Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Equilibrium Paths for von Mises Trusses in Finite Elasticity

Abstract

This paper deals with the equilibrium problem of von Mises trusses in nonlinear elasticity. A general loading condition is considered and the rods are regarded as hyperelastic bodies composed of a homogeneous isotropic material. Under the hypothesis of homogeneous deformations, the finite displacement fields and deformation gradients are derived. Consequently, the Piola-Kirchhoff and Cauchy stress tensors are computed by formulating the boundary-value problem. The equilibrium in the deformed configuration is then written and the stability of the equilibrium paths is assessed through the energy criterion. An application assuming a compressible Mooney-Rivlin material is performed. The equilibrium solutions for the case of vertical load present primary and secondary branches. Although, the stability analysis reveals that the only form of instability is the snap-through phenomenon. Finally, the finite theory is linearized by introducing the hypotheses of small displacement and strain fields. By doing so, the classical solution of the two-bar truss in linear elasticity is recovered.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Notes

  1. 1.

    Another phenomenon that may interest such mechanical systems is the bistable mechanism, which has been recently analyzed by means of elastica theory (see, e.g., [1, 6, 7, 9]).

  2. 2.

    Other studies on the behavior of rubber-like materials in which a similar stored energy function is assumed can be found in [14, 24, 28] and [31].

  3. 3.

    This is a convenient position that was used, for instance, in [12, 25] and [27].

References

  1. 1.

    Armanini, C., Dal Corso, F., Misseroni, D., Bigoni, D.: From the elastica compass to the elastica catapult: an essay on the mechanics of soft robot arm. Proc. R. Soc. A 473(2198), 20160, 870 (2017)

  2. 2.

    Bažant, Z.P., Cedolin, L.: Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories. World Scientific, Singapore (1991)

  3. 3.

    Bazzucchi, F., Manuello, A., Carpinteri, A.: Interaction between snap-through and Eulerian instability in shallow structures. Int. J. Non-Linear Mech. 88, 11–20 (2017)

  4. 4.

    Bellini, P.X.: The concept of snap-buckling illustrated by a simple model. Int. J. Non-Linear Mech. 7(6), 643–650 (1972)

  5. 5.

    Faa di Bruno, F.: Sullo sviluppo delle funzioni. Ann Sci. Mat. Fis. 6, 479–480 (1855)

  6. 6.

    Camescasse, B., Fernandes, A., Pouget, J.: Bistable buckled beam: elastica modeling and analysis of static actuation. Int. J. Solids Struct. 50(19), 2881–2893 (2013)

  7. 7.

    Cazzolli, A., Dal Corso, F.: Snapping of elastic strips with controlled ends. Int. J. Sol. Struct. 162, 285–303 (2019)

  8. 8.

    Ciarlet, P.G., Geymonat, G.: Sur les lois de comportement en élasticité non linéaire compressible. C. R. Acad. Sci. Paris Sér. II 295, 423–426 (1982)

  9. 9.

    Frazier, M.J., Kochmann, D.M.: Band gap transmission in periodic bistable mechanical systems. J. Sound Vib. 388, 315–326 (2017)

  10. 10.

    Gent, A.: A new constitutive relation for rubber. Rubber Chem. Technol. 69(1), 59–61 (1996)

  11. 11.

    Kwasniewski, L.: Complete equilibrium paths for Mises trusses. Int. J. Non-Linear Mech. 44(1), 19–26 (2009)

  12. 12.

    Lanzoni, L., Tarantino, A.M.: Damaged hyperelastic membranes. Int. J. Non-Linear Mech. 60, 9–22 (2014)

  13. 13.

    Lanzoni, L., Tarantino, A.M.: Equilibrium configurations and stability of a damaged body under uniaxial tractions. Z. Angew. Math. Phys. 66(1), 171–190 (2015)

  14. 14.

    Lanzoni, L., Tarantino, A.M.: A simple nonlinear model to simulate the localized necking and neck propagation. Int. J. Non-Linear Mech. 84, 94–104 (2016)

  15. 15.

    Lanzoni, L., Tarantino, A.M.: Finite anticlastic bending of hyperelastic solids and beams. J. Elast. 131(2), 137–170 (2018)

  16. 16.

    Ligaro, S.S., Valvo, P.S.: Large displacement analysis of elastic pyramidal trusses. Int. J. Solids Struct. 43(16), 4867–4887 (2006)

  17. 17.

    Mises, R.: Über die stabilitätsprobleme der elastizitätstheorie. Z. Angew. Math. Mech. 3(6), 406–422 (1923)

  18. 18.

    Mises, R., Ratzersdorfer, J.: Die Knicksicherheit von Fachwerken. Z. Angew. Math. Mech. 5(3), 218–235 (1925)

  19. 19.

    Ogden, R.W.: Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 567–583 (1972)

  20. 20.

    Pecknold, D., Ghaboussi, J., Healey, T.: Snap-through and bifurcation in a simple structure. J. Eng. Mech. 111(7), 909–922 (1985)

  21. 21.

    Psotny, M., Ravinger, J.: Von Mises truss with imperfection. Slov. J. Civ. Eng. 11, 1–7 (2003)

  22. 22.

    Rezaiee-Pajand, M., Naghavi, A.: Accurate solutions for geometric nonlinear analysis of eight trusses. Mech. Based Des. Struct. Mach. 39(1), 46–82 (2011)

  23. 23.

    Savi, M.A., Pacheco, P.M., Braga, A.M.: Chaos in a shape memory two-bar truss. Int. J. Non-Linear Mech. 37(8), 1387–1395 (2002)

  24. 24.

    Tarantino, A.M.: Thin hyperelastic sheets of compressible material: field equations, airy stress function and an application in fracture mechanics. J. Elast. 44(1), 37–59 (1996)

  25. 25.

    Tarantino, A.M.: Nonlinear fracture mechanics for an elastic Bell material. Q. J. Mech. Appl. Math. 50(3), 435–456 (1997)

  26. 26.

    Tarantino, A.M.: The singular equilibrium field at the notch-tip of a compressible material in finite elastostatics. Z. Angew. Math. Phys. 48(3), 370–388 (1997)

  27. 27.

    Tarantino, A.M.: On the finite motions generated by a mode I propagating crack. J. Elast. 57(2), 85–103 (1999)

  28. 28.

    Tarantino, A.M.: Crack propagation in finite elastodynamics. Math. Mech. Solids 10(6), 577–601 (2005)

  29. 29.

    Tarantino, A.M.: Homogeneous equilibrium configurations of a hyperelastic compressible cube under equitriaxial dead-load tractions. J. Elast. 92(3), 227 (2008)

  30. 30.

    Tarantino, A.M.: Equilibrium paths of a hyperelastic body under progressive damage. J. Elast. 114(2), 225–250 (2014)

  31. 31.

    Tarantino, A.M., Nobili, A.: Finite homogeneous deformations of symmetrically loaded compressible membranes. Z. Angew. Math. Phys. 58(4), 659–678 (2007)

  32. 32.

    Ziegler, H.: Principles of Structural Stability, vol. 35. Birkhäuser, Basel (2013)

Download references

Acknowledgements

The authors acknowledge funding from the Italian Ministry MIUR-PRIN voce COAN 5.50.16.01 code 2015JW9NJT.

Author information

Correspondence to Matteo Pelliciari.

Ethics declarations

Compliance with Ethical Standards

Conflict of Interest: The authors declare that they have no conflict of interest.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pelliciari, M., Tarantino, A.M. Equilibrium Paths for von Mises Trusses in Finite Elasticity. J Elast 138, 145–168 (2020). https://doi.org/10.1007/s10659-019-09731-1

Download citation

Keywords

  • Finite elasticity
  • Equilibrium
  • von Mises truss
  • Stability
  • Snap-through

Mathematics Subject Classification

  • 74B20
  • 74G05