys. Solids, 20 (1972) 401–413.

    Article  MATH  Google Scholar 

  19. Hill, R.: Some basic principles in the mechanics of solids without a natural time, J. Mech. Phys. Solids, 7 (1959) 209–225.

    Article  MATH  MathSciNet  Google Scholar 

  20. Movchan, A.A.: Stability of processes with respect to two metrics (in Russian), Prikl. Math. Mekh., 24 (1960) 988–1001.

    Google Scholar 

  21. Bazant, Z.P.: Stable states and stable paths of propagation of damage zones and interactive fractures, in: Cracking and Damage (Ed. J. Mazars and Z.P. Bazant), Elsevier, London 1989, 183–206.

    Google Scholar 

  22. Ericksen, J.L.: A thermo-kinetic view of elastic stability theory, Int. J. Solids Structures, 2 (1966) 573–580.

    Article  Google Scholar 

  23. Gurtin, M.E.: Thermodynamics and Stability, Arch. Rational Mech. Anal., 59 (1975) 63–96.

    Article  MATH  MathSciNet  Google Scholar 

  24. Petryk, H.: A consistent energy approach to defining stability of plastic deformation processes, in: Stability in the Mechanics of Continua, Proc. IUTAM Symp. (Ed. F.H. Schroeder), Springer, Berlin 1982, 262–272.

    Chapter  Google Scholar 

  25. Hill, R.: A general theory of uniqueness and stability in elastic-plastic solids, J. Mech. Phys. Solids, 6 (1958) 236–249.

    Article  MATH  Google Scholar 

  26. Petryk, H.: On the second-order work in plasticity, Arch. Mech., 43 (1991) 377–397.

    MATH  MathSciNet  Google Scholar 

  27. Nguyen, Q.S. and Radenkovic, D.: Stability of equilibrium in elastic plastic solids, Springer Lecture Notes in Mathematics, Vol. 503 (1976) 403–414.

    Article  Google Scholar 

  28. Petryk, H.: (to be published).

    Google Scholar 

  29. Knops, R.J. and Wilkes, E.W.: Theory of Elastic Stability, Handbuch der Physik VI a/3, Springer, Berlin 1973.

    Google Scholar 

  30. Koiter, W.T.: A basic open problem in the theory of elastic stability, Springer Lecture Notes in Mathematics, Vol. 503 (1976) 366–373.

    Article  MathSciNet  Google Scholar 

  31. Petryk, H.: The energy criteria of instability in time-independent inelastic solids, Arch. Mech. 43 (1991) 519–545.

    MATH  MathSciNet  Google Scholar 

  32. Petryk, H.: On constitutive inequalities and bifurcation in elastic-plastic solids with a yield-surface vertex, J. Mech. Phys. Solids, 37 (1989) 265–291.

    Article  MATH  MathSciNet  Google Scholar 

  33. Nguyen, Q.S. and Petryk, H.: A constitutive inequality for time-independent dissipative solids, C. R. Acad. Sci. Paris, t. 312, Serie II, (1991) 7–12.

    MATH  MathSciNet  Google Scholar 

  34. Morrey, C.B.: Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 2 (1952) 25–53.

    Article  MATH  MathSciNet  Google Scholar 

  35. Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 63 (1977) 337–403.

    Article  MATH  Google Scholar 

  36. Graves, L.M.: The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939) 656–660.

    Article  MathSciNet  Google Scholar 

  37. van Hove, L.: Sur l’extension de la condition de Legendre du calcul des variations aux intégrales multiples à plusieurs fonctions inconnues, Proc. Kon. Ned. Acad. Wet., 50 (1947) 18–23.

    MATH  Google Scholar 

  38. Petryk, H.: Material instability and strain-rate discontinuities in incrementally nonlinear solids, J. Mech. Phys. Solids, 40 (1992) 1227–1250.

    Article  MATH  MathSciNet  Google Scholar 

  39. Abeyratne, R. and Knowles, J.K.: On the driving traction acting on a surface of strain discontinuity in a continuum, J. Mech. Phys. Solids, 38 (1990) 345–360.

    Article  MathSciNet  Google Scholar 

  40. Petryk, H.: The energy criteria of instability of plastic flow, XVII IUTAM Congress, Grenoble (1988).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Polish Academy of Sciences, Warsaw, Poland

    H. Petryk

Authors
  1. H. Petryk
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Technical University, Berlin, Germany

    W. Muschik

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer-Verlag Wien

About this chapter

Cite this chapter

Petryk, H. (1993). Stability and Constitutive Inequalities in Plasticity. In: Muschik, W. (eds) Non-Equilibrium Thermodynamics with Application to Solids. International Centre for Mechanical Sciences, vol 336. Springer, Vienna. https://doi.org/10.1007/978-3-7091-4321-6_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-4321-6_5

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-82453-5

  • Online ISBN: 978-3-7091-4321-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation