Abstract
A number of new trends in material mechanics and engineering science can be traced back to the PhD work of Hussein Zbib at Michigan Technological University (MTU). In particular, the topics of shear bands and plastic instabilities found a new basis and direction, prompting distinguished researchers—whom he begun interacting with during his doctoral and post-doctoral years (see Appendix)—to turn their attention to gradient plasticity and make their own monumental contributions in this field. This article first provides a brief account of the initial attempts, I had the joy to share with him, on gradient mechanics theory and its implications to the problems of strain localization and size effects. It then continues with a brief exposition of topics that his “scientific family” has taken up in parallel with him or later on. Finally, it concludes with a sketch of ideas I discussed with him during his post-doctoral period at MTU and his tenure period as a faculty member and Chairman at Washington State University (WSU) which, unfortunately, he did not have the time to elaborate upon.
References
1.
Aifantis
, E. C.
, Walgraef
, D.
, and Zbib
, H., eds.
, 1998
, Material Instabilities
, Vol. 23, Nos. 2–3
, Elsevier Applied Science, London
, pp. 97
–305
.2.
Zbib
, H. M.
, and Aifantis
, E. C.
, 1988
, “On the Localization and Postlocalization Behavior of Plastic Deformation—I, II and III
,” Res. Mech.
, 23
(2–3
), pp. 261
–277
, 279–292, and 293–305.3.
Zbib
, H. M.
, 1987
, “Topics in Large Deformation Material Rotation, Anisotropy and Localization
,” Ph.D. thesis
, Michigan Technological University
, Houghton, MI
.4.
Aifantis
, E. C.
, 1984
, “On the Microstructural Origin of Certain Inelastic Models
,” ASME J. Eng. Mater. Technol.
, 106
(4
), pp. 326
–330
. 5.
Aifantis
, E. C.
, 1987
, “The Physics of Plastic Deformation
,” Int. J. Plast.
, 3
(3
), pp. 211
–247
. 6.
Aifantis
, E. C.
, and Serrin
, J. B.
, 1983
, “The Mechanical Theory of Fluid Interfaces and Maxwell’s Rule
,” J. Colloid Interface Sci.
, 96
(2
), pp. 517
–529
. 7.
Aifantis
, E. C.
, and Serrin
, J. B.
, 1983
, “Equilibrium Solutions in the Mechanical Theory of Fluid Microstructures
,” J. Colloid Interface Sci.
, 96
(2
), pp. 530
–547
. 8.
Liénard
, A.
, 1928
, “Etude des Oscillations Entretenues
,” Rev. générale l'électricité
, 23
(21
), pp. 901
–912
and 946–954. https://en.wikipedia.org/wiki/Li%C3%A9nard_equation9.
Drazin
, P. G.
, 1992
, Nonlinear Systems
, P. G.
Drazin
, ed., Cambridge University Press
, Cambridge
, pp. 170
–213
.10.
Sluys
, L. J.
, and de Borst
, R.
, 1994
, “Dispersive Properties of Gradient and Rate-Dependent Media
,” Mech. Mater.
, 183
(2
), pp. 131
–149
. 11.
Tomita
, Y.
, 1994
, “Simulations of Plastic Instabilities in Solid Mechanics
,” ASME Appl. Mech. Rev.
, 47
(6
), pp. 171
–205
. 12.
Carpinteri
, A., ed.
, 1995
, Size-Scale Effects in the Failure Mechanisms of Materials and Structures, Proc. IUTAM Symp.
, Chapman and Hall
, New York
.13.
de Borst
, R.
, and van der Giessen
, E., eds.
, 1998
, Material Instabilities in Solids, Proc. IUTAM Symp.
, Wiley
, Chichester
.14.
Muhlhaus
, H. B., ed.
, 1995
, Continuum Models for Materials With Microstructure
, Wiley
, New York
.15.
Aifantis
, E. C.
, 1992
, “On the Role of Gradients in the Localization of Deformation and Fracture
,” Int. J. Eng. Sci.
, 30
(10
), pp. 1279
–1299
. 16.
Triantafyllidis
, N.
, and Aifantis
, E. C.
, 1986
, “A Gradient Approach to Localization of Deformation—I. Hyperelastic Materials
,” J. Elast.
, 16
(3
), pp. 225
–238
. 17.
Triantafyllidis
, N.
, and Bardenhagen
, S.
, 1993
, “On Higher-Order Gradient Continuum Theories in 1-D. Nonlinear Elasticity. Derivation From and Comparison to the Corresponding Discrete Models
,” J. Elast.
, 33
(3
), pp. 259
–293
. 18.
Bardenhagen
, S.
, and Triantafyllidis
, N.
, 1994
, “Derivation of Higher-Order Gradient Continuum Theories in 2,3-D. Nonlinear Elasticity From Periodic Lattice Models
,” J. Mech. Phys. Solids
, 42
(1
), pp. 111
–139
. 19.
Altan
, S. B.
, and Aifantis
, E. C.
, 1992
, “On the Structure of the Mode III Crack-Tip in Gradient Elasticity
,” Scr. Metall. Mater.
, 26
(2
), pp. 319
–324
. 20.
Ru
, C. Q.
, and Aifantis
, E. C.
, 1993
, “A Simple Approach to Solve Boundary Value Problems in Gradient Elasticity
,” Acta Mech.
, 101
, pp. 59
–68
. 21.
Aifantis
, E. C.
, 2016
, “Internal Length Gradient (ILG) Material Mechanics Across Scales & Disciplines
,” Adv. Appl. Mech.
, 49
, pp. 1
–110
. 22.
Mindlin
, R. D.
, 1964
, “Micro-Structure in Linear Elasticity
,” Arch. Ration. Mech. Anal.
, 16
(1
), pp. 51
–78
. 23.
dell’Isola
, F.
, Sciarra
, G.
, and Vidoli
, S.
, 2009
, “Generalized Hooke’s Law for Isotropic Second Gradient Materials
,” Proc. R. Soc. A
, 465
(2107
), pp. 2177
–2196
. 24.
Auffray
, N.
, dell’Isola
, F.
, Eremeyev
, V. A.
, Madeo
, A.
, and Rosi
, G.
, 2015
, “Analytical Continuum Mechanics à la Hamilton-Piola: Least Action Principle for Second Gradient Continua and Capillary Fluids
,” Math. Mech. Solids
, 20
(4
), pp. 375
–417
. 25.
Bertram
, A.
, 2016
, “Finite Gradient Elasticity and Plasticity: A Constitutive Thermodynamic Framework
,” Continuum Mech. Thermodyn.
, 28
(3
), pp. 869
–883
. 26.
Bertram
, A.
, 2017
, Compendium on Gradient Materials
, 3rd ed., TU Berlin
, Magdeburg
.27.
Eringen
, A. C.
, 1985
, “Nonlocal Continuum Theory for Dislocations and Fracture,” The Mechanics of Dislocations
, E. C.
Aifantis
, and J. P.
Hirth
, eds., ASM
, Metals Park
, pp. 101
–110
.28.
Eringen
, A. C.
, 2001
, Nonlocal Continuum Field Theories
, Springer
, New York
.29.
Aifantis
, E. C.
, 2011
, “On the Gradient Approach—Relation to Eringen’s Nonlocal Theory
,” Int. J. Eng. Sci.
, 49
(12
), pp. 1367
–1377
. 30.
Coleman
, B. D.
, and Gurtin
, M. E.
, 1967
, “Thermodynamics With Internal State Variables
,” J. Chem. Phys.
, 47
(2
), pp. 597
–613
. 31.
Valanis
, K. C.
, 1967
, “The Viscoelastic Potential and Its Thermodynamic Foundations
,” J. Math. Phys.
, 47
(1–4
), pp. 262
–275
. 32.
Valanis
, K. C.
, 1996
, “A Gradient Theory of Internal Variables
,” Acta. Mech.
, 116
(1–4
), pp. 1
–14
. 33.
Valanis
, K. C.
, 1998
, “A Gradient Thermodynamic Theory of Self-Organization
,” Acta Mech.
, 127
(1–4
), pp. 1
–23
. 34.
Valanis
, K. C.
, 1998
, “Diffusion Potential and Well-Posedness in Non-Associative Plasticity
,” Int. J. Solids Struct.
, 35
(36
), pp. 5173
–5188
. 35.
Valanis
, K. C.
, 2000
, “Gradient Field Theory of Material Instabilities
,” Arch. Mech.
, 52
(4–5
), pp. 817
–838
.36.
Aifantis
, E. C.
, and Davison
, L., eds.
, 1984
, “Special Issue: Media With Microstructures and Wave Propagation
,” Int. J. Eng. Sci.
, 22
(8–10
), pp. 959
–1224
.37.
Aifantis
, E. C.
, 1984
, “Remarks on Media With Microstructure
,” Int. J. Eng. Sci.
, 22
(8–10
), pp. 961
–968
. 38.
Aifantis
, E. C.
, 1994
, “Gradient Effects at Macro, Micro, and Nano Scales
,” J. Mech. Behav. Mater.
, 5
(3
), pp. 355
–375
. 39.
Bammann
, D.
, and Aifantis
, E. C.
, 1989
, “A Damage Model for Ductile Metals
,” Nucl. Eng. Des.
, 116
(3
), pp. 355
–362
. 40.
Lesar
, R.
, Rickman
, J. M.
, Clarke
, D. R.
, Ruhle
, R.
, and Bravman
, J. C.
, eds., 2002
, Annual Review of Materials Research
, 32
, See, for Example, pp. 113
–140
and 163–194.41.
van der Waals
, J. D.
, 1873
, “Over de Continuited van den Gas-en Vloeistoftoestand
,” Ph.D. thesis
, University of Leiden
, Leiden
.42.
van der Waals
, J. D.
, 1895
, “Thèorie Thermodynamique de la Capillarité, Dans L’hypothèse D’une Variation Continue de Densité
,” Arch. Neerl. Sci. Exactes Nat.
, 28
, pp. 121
–209
.43.
Rowlinson
, J. S.
, 1979
, “Translation of J. D. van der Waals’ “The Thermodynamic Theory of Capillarity Under the Hypothesis of a Continuous Variation of Density”
,” J. Stat. Phys.
, 20
(2
), pp. 197
–200
. 44.
Ginzburg
, V. L.
, and Landau
, L. D.
, 1950
, “On the Theory of Superconductivity
,” Sov. Phys. JETP
, 20
, pp. 1064
–1082
. https://en.wikipedia.org/wiki/Ginzburg%E2%80%93Landau_theory45.
Ginzburg
, V. L.
, and Landau
, L. D.
, 2009
, “On the Theory of Superconductivity (Chapter 4),” On Superconductivity and Superfluidity
, V. L.
Ginzburg
, ed., Springer
, Berlin/Heidelberg
, pp. 113
–137
.46.
Ter Haar
, D.
, 1965
, Collected Papers of L.D. Landau
, Pergamon
, London
.47.
Cahn
, J. W.
, and Hilliard
, J. E.
, 1958
, “Free Energy of a Nonuniform System. I. Interfacial Free Energy
,” J. Chem. Phys.
, 28
(2
), pp. 258
–267
.48.
Cahn
, J. W.
, 1959
, “Free Energy of a Nonuniform System. II. Thermodynamic Basis
,” J. Chem. Phys.
, 30
(5
), pp. 1121
–1124
. 49.
Webb
, T.
, 1993
, “On the Theory of Stick-Slip Fracture
,” Ph.D. thesis
, Michigan Technological University
, Houghton, MI
.50.
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
. 51.
Fleck
, N. A.
, and Hutchinson
, J. W.
, 1997
, “Strain Gradient Plasticity,” Advances in Applied Mechanics
, J. W.
Hutchinson
, and T. W.
Wu
, eds., Academic Press
, New York
, pp. 295
–361
.52.
Fleck
, N. A.
, and Hutchinson
, J. W.
, 2001
, “A Reformulation of Strain Gradient Plasticity
,” J. Mech. Phys. Solids
, 49
(10
), pp. 2245
–2271
. 53.
Gao
, H.
, Huang
, Y.
, Nix
, W. D.
, and Hutchinson
, J. W.
, 1999
, “Mechanism-Based Strain Gradient Plasticity—I
,” J. Mech. Phys. Solids
, 47
(6
), pp. 1239
–1263
. 54.
Huang
, Y.
, Gao
, H.
, Nix
, W. D.
, and Hutchinson
, J. W.
, 2000
, “Mechanism-Based Strain Gradient Plasticity—II
,” J. Mech. Phys. Solids
, 48
(1
), pp. 99
–128
. 55.
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
. 56.
Gurtin
, M. E.
, Fried
, E.
, and Anand
, L.
, 2012
, The Mechanics and Thermodynamics of Continua
, Cambridge University Press
, New York
.57.
Tarasov
, V. E.
, and Aifantis
, E. C.
, 2019
, “On Fractional and Fractal Formulations of Gradient Linear and Nonlinear Elasticity
,” Acta Mech.
, 230
(6
), pp. 2043
–2070
. 58.
Aifantis
, E. C.
, 2019
, “Fractional Generalizations of Gradient Mechanics,” Applications in Physics, Part A
, Vol. 4, V. E.
Tarasov
, ed., De Gruyter
, Berlin
, pp. 241
–262
.59.
Tsagrakis
, I.
, and Aifantis
, E. C.
, 2002
, “Recent Developments in Gradient Plasticity. Part I: Formulation and Size Effects
,” ASME J. Eng. Mater. Technol.
, 124
(3
), pp. 352
–357
. [For an initial treatment and detailed review of related gradient theories up to that date the reader may consult: Aifantis, E. C., 1994, “Spatio-Temporal Instabilities in Deformation and Fracture,” Comput. Mater. Model., AD-Vol. 42/PVP-Vol. 294, A. K. Noor and A. Needleman, eds., ASME, New York, pp. 199–222.] 60.
Zhu
, H. T.
, and Zbib
, H. M.
, 1995
, “On the Role of Strain Gradients in Adiabatic Shear Banding
,” Acta Mech.
, 111
(1–2
), pp. 111
–124
. 61.
Aifantis
, E. C.
, 1999
, “Gradient Deformation Models at Nano, Micro, and Macro Scales
,” ASME J. Eng. Mater. Technol.
, 121
(2
), pp. 189
–202
. 62.
Tsagrakis
, I.
, and Aifantis
, E. C.
, “Shear Banding Instabilities in Bulk Metallic Glasses at Low Strain Rates: Gradient and Length Scale Effects
,” Rev. Adv. Mater. Sci.
, 48
, pp. 156
–169
.63.
Zbib
, H. M.
, and Aifantis
, E. C.
, 1988
, “On the Concept of Relative and Plastic Spins and Its Implications to Large Deformation Theories, Part I: Hypoelasticity and Vertex-Type Plasticity
,” Acta Mech.
, 75
(1–4
), pp. 15
–33
. 64.
Zbib
, H. M.
, and Aifantis
, E. C.
, 1988
, “On the Concept of Relative and Plastic Spins and Its Implications to Large Deformation Theories, Part II: Anisotropic Hardening Plasticity
,” Acta Mech.
, 75
(1–4
), pp. 35
–56
.65.
Webb
, T. W.
, and Alfantis
, E. C.
, 1995
, “Oscillatory Fracture in Polymeric Materials
,” Int. J. Solids Struct.
, 32
(17–18
), pp. 2725
–2743
. 66.
Webb
, T. W.
, and Aifantis
, E. C.
, 1989
, “Stick-Slip Peeling
,” ASME Winter Annual Meeting
, San Francisco, CA
, Paper No. 89-WA-EEP-7.67.
Aifantis
, K. E.
, 2005
, “Gradient Plasticity With Interfacial Effects and Experimental Confirmation Through Nano-Indentation
,” Ph.D. thesis
, University of Groningen
, Groningen
, The Netherlands.68.
Aifantis
, K. E.
, and Willis
, J. R.
, 2005
, “The Role of Interfaces in Enhancing the Yield Strength of Composites and Polycrystals
,” J. Mech. Phys. Solids
, 53
(5
), pp. 1047
–1070
. 69.
Aifantis
, K. E.
, Soer
, W. A.
, De Hosson
, J. T. M.
, and Willis
, J. R.
, 2006
, “Interfaces Within Strain Gradient Plasticity: Theory and Experiments
,” Acta Mater.
, 54
(19
), pp. 5077
–5085
. 70.
Aifantis
, K. E.
, and Ngan
, A. H. W.
, 2007
, “Modeling Dislocation—Grain Boundary Interactions Through Gradient Plasticity and Nanoindentation
,” Mater. Sci. Eng. A
, 459
(1–2
), pp. 251
–261
. 71.
Zhang
, X.
, and Aifantis
, K. E.
, 2011
, “Interpreting Strain Bursts and Size Effects in Micropillars Using Gradient Plasticity
,” Mater. Sci. Eng. A
, 528
(15
), pp. 5036
–5043
. 72.
Aifantis
, K. E.
, Konstantinidis
, A.
, and Forest
, S.
, 2012
, “Modeling Strain Localization Bands in Metal Foams
,” J. Comput. Theor. Nanosci.
, 7
(2
), pp. 1
–7
.73.
Konstantinidis
, A. A.
, Aifantis
, K. E.
, and De Hosson
, J. T. M.
, 2014
, “Capturing the Stochastic Mechanical Behavior of Micro and Nanopillars
,” Mater. Sci. Eng. A
, 597
, pp. 89
–94
. 74.
Konstantinidis
, A. A.
, and Aifantis
, K. E.
, 2019
, “Capturing Slip Band Formation in Ni3Al Nanocubes During Compression
,” Mater. Sci. Technol.
, 35
(5
), pp. 571
–576
. 75.
Zaiser
, M.
, and Hahner
, P.
, 1996
, “Random Aspects of Macroscopic Plastic Deformation
,” Philos. Mag. Lett.
, 73
(6
), pp. 369
–375
. 76.
Zaiser
, M.
, and Aifantis
, E. C.
, 2006
, “Randomness and Slip Avalanches in Gradient Plasticity
,” Int. J. Plast.
, 22
(8
), pp. 1432
–1455
. 77.
Zaiser
, M.
, and Seeger
, A.
, 2002
, “Long-Range Internal Stresses, Dislocation Patterning and Work Hardening in Crystal Plasticity,” Dislocations in Solids
, F. R. N.
Nabarro
, and M. S.
Duesbery
, eds., 11
, North-Holland
, Amsterdam
, pp. 1
–100
.78.
Aifantis
, E. C.
, 2021
, “Gradient Extension of Classical Material Models: From Nuclear & Condensed Matter Scales to Earth & Cosmological Scales,” Size-Dependent Continuum Mechanics Approaches
, E.
Ghavanloo
, S. A.
Fazelzadeh
, and F.
Marotti de Sciarra
, eds., Springer
, Cham
, pp. 417
–452
.79.
Fischbach
, E.
, Sudarsky
, D.
, Szafer
, A.
, Talmadge
, C.
, and Aronson
, S. H.
, 1986
, “Reanalysis of the Eötös Experiment
,” Phys. Rev. Lett.
, 56
(1
), pp. 3
–6
. 80.
Fischbach
, E.
, 2015
, “The Fifth Force: A Personal History
,” Eur. Phys. J. H
, 40
(4–5
), pp. 385
–467
. 81.
Zbib
, H. M.
, Rhee
, M.
, and Hirth
, J. P.
, 1998
, “On Plastic Deformation and the Dynamics of 3D Dislocations
,” Int. J. Mech. Sci.
, 40
(2–3
), pp. 113
–127
. 82.
Rhee
, M.
, Zbib
, H. M.
, Hirth
, J. P.
, Huang
, H.
, and de la Rubia
, T.
, 1998
, “Models for Long-/Short-Range Interactions and Cross Slip in 3D Dislocation Simulation of BCC Single Crystals
,” Modell. Simul. Mater. Sci. Eng.
, 6
(4
), pp. 467
–492
. 83.
Walgraef
, D.
, and Aifantis
, E. C.
, 1985
, “Dislocation Patterning in Fatigued Metals as a Result of Dynamical Instabilities
,” J. Appl. Phys.
, 58
(2
), pp. 688
–691
. 84.
Aifantis
, E. C.
, 1986
, “On the Dynamical Origin of Dislocation Patterns
,” Mater. Sci. Eng.
, 81
, pp. 563
–574
. 85.
Lepinoux
, J.
, and Kubin
, L. P.
, 1987
, “The Dynamic Organization of Dislocation Structures: A Simulation
,” Scr. Metall.
, 21
(6
), pp. 833
–838
. 86.
Gulluoglu
, A. N.
, Srolovitz
, D. J.
, LeSar
, R.
, and Lomdahl
, P. S.
, 1989
, “Dislocation Distributions in Two Dimensions
,” Scr. Metall.
, 23
(8
), pp. 1347
–1352
. 87.
Ghoniem
, N. M.
, and Amodeo
, R. J.
, 1990
, Patterns, Defects and Material Instabilities
, D.
Walgraef
, and N. M.
Ghoniem
, eds., Kluwer Academic Publishers
, Dordrecht
, pp. 303
–329
.88.
Kubin
, L. P.
, Canova
, G.
, Condat
, M.
, Devincre
, B.
, Pontikis
, V.
, and Brechet
, Y.
, 1992
, “Dislocation Microstructures in Two Dimensions: I. Relaxed Structures, Modelling Simulation
,” Mater. Sci. Eng.
, 1
, pp. 1
–17
.89.
Kubin
, L. P.
, 1993
, “Dislocation Patterning,” Materials Science and Technology
, Vol. 6
, R. W.
Cahn
, P.
Haasan
, and E. J.
Kramer
, eds., Weinheim VCH
, New York
, pp. 138
–190
.90.
Shizawa
, K.
, and Zbib
, H. M.
, 1997
, “A Thermodynamical Theory of Gradient Elastoplasticity With Dislocation Density Tensor. I: Fundamentals
,” Int. J. Plast.
, 15
(9
), pp. 899
–938
. 91.
Shizawa
, K.
, Kikuchi
, K.
, and Zbib
, H. M.
, 2001
, “A Strain-Gradient Thermodynamic Theory of Plasticity Based on Dislocation Density in Incompatibility Tensors
,” Mater. Sci. Eng. A
, 309–310
, pp. 416
–419
. 92.
Zbib
, H. M.
, Hamid
, M.
, Lyu
, H.
, and Mastorakos
, I.
, 2018
, “Multiscale Dislocation-Based Plasticity,” Mesoscale Models
, Vol. 587, S.
Mesarovic
, S.
Forest
, and H.
Zbib
, eds., CISM International Centre for Mechanical Sciences (Courses and Lectures), Springer
, Cham
, pp. 51
–85
.93.
Pontes
, J.
, Walgraef
, D.
, and Aifantis
, E. C.
, 2006
, “On Dislocation Patterning: Multiple Slip Effects in the Rate Equation Approach
,” Int. J. Plast.
, 22
(8
), pp. 1486
–1505
. 94.
Hamid
, M.
, Lyu
, H.
, and Zbib
, H.
, 2018
, “A Dislocation-Based Stress–Strain Gradient Plasticity Model for Strength and Ductility in Materials With Gradient Microstructures
,” Philos. Mag.
, 98
(32
), pp. 2896
–2916
. 95.
Steinmann
, P.
, Kergaßner
, A.
, Landkammer
, P.
, and Zbib
, H. M.
, 2019
, “A Novel Continuum Approach to Gradient Plasticity Based on the Complementing Concepts of Dislocation and Disequilibrium Densities
,” J. Mech. Phys. Solids
, 132
, p. 103680
. 96.
Zhou
, G.
, Jeong
, W.
, Homer
, E. R.
, Fullwood
, D. T.
, Lee
, M. G.
, Kim
, J. H.
, Lim
, H.
, Zbib
, H.
, and Wagoner
, R. H.
, 2020
, “A Predictive Strain-Gradient Model With no Undetermined Constants or Length Scales
,” J. Mech. Phys. Solids
, 145
, p. 104178
. 97.
Akarapu
, S.
, and Zbib
, H. M.
, 2006
, “Numerical Analysis of Plane Cracks in Strain-Gradient Elastic Materials
,” Int. J. Fract.
, 141
(3–4
), pp. 403
–430
. 98.
Coleman
, B. D.
, and Hodgdon
, M. L.
, 1988
, “On the Localization of Strain in Shearing Motions of Ductile Materials
,” Res. Mech.
, 23
, pp. 223
–238
.99.
Coleman
, B. D.
, and Newman
, D. C.
, 1989
, “On Adiabatic Shear Bands in Rigid-Plastic Materials
,” Acta Mech.
, 78
(3–4
), pp. 263
–279
. 100.
Coleman
, B. D.
, and Newman
, D. C.
, 1990
, “Mechanics of Neck Formation in the Cold Drawing of Elastic Films
,” Polym. Eng. Sci.
, 30
(20
), pp. 1299
–1302
. 101.
Coleman
, B. D.
, and Newman
, D. C.
, 1992
, “Rheology of Neck Formation in the Cold Drawing of Polymeric Fibers
,” J. Appl. Polym. Sci.
, 45
(6
), pp. 997
–1004
. 102.
Batra
, R. C.
, 1992
, “Analysis of Shear Bands in Simple Shearing Deformations of Nonpolar and Dipolar Viscoplastic Materials
,” ASME Appl. Mech. Rev.
, 45
(3S
), pp. S123
–S131
. 103.
Batra
, R. C.
, and Chen
, L.
, 1999
, “Shear Band Spacing in Gradient-Dependent Thermoviscoplastic Materials
,” Comput. Mech.
, 23
(1
), pp. 8
–19
. 104.
Chen
, L.
, and Batra
, R. C.
, 1999
, “Effect of Material Parameters on Shear Band Spacing in Work-Hardening Gradient Dependent Thermoviscoplastic Materials
,” Int. J. Plast.
, 15
(5
), pp. 551
–574
. 105.
Acharya
, A.
, and Bassani
, J. L.
, 1996
, “On Non-Local Flow Theories That Reserve the Classical Structure of Incremental Boundary Value Problems
,” Proceedings of the IUTAM Symposium on Micromechanics of Plasticity and Damage of Multiphase Materials
, Paris
, Aug. 29–Sept. 1
, Kluwer
, pp. 3
–9
.106.
Acharya
, A.
, Cherukuri
, H. P.
, and Govindarajan
, R. M.
, 1999
, “New Proposal in Gradient Plasticity: Theory and Application in 1-D Quasi-Statics and Dynamics
,” Mech. Cohesive-Frict. Mater.
, 4
(2
), pp. 153
–170
. 107.
Acharya
, A.
, and Bassani
, J. L.
, 2000
, “Lattice Incompatibility and a Gradient Theory of Crystal Plasticity
,” J. Mech. Phys. Solids
, 48
(8
), pp. 1565
–1595
. 108.
Acharya
, A.
, Bassani
, J. L.
, and Baudoin
, A.
, 2003
, “Geometrically Necessary Dislocations, Hardening, and a Simple Gradient Theory of Crystal Plasticity
,” Scr. Mater.
, 48
(2
), pp. 167
–172
. 109.
Acharya
, A.
, Tang
, H.
, Saigal
, S.
, and Bassani
, J. L.
, 2004
, “On Boundary Conditions and Plastic Strain-Gradient Discontinuity in Lower-Order Gradient Plasticity
,” J. Mech. Phys. Solids
, 52
(8
), pp. 1793
–1826
. 110.
Polizzotto
, C.
, and Borino
, G.
, 1998
, “A Thermodynamics-Based Formulation of Gradient-Dependent Plasticity
,” Eur. J. Mech. A Solids
, 17
(5
), pp. 741
–761
. 111.
Polizzotto
, C.
, 2003
, “Unified Thermodynamic Framework for Nonlocal/Gradient Continuum Theories
,” Eur. J. Mech. A Solids
, 22
(5
), pp. 651
–668
. 112.
Polizzotto
, C.
, 2009
, “A Nonlocal Strain Gradient Plasticity Theory for Finite Deformations
,” Int. J. Plast.
, 25
(7
), pp. 1280
–1300
. 113.
Polizzotto
, C.
, 2011
, “A Unified Residual-Based Thermodynamic Framework for Strain Gradient Theories of Plasticity
,” Int. J. Plast.
, 27
(3
), pp. 388
–413
. 114.
Polizzotto
, C.
, 2013
, “A Second Strain Gradient Elasticity Theory With Second Velocity Gradient Inertia—Part I: Constitutive Equations and Quasi-Static Behavior
,” Int. J. Solids Struct.
, 50
(24
), pp. 3749
–3765
. 115.
Polizzotto
, C.
, 2014
, “Surface Effects, Boundary Conditions and Evolution Laws Within Second Strain Gradient Plasticity
,” Int. J. Plast.
, 60
, pp. 197
–216
. 116.
Bammann
, D. J.
, 2001
, “A Model of Crystal Plasticity Containing a Natural Length Scale
,” Mater. Sci. Eng. A
, 309–310
, pp. 406
–410
. 117.
Hwang
, K. C.
, Jiang
, H.
, Huang
, Y.
, Gao
, H.
, and Hu
, N.
, 2002
, “A Finite Deformation Theory of Strain Gradient Plasticity
,” J. Mech. Phys. Solids
, 50
(1
), pp. 81
–99
. 118.
Hwang
, K. C.
, Jiang
, H.
, Huang
, Y.
, and Gao
, H.
, 2003
, “Finite Deformation Analysis of Mechanism-Based Strain Gradient Plasticity: Torsion and Crack tip Field
,” Int. J. Plast.
, 19
(2
), pp. 235
–251
. 119.
Hwang
, K. C.
, Guo
, Y.
, Jiang
, H.
, Huang
, Y.
, and Zhuang
, Z.
, 2004
, “The Finite Deformation Theory of Taylor-Based Nonlocal Plasticity
,” Int. J. Plast.
, 20
(4–5
), pp. 831
–839
. 120.
Huang
, Y.
, Qu
, S.
, Hwang
, K. C.
, Li
, M.
, and Gao
, H.
, 2004
, “A Conventional Theory of Mechanism-Based Strain Gradient Plasticity
,” Int. J. Plast.
, 20
(4–5
), pp. 753
–782
. 121.
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
. 122.
Abu Al-Rub
, R. K.
, and Voyiadjis
, G. Z.
, 2006
, “A Physically Based Gradient Plasticity Theory
,” Int. J. Plast.
, 22
(4
), pp. 654
–684
. 123.
Voyiadjis
, G. Z.
, and Faghihi
, D.
, 2012
, “Thermo-Mechanical Strain Gradient Plasticity With Energetic and Dissipative Length Scales
,” Int. J. Plast.
, 30–31
, pp. 218
–247
. 124.
Voyiadjis
, G. Z.
, and Faghihi
, D.
, 2013
, “Gradient Plasticity for Thermo-Mechanical Processes in Metals With Length and Time Scales
,” Philos. Mag.
, 93
(9
), pp. 1013
–1053
. 125.
Cardona
, J.-M.
, Forest
, S.
, and Sievert
, R.
, 1999
, “Towards a Theory of Second Grade Thermoelasticity
,” Extracta Math.
, 14
(2
), pp. 127
–140
.126.
Forest
, S.
, Cardona
, J. M.
, and Sievert
, R.
, 2000
, Continuum Thermomechanics, The Art and Science of Modelling Material Behavior, Paul Germain’s Anniversary Volume
, G.
Maugin
, R.
Drouot
, and F.
Sidoroff
, eds., Kluwer Academic Publishers
, Dordrecht
, pp. 163
–176
.127.
Forest
, S.
, 2008
, “The Micromorphic Approach to Plasticity and Diffusion
,” Proceedings of the International Conference on Continuum Models and Discrete Systems 11 (CMDS11)
, Paris, France
, July 30–Aug. 3, 2007
, pp. 105
–112
.128.
Forest
, S.
, 2009
, “The Micromorphic Approach for Gradient Elasticity, Viscoplasticity and Damage
,” ASCE J. Eng. Mech.
, 135
(3
), pp. 117
–131
. 129.
Forest
, S.
, and Sievert
, R.
, 2003
, “Elastoviscoplastic Constitutive Frameworks for Generalized Continua
,” Acta Mech.
, 160
(1–2
), pp. 71
–111
. 130.
Forest
, S.
, Cordero
, N. M.
, and Busso
, E. P.
, 2011
, “First vs. Second Gradient of Strain Theory for Capillarity Effects in an Elastic Fluid at Small Length Scales
,” Comput. Mater. Sci.
, 50
(4
), pp. 1299
–1304
. 131.
Menzel
, A.
, and Steinmann
, P.
, 2000
, “On the Continuum Formulation of Higher Gradient Plasticity for Single and Polycrystals
,” J. Mech. Phys. Solids
, 48
(8
), pp. 1777
–1796
. 132.
Menzel
, A.
, and Steinmann
, P.
, 1998
, “On the Formulation of Higher Gradient Single and Polycrystal Plasticity
,” J. Phys. IV
, 8
(Part 8), pp. 239
–247
.133.
Kirchner
, N.
, and Steinmann
, P.
, 2007
, “On the Material Setting of Gradient Hyperelasticity
,” Math. Mech. Solids
, 12
(5
), pp. 559
–580
. 134.
Sunyk
, R.
, and Steinmann
, P.
, 2003
, “On Higher Gradients in Continuum-Atomistic Modelling
,” Int. J. Solids Struct.
, 40
(24
), pp. 6877
–6896
. 135.
Sunyk
, R.
, and Steinmann
, P.
, 2006
, “Transition to Plasticity in Continuum-Atomistic Modelling
,” Multidiscipl. Model. Mater. Struct.
, 2
(3
), pp. 1
–38
. 136.
Geers
, M. G. D.
, Peerlings
, R. H. J.
, de Borst
, R.
, and Brekelmans
, W. A. M.
, 1998
, Material Instabilities in Solids
, R.
de Borst
, and E.
van der Giessen
, eds., John Wiley & Sons Ltd.
, Chichester
, pp. 405
–424
.137.
Peerlings
, R. H. J.
, de Borst
, R.
, Brekelmans
, W. A. M.
, and de Vree
, J. H. P.
, 1996
, “Gradient-Enhanced Damage for Quasi-Brittle Materials
,” Int. J. Numer. Methods Eng.
, 39
(19
), pp. 3391
–3403
. 138.
Peerlings
, R. H. J.
, de Borst
, R.
, Brekelmans
, W. A. M.
, and Geers
, M. G. D.
, 1998
, “Gradient-Enhanced Damage Modelling of Concrete Fracture
,” Mech. Cohesive-Frict. Mater.
, 3
(4
), pp. 323
–342
. 139.
Anand
, L.
, Aslan
, O.
, and Chester
, S. A.
, 2012
, “A Large-Deformation Gradient Theory for Elastic-Plastic Materials: Strain Softening and Regularization of Shear Bands
,” Int. J. Plast.
, 30–31
, pp. 116
–143
. 140.
Neff
, P.
, 2008
, “Remarks on Invariant Modelling in Finite Strain Gradient Plasticity
,” Tech. Mech.
, 28
(1
), pp. 13
–21
.141.
Neff
, P.
, Chelminski
, K.
, and Alber
, H.-D.
, 2009
, “Notes on Strain Gradient Plasticity: Finite Strain Covariant Modelling and Global Existence in the Infinitesimal Rate-Independent Case
,” Math. Models Methods Appl. Sci.
, 19
(2
), pp. 307
–346
. 142.
Bertram
, A.
, 2015
, “Finite Gradient Elasticity and Plasticity: A Constitutive Mechanical Framework
,” Continuum Mech. Thermodyn.
, 27
(6
), pp. 1039
–1058
. 143.
Bertram
, A.
, and Glüge
, R.
, 2016
, “Gradient Materials With Internal Constraints
,” Math. Mech. Complex Syst.
, 4
(1
), pp. 1
–15
. 144.
de Borst
, R.
, and Mühlhaus
, H.-B.
, 1992
, “Gradient-Dependent Plasticity: Formulation and Algorithmic Aspects
,” Int. J. Numer. Methods Eng.
, 35
(3
), pp. 521
–539
. 145.
Sluys
, L. J.
, de Borst
, R.
, and Mühlhaus
, H. B.
, 1993
, “Wave Propagation, Localization and Dispersion in a Gradient-Dependent Medium
,” Int. J. Solids Struct.
, 30
(9
), pp. 1153
–1171
. 146.
de Borst
, R.
, Sluys
, L. J.
, Muhlhaus
, H.-B.
, and Pamin
, J.
, 1993
, “Fundamental Issues in Finite Element Analyses of Localization of Deformation
,” Eng. Comput.
, 10
(2
), pp. 99
–121
. 147.
Belytschko
, T.
, and Lasry
, D.
, 1988
, “Localization limiters and numerical strategies for strain-softening materials
,” Strain Localization and Size Effect Due to Cracking and Damage
, Paris, France
, Sept. 6–9
, pp. 349
–362
.148.
Lasry
, D.
, and Belytschko
, T.
, 1988
, “Localization Limiters in Transient Problems
,” Int. J. Solids Struct.
, 24
(6
), pp. 581
–597
. 149.
Kulkarni
, M.
, and Belytschko
, T.
, 1991
, “On the Effect of Imperfections and Spatial Gradient Regularization in Strain Softening Viscoplasticity
,” Mech. Res. Commun.
, 18
(6
), pp. 335
–343
. 150.
Estrin
, Y.
, and Kubin
, L. P., eds.
, 1993
, “Viewpoint Set No. 21
,” Scripta Metall.
, 29
, pp. 1147
–1188
.151.
Estrin
, Y.
, and Kubin
, L. P.
, 1989
, “Collective Dislocations Behavior in Dilute Alloys and the Portevin-Le Chatelier Effect
,” J. Mech. Behav. Mater.
, 2
(3–4
), pp. 255
–292
. 152.
Hähner
, P.
, 1993
, “Modelling the Spatiotemporal Aspects of the Portevin-Le Chatelier Effect
,” Mater. Sci. Eng. A
, 164
(1–2
), pp. 23
–34
. 153.
Hähner
, P.
, 1993
, “Dislocation Dynamics and Instabilities of Plastic Deformation—Nonlinear Phenomena Far From Equilibrium
,” Mater. Sci. Forum
, 123–125
, pp. 701
–710
. 154.
Zaiser
, M.
, and Hahner
, P.
, 1997
, “Oscillatory Modes of Plastic Deformation: Theoretical Concepts
,” Phys. Stat. Sol. B
, 199
(2
), pp. 267
–330
. 155.
Hähner
, P.
, Ziegenbein
, A.
, Rizzi
, E.
, and Neuhäuser
, H.
, 2002
, “Spatiotemporal Analysis of Portevin-Le Chatelier Deformation Bands: Theory, Simulation, and Experiment
,” Phys. Rev. B
, 65
(13
), p. 134109
. Copyright © 2021 by ASME
You do not currently have access to this content.
