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Intermittency in Crystal Plasticity Informed by Lattice Symmetry

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Abstract

We develop a nonlinear, three-dimensional phase field model for crystal plasticity which accounts for the infinite and discrete symmetry group \(G\) of the underlying periodic lattice. This generates a complex energy landscape with countably-many \(G\)-related wells in strain space, whereon the material evolves by energy minimization under the loading through spontaneous slip processes inducing the creation and motion of dislocations without the need of auxiliary hypotheses. Multiple slips may be activated simultaneously, in domains separated by a priori unknown free boundaries. The wells visited by the strain at each position and time, are tracked by the evolution of a \(G\)-valued discrete plastic map, whose non-compatible discontinuities identify lattice dislocations. The main effects in the plasticity of crystalline materials at microscopic scales emerge in this framework, including the long-range elastic fields of possibly interacting dislocations, lattice friction, hardening, band-like vs. complex spatial distributions of dislocations. The main results concern the scale-free intermittency of the flow, with power-law exponents for the slip avalanche statistics which are significantly affected by the symmetry and the compatibility properties of the activated fundamental shears.

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Notes

  1. As for instance in [18], we do not include here any interface-penalizing gradient term in the free-energy functional because in all our simulations we keep the typical distance \(H\) between interpolation nodes always greater than or equal to the minimal physically relevant length \(h\) in the continuum, which in the present context is taken of the order the lattice cell spacing. To model plastic behavior of crystals at scales \(H\) larger than \(h\), besides including a gradient term in the phase field potential, the Schmidt tensors in (1) must also be suitably re-scaled (see for instance [35]). The corresponding scaling imposed on the potential \(\psi_{\mu}\) determines whether in the limit \(H\gg h\) the yield and Peierls stresses vanish or stay finite. In what follows we only consider plasticity at microscales with \(H\simeq h\).

References

  1. Acharya, A.: Jump condition on GND evolution as a constraint on slip transmission at grain boundaries. Philos. Mag. 87, 1349–1359 (2007)

    Article  ADS  Google Scholar 

  2. Alava, M.J., Laurson, L., Zapperi, S.: Review: Crackling noise in plasticity. Eur. Phys. J. Spec. Top. 223, 2353–2367 (2014)

    Article  Google Scholar 

  3. Ariza, M.P., Ortiz, M.: Discrete crystal elasticity and discrete dislocations in crystals. Arch. Ration. Mech. Anal. 178, 149–226 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Asaro, R.J.: Crystal plasticity. J. Appl. Mech. 50, 921–934 (1983)

    Article  ADS  MATH  Google Scholar 

  5. Balandraud, X., Barrera, N., Biscari, P., Grédiac, M., Zanzotto, G.: Strain intermittency in shape-memory alloys. Phys. Rev. B 91, 174111 (2015)

    Article  ADS  Google Scholar 

  6. Bhattacharya, K., Conti, S., Zanzotto, G., Zimmer, J.: Crystal symmetry and the reversibility of martensitic transformations. Nature 428, 55–59 (2004)

    Article  ADS  Google Scholar 

  7. Bilby, B.A., Bullough, R., de Grinberg, D.K.: General theory of surface dislocations. Discuss. Faraday Soc. 38, 61–68 (1964)

    Article  Google Scholar 

  8. Bortoloni, L., Cermelli, P.: Dislocation patterns and work-hardening in crystalline plasticity. J. Elast. 76, 113–138 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brinckmann, S., Kim, J.-Y., Greer, J.R.: Fundamental differences in mechanical behavior between two types of crystals at the nanoscale. Phys. Rev. Lett. 100, 155502 (2008)

    Article  ADS  Google Scholar 

  10. Cermelli, C., Gurtin, M.E.: The dynamics of solid-solid phase transitions 2. Incoherent interfaces. Arch. Ration. Mech. Anal. 127, 41–99 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cermelli, P., Gurtin, M.E.: On the characterization of geometrically necessary dislocations in finite plasticity. J. Mech. Phys. Solids 49, 1539–1568 (2001)

    Article  ADS  MATH  Google Scholar 

  12. Chaari, N., Clouet, E., Rodney, D.: First-principles study of secondary slip in zirconium. Phys. Rev. Lett. 112, 075504 (2014)

    Article  ADS  Google Scholar 

  13. Clauset, A., Shalizi, C.R., Newman, M.E.J.: Power-law distributions in empirical data. SIAM Rev. 51, 661–703 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Conti, S., Zanzotto, G.: A variational model for reconstructive phase transformations in crystals, and their relation to dislocations and plasticity. Arch. Ration. Mech. Anal. 173, 69–88 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cottrell, A.H., Bilby, B.A.: Dislocation theory of yielding and strain ageing of iron. Proc. Phys. Soc. A 62, 49–62 (1949)

    Article  ADS  Google Scholar 

  16. Csikor, F.F., Motz, C., Weygand, D., Zaiser, M., Zapperi, S.: Dislocation avalanches, strain bursts, and the problem of plastic forming at the micrometer scale. Science 318, 251–254 (2007)

    Article  ADS  Google Scholar 

  17. Dahmen, K.A., Ben-Zion, Y., Uhl, J.T.: Micromechanical model for deformation in solids with universal predictions for stress-strain curves and slip avalanches. Phys. Rev. Lett. 102, 175501 (2009)

    Article  ADS  Google Scholar 

  18. Denoual, C., Caucci, A.M., Soulard, L., Pellegrini, Y.-P.: Phase-field reaction-pathway kinetics of martensitic transformations in a model \(\mathrm{Fe}_{3}\) Ni alloy. Phys. Rev. Lett. 105, 035703 (2010)

    Article  ADS  Google Scholar 

  19. Devincre, B., Hoc, T., Kubin, L.: Dislocation mean free paths and strain hardening of crystals. Science 320, 1745–1748 (2008)

    Article  ADS  Google Scholar 

  20. Dezerald, L., Ventelon, L., Clouet, E., Denoual, C., Rodney, D., Willaime, F.: Ab initio modeling of the two-dimensional energy landscape of screw dislocations in bcc transition metals. Phys. Rev. B 89, 024104 (2014)

    Article  ADS  Google Scholar 

  21. Dimiduk, D.M., Woodward, C., LeSar, R., Uchic, M.D.: Scale-free intermittent flow in crystal plasticity. Science 312, 1188–1190 (2006)

    Article  ADS  Google Scholar 

  22. Ding, X., Zhao, Z., Lookman, T., Saxena, A., Salje, E.K.H.: High junction and twin boundary densities in driven dynamical systems. Adv. Mater. 24, 5385–5389 (2012)

    Article  Google Scholar 

  23. Ericksen, J.L.: Some phase transitions in crystals. Arch. Ration. Mech. Anal. 73, 99–124 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fedelich, B., Zanzotto, G.: Hysteresis in discrete systems of possibly interacting elements with a double-well energy. J. Nonlinear Sci. 2, 319–342 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Fressengeas, C., Beaudoin, A.J., Entemeyer, D., Lebedkina, T., Lebyodkin, M., Taupin, V.: Dislocation transport and intermittency in the plasticity of crystalline solids. Phys. Rev. B 79, 014108 (2009)

    Article  ADS  Google Scholar 

  26. Gu, R., Ngan, A.H.W.: Dislocation arrangement in small crystal volumes determines power-law size dependence of yield strength. J. Mech. Phys. Solids 61, 1531–1542 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  27. Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  28. Hirth, J., Lothe, J.: Theory of Dislocations. Wiley, New York (1982)

    Google Scholar 

  29. Ispanovity, P.D., Laurson, L., Zaiser, M., Groma, I., Zapperi, S., Alava, M.J.: Avalanches in 2D dislocation systems: Plastic yielding is not depinning. Phys. Rev. Lett. 112, 235501 (2014)

    Article  ADS  Google Scholar 

  30. Koslowski, M., Ortiz, M.: A multi-phase field model of planar dislocation networks. Model. Simul. Mater. Sci. Eng. 12, 1087–1097 (2004)

    Article  ADS  Google Scholar 

  31. Koslowski, M., LeSar, R., Thomson, R.: Dislocation structures and the deformation of materials. Phys. Rev. Lett. 93, 265503 (2004)

    Article  ADS  Google Scholar 

  32. Levitas, V.I.: Thermodynamically consistent phase field approach to phase transformations with interface stresses. Acta Mater. 61, 4305–4319 (2013)

    Article  Google Scholar 

  33. Levitas, V.I.: Phase-field theory for martensitic phase transformations at large strains. Int. J. Plast. 49, 85–118 (2013)

    Article  Google Scholar 

  34. Levitas, V.I.: Phase field approach to martensitic phase transformations with large strains and interface stresses. J. Mech. Phys. Solids 70, 154–189 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  35. Levitas, V.I., Javanbakht, M.: Advanced phase-field approach to dislocation evolution. Phys. Rev. B 86, 140101(R) (2012)

    Article  ADS  Google Scholar 

  36. Mamivand, M., Asle Zaeem, M., El Kadiri, H.: Shape memory effect and pseudoelasticity behavior in tetragonal zirconia polycrystals: A phase field study. Int. J. Plast. 60, 71–86 (2014)

    Article  Google Scholar 

  37. Miguel, M.-C., Vespignani, A., Zaiser, M., Zapperi, S.: Dislocation jamming and Andrade creep. Phys. Rev. Lett. 89, 165501 (2002)

    Article  ADS  Google Scholar 

  38. Miller, R.E., Rodney, D.: On the nonlocal nature of dislocation nucleation during nanoindentation. J. Mech. Phys. Solids 56, 1203–1223 (2008)

    Article  ADS  MATH  Google Scholar 

  39. Moretti, P., Miguel, M.-C., Zaiser, M., Zapperi, S.: Depinning transition of dislocation assemblies: Pileups and low-angle grain boundaries. Phys. Rev. B 69, 214103 (2004)

    Article  ADS  Google Scholar 

  40. Niemann, R., Baró, J., Heczko, O., Schultz, L., Fähler, S., Vives, E., Mañosa, Ll., Planes, A.: Tuning avalanche criticality: Acoustic emission during the martensitic transformation of a compressed Ni-Mn-Ga single crystal. Phys. Rev. B 86, 214101 (2012)

    Article  ADS  Google Scholar 

  41. Niemann, R., Kopecek, J., Heczko, O., Romberg, J., Schultz, L., Fähler, S., Vives, E., Mañosa, Ll., Planes, A.: Localizing sources of acoustic emission during the martensitic transformation. Phys. Rev. B 89, 214118 (2014)

    Article  ADS  Google Scholar 

  42. Perez-Reche, F.-J., Truskinovsky, L., Zanzotto, G.: Training-induced criticality in martensites. Phys. Rev. Lett. 99, 075501 (2007)

    Article  ADS  Google Scholar 

  43. Pitteri, M., Zanzotto, G.: Continuum Models for Phase Transitions and Twinning in Crystals. Chapman & Hall, London (2002)

    Book  Google Scholar 

  44. Planes, A., Mañosa, Ll., Vives, E.: Acoustic emission in martensitic transformations. J. Alloys Compd. 577, S699–S704 (2013)

    Article  Google Scholar 

  45. Reina, C., Conti, S.: Kinematic description of crystal plasticity in the finite kinematic framework: A micromechanical understanding of \(\mathbf {F}=\mathbf{F}_{\mathrm{r}} \mathbf{F}_{\mathrm{p}}\). J. Mech. Phys. Solids 67, 40–61 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  46. Rodney, D.: Activation enthalpy for kink-pair nucleation on dislocations: Comparison between static and dynamic atomic-scale simulations. Phys. Rev. B 76, 144108 (2007)

    Article  ADS  Google Scholar 

  47. Rodney, D., Proville, D.: Stress-dependent Peierls potential: Influence on kink-pair activation. Phys. Rev. B 79, 094108 (2009)

    Article  ADS  Google Scholar 

  48. Rodney, D., Le Bouar, Y., Finel, A.: Phase field methods and dislocations. Acta Mater. 51, 17–30 (2003)

    Article  Google Scholar 

  49. Salman, O.U., Truskinovsky, L.: Minimal integer automaton behind crystal plasticity. Phys. Rev. Lett. 106, 175503 (2011)

    Article  ADS  Google Scholar 

  50. Salman, O.U., Truskinovsky, L.: On the critical nature of plastic flow: One and two dimensional models. Int. J. Eng. Sci. 59, 219–254 (2012)

    Article  Google Scholar 

  51. Shen, C., Wang, Y.: Incorporation of \(\gamma\)-surface to phase field model of dislocations: Simulating dislocation dissociation in fcc crystals. Acta Mater. 52, 683–691 (2004)

    Article  Google Scholar 

  52. Song, Y., Chen, X., Dabade, V., Shield, T.W., James, R.D.: Enhanced reversibility and unusual microstructure of a phase-transforming material. Nature 502, 85–88 (2013)

    Article  ADS  Google Scholar 

  53. Steinbach, I., Pezzolla, F., Nestler, B., Seeßelberg, M., Prieler, R., Schmitz, G.J., Rezende, J.L.L.: A phase field concept for multiphase systems. Physica D 94, 135–147 (1996)

    Article  ADS  MATH  Google Scholar 

  54. Toth, L.Z., Szabo, S., Daroczi, L., Beke, D.L.: Calorimetric and acoustic emission study of martensitic transformation in single-crystalline \(\mathrm{Ni}_{2}\) MnGa alloys. Phys. Rev. B 90, 224103 (2014)

    Article  ADS  Google Scholar 

  55. Truskinovsky, L., Vainchtein, A.: The origin of nucleation peak in transformational plasticity. J. Mech. Phys. Solids 52, 1421–1446 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. Tsekenis, G., Uhl, J.T., Goldenfeld, N., Dahmen, K.A.: Determination of the universality class of crystal plasticity. Europhys. Lett. 101, 36003 (2013)

    Article  ADS  Google Scholar 

  57. Uchic, M.D., Shade, P.A., Dimiduk, D.M.: Plasticity of micrometer-scale single crystals in compression. Annu. Rev. Mater. Res. 39, 361–386 (2009)

    Article  ADS  Google Scholar 

  58. Wang, Y., Li, J.: Phase field modeling of defects and deformation. Acta Mater. 58, 1212–1235 (2010)

    Article  Google Scholar 

  59. Wang, Y.U., Jin, Y.M., Cuitiño, A.M., Khachaturyan, A.G.: Nanoscale phase field microelasticity theory of dislocations: Model and 3D simulations. Acta Mater. 49, 1847–1857 (2001)

    Article  Google Scholar 

  60. Wang, Y.U., Jin, Y.M., Khachaturyan, A.G.: Phase field microelasticity modeling of dislocation dynamics near free surface and in heteroepitaxial thin. Acta Mater. 51, 4209–4223 (2003)

    Article  Google Scholar 

  61. Weiss, J., Ben Rhouma, W., Richeton, T., Dechanel, S., Louchet, F., Truskinovsky, L.: From mild to wild fluctuations in crystal plasticity. Phys. Rev. Lett. 114, 105504 (2015)

    Article  ADS  Google Scholar 

  62. Zaiser, M.: Scale invariance in plastic flow of crystalline solids. Adv. Phys. 55, 185–245 (2006)

    Article  ADS  Google Scholar 

  63. Zhang, X., Pan, B., Shang, F.: Scale-free behavior of displacement bursts: Lower limit and scaling exponent. Europhys. Lett. 100, 16005 (2012)

    Article  Google Scholar 

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Acknowledgements

We acknowledge financial support from the Italian PRIN Contract 200959L72B004 and from a SAES Getters - Politecnico di Milano research contract.

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Authors and Affiliations

  1. Department of Physics, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy

    Paolo Biscari & Anna Zanzottera

  2. SAES Getters, Viale Italia 77, 20020, Lainate, Italy

    Marco Fabrizio Urbano

  3. DPG, Università di Padova, Via Venezia 8, 35131, Padova, Italy

    Giovanni Zanzotto

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  1. Paolo Biscari
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Correspondence to Paolo Biscari.

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Biscari, P., Urbano, M.F., Zanzottera, A. et al. Intermittency in Crystal Plasticity Informed by Lattice Symmetry. J Elast 123, 85–96 (2016). https://doi.org/10.1007/s10659-015-9548-z

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