×

Theoretical development of continuum dislocation dynamics for finite-deformation crystal plasticity at the mesoscale. (English) Zbl 1477.74088

Summary: The equations of dislocation transport at finite crystal deformation were developed, with a special emphasis on a vector density representation of dislocations. A companion thermodynamic analysis yielded a generalized expression for the driving force of dislocations that depend on Mandel (Cauchy) stress in the reference (spatial) configurations and the contribution of the dislocation core energy to the free energy of the crystal. Our formulation relied on several dislocation density tensor measures linked to the incompatibility of the plastic distortion in the crystal. While previous works develop such tensors starting from the multiplicative decomposition of the deformation gradient, we developed the tensor measures of the dislocation density and the dislocation flux from the additive decomposition of the displacement gradient and the crystal velocity fields. The two-point dislocation density measures defined by the referential curl of the plastic distortion and the spatial curl of the inverse elastic distortion and the associate dislocation currents were found to be more useful in deriving the referential and spatial forms of the transport equations for the vector density of dislocations. A few test problems showing the effect of finite deformation on the static dislocation fields are presented, with a particular attention to lattice rotation. The framework developed provides the theoretical basis for investigating crystal plasticity and dislocation patterning at the mesoscale, and it bears the potential for realistic comparison with experiments upon numerical solution.

MSC:

74N05 Crystals in solids
74A45 Theories of fracture and damage
74C99 Plastic materials, materials of stress-rate and internal-variable type
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Acharya, A., Microcanonical entropy and mesoscale dislocation mechanics and plasticity, J. Elast., 104, 23-44 (2011) · Zbl 1320.74010
[2] Acharya, A., Constitutive analysis of finite deformation field dislocation mechanics, J. Mech. Phys. Solids, 52, 301-316 (2004) · Zbl 1106.74315
[3] Acharya, A., A model of crystal plasticity based on the theory of continuously distributed dislocations, J. Mech. Phys. Solids, 49, 761-784 (2001) · Zbl 1017.74010
[4] Acharya, A.; Roy, A., 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, 1687-1710 (2006) · Zbl 1120.74328
[5] Acharya, A.; Roy, A.; Sawant, A., Continuum theory and methods for coarse-grained, mesoscopic plasticity, Scr. Mater., 54, 705-710 (2006)
[6] Anderson, P. M.; Hirth, J. P.; Lothe, J., Theory of Dislocations (2017), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1365.82001
[7] Arora, R.; Acharya, A., Dislocation pattern formation in finite deformation crystal plasticity, Int. J. Solids Struct. (2019)
[8] Arsenlis, A.; Cai, W.; Tang, M.; Rhee, M.; Oppelstrup, T.; Hommes, G.; Pierce, T. G.; Bulatov, V. V., Enabling strain hardening simulations with dislocation dynamics, Model. Simul. Mater. Sci. Eng., 15, 553-595 (2007)
[9] Bilby, B. A.; Bullough, R.; Gardner, L. R.T.; Smith, E., Continuous distributions of dislocations IV. Single glide and plane strain, Proc. R. Soc. A, 244, 538-557 (1958)
[10] Bilby, B. A.; Bullough, R.; Smith, E.; Gardner, L. R.T.; Smith, E., Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry, Proc. R. Soc. A: Math. Phys. Eng. Sci., 231, 263-273 (1955)
[11] Brenner, R.; Beaudoin, A. J.; Suquet, P.; Acharya, A., Numerical implementation of static field dislocation mechanics theory for periodic media, Philos. Mag., 94, 1764-1787 (2014)
[12] Cermelli, P.; Gurtin, M. E., On the characterization of geometrically necessary dislocations in finite plasticity, J. Mech. Phys. Solids, 49, 1539-1568 (2001) · Zbl 0989.74013
[13] de Wit, R., Continuous distribution of disclination loops, Phys. Status Solidi, 18, 669-681 (1973)
[14] de Wit, R., Theory of disclinations: III. Continuous and discrete disclinations in isotropic elasticity, J. Res. Natl. Bur. Stand. - A. Phys. Chem., 77, 359-368 (1973)
[15] Deng, J.; El-Azab, A., Temporal statistics and coarse graining of dislocation ensembles, Philos. Mag., 90, 3651-3678 (2010)
[16] Deng, J.; El-Azab, A., Mathematical and computational modelling of correlations in dislocation dynamics, Model. Simul. Mater. Sci. Eng., 17, Article 075010 pp. (2009)
[17] Deng, J.; El-Azab, A., Dislocation pair correlations from dislocation dynamics simulations, J. Comput. Mater. Des., 14, 295-307 (2007)
[18] Devincre, B.; Madec, R.; Monnet, G.; Queyreau, S.; Gatti, R.; Kubin, L., Modeling crystal plasticity with dislocation dynamics simulations: the “microMegas” code, Mech. Nano-Objects, 1, 81-99 (2011)
[19] El-Azab, A., Statistical mechanics of dislocation systems, Scr. Mater., 54, 723-727 (2006)
[20] El-Azab, A., Statistical mechanics treatment of the evolution of dislocation distributions in single crystals, Phys. Rev. B, 61, 11956-11966 (2000)
[21] El-Azab, A.; Po, G., Continuum dislocation dynamics: classical theory and contemporary models, Handbook of Materials Modeling, 1-23 (2018)
[22] Gurtin, M. E., A finite-deformation, gradient theory of single-crystal plasticity with free energy dependent on the accumulation of geometrically necessary dislocations, Int. J. Plast., 26, 1073-1096 (2010) · Zbl 1432.74047
[23] Gurtin, M. E., The Burgers vector and the flow of screw and edge dislocations in finite-deformation single-crystal plasticity, J. Mech. Phys. Solids, 54, 1882-1898 (2006) · Zbl 1120.74394
[24] Gurtin, M. E.; Anand, L.; Lele, S. P., Gradient single-crystal plasticity with free energy dependent on dislocation densities, J. Mech. Phys. Solids, 55, 1853-1878 (2007) · Zbl 1170.74013
[25] Gurtin, M. E.; Fried, E.; Anand, L., The Mechanics and Thermodynamics of Continua (2010), Cambridge University Press: Cambridge University Press Cambridge
[26] Gurtin, M. E.; Reddy, B. D., Some issues associated with the intermediate space in single-crystal plasticity, J. Mech. Phys. Solids, 95, 230-238 (2016)
[27] Hansen, N.; Huang, X., Microstructure and flow stress of polycrystals and single crystals, Acta Mater., 46, 1827-1836 (1998)
[28] Hauser, W., On the fundamental equations of electromagnetism, Am. J. Phys., 38, 80-85 (1970)
[29] Hochrainer, T., Thermodynamically consistent continuum dislocation dynamics, J. Mech. Phys. Solids, 88, 12-22 (2016)
[30] Hochrainer, T., Moving dislocations in finite plasticity: a topological approach, Z. Angew. Math. Mech., 93, 252-268 (2013) · Zbl 1277.74019
[31] Hochrainer, T.; Sandfeld, S.; Zaiser, M.; Gumbsch, P., Continuum dislocation dynamics: towards a physical theory of crystal plasticity, J. Mech. Phys. Solids, 63, 167-178 (2014)
[32] Hong, C.; Huang, X.; Winther, G., Dislocation content of geometrically necessary boundaries aligned with slip planes in rolled aluminium, Philos. Mag., 93, 3118-3141 (2013)
[33] Huang, X.; Hansen, N., Grain orientation dependence of microstructure in aluminium deformed in tension, Scr. Mater., 37, 1-7 (1997)
[34] Huang, X.; Winther, G., Dislocation structures. Part I. Grain orientation dependence, Philos. Mag., 87, 5189-5214 (2007)
[35] Hughes, D.. A.; Hansen, N., Deformation structures developing on fine scales, Philos. Mag., 83, 3871-3893 (2003)
[36] Hughes, D...; Liu, Q.; Chrzan, D. C.; Hansen, N., Scaling of microstructural parameters: misorientations of deformation induced boundaries, Acta Mater., 45, 105-112 (1997)
[37] Jakobsen, B., Formation and subdivision of deformation structures during plastic deformation, Science, 312, 889-892 (2006)
[38] Kashihara, K.; Tagami, M.; Inoko, F., Deformed structure and crystal orientation at deformation bands in <011> aluminum single crystals, Mater. Trans. JIM, 37, 564-571 (1996)
[39] Kawasaki, Y.; Takeuchi, T., Cell structures in copper single crystals deformed in the [001] and [111] axes, Scr. Metall., 14, 183-188 (1980)
[40] Kondo, K., On the geometrical and physical foundations of the theory of yielding, (Proceedings of the 2nd Japan National Congress for Applied Mechanics, 2 (1952)), 41-47
[41] Kosevich, A.M.M., 1965. Dynamical Theory of Dislocations. New York5, 579-609. https://doi.org/10.3367/UFNr.0084.196412c.0579
[42] Kossecka, E., De Wit, R., 1977. Disclination Kinematics. Arch Mech 29. · Zbl 0375.73094
[43] Kröner, E., Continuum theory of defects, Phsique des Defauts/Physics of Defects, Les houches, 215-315 (1981), North-Holland, session XXXV, 1980
[44] Kröner, E., 1960. General Continuum Theory of Dislocations and Proper Stresses4, 273-334.
[45] Kröner, E., Allgemeine kontinuumstheorie der versetzungen und eigenspannungen, Arch. Ration. Mech. Anal., 4, 273-334 (1959) · Zbl 0090.17601
[46] Kröner, E., Continuum theory of dislocations and self-stresses, Ergeb. Angew. Math., 5, 1-277 (1958)
[47] Langdon, N., Explicit expressions for stress field of a circular dislocation loop, Theor. Appl. Fract. Mech., 33, 219-231 (2000)
[48] Larson, B. C.; El-Azab, A.; Yang, W.; Tischler, J. Z.; Liu, W.; Ice, G. E., Experimental characterization of the mesoscale dislocation density tensor, Philos. Mag., 87, 1327-1347 (2007)
[49] Le, G. M.; Godfrey, A.; Hansen, N.; Liu, W.; Winther, G.; Huang, X., Influence of grain size in the near-micrometre regime on the deformation microstructure in aluminium, Acta Mater., 61, 7072-7086 (2013)
[50] Le, G. M.; Godfrey, A.; Hong, C. S.; Huang, X.; Winther, G., Orientation dependence of the deformation microstructure in compressed aluminum, Scr. Mater., 66, 359-362 (2012)
[51] Lee, E. H., Elastic-plastic deformation at finite strains, J. Appl. Mech., 36, 1 (1969) · Zbl 0179.55603
[52] Levine, L. E.; Geantil, P.; Larson, B. C.; Tischler, J. Z.; Kassner, M. E.; Liu, W.; Stoudt, M. R.; Tavazza, F., Disordered long-range internal stresses in deformed copper and the mechanisms underlying plastic deformation, Acta Mater., 59, 5803-5811 (2011)
[53] Lin, F. X.; Godfrey, A.; Jensen, D. J.; Winther, G., 3D EBSD characterization of deformation structures in commercial purity aluminum, Mater. Charact. (2010)
[54] Lin, P., El-Azab, A., 2019. Implementation of Annihilation and Junction Reactions in Vector Density-Based Continuum Dislocation Dynamics.
[55] Liu, Q.; Juul Jensen, D.; Hansen, N., Effect of grain orientation on deformation structure in cold-rolled polycrystalline aluminium, Acta Mater., 46, 5819-5838 (1998)
[56] Monavari, M.; Sandfeld, S.; Zaiser, M., Continuum representation of systems of dislocation lines: a general method for deriving closed-form evolution equations, J. Mech. Phys. Solids, 95, 575-601 (2016)
[57] Morro, A., Evolution equations and thermodynamic restrictions for dissipative solids, Math. Comput. Model. (2010) · Zbl 1205.80037
[58] Mura, T., Continuous distribution of moving dislocations, Philos. Mag., 24, 63-69 (1963)
[59] Nye, J. F., Some geometrical relations in dislocated crystals, Acta Metall., 1, 153-162 (1952)
[60] Oldroyd, J., On the formulation of rheological equations of state, Proc. R. Soc. Lond. Ser. A: Math. Phys. Sci., 200, 523-541 (1950) · Zbl 1157.76305
[61] Po, G.; Huang, Y.; Ghoniem, N., A continuum dislocation-based model of wedge microindentation of single crystals, Int. J. Plast., 114, 72-86 (2019)
[62] Po, G.; Mohamed, M. S.; Crosby, T.; Erel, C.; El-Azab, A.; Ghoniem, N., Recent progress in discrete dislocation dynamics and its applications to micro plasticity, JOM, 66, 2108-2120 (2014)
[63] Reddy, J. N., An Introduction to Continuum Mechanics (2013), Cambridge University Press: Cambridge University Press Cambridge
[64] Reed, R. P.; Clark, A. F., Materials at Low Temperatures, 590 (1983), American Society for Metals
[65] Rice, B.J.R., 1971. Inelastic Constitutive Relations for Solids: Theory and its Application19, 433-455. · Zbl 0235.73002
[66] Roy, A.; Acharya, A., 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, 1711-1743 (2006) · Zbl 1120.74333
[67] Sedláček, R.; Schwarz, C.; Kratochvíl, J.; Werner, E., Continuum theory of evolving dislocation fields, Philos. Mag., 87, 1225-1260 (2007)
[68] Sills, R. B.; Kuykendall, W. P.; Aghaei, A.; Cai, W., Fundamentals of Dislocation Dynamics Simulations, 53-87 (2016), Springer Series in Materials Science
[69] Steeds, J..., Dislocation arrangement in copper single crystals as a function of strain, Proc. R. Soc. Lond. Ser. A: Math. Phys. Sci., 292, 343-373 (1966)
[70] Stricker, M.; Sudmanns, M.; Schulz, K.; Hochrainer, T.; Weygand, D., Dislocation multiplication in stage II deformation of fcc multi-slip single crystals, J. Mech. Phys. Solids, 119, 319-333 (2018)
[71] Tagami, M.; Kashihara, K.; Okada, T.; Inoko, F., Effect of cross slips on multiple slips and recrystallization in <111> and <001> Al single crystals, Nippon Kinzoku Gakkaishi/Journal Japan Inst. Met. (2000)
[72] Ván, P., Objective time derivatives in nonequilibrium thermodynamics, Proc. Estonian Acad. Sci. (2008) · Zbl 1186.80002
[73] Weygand, D.; Friedman, L. H.; Giessen, E. Van der; Needleman, A.; Van Der Giessen, E.; Needleman, A., Aspects of boundary-value problem solutions with three-dimensional dislocation dynamics, Model. Simul. Mater. Sci. Eng., 10, 437-468 (2002)
[74] Xia, S.; Belak, J.; El-Azab, A., The discrete-continuum connection in dislocation dynamics: I. Time coarse graining of cross slip, Model. Simul. Mater. Sci. Eng., 24, Article 075007 pp. (2016)
[75] Xia, S.; El-Azab, A., Computational modelling of mesoscale dislocation patterning and plastic deformation of single crystals, Model. Simul. Mater. Sci. Eng., 23, Article 055009 pp. (2015)
[76] Xia, S.; El-Azab, A., A preliminary investigation of dislocation cell structure formation in metals using continuum dislocation dynamics, IOP Conf. Ser. Mater. Sci. Eng., 89, Article 012053 pp. (2015)
[77] Zaiser, M.; Hochrainer, T., Some steps towards a continuum representation of 3D dislocation systems, Scr. Mater., 54, 717-721 (2006)
[78] Zheng, X. H.; Zhang, H. W.; Huang, X.; Hansen, N.; Lu, K., Influence of strain rate on the orientation dependence of microstructure in nickel single crystals, Philos. Mag. Lett., 96, 52-59 (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.