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Elasto-plastic localised responses in one-dimensional structural models

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Abstract

This work complements recent developments concerning the buckling of beams lying on a nonlinear (non-convex) elastic foundation, and also reports on some investigations on the role of material nonlinearity. Two structural models are studied using a simple elasto-plastic constitutive relationship, and buckling problems are formulated as reversible fourth-order differential equations. It is demonstrated that modulated responses are possible under certain circumstances. Some numerical simulations are presented supporting the analytical findings.

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References

  1. A. D. Kerr. Elastic and viscoelastic foundation models. J. Appl. Mech. 64 (1964) 491–498.

    Google Scholar 

  2. J. C. Amazigo, B. Budiansky and G. F. Carrier. Asymptotic analysis of the buckling of imperfect columns on nonlinear elastic foundation. Int. J. Solids Structures 6 (1970) 1341–1356.

    Google Scholar 

  3. C. G. Lange and A. C. Newell. The post-buckling problem for thin elastic shells. SIAM J. Appl. Math. 21 (1971) 605–629.

    Google Scholar 

  4. M. K. Wadee, G. W. Hunt and A. I. M. Whiting. Asymptotic and Rayleigh-Ritz routes to localized buckling solutions in an elastic instability problem. Proc. R. Soc. London A 453 (1997) 2085–2107.

    Google Scholar 

  5. M. Potier-Ferry. Foundations of elastic post-buckling theory. In: J. Arbocz et al. (eds.), Buckling and Postbuckling, Berlin: Springer-Verlag (1987) pp. 1–79.

    Google Scholar 

  6. C. J. Budd, G.W. Hunt and R. Kuske. Asymptotics of cellular buckling close to Maxwell load. Proc. R. Soc. London A 457 (2001) 2935–2964.

    Google Scholar 

  7. G.W. Hunt, M. A. Peletier, A. R. Champneys, P. D. Woods, M. A. Wadee, C. J. Budd and G. J. Lord. Cellular buckling in long structures. Nonlinear Dynamics 21 (2000) 3–29.

    Google Scholar 

  8. M. K. Wadee and A. P. Bassom. Characterization of limiting homoclinic behaviour in a one-dimensional elastic buckling model. J. Mech. Phys. Solids 48 (2000) 2297–2313.

    Google Scholar 

  9. M. K. Wadee, C. D. Coman and A. P. Bassom. Solitary wave interaction phenomena in a strut buckling model incorporating restabilisation. Physica D 163 (2002) 26–48.

    Google Scholar 

  10. V. Tvergaard and A. Needleman. On the localization of buckling patterns. J. Appl. Mech. 47 (1980) 613–619.

    Google Scholar 

  11. G. W. Hunt, H. M. Bolt and J. M. T. Thompson. Structural localization phenomena and the dynamical phase-space analogy. Proc. R. Soc. London A 425 (1989) 245–267.

    Google Scholar 

  12. G. J. Lord, A. R. Champneys and G. W. Hunt. Computation of localized post-buckling in long axially compressed cylindrical shells. Phil. Trans. R. Soc. London A 355 (1997) 2137–2150.

    Google Scholar 

  13. J. M. T. Thompson and G. W. Hunt. A General Theory of Elastic Stability. London: Wiley (1973) 317 pp.

    Google Scholar 

  14. M. A. Biot. Mechanics of Incremental Deformation. New York: Wiley (1965) 504 pp.

    Google Scholar 

  15. A. R. Champneys, G. W. Hunt and J. M. T. Thompson. Localization and Solitary Waves in Mechanics. Singapore: World Scientific (1999) 396 pp.

    Google Scholar 

  16. G. W. Hunt, M. K. Wadee and N. Shiacolas. Localized elasticae for the strut on the linear foundation. J. Appl. Mech. 60 (1993) 1033–1038.

    Google Scholar 

  17. M. K. Wadee and A. P. Bassom. Effects of exponentially small terms in the perturbation approach to localized buckling. Proc. R. Soc. London A 455 (1999) 2351–2370.

    Google Scholar 

  18. G. A. Maugin. The Thermomechanics of Plasticity and Fracture. Cambridge: Cambridge University Press (1992) 350 pp.

    Google Scholar 

  19. J. Hutchinson. Plastic buckling. Adv. Appl. Mech. 14 (1974) 67–144.

    Google Scholar 

  20. S. R. Reid, T. X. Yu and J. L. Yang. An elastic-plastic hardening-softening cantilever beam subjected to a force pulse at its tip: a model for pipe whip. Proc. R. Soc. London A 454 (1998) 1031–1048.

    Google Scholar 

  21. L. G. Brazier. On the flexure of thin cylindrical shells and other sections. Proc. R. Soc. 116 (1927) 104–114.

    Google Scholar 

  22. S. Kyriakides. Propagating instabilities in structures. Adv. Appl. Mech. 30 (1993) 67–189.

    Google Scholar 

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

    Google Scholar 

  24. J. M. Gere and S. P. Timoshenko. Mechanics of Materials. London: Chapman & Hall (1991) 832 pp.

    Google Scholar 

  25. K. A. Lazopoulos. Beam buckling as a coexistence of phase phenomena. Eur. J. Mech. A/Solids 14 (1995) 589–604.

    Google Scholar 

  26. R. D. James. The equilibrium and post-buckling behaviour of an elastic curve governed by a non-convex energy. J. Elasticity 11 (1981) 239–269.

    Google Scholar 

  27. R. L. Fosdick and R. D. James. The elastica and the problem of pure bending for a non-convex stored energy function. J. Elasticity 11 (1981) 165–186.

    Google Scholar 

  28. Z. P. Bazant, T. B. Belytschko and T. P. Chang. Continuum theory for strain-softening. J. Engng. Mech. 110 (1984) 1666–1692.

    Google Scholar 

  29. G. Buttazzo, M. Giaquinta and S. Hildebrandt. One-dimensional Variational Problems. Oxford: Clarendon Press (1998) 262 pp.

    Google Scholar 

  30. I. M. Gelfand and S. V. Fomin. Calculus of Variations. New York: Dover Publications (1991) 232 pp.

    Google Scholar 

  31. N. Kikuchi and N. Triantafyllidis. On a certain class of elastic materials with non-elliptic energy densities. Quart. Appl. Math. 40 (1982) 241–248.

    Google Scholar 

  32. N. Triantafyllidis and S. Bardenhagen. On higher order gradient continuum theories in 1-D nonlinear elasticity; derivation and comparison to the corresponding discrete models. J. Elasticity. 33 (1993) 259–293.

    Google Scholar 

  33. E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede and X. Wang. AUTO97: continuation and bifurcation software for ordinary differential equations. Available via anonymous from ftp://ftp.cs.concordia.ca/pub/doedel/auto, 1997.

  34. R. D. James. Co-existence of phases in the one-dimensional static theory of elastic bars. Arch. Rat. Mech. Anal. 72 (1979) 99–140.

    Google Scholar 

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Coman, C.D., Bassom, A.P. & Wadee, M.K. Elasto-plastic localised responses in one-dimensional structural models. Journal of Engineering Mathematics 47, 83–100 (2003). https://doi.org/10.1023/A:1025856602682

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