Abstract
A simplified one dimensional rate dependent model for the evolution of plastic distortion is obtained from a three dimensional mechanically rigorous model of mesoscale field dislocation mechanics. Computational solutions of variants of this minimal model are investigated to explore the ingredients necessary for the development of microstructure. In contrast to prevalent notions, it is shown that microstructure can be obtained even in the absence of non-monotone equations of state. In this model, incorporation of wave propagative dislocation transport is vital for the modeling of spatial patterning. One variant gives an impression of producing stochastic behavior, despite being a completely deterministic model. The computations focus primarily on demanding macroscopic limit situations, where a convergence study reveals that the model-variant including non-monotone equations of state cannot serve as effective equations in the macroscopic limit; the variant without non-monotone ingredients, in all likelihood, can.
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References
Abeyaratne R, Chu C, James RD (1996) Kinetics of materials with wiggly energies: theory and application to the evolution of twinning microstructures in a Cu–Al–Ni shape memory alloy. Philos Mag A 73(2):457–497
Acharya A (2010) New inroads in an old subject: plasticity, from around the atomic to the macroscopic scale. J Mech Phys Solids 58(5):766–778
Acharya A, Beaudoin AJ (2000) Grain-size effect in viscoplastic polycrystals at moderate strains. J Mech Phys Solids 48(10):2213–2230
Acharya A, Roy A (2006) Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model of mesoscopic field dislocation mechanics: Part I. J Mech Phys Solids 54(8):1687–1710
Acharya A, Tang H, Saigal S, Bassani JL (2004) On boundary conditions and plastic strain-gradient discontinuity in lower-order gradient plasticity. J Mech Phys Solids 52(8):1793–1826
Acharya A, Tartar L (2011) On an equation from the theory of field dislocation mechanics. Boll dell’Unione Mat Ital 9:409–444
Aifantis EC (1984) On the microstructural origin of certain inelastic models. J Eng Mater Technol 106(4):326–330
Barenblatt GI (1996) Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics, vol 14. Cambridge University Press, Cambridge
Chen YS, Choi W, Papanikolaou S, Sethna JP (2010) Bending crystals: emergence of fractal dislocation structures. Phys Rev Lett 105(10):105501
Chen YS, Choi W, Papanikolaou S, Bierbaum M, Sethna JP (2013) Scaling theory of continuum dislocation dynamics in three dimensions: self-organized fractal pattern formation. Int J Plast 46:94–129
Choi W, Chen YS, Papanikolaou S, Sethna JP (2012) Is dislocation flow turbulent in deformed crystals? Comput Sci Eng 14(1):33–39
Das A, Acharya A, Zimmer J, Matthies K (2013) Can equations of equilibrium predict all physical equilibria? A case study from Field Dislocation Mechanics. Math Mech Solids 18(8):803–822
Dimiduk DM, Woodward C, LeSar R, Uchic MD (2006) Scale-free intermittent flow in crystal plasticity. Science 312(5777):1188–1190
Glazov M, Llanes LM, Laird C (1995) Self-organized dislocation structures (SODS) in fatigued metals. Phys Status Solidi (a) 149(1):297–321
Kreiss HO, Lorenz J (1989) Initial-boundary value problems and the Navier–Stokes equations, Vol. 47 of Classics in Applied Mathematics Series. SIAM, Philadelphia
Kurganov A, Noelle S, Petrova G (2001) Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations. SIAM J Sci Comput 23(3):707–740
Limkumnerd S, Sethna JP (2006) Mesoscale theory of grains and cells: crystal plasticity and coarsening. Phys Rev Lett 96(9):095503
Mughrabi H, Ackermann FU, Herz K (1979) Persistent slip bands in fatigued face-centered and body-centered cubic metals. Fatigue Mech ASTM STP 675:69–105
Ortiz M, Repetto EA (1999) Nonconvex energy minimization and dislocation structures in ductile single crystals. J Mech Phys Solids 47(2):397–462
Parthasarathy TA, Rao SI, Dimiduk DM, Uchic MD, Trinkle DR (2007) Contribution to size effect of yield strength from the stochastics of dislocation source lengths in finite samples. Scr Mater 56(4):313–316
Rice JR (1971) Inelastic constitutive relations for solids: an intenal-variable theory and its application to metal plasticity. J Mech Phys Solids 19(6):433–455
Roy A, Acharya A (2006) Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model of mesoscopic field dislocation mechanics: Part II. J Mech Phys Solids 54(8):1711–1743
Tartar L (2009) The general theory of homogenization. In: Rangarajan A, Vemuri B, Yuille AL (eds) Lecture notes of the unione matematica Italiana. Springer, Heidelberg
Xia S, El-Azab A (2015) Computational modelling of mesoscale dislocation patterning and plastic deformation of single crystals. Model Simul Mater Sci Eng 23(5):055009
Walgraef D, Aifantis EC (1985) On the formation and stability of dislocation patterns–I: one-dimensional considerations. Int J Eng Sci 23(12):1351–1358
Walgraef D, Aifantis EC (1985) On the formation and stability of dislocation patterns–II: two-dimensional considerations. Int J Eng Sci 23(12):1359–1364
Walgraef D, Aifantis EC (1985) On the formation and stability of dislocation patterns–III: three-dimensional considerations. Int J Eng Sci 23(12):1365–1372
Zhang X, Acharya A, Walkington NJ, Bielak J (2015) A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations. J Mech Phys Solids 84:145–195
Acknowledgments
AA acknowledges support in part from Grants NSF-CMMI-1435624, NSF-DMS-1434734, and ARO W911NF-15-1-0239. PS acknowledges the support of the Labex MEC and of A*Midex through Grants ANR-11- LABX-0092 and ANR-11-IDEX-0001-02.
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Appendix 1
Appendix 1
1.1 Simplified one-dimensional form of conservation law of Burgers vector content
The conservation law for Burgers vector content in mesoscale field dislocation mechanics is given by
where \({\varvec{\alpha }} \) is the mesoscale Nye tensor, \({\varvec{V}}\) is the mesoscale averaged dislocation velocity vector, and \({\varvec{L}}^{p}\) is the plastic strain rate produced by unresolved, at the mesoscale, and hence ‘statistical’ dislocations [4]. Here,
Consider a tensor field of the form
and we assume that all fields vary in only the \(x_3 \) direction. Then the only non-zero component of
is
We assume the ansatz
Then
Denoting \(U_{12}^p =\varphi \), \(x_3 =x, \;x_1 =y,\;x_2 =z\), \(V_3=V\), and \(L_{12}^p =L^{p}\) the conservation law reduces to the equation
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Das, A., Acharya, A. & Suquet, P. Microstructure in plasticity without nonconvexity. Comput Mech 57, 387–403 (2016). https://doi.org/10.1007/s00466-015-1249-8
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DOI: https://doi.org/10.1007/s00466-015-1249-8