Advertisement

Equilibrium Paths for von Mises Trusses in Finite Elasticity

  • Matteo PelliciariEmail author
  • Angelo Marcello Tarantino
Article
  • 27 Downloads

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.

Keywords

Finite elasticity Equilibrium von Mises truss Stability Snap-through 

Mathematics Subject Classification

74B20 74G05 

Notes

Acknowledgements

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

Compliance with Ethical Standards

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

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) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bažant, Z.P., Cedolin, L.: Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories. World Scientific, Singapore (1991) zbMATHGoogle Scholar
  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) ADSCrossRefGoogle Scholar
  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) ADSCrossRefGoogle Scholar
  5. 5.
    Faa di Bruno, F.: Sullo sviluppo delle funzioni. Ann Sci. Mat. Fis. 6, 479–480 (1855) Google Scholar
  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) CrossRefGoogle Scholar
  7. 7.
    Cazzolli, A., Dal Corso, F.: Snapping of elastic strips with controlled ends. Int. J. Sol. Struct. 162, 285–303 (2019) CrossRefGoogle Scholar
  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) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Frazier, M.J., Kochmann, D.M.: Band gap transmission in periodic bistable mechanical systems. J. Sound Vib. 388, 315–326 (2017) ADSCrossRefGoogle Scholar
  10. 10.
    Gent, A.: A new constitutive relation for rubber. Rubber Chem. Technol. 69(1), 59–61 (1996) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kwasniewski, L.: Complete equilibrium paths for Mises trusses. Int. J. Non-Linear Mech. 44(1), 19–26 (2009) ADSCrossRefGoogle Scholar
  12. 12.
    Lanzoni, L., Tarantino, A.M.: Damaged hyperelastic membranes. Int. J. Non-Linear Mech. 60, 9–22 (2014) ADSCrossRefGoogle Scholar
  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) MathSciNetCrossRefzbMATHGoogle Scholar
  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) ADSCrossRefGoogle Scholar
  15. 15.
    Lanzoni, L., Tarantino, A.M.: Finite anticlastic bending of hyperelastic solids and beams. J. Elast. 131(2), 137–170 (2018) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ligaro, S.S., Valvo, P.S.: Large displacement analysis of elastic pyramidal trusses. Int. J. Solids Struct. 43(16), 4867–4887 (2006) CrossRefzbMATHGoogle Scholar
  17. 17.
    Mises, R.: Über die stabilitätsprobleme der elastizitätstheorie. Z. Angew. Math. Mech. 3(6), 406–422 (1923) CrossRefzbMATHGoogle Scholar
  18. 18.
    Mises, R., Ratzersdorfer, J.: Die Knicksicherheit von Fachwerken. Z. Angew. Math. Mech. 5(3), 218–235 (1925) CrossRefzbMATHGoogle Scholar
  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) Google Scholar
  20. 20.
    Pecknold, D., Ghaboussi, J., Healey, T.: Snap-through and bifurcation in a simple structure. J. Eng. Mech. 111(7), 909–922 (1985) CrossRefGoogle Scholar
  21. 21.
    Psotny, M., Ravinger, J.: Von Mises truss with imperfection. Slov. J. Civ. Eng. 11, 1–7 (2003) Google Scholar
  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) CrossRefGoogle Scholar
  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) CrossRefzbMATHGoogle Scholar
  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) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tarantino, A.M.: Nonlinear fracture mechanics for an elastic Bell material. Q. J. Mech. Appl. Math. 50(3), 435–456 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  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) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Tarantino, A.M.: On the finite motions generated by a mode I propagating crack. J. Elast. 57(2), 85–103 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Tarantino, A.M.: Crack propagation in finite elastodynamics. Math. Mech. Solids 10(6), 577–601 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tarantino, A.M.: Homogeneous equilibrium configurations of a hyperelastic compressible cube under equitriaxial dead-load tractions. J. Elast. 92(3), 227 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tarantino, A.M.: Equilibrium paths of a hyperelastic body under progressive damage. J. Elast. 114(2), 225–250 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tarantino, A.M., Nobili, A.: Finite homogeneous deformations of symmetrically loaded compressible membranes. Z. Angew. Math. Phys. 58(4), 659–678 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ziegler, H.: Principles of Structural Stability, vol. 35. Birkhäuser, Basel (2013) Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.College of Civil EngineeringFuzhou UniversityFuzhouP.R. China
  2. 2.DIEFUniversità di Modena e Reggio EmiliaModenaItaly

Personalised recommendations