Skip to main content
SpringerLink
Log in

A numerical study on bifurcations in multi-void growth in nonlinear elasticity

  • Original Paper
  • Published:
International Journal of Fracture Aims and scope Submit manuscript

Abstract

A numerical study on a new bifurcation phenomenon in multi-void growth is carried out for a 2-D problem in nonlinear elasticity, where the reference configuration is y-axisymmetric with two pre-existing small voids. The numerical experiments show that, under large radially-symmetric displacement boundary conditions, other than the y-axisymmetric equilibrium solution one would normally expect, there are two energetically more favorable non-y-axisymmetric equilibrium solutions in which the growth of one void is overwhelmingly dominant. The relationships between the critical bifurcation boundary displacement and the compressibility of the material as well as some geometric parameters are illustrated by numerical results. The numerical experiments also show that the existence of the secondary bifurcation will effectively expedite the onset of fractures.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  • Ball JM (1982) Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos Trans R Soc Lond A 306:557–611

    Article  Google Scholar 

  • Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer, New York

    Book  Google Scholar 

  • Gent AN, Lindley PB (1958) Internal rupture of bonded rubber cylinders in tension. Proc R Soc Lond A 249:195–205

    Article  Google Scholar 

  • Henao D, Serfaty S (2013) Energy estimates and cavity interaction for a critical-exponent cavitation model. Commun Pure Appl Math 66(7):1028–1101

    Article  Google Scholar 

  • Henao D, Mora-Corral C, Xu X (2016) A numerical study of void coalescence and fracture in nonlinear elasticity. Comput Methods Appl Mech Eng 303:163–184

    Article  Google Scholar 

  • Huang W, Li Z (2018) A mixed finite element method for multi-cavity computation in incompressible nonlinear elasticity. J Comp Math. arXiv:1801.02314v1

  • Huang W, Li Z (XXXX) A locking-free DP-Q2-P1 MFEM for incompressible nonlinear elasticity problems. Numer Math Theory Methods Appl

  • Kumar A, Ravi-Chandar K, Lopez-Pamies O (2018) The configurational-forces view of the nucleation and propagation of fracture and healing in elastomers as a phase transition. Int J Fract 213:1–16

    Article  CAS  Google Scholar 

  • Kundu S, Crosby A (2009) Cavitation and fracture behavior of polyacrylamide hydrogels. Soft Matter 5(20):3963–3968

    Article  CAS  Google Scholar 

  • Lian Y, Li Z (2012) Position and size effects on voids growth in nonlinear elasticity. Int J Fract 173:147–161

    Article  Google Scholar 

  • Lopez-Pamies O, Idiart MI, Nakamura T (2011) Cavitationin elastomeric solids: I—a defect-growth theory. J Mech Phys Solids 59:1464–1487

    Article  Google Scholar 

  • Needleman A (1972) Void Growth in an elastic–plastic medium. J Appl Mech 39(4):964–970

    Article  Google Scholar 

  • Poulain X, Lefèvre V, Lopez-Pamies O, Ravi-Chandar K (2017) Damage in elastomers: nucleation and growth of cavities, micro-cracks, and macro-cracks. Int J Fract 205:1–21

    Article  CAS  Google Scholar 

  • Su C, Li Z (2015) Error analysis of a dual-parametric bi-quadratic FEM in cavitation computation in elasticity. SIAM J Numer Anal 53(3):1629–1649

    Article  Google Scholar 

  • Su C, Li Z (2018) A meshing strategy for a quadratic iso-parametric FEM in cavitation computation in nonlinear elasticity. J Comput Math Appl 330:630–647

    Article  Google Scholar 

  • Tvergaard V (1996) Effect of void size difference on growth and cavitation instabilities. J Mech Phys Solids 44(8):1237–1253

    Article  CAS  Google Scholar 

  • Zienkiewicz OC, Taylor RL (1989) The finite element method: basic formulation and linear problems, vol 1, 4th edn. McGraw-Hill, London

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Weijie Huang & Zhiping Li

Authors
  1. Weijie Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhiping Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhiping Li.

Additional information

Publisher's Note

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

The research was supported by the NSFC Projects 11171008 and 11571022.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, W., Li, Z. A numerical study on bifurcations in multi-void growth in nonlinear elasticity. Int J Fract 214, 129–137 (2018). https://doi.org/10.1007/s10704-018-0323-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10704-018-0323-6

Keywords

Search

Navigation