The mechanical and thermal behavior of small volume metallic compounds on the fast transient time are addressed in this work through developing a thermodynamically consistent nonlocal framework. In this regard, an enhanced gradient plasticity theory is coupled with the application of the micromorphic approach to the temperature variable. The yield function of the VA–FCC (Voyiadjis Abed Face Centered Cubic) model based on the concept of thermal activation energy and the dislocations interaction mechanisms including nonlinear hardening is taken into consideration in the derivation. The effect of the material microstructural interface between two materials is also incorporated in the formulation with both temperature and rate effects. In order to accurately address the strengthening and hardening mechanisms, the theory is developed based on the decomposition of the mechanical state variables into energetic and dissipative counterparts which provided the constitutive equations to have both energetic and dissipative gradient length scales for the bulk material and the interface. Moreover, the nonlocal evolution of temperature is addressed by incorporating the microstructural interaction effect in the fast transient process using two time scales in the microscopic heat equation.
Issue Section:
Research Papers
References
1.
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.2007.01.0102.
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.
Narayan
, J.
, Godbole
, V. P.
, and White
, C. W.
, 1991
, “Laser Method for Synthesis and Processing of Continuous Diamond Films on Nondiamond Substrates
,” Science
, 252
(5004
), pp. 416
–418
.10.1126/science.252.5004.4165.
Sze
, S. M.
, and Ng
, K. K.
, 2007
, Physics of Semiconductor Devices
, 3rd ed., Wiley-Interscience
, Hoboken, NJ.
6.
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)00020-87.
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.5938.
Joshi
, A. A.
, and Majumdar
, A.
, 1993
, “Transient Ballistic and Diffusive Phonon Heat-Transport in Thin-Films
,” J. Appl. Phys.
, 74
(1
), pp. 31
–39
.10.1063/1.3541119.
Torii
, S.
, and Yang
, W. J.
, 2005
, “Heat Transfer Mechanisms in Thin Film With Laser Heat Source
,” Int. J. Heat Mass Transfer
, 48
(3–4
), pp. 537
–544
.10.1016/j.ijheatmasstransfer.2004.09.01110.
Tzou
, D. Y.
, 1995
, “Experimental Support for the Lagging Behavior in Heat Propagation
,” J. Thermophys. Heat Transfer
, 9
(4
), pp. 686
–693
.10.2514/3.72511.
Fujimoto
, J. G.
, Liu
, J. M.
, Ippen
, E. P.
, and Bloembergen
, N.
, 1984
, “Femtosecond Laser Interaction With Metallic Tungsten and Nonequilibrium Electron and Lattice Temperatures
,” Phys. Rev. Lett.
, 53
(19
), pp. 1837
–1840
.10.1103/PhysRevLett.53.183712.
Brorson
, S. D.
, Kazeroonian
, A.
, Moodera
, J. S.
, Face
, D. W.
, Cheng
, T. K.
, Ippen
, E. P.
, Dresselhaus
, M. S.
, and Dresselhaus
, G.
, 1990
, “Femtosecond Room-Temperature Measurement of the Electron-Phonon Coupling Constant-Lambda in Metallic Superconductors
,” Phys. Rev. Lett.
, 64
(18
), pp. 2172
–2175
.10.1103/PhysRevLett.64.217213.
Elsayed-Ali
, H. E.
, Juhasz
, T.
, Smith
, G. O.
, and Bron
, W. E.
, 1991
, “Femtosecond Thermoreflectivity and Thermotransmissivity of Polycrystalline and Single-Crystalline Gold-Films
,” Phys. Rev. B
, 43
(5
), pp. 4488
–4491
.10.1103/PhysRevB.43.448814.
Groeneveld
, R. H. M.
, Sprik
, R.
, and Lagendijk
, A.
, 1990
, “Ultrafast Relaxation of Electrons Probed by Surface-Plasmons at a Thin Silver Film
,” Phys. Rev. Lett.
, 64
(7
), pp. 784
–787
.10.1103/PhysRevLett.64.78415.
Voyiadjis
, G. Z.
, and Abed
, F. H.
, 2005
, “Effect of Dislocation Density Evolution on the Thermomechanical Response of Metals With Different Crystal Structures at Low and High Strain Rates and Temperatures
,” Arch. Mech.
, 57
(4
), pp. 299
–343
.16.
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.2004.02.00317.
Voyiadjis
, G. Z.
, and Abed
, F. H.
, 2006
, “Implicit Algorithm for Finite Deformation Hypoelastic-Viscoplasticity in FCC Metals
,” Int. J. Numer. Methods Eng.
, 67
(7
), pp. 933
–959
.10.1002/nme.165518.
Voyiadjis
, G. Z.
, and Abed
, F. H.
, 2007
, “Transient Localizations in Metals Using Microstructure-Based Yield Surfaces
,” Model. Simul. Mater. Sci. Eng.
, 15
(1
), pp. S83
–S95
.10.1088/0965-0393/15/1/S0819.
Abed
, F. H.
, and Voyiadjis
, G. Z.
, 2005
, “A Consistent Modified Zerilli–Armstrong Flow Stress Model for BCC and FCC Metals for Elevated Temperatures
,” Acta Mech.
, 175
(1–4
), pp. 1
–18
.10.1007/s00707-004-0203-120.
Abed
, F. H.
, and Voyiadjis
, G. Z.
, 2007
, “Thermodynamic Consistent Formulations of Viscoplastic Deformations in FCC Metals
,” ASCE J. Eng. Mech.
, 133
(1
), pp. 76
–86
.10.1061/(ASCE)0733-9399(2007)133:1(76)21.
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
.10.1016/j.jmps.2004.12.00322.
Gurtin
, M. E.
, 2008
, “A Theory of Grain Boundaries That Accounts Automatically for Grain Misorientation and Grain-Boundary Orientation
,” J. Mech. Phys. Solids
, 56
(2
), pp. 640
–662
.10.1016/j.jmps.2007.05.00223.
Gurtin
, M. E.
, 2003
, “On a Framework for Small-Deformation Viscoplasticity: Free Energy, Microforces, Strain Gradients
,” Int. J. Plast.
, 19
(1
), pp. 47
–90
.10.1016/S0749-6419(01)00018-324.
Sun
, S.
, Adams
, B. L.
, and King
, W. E.
, 2000
, “Observations of Lattice Curvature Near the Interface of a Deformed Aluminium Bicrystal
,” Philos. Mag. A
, 80
(1
), pp. 9
–25
.10.1080/0141861000821203825.
Soer
, W. A.
, and De Hosson
, J. T. M.
, 2005
, “Detection of Grain-Boundary Resistance to Slip Transfer Using nano-indentation
,” Mater. Lett.
, 59
(24–25
), pp. 3192
–3195
.10.1016/j.matlet.2005.03.07526.
Wang
, M. G.
, and Ngan
, A. H. W.
, 2004
, “Indentation Strain Burst Phenomenon Induced by Grain Boundaries in Niobium
,” J. Mater. Res.
, 19
(8
), pp. 2478
–2486
.10.1557/JMR.2004.031627.
Britton
, T. B.
, Randman
, D.
, and Wilkinson
, A. J.
, 2009
, “Nanoindentation Study of Slip Transfer Phenomenon at Grain Boundaries
,” J. Mater. Res.
, 24
(3
), pp. 607
–615
.10.1557/jmr.2009.008828.
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)00491-229.
Gurtin
, M. E.
, and Needleman
, A.
, 2005
, “Boundary Conditions in Small-Deformation, Single-Crystal Plasticity that Account for the Burgers Vector
,” J. Mech. Phys. Solids
, 53
(1
), pp. 1
–31
.10.1016/j.jmps.2004.06.00630.
Aifantis
, K. E.
, and Willis
, J. R.
, 2006
, “Scale Effects Induced by Strain-Gradient Plasticity and Interfacial Resistance in Periodic and Randomly Heterogeneous Media
,” Mech. Mater.
, 38
(8–10
), pp. 702
–716
.10.1016/j.mechmat.2005.06.01031.
Fredriksson
, P.
, and Gudmundson
, P.
, 2007
, “Competition Between Interface and Bulk Dominated Plastic Deformation in Strain Gradient Plasticity
,” Model. Simul. Mater. Sci. Eng.
, 15
(1
), pp. S61
–S69
.10.1088/0965-0393/15/1/S0632.
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.2008.09.01033.
Voyiadjis
, G. Z.
, and Faghihi
, D.
, 2011
, “Variable (Intrinsic) Material Length Scale for Face-Centred Cubic Metals Using NanoIndentation
,” Proc. Inst. Mech. Eng.
, Part N, 224
, pp. 123
–147
.10.1177/174034991141364734.
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.2011.07.00235.
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)36.
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.2010.07.00937.
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/s00161-009-0107-338.
Zaiser
, M.
, Glazov
, M.
, Lalli
, L. A.
, and Richmond
, O.
, 1999
, “On the Relations Between Strain and Strain-Rate Softening Phenomena in Some Metallic Materials: A Computational Study
,” Comput. Mater. Sci.
, 15
(1
), pp. 35
–49
.10.1016/S0927-0256(98)00131-139.
Hirth
, J. P.
, and Nix
, W. D.
, 1969
, “An Analysis of Thermodynamics of Dislocation Glide
,” Phys. Status Solidi
, 35
(1
), pp. 177
–188
.10.1002/pssb.1969035011640.
Bammann
, D. J.
, and Aifantis
, E. C.
, 1982
, “On a Proposal for a Continuum With Microstructure
,” Acta Mech.
, 45
(1–2
), pp. 91
–121
.10.1007/BF0129557341.
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)90005-542.
Kocks
, U. F.
, 2001
, “Realistic Constitutive Relations for Metal Plasticity
,” Mater. Sci. Eng. A
, 317
(1–2
), pp. 181
–187
.10.1016/S0921-5093(01)01174-143.
Zerilli
, F. J.
, and Armstrong
, R. W.
, 1987
, “Dislocation-Mechanics-Based Constitutive Relations for Material Dynamics Calculations
,” J. Appl. Phys.
, 61
(5
), pp. 1816
–1825
.10.1063/1.33802444.
Nemat-Nasser
, S.
, Isaacs
, J. B.
, and Liu
, M. Q.
, 1998
, “Microstructure of High-Strain, High-Strain-Rate Deformed Tantalum
,” Acta Mater.
, 46
(4
), pp. 1307
–1325
.10.1016/S1359-6454(97)00746-545.
Nemat-Nasser
, S.
, and Li
, Y. L.
, 1998
, “Flow Stress of FCC Polycrystals With Application to OFHC Cu
,” Acta Mater.
, 46
(2
), pp. 565
–577
.10.1016/S1359-6454(97)00230-946.
Orowan
, E.
, 1934
, “Mechanical Strength Properties and Real Structure of Crystals
,” Z. Kristallogr.
, 89
(3/4
), pp. 327
–343
.47.
Barlat
, F.
, Glazov
, M. V.
, Brem
, J. C.
, and Lege
, D. J.
, 2002
, “A Simple Model for Dislocation Behavior, Strain and Strain Rate Hardening Evolution in Deforming Aluminum Alloys
,” Int. J. Plast.
, 18
(7
), 919
–939
.10.1016/S0749-6419(01)00015-848.
Klepaczko
, J. R.
, 1988
, “A General Approach to Rate Sensitivity and Constitutive Modeling of FCC and BCC Metals
,” Impact Effects of Fast Transient Loadings
, Balkema, Leiden
, The Netherlands
, pp. 3
–10
.49.
Kubin
, L. P.
, and Estrin
, Y.
, 1990
, “Evolution of Dislocation Densities and the Critical Conditions for the Portevin-Lechatelier Effect
,” Acta Metall. Mater.
, 38
(5
), pp. 697
–708
.10.1016/0956-7151(90)90021-850.
Klepaczko
, J. R.
, 1988
, “A General Approach to Rate Sensitivity and Constitutive Modeling of FCC and BCC Metals,” Impact: Effects of Fast Transient Loadings, A. A.
Balkema
, Leiden
, The Netherlands
, p. 3
.51.
Taylor
, G. I.
, 1938
, “Plastic Strain in Metals
,” J. Instit. Metals
, 62
, pp. 307
–324
.52.
Abu Al-Rub
, R. K.
, and Faruk
, A. N. M.
, 2011
, “Coupled Interfacial Energy and Temperature Effects on Size-Dependent Yield Strength and Strain Hardening of Small Metallic Volumes
,” ASME J. Eng. Mater. Technol.
, 133
(1
), p. 011017
.10.1115/1.400265153.
Cahn
, J. W.
, and Hilliard
, J. E.
, 1959
, “Free Energy of a Nonuniform System. 3. Nucleation in a 2-Component Incompressible Fluid
,” J. Chem. Phys.
, 31
(3
), pp. 688
–699
.10.1063/1.173044754.
Cahn
, J. W.
, and Hilliard
, J. E.
, 1958
, “Free Energy of a Nonuniform System. 1. Interfacial Free Energy
,” J. Chem. Phys.
, 28
(2
), pp. 258
–267
.10.1063/1.174410255.
Meyers
, M. A.
, and Chawla
, K. K.
, 2009
, Mechanical Behavior of Materials
, 2nd ed., Cambridge University Press
, New York
.56.
Chung
, Y.-W.
, 2007
, Introduction to Materials Science and Engineering
, CRC Press
, Boca Raton, FL
.57.
Amiri
, M.
, and Khonsari
, M. M.
, 2010
, “Life Prediction of Metals Undergoing Fatigue Load Based on Temperature Evolution
,” Mater. Sci. Eng. A
, 527
(6
), pp. 1555
–1559
.10.1016/j.msea.2009.10.02558.
Amiri
, M.
, and Khonsari
, M. M.
, 2010
, “Rapid Determination of Fatigue Failure Based on Temperature Evolution: Fully Reversed Bending Load
,” Int. J. Fatigue
, 32
(2
), pp. 382
–389
.10.1016/j.ijfatigue.2009.07.01559.
Amiri
, M.
, and Khonsari
, M. M.
, 2011
, “On the Role of Entropy Generation in Processes Involving Fatigue
,” Entropy
, 14
(1
), pp. 24
–31
.10.3390/e1401002460.
Naderi
, M.
, Amiri
, M.
, and Khonsari
, M. M.
, 2010
, “On the Thermodynamic Entropy of Fatigue Fracture
,” Proc. R. Soc. London, Ser. A
, 466
(2114
), pp. 423
–438
.10.1098/rspa.2009.034861.
Kaganov
, M. I.
, Lifshitz
, I. M.
, and Tanatarov
, L. V.
, 1956
, “Relaxation Between Electrons and the Crystalline Lattice
,” Zh. Eksp. Teor. Fiz.
, 31
, pp. 232
–237
.62.
Qiu
, T. Q.
, and Tien
, C. L.
, “Short-Pulse Laser-Heating on Metals
,” Int. J. Heat Mass Transfer
, 35
(3
), pp. 719
–726
.10.1016/0017-9310(92)90131-B63.
Qiu
, T. Q.
, and Tien
, C. L.
, 1993
, “Heat-Transfer Mechanisms During Short-Pulse Laser-Heating of Metals
,” ASME J. Heat Transfer
, 115
(4
), pp. 835
–841
.10.1115/1.291137764.
Voyiadjis
, G. Z.
, and Faghihi
, D.
, 2011
, “Thermo-Mechanical Strain Gradient Plasticity With Energetic and Dissipative Length Scales
,” Int. J. Plast.
, 30–31
, pp. 218
–247
.10.1016/j.ijplas.2011.10.00765.
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)90112-066.
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-X67.
Naderi
, M.
, and Khonsari
, M. M.
, 2010
, “An Experimental Approach to Low-Cycle Fatigue Damage Based on Thermodynamic Entropy
,” Int. J. Solids Struct.
, 47
(6
), pp. 875
–880
.10.1016/j.ijsolstr.2009.12.00568.
Naderi
, M.
, and Khonsari
, M. M.
, 2010
, “A Thermodynamic Approach to Fatigue Damage Accumulation Under Variable Loading
,” Mater. Sci. Eng. A
, 527
(23
), pp. 6133
–6139
.10.1016/j.msea.2010.05.01869.
Rajagopal
, K. R.
, and Srinivasa
, A. R.
, 2004
, “On the Thermomechanics of Materials That Have Multiple Natural Configurations—Part I: Viscoelasticity and Classical Plasticity
,” Z. Angew. Math. Phys.
, 55
(5
), pp. 861
–893
.10.1007/s00033-004-4019-670.
Rajagopal
, K. R.
, and Srinivasa
, A. R.
, 2004
, “On the Thermomechanics of Materials That Have Multiple Natural Configurations—Part II: Twinning and Solid to Solid Phase Transformation
,” Z. Angew. Math. Phys.
, 55
(6
), pp. 1074
–1093
.10.1007/s00033-004-4020-071.
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.2008.12.00272.
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.2009.03.00773.
Rao
, I. J.
, and Rajagopal
, K. R.
, 2000
, “Phenomenological Modeling of Polymer Crystallization Using the Notion of Multiple Natural Configurations
,” Interfaces Free Boundaries
, 2
, pp. 73
–94
.10.4171/IFB/1374.
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.2005.03.00375.
Bardella
, L.
, 2007
, “Some Remarks on the Strain Gradient Crystal Plasticity Modelling, With Particular Reference to the Material Length Scales Involved
,” Int. J. Plast.
, 23
(2
), pp. 296
–322
.10.1016/j.ijplas.2006.05.00476.
Reddy
, B. D.
, Ebobisse
, F.
, and McBride
, A.
, 2008
, “Well-Posedness of a Model of Strain Gradient Plasticity for Plastically Irrotational Materials
,” Int. J. Plast.
, 24
(1
), pp. 55
–73
.10.1016/j.ijplas.2007.01.01377.
Shu
, J. Y.
, Fleck
, N. A.
, Van der Giessen
, E.
, and Needleman
, A.
, 2001
, “Boundary Layers in Constrained Plastic Flow: Comparison of Nonlocal and Discrete Dislocation Plasticity
,” J. Mech. Phys. Solids
, 49
(6
), pp. 1361
–1395
.10.1016/S0022-5096(00)00074-078.
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.2005.08.00379.
Bardella
, L.
, and Giacomini
, A.
, 2008
, “Influence of Material Parameters and Crystallography on the Size Effects Describable by Means of Strain Gradient Plasticity
,” J. Mech. Phys. Solids
, 56
(9
), pp. 2906
–2934
.10.1016/j.jmps.2008.04.00180.
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)00086-081.
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-X82.
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.2003.10.00783.
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.2004.12.01084.
Kysar
, J. W.
, Saito
, Y.
, Oztop
, M. S.
, Lee
, D.
, and Huh
, W. T.
, 2010
, “Experimental Lower Bounds on Geometrically Necessary Dislocation Density
,” Int. J. Plast.
, 26
(8
), pp. 1097
–1123
.10.1016/j.ijplas.2010.03.00985.
Gudmundson
, P.
, 2004
, “A Unified Treatment of Strain Gradient Plasticity
,” J. Mech. Phys. Solids
, 52
(6
), pp. 1379
–1406
.10.1016/j.jmps.2003.11.00286.
Voyiadjis
, G. Z.
, Almasri
, A. H.
, and Park
, T.
, 2010
, “Experimental Nanoindentation of BCC Metals
,” Mech. Res. Commun.
, 37
(3
), pp. 307
–314
.10.1016/j.mechrescom.2010.02.00187.
Voyiadjis
, G. Z.
, and Peters
, R.
, 2010
, “Size Effects in Nanoindentation: An Experimental and Analytical Study
,” Acta Mech.
, 211
(1–2
), pp. 131
–153
.10.1007/s00707-009-0222-z88.
Almasri
, A. H.
, and Voyiadjis
, G. Z.
, 2010
, “Nano-Indentation in FCC Metals: Experimental Study
,” Acta Mech.
, 209
(1–2
), pp. 1
–9
.10.1007/s00707-009-0151-x89.
Ohmura
, T.
, Minor
, A. M.
, Stach
, E. A.
, and Morris
, J. W.
, 2004
, “Dislocation-Grain Boundary Interactions in Martensitic Steel Observed Through In Situ Nanoindentation in a Transmission Electron Microscope
,” J. Mater. Res.
, 19
(12
), pp. 3626
–3632
.10.1557/JMR.2004.047490.
Voyiadjis
, G. Z.
, Faghihi
, D.
, and Zhang
, C.
, 2011
, “Analytical and Experimental Detemination of Rate, and Temperature Dependent Length Scales Using Nanoindentation Experiments
,” ASCE J. Nanomech. Micromech.
, 1
(1
), pp. 24
–40
.10.1061/(ASCE)NM.2153-5477.000002791.
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)90534-692.
Bisson
, J. F.
, Yagi
, H.
, Yanagitani
, T.
, Kaminskii
, A.
, Barabanenkov
, Y. N.
, and Ueda
, K. I.
, 2007
, “Influence of the Grain Boundaries on the Heat Transfer in Laser Ceramics
,” Opt. Rev.
, 14
(1
), pp. 1
–13
.10.1007/s10043-007-0001-993.
Cahill
, D. G.
, Ford
, W. K.
, Goodson
, K. E.
, Mahan
, G. D.
, Majumdar
, A.
, Maris
, H. J.
, Merlin
, R.
, and Phillpot
, S. R.
, 2003
, “Nanoscale Thermal Transport
,” J. Appl. Phys.
, 93
(2
), pp. 793
–818
.10.1063/1.152430594.
Swartz
, E. T.
, and Pohl
, R. O.
, 1989
, “Thermal Boundary Resistance
,” Rev. Mod. Phys.
, 61
(3
), pp. 605
–668
.10.1103/RevModPhys.61.60595.
Voyiadjis
, G. Z.
, and Deliktas
, B.
, 2009
, “Theoretical and Experimental Characterization for the Inelastic Behavior of the Micro-/Nanostructured Thin Films Using Strain Gradient Plasticity With Interface Energy
,” ASME J. Eng. Mater. Technol.
, 131
(4
), p. 041202
.10.1115/1.318377496.
Shampine
, L. F.
, 2003
, “Singular Boundary Value Problems for ODEs
,” Appl. Math. Comput.
, 138
(1
), pp. 99
–112
.10.1016/S0096-3003(02)00111-X97.
Kierzenka
, J.
, and Shampine
, L. F.
, 2001
, “A BVP Solver Based on Residual Control and the MATLAB PSE
,” ACM Trans.
, 27
(3
), pp. 299
–316
.10.1145/502800.50280198.
Shampine
, L. F.
, Reichelt
, M. W.
, and Kierzenka
, J.
, 2000
, “Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB With bvp4c
,” Tutorial Notes
, available at http://www.mathworks.com/bvp_tutorial99.
Zhang
, J.
, and Zhao
, J. J.
, 2001
, “High Accuracy Stable Numerical Solution of 1D Microscale Heat Transport Equation
,” Commun. Numer. Methods Eng.
, 17
(11
), pp. 821
–832
.10.1002/cnm.453100.
Elsayed-Ali
, H. E.
, Norris
, T. B.
, Pessot
, M. A.
, and Mourou
, G. A.
, 1987
, “Time-Resolved Observation of Electron-Phonon Relaxation in Copper
,” Phys. Rev. Lett.
, 58
(12
), pp. 1212
–1215
.10.1103/PhysRevLett.58.1212Copyright © 2013 by ASME
You do not currently have access to this content.
