Grain size dependence of polycrystalline plasticity modeling in cylindrical indentation. (English) Zbl 1481.74096

Summary: Grain boundary strengthening effect for polycrystalline copper is studied numerically using crystal plasticity in conjunction with cylindrical indentation simulations under the plane strain condition. In order to compare with an isotropic, heterogeneous continuum model a new constitutive relation is developed. This new nonlocal continuum model that encompasses the heterogeneity in yield strength based on the exponentiated Weibull function can predict the plastic properties of materials in the micron length scale. The spatial description of the deformation gradient two-point tensor is utilized to capture the intrinsic size effect in line with the subsequent deformation measures. Moreover, the total geometrically necessary dislocation density is obtained from the non-zero components of Nye dislocation density tensor. From the simulation, the relationship between the effective Green-Lagrange strain and effective stress measures is explained using the persistent long-range order and intermittent short-range order. The observation derived from the analogy between the cylindrical indentation and the progress in cylindrical voids describes how different average grain sizes of polycrystalline materials are compared with the behavior of isotropic materials. The trajectories of directions of both principal stretch and maximum shear strain explain that the internal stresses induced by cylindrical indentation either hinder or reinforce the dislocation flow, forming strain localization sporadically. The grain size dependence of polycrystalline modeling incorporates the Hall-Petch strengthening as well as the homogenization of anisotropic polycrystalline metal into the isotropic effective medium. This is a physically-based model that is used to model failure characterization in metals.


74C20 Large-strain, rate-dependent theories of plasticity
74E15 Crystalline structure
74M15 Contact in solid mechanics
74A20 Theory of constitutive functions in solid mechanics
74Q15 Effective constitutive equations in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
Full Text: DOI


[1] Abaqus/Standard: Version 2019 (2019) Hibbitt, Karlsson and Sorensen. User’s manual. SIMULIA, Johnston, RI, USA
[2] Akyuz, FA; Merwin, JE, Solution of nonlinear problems of elastoplasticity by finite elementmethod, AIAA J, 6, 10, 1825-1831 (1968) · Zbl 0187.48101
[3] Almasri, AH; Voyiadjis, GZ, Nano-indentation in FCC metals: experimental study, Acta Mech, 209, 1-2, 1 (2010) · Zbl 1381.74006
[4] Armstrong, R., On size effects in polycrystal plasticity, J Mech Phys Solids, 9, 3, 196-199 (1961)
[5] Asaro, R.; Needleman, A., Overview no. 42 texture development and strain hardening in rate dependent polycrystals, Acta Metall, 33, 6, 923-953 (1985)
[6] Asaro, R.; Rice, J., Strain localization in ductile single crystals, J Mech Phys Solids, 25, 5, 309-338 (1977) · Zbl 0375.73097
[7] Asaro, RJ, Micromechanics of crystals and polycrystals, Adv Appl Mech, 23, 1, 115 (1983)
[8] Asaro, RJ, Crystal Plasticity, J Appl Mech, 50, 4, 921-934 (1983) · Zbl 0557.73033
[9] Ashby, M., The deformation of plastically non-homogeneous materials, Philos Mag J Theor Exp Appl Phys, 21, 170, 399-424 (1970)
[10] Bachmann, F.; Hielscher, R.; Schaeben, H., Texture analysis with mtex—free and open source software toolbox, Solid State Phenom, 160, 63-68 (2010)
[11] Bassani, JL; Wu, TY, Latent hardening in single crystals. II. Analytical characterization and predictions, Proc R Soc Lond Ser A Math Phys Sci, 435, 1893, 21-41 (1991) · Zbl 0731.73021
[12] Bishop, RF; Hill, R.; Mott, NF, The theory of indentation and hardness tests, Proc Phys Soc, 57, 3, 147-159 (1945)
[13] Bolshakov, A.; Oliver, W.; Pharr, G., Influences of stress on the measurement of mechanical properties using nanoindentation: part II. Finite element simulations, J Mater Res, 11, 3, 760-768 (1996)
[14] Borg, U.; Kysar, JW, Strain gradient crystal plasticity analysis of a single crystal containing a cylindrical void, Int J Solids Struct, 44, 20, 6382-6397 (2007) · Zbl 1166.74324
[15] Borg, U.; Niordson, CF; Kysar, JW, Size effects on void growth in single crystals with distributed voids, Int J Plast, 24, 4, 688-701 (2008) · Zbl 1131.74009
[16] Budiansky B, Te Wu T (1961) Theoretical prediction of plastic strains of polycrystals. Contract Nonr 1866(02) Technical report. Division of Engineering and Applied Physics, Harvard University
[17] Cheng, YT; Cheng, CM, Scaling approach to conical indentation in elastic-plastic solids with work hardening, J Appl Phys, 84, 3, 1284-1291 (1998)
[18] Cottrell, AH; Bilby, B., Dislocation theory of yielding and strain ageing of iron, Proc Phys Soc Sect A, 62, 1, 49 (1949)
[19] Dahlberg, C.; Saito, Y.; Öztop, M.; Kysar, J., Geometrically necessary dislocation density measurements associated with different angles of indentations, Int J Plast, 54, 81-95 (2014)
[20] Dahlberg, CF; Saito, Y.; Öztop, M.; Kysar, J., Geometrically necessary dislocation density measurements at a grain boundary due to wedge indentation into an aluminum bicrystal, J Mech Phys Solids, 105, 131-149 (2017)
[21] Dao, M.; Chollacoop, NV; Van Vliet, K.; Venkatesh, T.; Suresh, S., Computational modeling of the forward and reverse problems in instrumented sharp indentation, Acta Mater, 49, 19, 3899-3918 (2001)
[22] Davis, J.; Committee, A., Copper and copper alloys (2001), Cleveland: ASM International, Cleveland
[23] Dumas, G.; Baronet, C., Elastoplastic indentation of a half-space by an infinitely long rigid circular cylinder, Int J Mech Sci, 13, 6, 519-530 (1971)
[24] Dunne, F.; Petrinic, N., Introduction to computational plasticity (2005), Oxford: Oxford University Press on Demand, Oxford · Zbl 1081.74002
[25] Durst, K.; Backes, B.; Göken, M., Indentation size effect in metallic materials: correcting for the size of the plastic zone, Scr Mater, 52, 11, 1093-1097 (2005)
[26] Eshelby, J.; Frank, F.; Nabarro, F., Xli. The equilibrium of linear arrays of dislocations, Lond Edinb Dublin Philos Mag J Sci, 42, 327, 351-364 (1951) · Zbl 0042.22704
[27] Faghihi, D.; Voyiadjis, GZ, Determination of nanoindentation size effects and variable material intrinsic length scale for body-centered cubic metals, Mech Mater, 44, 189-211 (2012)
[28] Gan, YX; Kysar, JW, Cylindrical void in a rigid-ideally plastic single crystal III: hexagonal close-packed crystal, Int J Plast, 23, 4, 592-619 (2007) · Zbl 1190.74004
[29] Gan, YX; Kysar, JW; Morse, TL, Cylindrical void in a rigid-ideally plastic single crystal II: experiments and simulations, Int J Plast, 22, 1, 39-72 (2006) · Zbl 1148.74311
[30] Gertsman, V.; Hoffmann, M.; Gleiter, H.; Birringer, R., The study of grain size dependence of yield stress of copper for a wide grain size range, Acta Metall Mater, 42, 10, 3539-3544 (1994)
[31] Gomez, J.; Basaran, C., Nanoindentation of Pb/Sn solder alloys; experimental and finite element simulation results, Int J Solids Struct, 43, 6, 1505-1527 (2006) · Zbl 1120.74302
[32] Hall, E., The deformation and ageing of mild steel: III discussion of results, Proc Phys Soc Sect B, 64, 9, 747 (1951)
[33] Hardy, C.; Baronet, CN; Tordion, GV, The elasto-plastic indentation of a half-space by a rigid sphere, Int J Numer Methods Eng, 3, 4, 451-462 (1971)
[34] Hill, R., Generalized constitutive relations for incremental deformation of metal crystals by multislip, J Mech Phys Solid, 14, 2, 95-102 (1966)
[35] Hill, R., On constitutive inequalities for simple materials—I, J Mech Phys Solids, 16, 4, 229-242 (1968) · Zbl 0162.28702
[36] Hill, R.; Rice, J., Constitutive analysis of elastic-plastic crystals at arbitrary strain, J Mech Phys Solids, 20, 6, 401-413 (1972) · Zbl 0254.73031
[37] Hockett, J.; Sherby, O., Large strain deformation of polycrystalline metals at low homologous temperatures, J Mech Phys Solids, 23, 2, 87-98 (1975)
[38] Honneff, H.; Mecking, H., Analysis of the deformation texture at different rolling conditions, Proc ICOTOM, 6, 347-355 (1981)
[39] Huang Y (1991) A user-material subroutine incorporating single crystal plasticity in the abaqus finite element program. Mech report 178. Reportnumber: Report Mech-178
[40] Hutchinson, JW; Hill, R., Elastic-plastic behaviour of polycrystalline metals and composites, Proc R Soc Lond A Math Phys Sci, 319, 1537, 247-272 (1970)
[41] Hutchinson, JW; Hill, R., Bounds and self-consistent estimates for creep of polycrystalline materials, Proc R Soc Lond A Math Phys Sci, 348, 1652, 101-127 (1976) · Zbl 0319.73059
[42] Johnson, K., The correlation of indentation experiments, J Mech Phys Solids, 18, 2, 115-126 (1970)
[43] Johnson, K., Contact mechanics (1987), Cambridge: Cambridge University Press, Cambridge
[44] Juul, K.; Nellemann, C.; Nielsen, K.; Niordson, C.; Kysar, J., Wedge indentation of single crystalline monazite: a numerical investigation, Int J Plast, 112, 36-51 (2019)
[45] Juul, K.; Niordson, C.; Nielsen, K.; Kysar, J., A novel numerical framework for self-similarity in plasticity: wedge indentation in single crystals, J Mech Phys Solids, 112, 667-684 (2018)
[46] Kocks, U., The relation between polycrystal deformation and single-crystal deformation, Metall Mater Trans B, 1, 5, 1121-1143 (1970)
[47] Kocks, U.; Chandra, H., Slip geometry in partially constrained deformation, Acta Metall, 30, 3, 695-709 (1982)
[48] Koiter, WT, Stress-strain relations, uniqueness and variational theorems for elastic-plastic materials with a singular yield surface, Q Appl Math, 11, 3, 350-354 (1953) · Zbl 0053.14003
[49] Kratochvil, J., Finite-strain theory of crystalline elastic-inelastic materials, J Appl Phys, 42, 3, 1104-1108 (1971)
[50] Kysar J (1997) Addendum to a user-material subroutine incorporating single crystal plasticity in the abaqus finite element program. Mech report 178
[51] Kysar, J.; Saito, Y.; Oztop, M.; Lee, D.; Huh, W., Experimental lower bounds on geometrically necessary dislocation density, Int J Plast, 26, 8, 1097-1123 (2010) · Zbl 1426.74016
[52] Kysar, JW; Gan, YX; Mendez-Arzuza, G., Cylindrical void in a rigid-ideally plastic single crystal. Part I: anisotropic slip line theory solution for face-centered cubic crystals, Int J Plast, 21, 8, 1481-1520 (2005) · Zbl 1148.74318
[53] Kysar, JW; Gan, YX; Morse, TL; Chen, X.; Jones, ME, High strain gradient plasticity associated with wedge indentation into face-centered cubic single crystals: geometrically necessary dislocation densities, J Mech Phys Solids, 55, 7, 1554-1573 (2007)
[54] Lebensohn, RA, N-site modeling of a 3d viscoplastic polycrystal using fast fourier transform, Acta Mater, 49, 14, 2723-2737 (2001)
[55] Lee, C.; Masaki, S.; Kobayashi, S., Analysis of ball indentation, Int J Mech Sci, 14, 7, 417-426 (1972)
[56] Lee, EH, Elastic-plastic deformation at finite strains, J Appl Mech, 36, 1, 1-6 (1969) · Zbl 0179.55603
[57] Mandel, J., Generalisation de la theorie de plasticite de W.T. Koiter, Int J Solids Struct, 1, 3, 273-295 (1965)
[58] Mandel, J., Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques, Int J Solids Struct, 9, 6, 725-740 (1973) · Zbl 0255.73004
[59] Marsh, DM; Cottrell, AH, Plastic flow in glass, Proc R Soc Lond Ser A Math Phys Sci, 279, 1378, 420-435 (1964)
[60] Meyers, M.; Pak, HR, Observation of an adiabatic shear band in titanium by high-voltage transmission electron microscopy, Acta Metall, 34, 12, 2493-2499 (1986)
[61] Meyers, MA; Ashworth, E., A model for the effect of grain size on the yield stress of metals, Philos Mag A, 46, 5, 737-759 (1982)
[62] Mudholkar, GS; Srivastava, DK, Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Trans Reliab, 42, 2, 299-302 (1993) · Zbl 0800.62609
[63] Mulhearn, T., The deformation of metals by Vickers-type pyramidal indenters, J Mech Phys Solids, 7, 2, 85-88 (1959)
[64] Needleman, A., On finite element formulations for large elastic-plastic deformations, Comput Struct, 20, 1, 247-257 (1985)
[65] Ogden, R., Non-linear elastic deformations (2013), Mineola: Dover Publications, Mineola
[66] Peirce, D.; Asaro, R.; Needleman, A., An analysis of nonuniform and localized deformation in ductile single crystals, Acta Metall, 30, 6, 1087-1119 (1982)
[67] Peirce, D.; Asaro, R.; Needleman, A., Material rate dependence and localized deformation in crystalline solids, Acta Metall, 31, 12, 1951-1976 (1983)
[68] Peirce, D.; Shih, C.; Needleman, A., A tangent modulus method for rate dependent solids, Comput Struct, 18, 5, 875-887 (1984) · Zbl 0531.73057
[69] Petch, N., The cleavage strength of polycrystals, J Iron Steel Inst, 174, 25-28 (1953)
[70] Quey, R.; Dawson, P.; Barbe, F., Large-scale 3d random polycrystals for the finite element method: generation, meshing and remeshing, Comput Methods Appl Mech Eng, 200, 17, 1729-1745 (2011) · Zbl 1228.74093
[71] Reuber, C.; Eisenlohr, P.; Roters, F.; Raabe, D., Dislocation density distribution around an indent in single-crystalline nickel: comparing nonlocal crystal plasticity finite-element predictions with experiments, Acta Mater, 71, 333-348 (2014)
[72] Rice, J., Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity, J Mech Phys Solids, 19, 6, 433-455 (1971) · Zbl 0235.73002
[73] Rice, JR, Tensile crack tip fields in elastic-ideally plastic crystals, Mech Mater, 6, 4, 317-335 (1987)
[74] Rollett, AD; Kocks, U., A review of the stages of work hardening, Solid State Phenom, 35, 1-18 (1993)
[75] Roters, F.; Eisenlohr, P.; Hantcherli, L.; Tjahjanto, D.; Bieler, T.; Raabe, D., Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications, Acta Mater, 58, 4, 1152-1211 (2010)
[76] Rousseau, T.; Song, Y.; Wang, W.; Rastogi, S.; Voyiadjis, GZ; Kysar, JW, Order in polycrystalline plasticity deformation fields: short-range intermittency and long-range persistency, Int J Plast, 128, 102674 (2020)
[77] Saito, Y.; Kysar, J., Wedge indentation into elastic-plastic single crystals, 1: asymptotic fields for nearly-flat wedge, Int J Plast, 27, 10, 1640-1657 (2011) · Zbl 1426.74239
[78] Saito, Y.; Oztop, MS; Kysar, JW, Wedge indentation into elastic-plastic single crystals 2 simulations for face-centered cubic crystals, Int J Plast, 28, 1, 70-87 (2012)
[79] Samuels, L.; Mulhearn, T., An experimental investigation of the deformed zone associated with indentation hardness impressions, J Mech Phys Solids, 5, 2, 125-134 (1957)
[80] Sarac, A.; Oztop, M.; Dahlberg, C.; Kysar, J., Spatial distribution of the net burgers vector density in a deformed single crystal, Int J Plast, 85, 110-129 (2016)
[81] Sarris, E.; Constantinides, G., Finite element modeling of nanoindentation on C-S-H: effect of pile-up and contact friction, Cement Concr Compos, 36, 78-84 (2013)
[82] Seth B (1961) Generalized strain measure with applications to physical problems. Tech. Rep. MRC Technical Summary Report 248, Wisconsin Univ-Madison Mathematics Research Center, United States Army, University of Wisconsin
[83] Simo, J.; Hughes, T., Computational inelasticity (2006), New York: Springer, New York · Zbl 0934.74003
[84] Smallman, R.; Green, D., The dependence of rolling texture on stacking fault energy, Acta Metall, 12, 2, 145-154 (1964)
[85] Taylor, GI, Plastic strain in metals, J Inst Met, 62, 307-324 (1938)
[86] Taylor GI (1938b) Analysis of plastic strain in a cubic crystal. Stephen Timoshenko 60th anniversary volume, pp 218-224
[87] Taylor, GI; Elam, CF, Bakerian lecture: the distortion of an aluminium crystal during a tensile test, Proc R Soc Lond Ser A Contain Pap Math Phys Charac, 102, 719, 643-667 (1923)
[88] Taylor, GI; Elam, CF, The plastic extension and fracture of aluminium crystals, Proc R Soc Lond Ser A Contain Pap Math Phys Charac, 108, 745, 28-51 (1925)
[89] Tjahjanto, D.; Eisenlohr, P.; Roters, F., A novel grain cluster-based homogenization scheme, Model Simul Mater Sci Eng, 18, 1, 015006 (2009)
[90] Van Houtte, P.; Delannay, L.; Kalidindi, S., Comparison of two grain interaction models for polycrystal plasticity and deformation texture prediction, Int J Plast, 18, 3, 359-377 (2002) · Zbl 1147.74311
[91] Van Houtte, P.; Delannay, L.; Samajdar, I., Quantitative prediction of cold rolling textures in low-carbon steel by means of the lamel model, Texture Stress Microstruct, 31, 3, 109-149 (1999)
[92] Voyiadjis, G.; Huang, W., A modelling of single crystal plasticity with backstress evolution, Eur J Mech A Solids, 15, 4, 553-573 (1996) · Zbl 0876.73026
[93] Voyiadjis, G.; Yaghoobi, M., Size effects in plasticity: from macro to nano (2019), Amsterdam: Elsevier Science, Amsterdam
[94] Voyiadjis, GZ; Almasri, AH, Variable material length scale associated with nanoindentation experiments, J Eng Mech, 135, 3, 139-148 (2009)
[95] Voyiadjis, GZ; Deliktas, B., Modeling of strengthening and softening in inelastic nanocrystalline materials with reference to the triple junction and grain boundaries using strain gradient plasticity, Acta Mech, 213, 1-2, 3-26 (2010) · Zbl 1272.74073
[96] Voyiadjis, GZ; Faghihi, D.; Zhang, C., Analytical and experimental determination of rate-and temperature-dependent length scales using nanoindentation experiments, J Nanomech Micromech, 1, 1, 24-40 (2011)
[97] Voyiadjis, GZ; Peters, R., Size effects in nanoindentation: an experimental and analytical study, Acta Mech, 211, 1-2, 131-153 (2010) · Zbl 1397.74156
[98] Voyiadjis, GZ; Yaghoobi, M., Large scale atomistic simulation of size effects during nanoindentation: dislocation length and hardness, Mater Sci Eng A, 634, 20-31 (2015)
[99] Voyiadjis, GZ; Zhang, C., The mechanical behavior during nanoindentation near the grain boundary in a bicrystal fcc metal, Mater Sci Eng A, 621, 218-228 (2015)
[100] Wang, Y.; Raabe, D.; Klüber, C.; Roters, F., Orientation dependence of nanoindentation pile-up patterns and of nanoindentation microtextures in copper single crystals, Acta Mater, 52, 8, 2229-2238 (2004)
[101] Weibull, W., Wide applicability, J Appl Mech, 103, 730, 293-297 (1951) · Zbl 0042.37903
[102] Wu, TY; Bassani, JL; Laird, C., Latent hardening in single crystals-I. Theory and experiments, Proc R Soc Lond Ser A Math Phys Sci, 435, 1893, 1-19 (1991) · Zbl 0731.73020
[103] Yaghoobi, M.; Voyiadjis, GZ, Effect of boundary conditions on the MD simulation of nanoindentation, Comput Mater Sci, 95, 626-636 (2014)
[104] Yaghoobi, M.; Voyiadjis, GZ, Atomistic simulation of size effects in single-crystalline metals of confined volumes during nanoindentation, Comput Mater Sci, 111, 64-73 (2016)
[105] Yuan, R.; Beyerlein, IJ; Zhou, C., Coupled crystal orientation-size effects on the strength of nano crystals, Sci Rep, 6, 1, 1-9 (2016)
[106] Zhang, C.; Voyiadjis, GZ, Rate-dependent size effects and material length scales in nanoindentation near the grain boundary for a bicrystal fcc metal, Mater Sci Eng A, 659, 55-62 (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.