# A repeated yielding model under periodic perturbation

## Abstract

The Ananthakrishna model, seeking to explain the phenomenon of repeated yielding of materials, is studied with or without periodic perturbation. For the unforced model, Hopf bifurcation, degenerate Hopf bifurcation and saddle-node bifurcation are detected. For the periodically forced model, two elementary periodic mechanisms are analyzed corresponding to five bifurcation cases of the unforced one. Rich dynamical behaviors arise, including stable and unstable periodic solutions of different periods, quasi-periodic solutions, chaos through torus destruction or cascade of period doublings. Moreover, even small change of a parameter can lead to bifurcation of different periodic solutions. Finally, according to the forced Ananthakrishna model, four types of stress–time curves are simulated, which can well interpret various experimental phenomena of repeated yielding.

## Keywords

Ananthakrishna model Bifurcation Attractor Periodically forced Repeated yielding## Notes

### Acknowledgements

This research is supported by the National Natural Science Foundation of China (11771407), the Innovative Research Team of Science and Technology in Henan Province (17IRTSTHN007) and the National Key R&D Program of China (2017YFB0702500).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.

## References

- 1.Zhang, Y., Liu, J.P., Chen, S.Y., Xie, X., Liaw, P.K., Dahmen, K.A., Qiao, J.W., Wang, Y.L.: Serration and noise behaviors in materials. Prog. Mater. Sci.
**90**, 358–460 (2017)CrossRefGoogle Scholar - 2.Lee, S.Y., Lee, S.I., Hwang, B.: Effect of strain rate on tensile and serration behaviors of an austenitic Fe–22Mn–0.7C twinning-induced plasticity steel. Mater. Sci. Eng. A
**711**, 22–28 (2018)CrossRefGoogle Scholar - 3.Serajzadeh, S., Akhgar, J.M.: A study on strain ageing during and after warm rolling of a carbon steel. Mater. Lett.
**62**(6–7), 946–948 (2008)CrossRefGoogle Scholar - 4.Hu, Q., Zhang, Q.C., Cao, P.T., Fu, S.H.: Thermal analyses and simulations of the type A and type B Portevin-Le Chatelier effects in an Al–Mg alloy. Acta Mater.
**60**(4), 1647–1657 (2012)CrossRefGoogle Scholar - 5.Cai, Y.L., Yang, S.L., Wang, Y.H., Fu, S.H., Zhang, Q.C.: Characterization of the deformation behaviors associated with the serrated flow of a 5456 Al-based alloy using two orthogonal digital image correlation systems. Mater. Sci. Eng. A
**664**, 155–164 (2016)CrossRefGoogle Scholar - 6.Raia, R.K., Sahub, J.K.: Mechanism of serrated flow in a cast nickel base superalloy. Mater. Lett.
**210**, 298–300 (2018)CrossRefGoogle Scholar - 7.Moshtaghin, R.S., Asgari, S.: The characteristics of serrated flow in superalloy IN738LC. Mater. Sci. Eng. A
**486**(1–2), 376–380 (2008)CrossRefGoogle Scholar - 8.Sarmah, R., Ananthakrishna, G., Sun, B.A., Wang, W.H.: Hidden order in serrated flow of metallic glasses. Acta Mater.
**59**(11), 4482–4493 (2011)CrossRefGoogle Scholar - 9.Ren, J.L., Chen, C., Wang, G., Mattern, N., Eckert, J.: Dynamics of serrated flow in a bulk metallic glass. AIP Adv.
**1**(3), 2158–3226 (2011)CrossRefGoogle Scholar - 10.Chen, S.Y., Yu, L.P., Ren, J.L., Xie, X., Li, X.P., Xu, Y., Zhao, G.F., Li, P.Z., Yang, F.Q., Ren, Y., Liaw, P.K.: Self-similar random process and chaotic behavior in serrated flow of high-entropy alloys. Sci. Rep. UK
**6**, 29798 (2016)CrossRefGoogle Scholar - 11.Guo, X.X., Xie, X., Ren, J.L., Laktionova, M., Tabachnikova, E., Yu, L.P., Cheung, W.S., Dahmen, K.A., Liaw, P.K.: Plastic dynamics of the \(Al_{0.5}CoCrCuFeNi\) high entropy alloy at cryogenic temperatures: Jerky flow, stair-like fluctuation, scaling behavior, and non-chaotic state. Appl. Phys. Lett.
**111**(25), 251905 (2017)CrossRefGoogle Scholar - 12.Ren, J.L., Chen, C., Wang, G., Liaw, P.K.: Transition of temporal scaling behavior in percolation assisted shear-branching structure during plastic deformation. Sci. Rep. UK
**7**, 45083 (2017)CrossRefGoogle Scholar - 13.Chen, C., Ren, J.L., Wang, G., Dahmen, K.A., Liaw, P.K.: Scaling behavior and complexity of plastic deformation for a bulk metallic glass at cryogenic temperatures. Phys. Rev. E
**92**(1), 012113 (2015)CrossRefGoogle Scholar - 14.Ren, J.L., Chen, C., Liu, Z.Y., Li, R., Wang, G.: Plastic dynamics transition between chaotic and self-organized critical states in a glassy metal via a multifractal intermediate. Phys. Rev. B
**86**(13), 134303 (2012)CrossRefGoogle Scholar - 15.Ren, J.L., Chen, C., Cheung, W.S., Sun, B.A., Mattern, N., Siegmund, S., Eckert, J.: Various sizes of sliding event bursts in the plastic flow of metallic glasses based on a spatiotemporal dynamic model. J. Appl. Phys.
**116**(3), 033520 (2014)CrossRefGoogle Scholar - 16.Ananthakrishna, G., Sahoo, D.: A model based on nonlinear oscillations to explain jumps on creep curves. J. Phys. D Appl. Phys.
**14**(11), 2081–2090 (1981)CrossRefGoogle Scholar - 17.Valsakumar, M.C., Ananthakrishna, G.: A model based on nonlinear oscillations to explain jumps on creep curves: II. Approximate solutions. J. Phys. D Appl. Phys.
**16**(6), 1055–1068 (1983)CrossRefGoogle Scholar - 18.Ananthakrishna, G., Valsakumar, M.C.: Repeated yield drop phenomenon: a temporal dissipative structure. Phys. D Appl. Phys.
**15**(12), L171–L175 (1982)CrossRefGoogle Scholar - 19.Ananthakrishna, G.: Dislocation dynamics and cooperative behaviour of dislocations. Solid State Phenom.
**3–4**, 357–367 (1991)CrossRefGoogle Scholar - 20.Ananthakrishna, G., Valsakumar, M.C.: Chaotic flow in a model for repeated yielding. Phys. Lett. A
**95**(2), 69–71 (1983)MathSciNetCrossRefGoogle Scholar - 21.Satyanarayana, S.V.M., Sridhar, V., Koka, S.: Characterization of chaos in a serrated plastic flow model. Pramana-J. Phys.
**48**(4), 871–882 (1997)CrossRefGoogle Scholar - 22.Cottrell, A.H., Bilby, B.A.: Dislocation theory of yielding and strain ageing of iron. Proc. R. Soc. Lond. Ser. A
**62**(1), 49–62 (1949)CrossRefGoogle Scholar - 23.Li, X.P., Ren, J.L., Campbell, S.A., Wolkowicz, G.S.K., Zhu, H.P.: How seasonal forcing influences the complexity of a predated-pery system. Discrete Contin. Dyn. B
**23**(2), 785–807 (2018)MathSciNetGoogle Scholar - 24.Ren, J.L., Li, X.P.: Bifurcations in a seasonally forced predator-prey model with generalized Holling type IV functional response. Int. J. Bifurc. Chaos
**26**(12), 1650203 (2016)MathSciNetCrossRefMATHGoogle Scholar - 25.Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 2nd edn. Springer, New York (2004)CrossRefMATHGoogle Scholar
- 26.Gillis, P.P., Gilman, J.J.: Dynamical dislocation theory of crystal plasticity. I. The yield stress. J. Appl. Phys.
**36**(11), 3370–3380 (1965)CrossRefGoogle Scholar - 27.Sahoo, D., Ananthakrishna, G.: A phenomenological dislocation transformation model for the mobile fraction in simple systems. J. Phys. D Appl. Phys.
**15**(8), 1439–1449 (1982)CrossRefGoogle Scholar - 28.Cottrell, A.H.: A note on the PorteVin-Le Chatelier effect. Philos. Mag.
**44**(355), 829–832 (1953)CrossRefGoogle Scholar - 29.Bekele, M., Ananthakrishna, G.: High-order amplitude equation for steps on the creep curve. Phys. Rev. E
**56**(6), 6917–6928 (1997)CrossRefGoogle Scholar - 30.Rajesh, S., Ananthakrishna, G.: Relaxation oscillations and negative strain rate sensitivity in the Portevin-Le Chatelier effect. Phys. Rev. E
**61**(4), 3664–3674 (2000)CrossRefGoogle Scholar