The mechanical responses of small volume metallic compounds are addressed in this work through developing a nonlocal continuum theory. In this regard, a thermodynamic-based higher-order strain-gradient plasticity framework for coupled thermoviscoplasticity modeling is presented. The concept of thermal activation energy and the dislocations interaction mechanisms are taken into consideration to describe the choice of thermodynamic potentials such as Helmholtz free energy and rate of dissipation. The theory is developed based on the decomposition of the thermodynamic conjugate forces into energetic and dissipative counterparts, which provides the constitutive equations to have both energetic and dissipative gradient length scales. The derived constitutive model is calibrated against the experimental data of bulge test conducted on thin films.
Issue Section:
Research Papers
References
1.
Fleck
, N. A.
, Muller
, G. M.
, Ashby
, M. F.
, and Hutchinson
, J. W.
, 1994
, “Strain Gradient Plasticity—Theory and Experiment
,” Acta Metall. Mater.
, 42
(2
), pp. 475
–487
.10.1016/0956-7151(94)90502-92.
Espinosa
, H. D.
, Prorok
, B. C.
, and Peng
, B.
, 2004
, “Plasticity Size Effects in Free-Standing Submicron Polycrystalline FCC Films Subjected to Pure Tension
,” J. Mech. Phys. Solids
, 52
(3
), pp. 667
–689
.10.1016/j.jmps.2003.07.0013.
Vlassak
, J. J.
, Xiang
, Y.
, and Chen
, X.
, 2005
, “Plane-Strain Bulge Test for Thin Films
,” J. Mater. Res.
, 20
(9
), pp. 2360
–2370
.10.1557/jmr.2005.03134.
Chen
, X.
, Ogasawara
, N.
, Zhao
, M. H.
, and Chiba
, N.
, 2007
, “On the Uniqueness of Measuring Elastoplastic Properties From Indentation: The Indistinguishable Mystical Materials
,” J. Mech. Phys. Solids
, 55
(8
), pp. 1618
–1660
.10.1016/j.jmps.2007.01.0105.
Ma
, Q.
, and Clarke
, D. R.
, 1995
, “Size-Dependent Hardness of Silver Single-Crystals
,” J. Mater. Res.
, 10
(4
), pp. 853
–863
.10.1557/JMR.1995.08536.
Chen
, K.
, Meng
, W. J.
, Mei
, F. H.
, Hiller
, J.
, and Miller
, D. J.
, 2011
, “From Micro- to Nano-Scale Molding of Metals: Size Effect During Molding of Single Crystal Al With Rectangular Strip Punches
,” Acta Mater.
, 59
(3
), pp. 1112
–1120
.10.1016/j.actamat.2010.10.0447.
Chen
, X.
, and Vlassak
, J. J.
, 2001
, “Numerical Study on the Measurement of Thin Film Mechanical Properties by Means of Nanoindentation
,” J. Mater. Res.
, 16
(10
), pp. 2974
–2982
.10.1557/JMR.2001.04088.
Arsenlis
, A.
, and Parks
, D. M.
, 1999
, “Crystallographic Aspects of Geometrically-Necessary and Statistically-Stored Dislocation Density
,” Acta Mater.
, 47
(5
), pp. 1597
–1611
.10.1016/S1359-6454(99)00020-89.
Bittencourt
, E.
, Needleman
, A.
, Gurtin
, M. E.
, and Van der Giessen
, E.
, 2003
, “A Comparison of Nonlocal Continuum and Discrete Dislocation Plasticity Predictions
,” J. Mech. Phys. Solids
, 51
(2
), pp. 281
–310
.10.1016/S0022-5096(02)00081-910.
Nicola
, L.
, Van der Giessen
, E.
, and Needleman
, A.
, 2003
, “Discrete Dislocation Analysis of Size Effects in Thin Films
,” J. Appl. Phys.
, 93
(10
), pp. 5920
–5928
.10.1063/1.156647111.
Aifantis
, E. C.
, 1999
, “Gradient Deformation Models at Nano, Micro, and Macro Scales
,” ASME J. Eng. Mater. Technol.
, 121
(2
), pp. 189
–202
.10.1115/1.281236612.
Huang
, Y.
, Gao
, H.
, Nix
, W. D.
, and Hutchinson
, J. W.
, 2000
, “Mechanism-Based Strain Gradient Plasticity—II. Analysis
,” J. Mech. Phys. Solids
, 48
(1
), pp. 99
–128
.10.1016/S0022-5096(99)00022-813.
Stolken
, J. S.
, and Evans
, A. G.
, 1998
, “A Microbend Test Method for Measuring the Plasticity Length Scale
,” Acta Mater.
, 46
(14
), pp. 5109
–5115
.10.1016/S1359-6454(98)00153-014.
Xiang
, Y.
, and Vlassak
, J. J.
, 2005
, “Bauschinger Effect in Thin Metal Films
,” Scr. Mater.
, 53
(2
), pp. 177
–182
.10.1016/j.scriptamat.2005.03.04815.
Xiang
, Y.
, and Vlassak
, J. J.
, 2006
, “Bauschinger and Size Effects in Thin-Film Plasticity
,” Acta Mater.
, 54
(20
), pp. 5449
–5460
.10.1016/j.actamat.2006.06.05916.
Needleman
, A.
, Nicola
, L.
, Xiang
, Y.
, Vlassak
, J. J.
, and Van der Giessen
, E.
, 2006
, “Plastic Deformation of Freestanding Thin Films: Experiments and Modeling
,” J. Mech. Phys. Solids.
54
(10
), pp. 2089
–2110
.10.1016/j.jmps.2006.04.00517.
Nicola
, L.
, Xiang
, Y.
, Vlassak
, J. J.
, Van der Giessen
, E.
, and Needleman
, A.
, 2006
, “Plastic Deformation of Freestanding Thin Films: Experiments and Modeling
,” J. Mech. Phys. Solids
, 54
(10
), pp. 2089
–2110
.10.1016/j.jmps.2006.04.00518.
Muhlhaus
, H. B.
, and Aifantis
, E. C.
, 1991
, “A Variational Principle for Gradient Plasticity
,” Int. J. Solids Struct.
, 28
(7
), pp. 845
–857
.10.1016/0020-7683(91)90004-Y19.
Zbib
, H. M.
, and Aifantis
, E. C.
, 1992
, “On the Gradient-Dependent Theory of Plasticity and Shear Banding
,” Acta Mech.
, 92
(1–4
), pp. 209
–225
.10.1007/BF0117417720.
Fleck
, N. A.
, and Hutchinson
, J. W.
, 1993
, “A Phenomenological Theory for Strain Gradient Effects in Plasticity
,” J. Mech. Phys. Solids
, 41
(12
), pp. 1825
–1857
.10.1016/0022-5096(93)90072-N21.
Zbib
, H. M.
, de la Rubia
, T. D.
, and Bulatov
, V.
, 2002
, “A Multiscale Model of Plasticity Based on Discrete Dislocation Dynamics
,” ASME J. Eng. Mater. Technol.
, 124
(1
), pp. 78
–87
.10.1115/1.142135122.
Khraishi
, T. A.
, and Zbib
, H. M.
, 2002
, “Dislocation Dynamics Simulations of the Interaction Between a Short Rigid Fiber and a Glide Circular Dislocation Pile-Up
,” Comput. Mater. Sci.
, 24
(3
), pp. 310
–322
.10.1016/S0927-0256(01)00253-123.
Schiotz
, J.
, Di Tolla
, F. D.
, and Jacobsen
, K. W.
, 1998
, “Softening of Nanocrystalline Metals at Very Small Grains
,” Nature
, 391
, pp. 561
–563
.10.1038/3532824.
Niordson
, C. F.
, and Hutchinson
, J. W.
, 2003
, “Non-Uniform Plastic Deformation of Micron Scale Objects
,” Int. J. Numer. Methods Eng.
56
(7
), pp. 961
–975
.10.1002/nme.59325.
Aifantis
, E. C.
, 1984
, “On the Microstructural Origin of Certain Inelastic Models
,” ASME J. Eng. Mater. Technol.
106
(4
), pp. 326
–330
.10.1115/1.322572526.
Ziegler
, H.
, and Wehrli
, C.
, 1987
, “The Derivation of Constitutive Relations From the Free-Energy and the Dissipation Function
,” Adv. Appl. Mech.
, 25
, pp. 183
–238
.10.1016/S0065-2156(08)70278-327.
Ziegler
, H.
, 1962
, Some Extremum Principles in Irreversible Thermodynamics, With Application to Continuum Mechanics
, Swiss Federal Institute of Technology
, Zurich, Switzerland
.28.
Ziegler
, H.
, 1958
, “An Attempt to Generalize Onsager's Principle, and Its Significance for Rheological Problems
,” Z. Angew. Math. Phys.
, 9
(5
), pp. 748
–763
.10.1007/BF0242479329.
Gurtin
, M. E.
, and Reddy
, B. D.
, 2009
, “Alternative Formulations of Isotropic Hardening for Mises Materials, and Associated Variational Inequalities
,” Continuum Mech. Thermodyn.
, 21
(3
), pp. 237
–250
.10.1007/s00161-009-0107-330.
Faghihi
, D.
, and Voyiadjis
, G. Z.
, 2012
, “Determination of Nanoindentation Size Effects and Variable Material Intrinsic Length Scale for Body-Centered Cubic Metals
,” Mech. Mater.
, 44
(0
), pp. 189
–211
.10.1016/j.mechmat.2011.07.00231.
Shizawa
, K.
, and Zbib
, H. M.
, 1999
, “A Thermodynamical Theory of Gradient Elastoplasticity With Dislocation Density Tensor. I: Fundamentals
,” Int. J. Plast.
, 15
(9
), pp. 899
–938
.10.1016/S0749-6419(99)00018-232.
Reddy
, B. D.
, 2012
, “Erratum to: The Role of Dissipation and Defect Energy in Variational Formulations of Problems in Strain-Gradient Plasticity, Part 1, Polycrystalline Plasticity
,” Continuum Mech. Thermodyn.
, 24
(1
), p. 79
.10.1007/s00161-011-0208-733.
Reddy
, B. D.
, 2011
, “The Role of Dissipation and Defect Energy in Variational Formulations of Problems in Strain-Gradient Plasticity. Part 2: Single-Crystal Plasticity
,” Continuum Mech. Thermodyn.
, 23
(6
), pp. 551
–572
.10.1007/s00161-011-0195-834.
Fleck
, N. A.
, and Willis
, J. R.
, 2009
, “A Mathematical Basis for Strain-Gradient Plasticity Theory—Part I: Scalar Plastic Multiplier
,” J. Mech. Phys. Solids
, 57
(1
), pp. 161
–177
.10.1016/j.jmps.2008.09.01035.
Fredriksson
, P.
, and Gudmundson
, P.
, 2007
, “Modelling of the Interface Between a Thin Film and a Substrate Within a Strain Gradient Plasticity Framework
,” J. Mech. Phys. Solids
, 55
(5
), pp. 939
–955
.10.1016/j.jmps.2006.11.00136.
Voyiadjis
, G. Z.
, and Deliktas
, B.
, 2009
, “Mechanics of Strain Gradient Plasticity With Particular Reference to Decomposition of the State Variables Into Energetic and Dissipative Components
,” Int. J. Eng. Sci.
, 47
(11-12
), pp. 1405
–1423
.10.1016/j.ijengsci.2009.05.01337.
Gurtin
, M. E.
, 2010
, “A Finite-Deformation, Gradient Theory of Single-Crystal Plasticity With Free Energy Dependent on the Accumulation of Geometrically Necessary Dislocations
,” Int. J. Plast.
, 26
(8
), pp. 1073
–1096
.10.1016/j.ijplas.2010.02.00238.
Hansen
, N. R.
, and Schreyer
, H. L.
, 1994
, “A Thermodynamically Consistent Framework for Theories of Elastoplasticity Coupled With Damage
,” Int. J. Solids Struct.
, 31
(3
), pp. 359
–389
.10.1016/0020-7683(94)90112-039.
Lemaitre
, J.
, 1985
, “Coupled Elasto-Plasticity and Damage Constitutive-Equations
,” Comput. Methods Appl. Mech. Eng.
, 51
(1–3
), pp. 31
–49
.10.1016/0045-7825(85)90026-X40.
Nemat-Nasser
, S.
, and Guo
, W. G.
, 2000
, “Flow Stress of Commercially Pure Niobium Over a Broad Range of Temperatures and Strain Rates
,” Mater. Sci. Eng., A
, 284
(1–2
), pp. 202
–210
.10.1016/S0921-5093(00)00740-141.
Voyiadjis
, G. Z.
, and Abed
, F. H.
, 2005
, “Microstructural Based Models for BCC and FCC Metals With Temperature and Strain Rate Dependency
,” Mech. Mater.
, 37
(2–3
), pp. 355
–378
.10.1016/j.mechmat.2004.02.00342.
Taylor
, G. I.
, and Quinney
, H.
, 1934
, “The Latent Energy Remaining in a Metal After Cold Working
,” Proc. R. Soc. London, Ser. A
, 143
(849
), pp. 307
–326
.10.1098/rspa.1934.000443.
Hodowany
, J.
, Ravichandran
, G.
, Rosakis
, A. J.
, and Rosakis
, P.
, 2000
, “Partition of Plastic Work Into Heat and Stored Energy in Metals
,” Exp. Mech.
, 40
(2
), pp. 113
–123
.10.1007/BF0232503644.
Mason
, J. J.
, Rosakis
, A. J.
, and Ravichandran
, G.
, 1994
, “On the Strain and Strain-Rate Dependence of the Fraction of Plastic Work Converted to Heat—An Experimental-Study Using High-Speed Infrared Detectors and the Kolsky Bar
,” Mech. Mater.
, 17
(2–3
), pp. 135
–145
.10.1016/0167-6636(94)90054-X45.
Oliferuk
, W.
, and Maj
, M.
, 2009
, “Stress-Strain Curve and Stored Energy During Uniaxial Deformation of Polycrystals
,” Eur. J. Mech. A-Solids
, 28
(2
), pp. 266
–272
.10.1016/j.euromechsol.2008.06.00346.
Rosakis
, P.
, Rosakis
, A. J.
, Ravichandran
, G.
, and Hodowany
, J.
, 2000
, “A Thermodynamic Internal Variable Model for the Partition of Plastic Work Into Heat and Stored Energy in Metals
,” J. Mech. Phys. Solids
, 48
(3
), pp. 581
–607
.10.1016/S0022-5096(99)00048-447.
Rusinek
, A.
, and Klepaczko
, J. R.
, 2009
, “Experiments on Heat Generated During Plastic Deformation and Stored Energy for TRIP Steels
,” Mater. Des.
, 30
(1
), pp. 35
–48
.10.1016/j.matdes.2008.04.04848.
Stainier
, L.
, and Ortiz
, M.
, 2010
, “Study and Validation of a Variational Theory of Thermo-Mechanical Coupling in Finite Visco-Plasticity
,” Int. J. Solids Struct.
, 47
(5
), pp. 705
–715
.10.1016/j.ijsolstr.2009.11.01249.
Mroz
, Z.
, and Oliferuk
, W.
, 2002
, “Energy Balance and Identification of Hardening Moduli in Plastic Deformation Processes
,” Int. J. Plast.
, 18
(3
), pp. 379
–397
.10.1016/S0749-6419(00)00103-050.
Ristinmaa
, M.
, Wallin
, M.
, and Ottosen
, N. S.
, 2007
, “Thermodynamic Format and Heat Generation of Isotropic Hardening Plasticity
,” Acta Mech.
, 194
(1-4
), pp. 103
–121
.10.1007/s00707-007-0448-651.
Aravas
, N.
, Kim
, K. S.
, and Leckie
, F. A.
, 1990
, “On the Calculations of the Stored Energy of Cold Work
,” ASME J. Eng. Mater. Technol.
, 112
(4
), pp. 465
–470
.10.1115/1.290335852.
Zehnder
, A. T.
, 1991
, “A Model for the Heating Due to Plastic Work
,” Mech. Res. Commun.
, 18
(1
), pp. 23
–28
.10.1016/0093-6413(91)90023-P53.
Longère
, P.
, and Dragon
, A.
, 2008
, “Evaluation of the Inelastic Heat Fraction in the Context of Microstructure-Supported Dynamic Plasticity Modelling
,” Int. J. Impact Eng.
, 35
(9
), pp. 992
–999
.10.1016/j.ijimpeng.2007.06.00654.
Saanouni
, K.
, Forster
, C. H.
, and Hatira
, F. B.
, 1994
, “On the Anelastic Flow With Damage
,” Int. J. Damage Mech.
, 3
(2
), pp. 140
–169
.10.1177/10567895940030020355.
Gurtin
, M. E.
, and Anand
, L.
, 2009
, “Thermodynamics Applied to Gradient Theories Involving the Accumulated Plastic Strain: The Theories of Aifantis and Fleck and Hutchinson and Their Generalization
,” J. Mech. Phys. Solids
, 57
(3
), pp. 405
–421
.10.1016/j.jmps.2008.12.00256.
Gurtin
, M. E.
, and Anand
, L.
, 2005
, “A Theory of Strain-Gradient Plasticity for Isotropic, Plastically Irrotational Materials. Part II: Finite Deformations
,” Int. J. Plast.
, 21
(12
), pp. 2297
–2318
.10.1016/j.ijplas.2005.01.00657.
Gurtin
, M. E.
, 2004
, “A Gradient Theory of Small-Deformation Isotropic Plasticity That Accounts for the Burgers Vector and for Dissipation Due to Plastic Spin
,” J. Mech. Phys. Solids
, 52
(11
), pp. 2545
–2568
.10.1016/j.jmps.2004.04.01058.
Bardella
, L.
, 2010
, “Size Effects in Phenomenological Strain Gradient Plasticity Constitutively Involving the Plastic Spin
,” Int. J. Eng. Sci.
, 48
(5
), pp. 550
–568
.10.1016/j.ijengsci.2010.01.00359.
Nye
, J. F.
, 1953
, “Some Geometrical Relations in Dislocated Crystals
,” Acta Metall.
, 1
(2
), pp. 153
–162
.10.1016/0001-6160(53)90054-660.
Fleck
, N. A.
, and Hutchinson
, J. W.
, 1997
, “Strain Gradient Plasticity
,” Adv. Appl. Mech.
, 33
, pp. 295
–361
.10.1016/S0065-2156(08)70388-061.
Bassani
, J. L.
, 2001
, “Incompatibility and a Simple Gradient Theory of Plasticity
,” J. Mech. Phys. Solids
, 49
(9
), pp. 1983
–1996
.10.1016/S0022-5096(01)00037-062.
Gurtin
, M. E.
, 2000
, “On the Plasticity of Single Crystals: Free Energy, Microforces, Plastic-Strain Gradients
,” J. Mech. Phys. Solids
, 48
(5
), pp. 989
–1036
.10.1016/S0022-5096(99)00059-963.
Gudmundson
, P.
, 2004
, “A Unified Treatment of Strain Gradient Plasticity
,” J. Mech. Phys. Solids
, 52
(6
), pp. 1379
–1406
.10.1016/j.jmps.2003.11.00264.
Gurtin
, M. E
, and Anand
, L.
, 2005
, “A Theory of Strain-Gradient Plasticity for Isotropic, Plastically Irrotational Materials. Part I: Small Deformations
,” J. Mech. Phys. Solids
, 53
(7
), pp. 1624
–1649
.10.1016/j.jmps.2004.12.00865.
Fleck
, N. A.
, and Willis
, J. R.
, 2009
, “A Mathematical Basis for Strain-Gradient Plasticity Theory. Part II: Tensorial Plastic Multiplier
,” J. Mech. Phys. Solids
, 57
(7
), pp. 1045
–1057
.10.1016/j.jmps.2009.03.00766.
Voyiadjis
, G. Z.
, and Faghihi
, D.
, 2012
, “Thermo-Mechanical Strain Gradient Plasticity With Energetic and Dissipative Length Scales
,” Int. J. Plast.
, 30–31
(0
), pp. 218
–247
.10.1016/j.ijplas.2011.10.00767.
Fleck
, N. A.
, and Hutchinson
, J. W.
, 2001
, “A Reformulation of Strain Gradient Plasticity
,” J. Mech. Phys. Solids
, 49
(10
), pp. 2245
–2271
.10.1016/S0022-5096(01)00049-768.
Lele
, S. P.
, and Anand
, L.
, 2008
, “A Small-Deformation Strain-Gradient Theory for Isotropic Viscoplastic Materials
,” Philos. Mag.
, 88
(30-32
), pp. 3655
–3689
.10.1080/1478643080208703169.
Lele
, S. P.
, and Anand
, L.
, 2009
, “A Large-Deformation Strain-Gradient Theory for Isotropic Viscoplastic Materials
,” Int. J. Plast.
, 25
(3
), pp. 420
–453
.10.1016/j.ijplas.2008.04.00370.
Cermelli
, P.
, and Gurtin
, M. E.
, 2002
, “Geometrically Necessary Dislocations in Viscoplastic Single Crystals and Bicrystals Undergoing Small Deformations
,” Int. J. Solids Struct.
, 39
(26
), pp. 6281
–6309
.10.1016/S0020-7683(02)00491-271.
Gurtin
, M. E.
, and Ohno
, N.
, 2011
, “A Gradient Theory of Small-Deformation, Single-Crystal Plasticity That Accounts for GND-Induced Interactions Between Slip Systems
,” J. Mech. Phys. Solids
, 59
(2
), pp. 320
–343
.10.1016/j.jmps.2010.10.00572.
Rusinek
, A.
, Zaera
, R.
, and Klepaczko
, J. R.
, 2007
, “Constitutive Relations in 3-D for a Wide Range of Strain Rates and Temperatures—Application to Mild Steels
,” Int. J. Solids Struct.
, 44
(17
), pp. 5611
–5634
.10.1016/j.ijsolstr.2007.01.01573.
Farren
, W. S.
, and Taylor
, G. I.
, 1925
, “The Heat Developed During Plastic Extension of Metals
,” Proc. R. Soc. London, Ser. A
, 107
(743
), pp. 422
–451
.10.1098/rspa.1925.003474.
Taylor
, G. I.
, and Quinney
, H.
, 1932
, “The Plastic Distortion of Metals
,” Philos. Trans. R. Soc. London, Ser. A
, 230
, pp. 323
–362
.10.1098/rsta.1932.000975.
Mollica
, F.
, Rajagopal
, K. R.
, and Srinivasa
, A. R.
, 2001
, “The Inelastic Behavior of Metals Subject to Loading Reversal
,” Int. J. Plast.
, 17
(8
), pp. 1119
–1146
.10.1016/S0749-6419(00)00082-676.
Oliferuk
, W.
, Swiatnicki
, W. A.
, and Grabski
, M. W.
, 1993
, “Rate of Energy-Storage and Microstructure Evolution During the Tensile Deformation of Austenitic Steel
,” Mater. Sci. Eng. A
, 161
(1
), pp. 55
–63
.10.1016/0921-5093(93)90475-T77.
Oliferuk
, W.
, Swiatnicki
, W. A.
, and Grabski
, M. W.
, 1995
, “Effect of the Grain-Size on the Rate of Energy-Storage During the Tensile Deformation of an Austenitic Steel
,” Mater. Sci. Eng. A
, 197
(1
), pp. 49
–58
.10.1016/0921-5093(94)09766-678.
Gurtin
, M. E.
, 2002
, “A Gradient Theory of Single-Crystal Viscoplasticity That Accounts for Geometrically Necessary Dislocations
,” J. Mech. Phys. Solids
, 50
(1
), pp. 5
–32
.10.1016/S0022-5096(01)00104-179.
Hasegawa
, T.
, Yakou
, T.
, and Karashima
, S.
, 1975
, “Deformation Behavior and Dislocation-Structures Upon Stress Reversal in Polycrystalline Aluminum
,” Mater. Sci. Eng.
, 20
(3
), pp. 267
–276
.10.1016/0025-5416(75)90159-780.
Groma
, I.
, Csikor
, F. F.
, and Zaiser
, M.
, 2003
, “Spatial Correlations and Higher-Order Gradient Terms in a Continuum Description of Dislocation Dynamics
,” Acta Mater.
, 51
(5
), pp. 1271
–1281
.10.1016/S1359-6454(02)00517-781.
Bardella
, L.
, 2006
, “A Deformation Theory of Strain Gradient Crystal Plasticity That Accounts for Geometrically Necessary Dislocations
,” J. Mech. Phys. Solids
, 54
(1
), pp. 128
–160
.10.1016/j.jmps.2005.08.00382.
Ohno
, N.
, and Okumura
, D.
, 2007
, “Higher-Order Stress and Grain Size Effects Due to Self-Energy of Geometrically Necessary Dislocations
,” J. Mech. Phys. Solids
, 55
(9
), pp. 1879
–1898
.10.1016/j.jmps.2007.02.00783.
Garroni
, A.
, Leoni
, G.
, and Ponsiglione
, M.
, 2010
, “Gradient Theory for Plasticity Via Homogenization of Discrete Dislocations
,” J. Eur. Math. Soc.
, 12
(5
), pp. 1231
–1266
.10.4171/JEMS/22884.
Lubarda
, V. A.
, 2008
, “On the Gibbs Conditions of Stable Equilibrium, Convexity and the Second-Order Variations of Thermodynamic Potentials in Nonlinear Thermoelasticity
,” Int. J. Solids Struct.
, 45
(1
), pp. 48
–63
.10.1016/j.ijsolstr.2007.07.01085.
Callen
, H. B.
, 1960
, Thermodynamics
, John Wiley
, New York
.86.
Kuhlmann-Wilsdorf
, D.
, 1989
, “Theory of Plastic-Deformation—Properties of Low-Energy Dislocation-Structures
,” Mater. Sci. Eng. A
, 113
, pp. 1
–41
.10.1016/0921-5093(89)90290-687.
Kuhlmann-Wilsdorf
, D.
, 1999
, “The Theory of Dislocation-Based Crystal Plasticity
,” Philos. Mag. A
, 79
(4
), pp. 955
–1008
.10.1080/0141861990821034288.
Gurtin
, M. E.
, 2008
, “A Finite-Deformation, Gradient Theory of Single-Crystal Plasticity With Free Energy Dependent on Densities of Geometrically Necessary Dislocations
,” Int. J. Plast.
, 24
(4
), pp. 702
–725
.10.1016/j.ijplas.2007.07.01489.
Dascalu
, C.
, and Maugin
, G. A.
, 1993
, “Material Forces and Energy-Release Rates in Homogeneous Elastic Bodies With Defects
,” C. R. Acad. Sci. Ser. II
, 317
(9
), pp. 1135
–1140
.90.
Maugin
, G. A.
, and Trimarco
, C.
, 1995
, “On Material and Physical Forces in Liquid-Crystals
,” Int. J. Eng. Sci.
, 33
(11
), pp. 1663
–1678
.10.1016/0020-7225(95)00025-S91.
Zaiser
, M.
, and Aifantis
, E. C.
, 2006
, “Randomness and Slip Avalanches in Gradient Plasticity
,” Int. J. Plast.
, 22
(8
), pp. 1432
–1455
.10.1016/j.ijplas.2005.07.01092.
Shishvan
, S. S.
, Nicola
, L.
, and Van der Giessen
, E.
, 2010
, “Bauschinger Effect in Unpassivated Freestanding Thin Films
,” J. Appl. Phys.
, 107
(9
), p. 093529
.10.1063/1.340750593.
Cleveringa
, H. H. M.
, Van der Giessen
, E.
, and Needleman
, A.
, 1999
, “A Discrete Dislocation Analysis of Residual Stresses in a Composite Material
,” Philos. Mag. A
, 79
(4
), pp. 893
–920
.10.1080/0141861990821033894.
Caillard
, D.
, and Martin
, J. L.
, 2003
, Thermally Activated Mechanisms in Crystal Plasticity
(Pergamon Materials Series), Elsevier, Oxford, UK.95.
Davoudi
, K. M.
, Nicola
, L.
, and Vlassak
, J. J.
, 2012
, “Dislocation Climb in Two-Dimensional Discrete Dislocation Dynamics
,” J. Appl. Phys.
, 111
(10
), p. 103522
.10.1063/1.471843296.
Coleman
, B. D.
, and Gurtin
, M. E.
, 1967
, “Thermodynamics With Internal State Variables
,” J. Chem. Phys.
, 47
(2
), pp. 597
–613.10.1063/1.171193797.
Gurtin
, M. E.
, Anand
, L.
, and Lele
, S. P.
, 2007
, “Gradient Single-Crystal Plasticity With Free Energy Dependent on Dislocation Densities
,” J. Mech. Phys. Solids
, 55
(9
), pp. 1853
–1878
.10.1016/j.jmps.2007.02.00698.
Ohno
, N.
, Okumura
, D.
, and Shibata
, T.
, 2008
, “Grain-Size Dependent Yield Behavior Under Loading, Unloading and Reverse Loading
,” Int. J. Mod. Phys. B
, 22
(31–32
), pp. 5937
–5942
.10.1142/S021797920805140699.
Dahlberg
, C.
, and Faleskog
, J.
, 2012
, “An Improved Strain Gradient Plasticity Formulation With Energetic Interfaces: Theory and A Fully Implicit Finite Element Formulation
,” Comput. Mech.
, pp. 1
–19
.10.1007/s00466-012-0743-5100.
Roy
, A.
, Peerlings
, R. H. J.
, Geers
, M. G. D.
, and Kasyanyuk
, Y.
, 2008
, “Continuum Modeling of Dislocation Interactions: Why Discreteness Matters?
,” Mater. Sci. Eng. A
, 486
(1–2
), pp. 653
–661
.10.1016/j.msea.2007.09.074101.
Kröner
, E.
, 2001
, “Benefits and Shortcomings of the Continuous Theory of Dislocations
,” Int. J. Solids Struct.
, 38
(6–7
), pp. 1115
–1134
.10.1016/S0020-7683(00)00077-9102.
Forest
, S.
, and Amestoy
, M.
, 2008
, “Hypertemperature in Thermoelastic Solids
,” C. R. Mec.
, 336
(4
), pp. 347
–353
.10.1016/j.crme.2008.01.007103.
Forest
, S.
, 2009
, “Micromorphic Approach for Gradient Elasticity, Viscoplasticity, and Damage
,” ASCE J. Eng. Mech.
135
(3
), pp. 117
–131
.10.1061/(ASCE)0733-9399(2009)135:3(117)104.
Forest
, S.
, and Aifantis
, E. C.
, 2010
, “Some Links Between Recent Gradient Thermo-Elasto-Plasticity Theories and the Thermomechanics of Generalized Continua
,” Int. J. Solids Struct.
, 47
(25–26
), pp. 3367
–3376
.10.1016/j.ijsolstr.2010.07.009105.
Arsenlis
, A.
, Parks
, D. M.
, Becker
, R.
, and Bulatov
, V. V.
, 2004
, “On the Evolution of Crystallographic Dislocation Density in Non-Homogeneously Deforming Crystals
,” J. Mech. Phys. Solids
, 52
(6
), pp. 1213
–1246
.10.1016/j.jmps.2003.12.007106.
Geers
, M. G. D.
, Brekelmans
, W. A. M.
, and Bayley
, C. J.
, 2007
, “Second-Order Crystal Plasticity: Internal Stress Effects and Cyclic Loading
,” Modell. Simul. Mater. Sci. Eng.
, 15
(1
), pp. S133
–S145
.10.1088/0965-0393/15/1/S12107.
Geers
, M. G. D.
, Peerlings
, R. H. J.
, Hoefnagels
, J. P. M.
, and Kasyanyuk
, Y.
, 2009
, “On a Proper Account of First- and Second-Order Size Effects in Crystal Plasticity
,” Adv. Eng. Mater.
, 11
(3
), pp. 143
–147
.10.1002/adem.200800287108.
Limkumnerd
, S.
, and Van der Giessen
, E.
, 2008
, “Study of Size Effects in Thin Films by Means of a Crystal Plasticity Theory Based on DiFT
,” J. Mech. Phys. Solids
, 56
(11
), pp. 3304
–3314
.10.1016/j.jmps.2008.06.004109.
Abu Al-Rub
, R. K.
, Voyiadjis
, G. Z.
, and Bammann
, D. J.
, 2007
, “A Thermodynamic Based Higher-Order Gradient Theory for Size Dependent Plasticity
,” Int. J. Solids Struct.
, 44
(9
), pp. 2888
–2923
.10.1016/j.ijsolstr.2006.08.034110.
Abu Al-Rub
, R. K.
, 2008
, “Interfacial Gradient Plasticity Governs Scale-Dependent Yield Strength and Strain Hardening Rates in Micro/Nano Structured Metals
,” Int. J. Plast.
, 24
(8
), pp. 1277
–1306
.10.1016/j.ijplas.2007.09.005111.
Voyiadjis
, G. Z.
, Pekmezi
, G.
, and Deliktas
, B.
, 2010
, “Nonlocal Gradient-Dependent Modeling of Plasticity With Anisotropic Hardening
,” Int. J. Plast.
, 26
(9
), pp. 1335
–1356
.10.1016/j.ijplas.2010.01.015112.
Kuroda
, M.
, and Tvergaard
, V.
, 2008
, “On the Formulations of Higher-Order Strain Gradient Crystal Plasticity Models
,” J. Mech. Phys. Solids
, 56
(4
), pp. 1591
–1608
.10.1016/j.jmps.2007.07.015113.
Vlassak
, J. J.
, and Nix
, W. D.
, 1992
, “A New Bulge Test Technique for the Determination of Young Modulus and Poisson Ratio of Thin-Films
,” J. Mater. Res.
, 7
(12
), pp. 3242
–3249
.10.1557/JMR.1992.3242114.
Xiang
, Y.
, Chen
, X.
, and Vlassak
, J. J.
, 2005
, “Plane-Strain Bulge Test for Thin Films
,” J. Mater. Res.
, 20
(9
), pp. 2360
–2370
.10.1557/jmr.2005.0313115.
Xiang
, Y.
, Tsui
, T. Y.
, and Vlassak
, J. J.
, 2006
, “The Mechanical Properties of Freestanding Electroplated Cu Thin Films
,” J. Mater. Res.
, 21
(6
), pp. 1607
–1618
.10.1557/jmr.2006.0195116.
Nix
, W. D
, and Gao
, H. J.
, 1998
, “Indentation Size Effects in Crystalline Materials: A Law for Strain Gradient Plasticity
,” J. Mech. Phys. Solids
, 46
(3
), pp. 411
–425
.10.1016/S0022-5096(97)00086-0117.
Swadener
, J. G.
, George
, E. P.
, and Pharr
, G. M.
, 2002
, “The Correlation of the Indentation Size Effect Measured With Indenters of Various Shapes
,” J. Mech. Phys. Solids
, 50
(4
), pp. 681
–694
.10.1016/S0022-5096(01)00103-X118.
Abu Al-Rub
, R. K
, and Voyiadjis
, G. Z.
, 2004
, “Analytical and Experimental Determination of the Material Intrinsic Length Scale of Strain Gradient Plasticity Theory From Micro- and Nano-Indentation Experiments
,” Int. J. Plast.
, 20
(6
), pp. 1139
–1182
.10.1016/j.ijplas.2003.10.007119.
Voyiadjis
, G. Z
, and Abu Al-Rub
, R. K.
, 2005
, “Gradient Plasticity Theory With a Variable Length Scale Parameter
,” Int. J. Solids Struct.
, 42
(14
), pp. 3998
–4029
.10.1016/j.ijsolstr.2004.12.010120.
Anand
, L.
, Gurtin
, M. E.
, Lele
, S. P.
, and Gething
, C.
, 2005
, “A One-Dimensional Theory of Strain-Gradient Plasticity: Formulation, Analysis, Numerical Results
,” J. Mech. Phys. Solids
, 53
(8
), pp. 1789
–1826
.10.1016/j.jmps.2005.03.003121.
Bauschinger
, J.
, 1881
, “Tension Prisma Stabe
,” Civilingenieur
, 27
, pp. 289
–301
.122.
Yefimov
, S.
, Groma
, I.
, and Van der Giessen
, E.
, 2004
, “A Comparison of a Statistical-Mechanics Based Plasticity Model With Discrete DislocationPlasticity Calculations
,” J. Mech. Phys. Solids
, 52
(2
), pp. 279
–300
.10.1016/S0022-5096(03)00094-2123.
Ashby
, M. F.
, 1970
, “Deformation of Plastically Non-Homogeneous Materials
,” Philos. Mag.
, 21
(170
), pp. 399
–424
.10.1080/14786437008238426124.
Voyiadjis
, G.
, and Abu Al-Rub
, R. K.
, 2002
, “Length Scales in Gradient Plasticity Theory
,” Proceedings of the IUTAM Symposium on Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, Morocco, October 20–25, pp. 167
–174
.125.
Lee
, T. C.
, Robertson
, I. M.
, and Birnbaum
, H. K.
, 1989
, “Prediction of Slip Transfer Mechanisms Across Grain-Boundaries
,” Scr. Metall.
, 23
(5
), pp. 799
–803
.10.1016/0036-9748(89)90534-6126.
Lee
, T. C.
, Robertson
, I. M.
, and Birnbaum
, H. K.
, 1990
, “An In Situ Transmission Electron-Microscope Deformation Study of the Slip Transfer Mechanisms in Metals
,” Metall. Trans. A
, 21
(9
), pp. 2437
–2447
.10.1007/BF02646988127.
Kocks
, U. F.
, Argon
, A. S.
, and Ashby
, M. F.
, 1975
, “Thermodynamics and Kinetics of Slip
,” Prog. Mater. Sci.
, 19
, pp. 1
–281
.10.1016/0079-6425(75)90005-5128.
Hirth
, J. P.
, and Lothe
, J.
, 1982
, Theory of Dislocations
, 2nd ed., Krieger Publishing, Malabar, FL, p. 857
.129.
Meyers
, M. A.
, and Chawla
, K. K.
, 2009
, Mechanical Behavior of Materials
, Vol. xxii, 2nd ed., Cambridge University Press
, Cambridge, New York
, p. 856
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