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A regularization study of some softening beam problems with an implicit gradient plasticity model

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Abstract

The modelling of plastic beams experiencing softening is studied. The homogeneous cantilever beam loaded by a concentrated force at its extremity is considered. This simple structural problem with gradient bending moment allows an analytical treatment of the evolution problem. A gradient plasticity model is developed in order to overcome Wood’s paradox. Surprisingly, explicit gradient plasticity models do not eliminate this paradox, since the beam response is found to be not continuous with respect to the loading parameter. A new implicit gradient plasticity model is used in this paper. It is shown that the new regularized problem is well-posed. Closed-form solutions of the elastoplastic deflection are finally derived. These results are valid for the beam bending problem, but also for the simple analogy of the bar subjected to distributed axial force.

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References

  1. Wood RH (1968) Some controversial and curious developments in the plastic theory of structures. In: Heyman J, Leckie FA (eds) Engineering plasticity. Cambridge University Press, UK, pp 665–691

    Google Scholar 

  2. Bažant ZP (1976) Instability, ductility and size effect in strain-softening concrete. J Eng Mech ASCE 102: 331–344

    Google Scholar 

  3. Jirásek M, Bažant ZP (2002) Inelastic analysis of structures. Wiley

  4. Eringen AC (1981) On nonlocal plasticity. Int J Eng Sci 19: 1461–1474

    Article  MATH  MathSciNet  Google Scholar 

  5. Eringen AC (1983) Theories of non-local plasticity. Int J Eng Sci 21: 741–751

    Article  MATH  Google Scholar 

  6. Pijaudier-Cabot G, Bažant ZP (1987) Nonlocal damage theory. J Eng Mech 113: 1512–1533

    Article  Google Scholar 

  7. Aifantis EC (1984) On the microstructural origin of certain inelastic models. J Eng Mater Technol ASME 106: 326–330

    Article  Google Scholar 

  8. Zbib H, Aifantis EC (1988) On the localization and post-localization behaviour of plastic deformation. I, II, III. Res Mech 23: 261–277;279–292;293–305

  9. Mühlhaus HB, Aifantis EC (1991) A variational principle for gradient plasticity. Int J Solids Struct 28: 845–857

    Article  MATH  Google Scholar 

  10. de Borst R, Mühlhaus HB (1992) Gradient-dependent plasticity: formulation and algorithmic aspects. Int J Numer Methods Eng 35: 521–539

    Article  MATH  Google Scholar 

  11. Peerlings RHJ, de Borst R, Brekelmans WAM, de Vree JHP (1996) Gradient-enhanced damage for quasi-brittle materials. Int J Numer Methods Eng 39: 3391–3403

    Article  MATH  Google Scholar 

  12. Engelen RAB, Geers MGD, Baaijens FPT (2003) Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour. Int J Plasticity 19: 403–433

    Article  MATH  Google Scholar 

  13. Engelen RAB, Fleck NA, Peerlings RHJ, Geers MGD (2006) An evaluation of higher-order plasticity theories for predicting size effects and localisation. Int J Solids Struct 43: 1857–1877

    Article  MATH  Google Scholar 

  14. Huerta A, Pijaudier-Cabot G (1994) Discretization influence on regularization by two localization limiters. J Eng Mech 120(6): 1198–1218

    Article  Google Scholar 

  15. Bažant ZP, Jirásek M (2002) Nonlocal integral formulations of plasticity and damage: Survey of progress. J Eng Mech 128: 1119–1149

    Article  Google Scholar 

  16. Fleck NA, Hutchinson JW (2001) A reformulation of strain gradient plasticity. J Mech Phys Solids 49: 2245–2271

    Article  MATH  ADS  Google Scholar 

  17. Benvenuti E, Borino G, Tralli A (2002) A thermodynamically consistent nonlocal formulation for damaging materials. Eur J Mech A/Solids 21: 535–553

    Article  MATH  MathSciNet  Google Scholar 

  18. Jirásek M, Rolshoven S (2003) Comparison of integral-type nonlocal plasticity models for strain-softening materials. Int J Eng Sci 41: 1553–1602

    Article  Google Scholar 

  19. Polizzotto C (2007) Strain-gradient elastic-plastic material models and assessment of the higher order boundary conditions. Eur J Mech A/Solids 26: 189–211

    Article  MATH  MathSciNet  Google Scholar 

  20. Bažant ZP, Zubelewicz A (1988) Strain-softening bar and beam: exact non-local solution. Int J Solids Struct 24(7): 659–673

    Article  MATH  Google Scholar 

  21. Royer-Carfagni G (2001) Can a moment–curvature relationship describe the flexion of softening beams. Eur J Mech A/Solids 20: 253–276

    Article  MATH  MathSciNet  Google Scholar 

  22. Challamel N (2003) Une approche de plasticité au gradient en construction métallique. Comptes-Rendus Mécanique 331(9): 647–654

    Article  Google Scholar 

  23. Challamel N, Hjiaj M (2005) Non-local behavior of plastic softening beams. Acta Mech 178: 125–146

    Article  MATH  Google Scholar 

  24. Pamin J (1994) Gradient-dependent plasticity in numerical simulation of localization phenomena, Dissertation, Delft University of Technology, Delft. Available from http://www.library.tudelft.nl/dissertations.

  25. Galileo, Discorsi e Dimonstrazioni Matematiche, intorno à due nuove Scienze, 1638, In: Sur les épaules des géants – les plus grands textes de physique et d’astronomie (Hawkings, S.), pp 154–182. Dunod, 2002

  26. Timoshenko SP (1983) History of strength of materials. Dover Publications.

  27. Salençon J (1990) An introduction to the yield design theory and its application to soil mechanics. Eur J Mech A/Solids 9(5): 477–500

    MATH  Google Scholar 

  28. Challamel N, Pijaudier-Cabot G (2006) Stability and dynamics of a plastic softening oscillator. Int J Solids Struct 43: 5867–5885

    Article  MATH  Google Scholar 

  29. Challamel N (2007) Lateral-torsional buckling of beams under combined loading—a reappraisal of the Papkovitch-Schaefer theorem. Int J Struct Stab Dyn 7(1): 55–79

    Article  MathSciNet  Google Scholar 

  30. Challamel N, Andrade A, Camotim D (2007) An analytical study on the lateral–torsional buckling of linearly tapered cantilever strip beams. Int J Struct Stab Dyn 7(3): 441–456

    Article  Google Scholar 

  31. Peerlings RHJ (2007) On the role of moving elastic-plastic boundaries in strain gradient plasticity. Model Simul Mater Sci Eng 15: 109–120

    Article  ADS  Google Scholar 

  32. de Borst R, Pamin J (1996) Some novel developments in finite element procedures for gradient-dependent plasticity. Int J Numer Methods Eng 39: 2477–2505

    Article  MATH  Google Scholar 

  33. Vermeer PA, Brinkgreve RBJ (1994) In: Chambon R, Desrues J, Vardoulakis I (eds) A new effective non-local strain measure for softening plasticity. Rotterdam, Balkema, pp 89–100

  34. Tikhonov AN, Arsenine VY (1977) Solutions to ill-posed problems. Winston-Wiley, New-York

    Google Scholar 

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  1. Laboratoire de Génie Civil et de Génie Mécanique, INSA de Rennes, Université Européenne de Bretagne, 20, avenue des Buttes de Coësmes, 35043, Rennes cedex, France

    Noël Challamel

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Challamel, N. A regularization study of some softening beam problems with an implicit gradient plasticity model. J Eng Math 62, 373–387 (2008). https://doi.org/10.1007/s10665-008-9233-3

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  • DOI: https://doi.org/10.1007/s10665-008-9233-3

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