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Research Papers

Overview of Enhanced Continuum Theories for Thermal and Mechanical Responses of the Microsystems in the Fast-Transient Process

[+] Author and Article Information
George Z. Voyiadjis

Boyd Professor
Computational Solid Mechanics Laboratory,
Department of Civil and
Environmental Engineering,
Louisiana State University,
Baton Rouge, LA 70803
e-mail: voyiadjis@eng.lsu.edu

Danial Faghihi

Computational Solid Mechanics Laboratory,
Department of Civil and
Environmental Engineering,
Louisiana State University,
Baton Rouge, LA 70803

Referred to quasi-conservative by Ziegler and Wehrli [53].

While in the context of strain gradient theory, the stored energy can be due to both homogeneous and nonhomogenous plastic deformation—in spirit of the conventional plasticity—the term “gradient independent stored energy of cold work” refers to the stored energy connected with the homogenous plastic deformation in this paper.

According to Gurtin [31] “the plastic spin vanishes identically when the free energy is independent of curl of plastic strain, but not generally otherwise”. He noted that sufficiently far from boundaries solutions should not be affected by plastic spin, but close to microscopically hard boundaries, boundary layers characterized by a large Burgers vector and large plastic spin should form.

Using direct notation, Eq. (6) reads as α=(curlɛp)T. It should be noted that the transpose of α is referred to as Burgers tensor, G, in the works of Gurtin (e.g., [31]), therefore: G=(curlɛp).

The examples of such micro-free and micro-clamped boundary conditions can be found in thin films with unpassivated and passivated surfaces (e.g., [167]).

In this formulation the term “elastic energy” is used to imply the recoverable energy associated with the stretching of the atomic bonds in a crystal lattice. The term “defect energy” is used to denote the energy stored and dissipated in the lattice due to the presence of defects such as dislocations (both SSDs and GNDs associated with plastic strain and gradient of plastic train) and entanglements.

It should be noted that TY1 is assumed to be the melting temperature of the specific material by some other researchers in order not to introduce an additional material parameter. However, a general constant is included here as a normalizing parameter that needs to be calibrated against careful experimental data.

Gurtin and Reddy [70] showed that “the classical isotropic hardening rule, which is dissipative in nature, may equally well be characterized via a defect energy and, what is striking, this energetically based hardening rule mimics dissipative behavior by describing loading processes that are irreversible.” As it will be shown later in this paper, such manipulation is taken into consideration here in order to derive the plastic strain gradient independent stored energy of cold work with no additional material parameters that required to be calibrated against experimental data [156].

Specifically, densities of edge and screw GNDs according to Gurtin (e.g., [30, 31]). It should be noted that the kinematic hardening in the current theory does not account for certain interval of the plastic strain during the loading reversal where dislocation density remains constant (Hasegawa et al., 1975). This is because at the onset of reverse loading, the dislocation structure is first annihilated and then rebuilt in the slip systems that correspond to the reverse direction [154].

Such correlation between GNDs will quickly decrease to zero within a few dislocation distances [173].

Starting from Eq. (17) and considering virtual variations of the temperature and entropy states—which obeys the energy equation—the relation between second-order variation of Ψ and e can be derived such as 2Ψ/T2(δT)2=-2e/s2(δs)2. From this relation, one can conclude that at the state where e is a convex function of entropy, Ψ is a concave function of temperature 2e/s2>0=>2Ψ/T2<0 [175]. Moreover, according to [176] one can assume that entropy is a monotonically increasing function of temperature, thus: s/T=-2Ψ/T2.

Conventionally, dislocations impinging transversely on a slip plane are termed forest dislocations and are thought to be responsible for the second stage of hardening [177,178].

e.g., [179] and [180] for an inclusive treatment of the configurational (material) forces.

Assuming the interaction between only two GNDs (or pile-ups), this stress acts on them in the opposite direction of slip.

As it is discussed by Bardella [160], ensuring m1>0 and m2>0 not only preserve the convexity of Daccp and Daccg, respectively, but also abolishes the requirement to implement any yield criterion in the moving elastic–plastic boundaries (i.e., imposing any higher-order boundary condition at the internal surfaces between elastic and plastic domains).

For further details regarding the effect of the interaction coefficient value, see Dahlberg and Faleskog [187].

According to the second paragraph after Eq. (39).

Both Kuroda and Tvergaard [195] and Gurtin and Ohno [85] discussed the crystal plasticity case which can also be applied to the continuum plasticity.

Gurtin [204]—Eq. (3.14), p. 647.

Where ΨI is convex with respect to ɛijpI.

Corresponding to the condition that dislocations are nucleated within the bulk and did not reach the interface yet.

Since it is not possible to program a priori the unloading point exactly at the onset of the strain burst, hence there are no experimental observations to prove the recoverable nature of this energy. However, this can be interpreted as the vanishing of the repulsive stress of pile-up by removing the applied load.

Obviously, the calibrated parameters will be more accurate by means of a 3D finite element modeling of the indentation using the gradient theory proposed by Faghihi and Voyiadjis [156]. However, such a calibration along with implementation of the theory to 3D finite element model is far too complex for practical purposes.

Detailed set of indentation tests at various temperatures and strain rates are required to calibrate the full model parameters which does not currently exist in the literature. The experiments at elevated temperatures are more challenging since specialized equipment and sample preparation is necessary and some indenter parts are not vacuum-compatible. One may use the hardening-softening observed in the experiments (e.g., Voyiadjis et al. [247] and Faghihi and Voyiadjis [206] and the references herein) without accounting for the exact distance between the indenter and the grain boundary to determine the average rate and temperature dependent grain boundary parameters of the material.

The unloading (i.e., reverse straining) is only applied for the stress–strain curves and the other results are extracted at the end of loading.

This agrees with the experimental observations of Xiang and Vlassak [20] for the passivated thin film.

The thermal softening will be more pronounced in the absence of the energetic length scale. However, a large value of the backstress is considered here in order to investigate the capability of the model to address the effect of temperature rise on overcoming the kinematic hardening mechanisms.

r=1 is considered for the results presented in this section.

In order to emphasize the variation, for the results of the stress-strain curves extracted by assuming Y=YI=20(MPa) and 0.03% applied stress.

It should be noted that since the experiments are conducted under the low strain rate, the isothermal condition is preserved for the obtained results. For simplicity, the effect of the gradient independent stored energy of cold work is not incorporated and the parameters TY1 and TY2 are assumed to be the Al melting temperature.

The average grain sizes of the films with 1.00, 1.90, and 4.20μm are 1.5±0.05, 1.51±0.04, and 1.5±0.05μm, respectively.

A well-adhered passivating layer prevents dislocation from exiting the film and results in significant plastic strain gradient. The plastic flow constrain due to the presence of the passivation layer, cannot be described neither by the classical plasticity nor the first-order gradient theories.

1Corresponding author.

2Present address: Institute for Computational Engineering and Science, The University of Texas at Austin, Austin, TX 78712.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received July 17, 2014; final manuscript received July 28, 2014; published online August 22, 2014. Assoc. Editor: Mohammed Zikry.

J. Eng. Mater. Technol 136(4), 041003 (Aug 22, 2014) (36 pages) Paper No: MATS-14-1145; doi: 10.1115/1.4028121 History: Received July 17, 2014; Revised July 28, 2014

The recently growing demand for production and applications of microscale devices and systems has motivated research on the behavior of small volume materials. The computational models have become one of great interests in order to advance the manufacturing of microdevices and to reduce the time to insert new product in applications. Among the various numerical and computational techniques, still the approaches in the context of continuum theories are more preferable due to their minimum computational cost to simulation on realistic time and material structures. This paper reviews the methods to address the thermal and mechanical responses of microsystems. The focus is on the recent developments on the enhanced continuum theories to address the phenomena such as size and boundary effects as well as microscale heat transfer. The thermodynamic consistency of the theories is discussed and microstructural mechanisms are taken into account as physical justification of the framework. The presented constitutive model is calibrated using an extensive set of microscale experimental measurements of thin metal films over a wide range of size and temperature of the samples. An energy based approach is presented to extract the first estimate of the interface model parameters from results of nanoindentation test.

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References

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. [CrossRef]
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. [CrossRef]
Vlassak, J. J., Xiang, Y., and Chen, X., 2005, “Plane-Strain Bulge Test for Thin Films,” J. Mater. Res., 20(9), pp. 2360–2370. [CrossRef]
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. [CrossRef]
Ma, Q., and Clarke, D. R., 1995, “Size-Dependent Hardness of Silver Single-Crystals,” J. Mater. Res., 10(4), pp. 853–863. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Voyiadjis, G. Z., and Peters, R., 2010, “Size Effects in Nanoindentation: An Experimental and Analytical Study,” Acta Mech., 211(1–2), pp. 131–153. [CrossRef]
Voyiadjis, G. Z., Faghihi, D., and Zhang, C., 2011, “Analytical and Experimental Determination of Rate- and Temperature-Dependent Length Scales Using Nanoindentation Experiments,” J. Nanomech. Micromech., 1(1), pp. 24–40. [CrossRef]
Sze, S. M., and Ng, K. K., 2007, Physics of Semiconductor Devices, 3rd ed., Wiley-Interscience, Hoboken, NJ.
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. [CrossRef] [PubMed]
Hall, E. O., 1951, “The Deformation and Ageing of Mild Steel: III Discussion and Results,” Proc. Phys. Soc. B, 64, pp. 747–753. [CrossRef]
Petch, N. J., 1953, “The Cleavage Strength of Polycrystals,” J. Iron Steel Inst., 174, pp. 25–28.
Arsenlis, A., and Parks, D. M., 1999, “Crystallographic Aspects of Geometrically-Necessary and Statistically-Stored Dislocation Density,” Acta Mater., 47(5), pp. 1597–1611. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Aifantis, E. C., 1999, “Gradient Deformation Models at Nano, Micro, and Macro Scales,” ASME J. Eng. Mater. Technol., 121(2), pp. 189–202. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Xiang, Y., and Vlassak, J. J., 2005, “Bauschinger Effect in Thin Metal Films,” Scr. Mater., 53(2), pp. 177–182. [CrossRef]
Xiang, Y., and Vlassak, J. J., 2006, “Bauschinger and Size Effects in Thin-Film Plasticity,” Acta Mater., 54(20), pp. 5449–5460. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Muhlhaus, H. B., and Aifantis, E. C., 1991, “A Variational Principle for Gradient Plasticity,” Int. J. Solids Struct., 28(7), pp. 845–857. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Elkhodary, K. I., and Zikry, M. A., 2011, “A Fracture Criterion for Finitely Deforming Crystalline Solids—The Dynamic Fracture of Single Crystals,” J. Mech. Phys. Solids, 59(10), pp. 2007–2022. [CrossRef]
Labarbera, D., and Zikry, M. A., 2013, “Microstructural Behavior of Energetic Crystalline Aggregates,” MRS Online Proc. Libr., 1526, p. mrsf12-1526-tt06-07. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Schiotz, J., Tolla, F. D. D., and Jacobsen, K. W., 1998, “Softening of Nanocrystalline Metals at Very Small Grains,” Nature, 391(6667), pp. 561–563. [CrossRef]
Lidorikis, E., Bachlechner, M. E., Kalia, R. K., Nakano, A., Vashishta, P., and Voyiadjis, G. Z., 2001, “Coupling Length Scales for Multiscale Atomistics-Continuum Simulations: Atomistically Induced Stress Distributions in Si/Si3N4 Nanopixels,” Phys. Rev. Lett., 8708(8), p. 086104. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Niordson, C. F., and Hutchinson, J. W., 2003, “Non-Uniform Plastic Deformation of Micron Scale Objects,” Int. J. Numer. Meth. Eng., 56(7), pp. 961–975. [CrossRef]
Aifantis, E. C., 1984, “On the Microstructural Origin of Certain Inelastic Models,” ASME J. Eng. Mater. Technol., 106(4), pp. 326–330. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Han, C. S., Gao, H. J., Huang, Y. G., and Nix, W. D., 2005, “Mechanism-Based Strain Gradient Crystal Plasticity—I. Theory,” J. Mech. Phys. Solids, 53(5), pp. 1188–1203. [CrossRef]
Han, C. S., Gao, H. J., Huang, Y. G., and Nix, W. D., 2005, “Mechanism-Based Strain Gradient Crystal Plasticity—II. Analysis,” J. Mech. Phys. Solids, 53(5), pp. 1204–1222. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Evans, A. G., and Hutchinson, J. W., 2009, “A Critical Assessment of Theories of Strain Gradient Plasticity,” Acta Mater., 57(5), pp. 1675–1688. [CrossRef]
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. [CrossRef]
Fleck, N. A., and Hutchinson, J. W., 2001, “A Reformulation of Strain Gradient Plasticity,” J. Mech. Phys. Solids, 49(10), pp. 2245–2271. [CrossRef]
Hirth, J. P., 1972, “Influence of Grain-Boundaries on Mechanical Properties,” Metall. Trans., 3(12), pp. 3047–3067. [CrossRef]
Polcarova, M., Gemperlova, J., Bradler, J., Jacques, A., George, A., and Priester, L., 1998, “In-Situ Observation of Plastic Deformation of Fe-Si Bicrystals by White-Beam Synchrotron Radiation Topography,” Philos. Mag. A, 78(1), pp. 105–130. [CrossRef]
Shen, Z., Wagoner, R. H., and Clark, W. A. T., 1988, “Dislocation and Grain-Boundary Interactions in Metals,” Acta Metall., 36(12), pp. 3231–3242. [CrossRef]
Clark, W. A. T., Wagoner, R. H., Shen, Z. Y., Lee, T. C., Robertson, I. M., and Birnbaum, H. K., 1992, “On the Criteria for Slip Transmission Across Interfaces in Polycrystals,” Scr. Metall. Mater., 26(2), pp. 203–206. [CrossRef]
Dehosson, J. T. M., and Pestman, B. P., 1993, “Interactions Between Lattice Dislocations and Grain-Boundaries in L12 Ordered Compounds Investigated by In Situ Transmission Electron Microscopy and Computer Modeling Experiments,” Mater. Sci. Eng., A, 164(1–2), pp. 415–420. [CrossRef]
Pestman, B. J., and Dehosson, J. T. M., 1992, “Interactions Between Lattice Dislocations and Grain-Boundaries in Ni3Al Investigated by Means of In Situ TEM and Computer Modeling Experiments,” Acta Metall. Mater., 40(10), pp. 2511–2521. [CrossRef]
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. [CrossRef]
Ziegler, H., 1963, “Some Extremum Principles in Irreversible Thermodynamics With Application to Continuum Mechanics,” Progress in Solid Mechanics, Vol. 4, I. N. Sneddon and R. Hill, eds., North-Holland, Amsterdam, The Netherlands, pp. 93–193.
Ziegler, H., 1958, “An Attempt to Generalize Onsager's Principle, and Its Significance for Rheological Problems,” Z. Angew. Math. Phys. (ZAMP), 9(5), pp. 748–763. [CrossRef]
Collins, I. F., and Houlsby, G. T., 1997, “Application of Thermomechanical Principles to the Modelling of Geotechnical Materials,” Proc. R. Soc. London, Ser. A, 453(1964), pp. 1975–2001. [CrossRef]
Fremond, M., and Nedjar, B., 1996, “Damage, Gradient of Damage and Principle of Virtual Power,” Int. J. Solids Struct., 33(8), pp. 1083–1103. [CrossRef]
Nedjar, B., 2001, “Elastoplastic-Damage Modelling Including the Gradient of Damage: Formulation and Computational Aspects,” Int. J. Solids Struct., 38(30–31), pp. 5421–5451. [CrossRef]
Voyiadjis, G. Z., and Deliktas, B., 2000, “Multi-Scale Analysis of Multiple Damage Mechanisms Coupled With Inelastic Behavior of Composite Materials,” Mech. Res. Commun., 27(3), pp. 295–300. [CrossRef]
Voyiadjis, G. Z., Taqieddin, Z. N., and Kattan, P. I., 2008, “Anisotropic Damage-Plasticity Model for Concrete,” Int. J. Plast., 24(10), pp. 1946–1965. [CrossRef]
Beheshti, A., and Khonsari, M., 2010, “A Thermodynamic Approach for Prediction of Wear Coefficient Under Unlubricated Sliding Condition,” Tribol. Lett., 38(3), pp. 347–354. [CrossRef]
Lodygowski, A., Voyiadjis, G. Z., Deliktas, B., and Palazotto, A., 2011, “Non-Local and Numerical Formulations for Dry Sliding Friction and Wear at High Velocities,” Int. J. Plast., 27(7), pp. 1004–1024. [CrossRef]
Aghdam, A. B., Beheshti, A., and Khonsari, M. M., 2012, “On the Fretting Crack Nucleation With Provision for Size Effect,” Tribol. Int., 47, pp. 32–43. [CrossRef]
Darabi, M. K., Abu, Al.-R. R. K., Masad, E. A., and Little, D. N., 2012, “A Thermodynamic Framework for Constitutive Modeling of Time- and Rate-Dependent Materials. Part II: Numerical Aspects and Application to Asphalt Concrete,” Int. J. Plast., 35, pp. 67–99. [CrossRef]
Darabi, M. K., Abu, Al.-R. R. K., Masad, E. A., and Little, D. N., 2012, “Thermodynamic-Based Model for Coupling Temperature-Dependent Viscoelastic, Viscoplastic, and Viscodamage Constitutive Behavior of Asphalt Mixtures,” Int. J. Numer. Anal. Methods Geomech.,”36(7), pp. 817–854. [CrossRef]
Abu, Al.-R. R. K., and Darabi, M. K., 2012, “A Thermodynamic Framework for Constitutive Modeling of Time- and Rate-Dependent Materials. Part I: Theory,” Int. J. Plast., 34, pp. 61–92. [CrossRef]
Voyiadjis, G. Z., Shojaei, A., and Li, G., 2011, “A Thermodynamic Consistent Damage and Healing Model for Self Healing Materials,” Int. J. Plast., 27(7), pp. 1025–1044. [CrossRef]
Voyiadjis, G. Z., Shojaei, A., Li, G. Q., and Kattan, P. I., 2012, “A Theory of Anisotropic Healing and Damage Mechanics of Materials,” Proc. R. Soc. A, 468(2137), pp. 163–183. [CrossRef]
Voyiadjis, G. Z., Shojaei, A., and Li, G., 2012, “A Generalized Coupled Viscoplastic–Viscodamage–Viscohealing Theory for Glassy Polymers,” Int. J. Plast., 28(1), pp. 21–45. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Reddy, B. D., 2012, “The Role of Dissipation and Defect Energy in Variational Formulations of Problems in Strain-Gradient Plasticity. Part 1: Polycrystalline Plasticity,” Continuum Mech. Thermodyn., 23(6), pp. 527–549. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Gurtin, M. E., 2003, “On a Framework for Small-Deformation Viscoplasticity: Free Energy, Microforces, Strain Gradients,” Int. J. Plast., 19(1), pp. 47–90. [CrossRef]
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. [CrossRef]
Gurtin, M. E., 2006, “The Burgers Vector and the Flow of Screw and Edge Dislocations in Finite-Deformation Single-Crystal Plasticity,” J. Mech. Phys. Solids, 54(9), pp. 1882–1898. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Gurtin, M. E., and Anand, L., 2007, “A Gradient Theory for Single-Crystal Plasticity,” Modell. Simul. Mater. Sci. Eng., 15(1), pp. S263–S270. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Gudmundson, P., 2004, “A Unified Treatment of Strain Gradient Plasticity,” J. Mech. Phys. Solids, 52(6), pp. 1379–1406. [CrossRef]
Hutchinson, J. W., 2012, “Generalizing J2 Flow Theory: Fundamental Issues in Strain Gradient Plasticity,” Acta Mech. Sin., 28(4), pp. 1078–1086. [CrossRef]
Qiu, T. Q., and Tien, C. L., 1992, “Short-Pulse Laser-Heating on Metals,” Int. J. Heat Mass Transfer, 35(3), pp. 719–726. [CrossRef]
Evers, L. P., Brekelmans, W. A. M., and Geers, M. G. D., 2004, “Non-Local Crystal Plasticity Model With Intrinsic SSD and GND Effects,” J. Mech. Phys. Solids, 52(10), pp. 2379–2401. [CrossRef]
deBorst, R., and Pamin, J., 1996, “Some Novel Developments in Finite Element Procedures for Gradient-Dependent Plasticity,” Int. J. Numer. Methods Eng., 39(14), pp. 2477–2505. [CrossRef]
Papanastasiou, P. C., and Vardoulakis, I. G., 1992, “Numerical Treatment of Progressive Localization in Relation to Borehole Stability,” Int. J. Numer. Anal. Methods Geomech., 16(6), pp. 389–424. [CrossRef]
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. [CrossRef]
Voyiadjis, G. Z., and Deliktas, B., 2009, “Formulation of Strain Gradient Plasticity With Interface Energy in a Consistent Thermodynamic Framework,” Int. J. Plast., 25(10), pp. 1997–2024. [CrossRef]
Polizzotto, C., 2009, “A Link Between the Residual-Based Gradient Plasticity Theory and the Analogous Theories Based on the Virtual Work Principle,” Int. J. Plast., 25(11), pp. 2169–2180. [CrossRef]
Polizzotto, C., 2009, “A Nonlocal Strain Gradient Plasticity Theory for Finite Deformations,” Int. J. Plast., 25(7), pp. 1280–1300. [CrossRef]
Polizzotto, C., 2010, “Shakedown Analysis for a Class of Strengthening Materials Within the Framework of Gradient Plasticity,” Int. J. Plast., 26(7), pp. 1050–1069. [CrossRef]
Polizzotto, C., 2011, “A Unified Residual-Based Thermodynamic Framework for Strain Gradient Theories of Plasticity,” Int. J. Plast., 27(3), pp. 388–413. [CrossRef]
Mikkelsen, L. P., 1997, “Post-Necking Behaviour Modelled by a Gradient Dependent Plasticity Theory,” Int. J. Solids Struct., 34(35–36), pp. 4531–4546. [CrossRef]
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. [CrossRef]
Voyiadjis, G. Z., Abu Al-Rub, R. K., and Palazotto, A. N., 2006, “On the Small and Finite Deformation Thermo-Elasto-Viscoplasticity Theory for Strain Localization Problems: Algorithmic and Computational Aspects,” Eur. J. Comput. Mech., 15(7–8), pp. 945–987. [CrossRef]
Voyiadjis, G. Z., Abu Al-Rub, R. K., and Palazotto, A. N., 2004, “Thermodynamic Framework for Coupling of Non-Local Viscoplasticity and Non-Local Anisotropic Viscodamage for Dynamic Localization Problems Using Gradient Theory,” Int. J. Plast., 20(6), pp. 981–1038. [CrossRef]
Abu Al-Rub, R. K., Darabi, M. K., and Masad, E. A., 2010, “A Straightforward Numerical Technique for Finite Element Implementation of Nonlocal Gradient-Dependent Continuum Damage Mechanics Theories,” Int. J. Theor. Appl. Multiscale Mech., 1(4), pp. 352–385. [CrossRef]
Fredriksson, P., and Gudmundson, P., 2005, “Size-Dependent Yield Strength of Thin Films,” Int. J. Plast., 21(9), pp. 1834–1854. [CrossRef]
Fredriksson, P., and Gudmundson, P., 2007, “Competition Between Interface and Bulk Dominated Plastic Deformation in Strain Gradient Plasticity,” Modell. Simul. Mater. Sci. Eng., 15(1), pp. S61–S69. [CrossRef]
Fredriksson, P., Gudmundson, P., and Mikkelsen, L. P., 2009, “Finite Element Implementation and Numerical Issues of Strain Gradient Plasticity With Application to Metal Matrix Composites,” Int. J. Solids Struct., 46(22–23), pp. 3977–3987. [CrossRef]
Fredriksson, P., and Larsson, P. L., 2008, “Wedge Indentation of Thin Films Modelled by Strain Gradient Plasticity,” Int. J. Solids Struct., 45(21), pp. 5556–5566. [CrossRef]
Niordson, C. F., and Redanz, P., 2004, “Size-Effects in Plane Strain Sheet-Necking,” J. Mech. Phys. Solids, 52(11), pp. 2431–2454. [CrossRef]
Niordson, C. F., and Tvergaard, V., 2005, “Instabilities in Power Law Gradient Hardening Materials,” Int. J. Solids Struct., 42(9–10), pp. 2559–2573. [CrossRef]
Niordson, C. F., 2008, “On Higher-Order Boundary Conditions at Elastic-Plastic Boundaries in Strain-Gradient Plasticity,” Philos. Mag., 88(30–32), pp. 3731–3745. [CrossRef]
Legarth, B. N., and Niordson, C. F., 2010, “Debonding Failure and Size Effects in Micro-Reinforced Composites,” Int. J. Plast., 26(1), pp. 149–165. [CrossRef]
Azizi, R., Niordson, C. F., and Legarth, B. N., 2011, “Size-Effects on Yield Surfaces for Micro Reinforced Composites,” Int. J. Plast., 27(11), pp. 1817–1832. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef] [PubMed]
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. [CrossRef]
Kaganov, M. I., Lifshitz, I. M., and Tanatarov, L. V., 1956, “Relaxation Between Electrons and the Crystalline Lattice,” J Exp. Theoretical Physics (Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki), 31, pp. 232–237.
Voyiadjis, G. Z., Faghihi, D., and Zhang, Y., 2014, “A Theory for Grain Boundaries With Strain-Gradient Plasticity,” Int. J. Solids Struct., 51(10), pp. 1872–1889. [CrossRef]
Gan, M., and Tomar, V., 2010, “Role of Length Scale and Temperature in Indentation Induced Creep Behavior of Polymer Derived Si–C–O Ceramics,” Mater. Sci. Eng., A, 527(29–30), pp. 7615–7623. [CrossRef]
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. [CrossRef]
Voyiadjis, G. Z., and Faghihi, D., 2010, “Variable (Intrinsic) Material Length Scale for Face-Centred Cubic Metals Using Nano-Indentation,” Proc. Inst. Mech. Eng., 224(3), pp. 123–147. [CrossRef]
Faghihi, D., Voyiadjis, G. Z., and Park, T., 2013, “Coupled Thermomechanical Modeling of Small Volume FCC Metals,” ASME J. Eng. Mater. Technol., 135(2), p. 021003. [CrossRef]
Faghihi, D., 2012, “Continuum and Crystal Strain Gradient Plasticity With Energetic and Dissipative Length Scales,” Doctoral dissertation, Louisiana State University, Baton Rouge, LA.
NSF Blue Ribbon Panel on Simulation-Based Engineering Science, 2006, “Simulation-Based Engineering Science: Revolutionizing Engineering Science Through Simulation,” National Science Foundation, Arlington, VA, available at: http://www.nsf.gov/pubs/reports/sbes_final_report.pdf
Owolabi, G. M., Odeshi, A. G., Singh, M. N. K., and Bassim, M. N., 2007, “Dynamic Shear Band Formation in Aluminum 6061-T6 and Aluminum 6061-T6/Al2O3 Composites,” Mater. Sci. Eng., A, 457(1–2), pp. 114–119. [CrossRef]
Odeshi, A. G., Bassim, M. N., Al-Ameeri, S., and Li, Q., 2005, “Dynamic Shear Band Propagation and Failure in AISI 4340 Steel,” J. Mater. Process. Technol., 169(2), pp. 150–155. [CrossRef]
Mason, J. J., Rosakis, A. J., and Ravichandran, G., 1994, “Full-Field Measurements of the Dynamic Deformation Field Around a Growing Adiabatic Shear-Band at the Tip of a Dynamically Loaded Crack or Notch,” J. Mech. Phys. Solids, 42(11), pp. 1679–1697. [CrossRef]
Mason, C., and Worswick, M. J., 2001, “Adiabatic Shear in Annealed and Shock-Hardened Iron and in Quenched and Tempered 4340 Steel,” Int. J. Fract., 111(1), pp. 29–51. [CrossRef]
Kalkman, A. J., Verbruggen, A. H., and Janssen, G. C. A. M., 2003, “High-Temperature Bulge-Test Setup for Mechanical Testing of Free-Standing Thin Films,” Rev. Sci. Instrum., 74(3), pp. 1383–1385. [CrossRef]
Chen, D., Sixta, M. E., Zhang, X. F., De Jonghe, L. C., and Ritchie, R. O., 2000, “Role of the Grain-Boundary Phase on the Elevated-Temperature Strength, Toughness, Fatigue and Creep Resistance of Silicon Carbide Sintered With Al, B and C,” Acta Mater., 48(18–19), pp. 4599–4608. [CrossRef]
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. [CrossRef]
Lemaitre, J., 1985, “Coupled Elasto-Plasticity and Damage Constitutive-Equations,” Comput. Meth. Appl. Mech. Eng., 51(1–3), pp. 31–49. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Naderi, M., Amiri, M., and Khonsari, M. M., 2009, “On the Thermodynamic Entropy of Fatigue Fracture,” Proc. R. Soc. London, Ser. A, 466(2114), pp. 423–438. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Zehnder, A. T., Babinsky, E., and Palmer, T., 1998, “Hybrid Method for Determining the Fraction of Plastic Work Converted to Heat,” Exp. Mech., 38(4), pp. 295–302. [CrossRef]
Jovic, C., Wagner, D., Herve, P., Gary, G., and Lazzarotto, L., 2006, “Mechanical Behaviour and Temperature Measurement During Dynamic Deformation on Split Hopkinson Bar of 304L Stainless Steel and 5754 Aluminium Alloy,” J. Phys. IV, 134, pp. 1279–1285. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Miller, R. E., and Tadmor, E. B., 2002, “The Quasicontinuum Method: Overview, Applications and Current Directions,” J. Comput.-Aided Mater. Des., 9(3), pp. 203–239. [CrossRef]
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. [CrossRef]
Zehnder, A. T., 1991, “A Model for the Heating Due to Plastic Work,” Mech. Res. Commun., 18(1), pp. 23–28. [CrossRef]
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. [CrossRef]
Benzerga, A. A., Brechet, Y., Needleman, A., and Van der Giessen, E., 2005, “The Stored Energy of Cold Work: Predictions From Discrete Dislocation Plasticity,” Acta Mater.53(18), pp. 4765–4779. [CrossRef]
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. [CrossRef]
Faghihi, D., and Voyiadjis, G. Z., 2012, “Thermal and Mechanical Responses of BCC Metals to the Fast-Transient Process in Small Volumes,” J. Nanomech. Micromech.2(3), pp. 29–41. [CrossRef]
Faghihi, D., and Voyiadjis, G. Z., 2013, “A Thermodynamic Consistent Model for Coupled Strain-Gradient Plasticity With Temperature,” ASME J. Eng. Mater. Technol., 136(1), p.011002. [CrossRef]
Voyiadjis, G. Z., and Faghihi, D., 2012, “Microstructure to Macro-Scale Using Gradient Plasticity With Temperature and Rate Dependent Length Scale,” Procedia IUTAM, 3, pp. 205–227. [CrossRef]
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. [CrossRef]
Voyiadjis, G. Z., and Faghihi, D., 2012, “Gradient Plasticity for Thermo-Mechanical Processes in Metals With Length and Time Scales,” Philos. Mag., 93(9), pp. 1013–1053. [CrossRef]
Bardella, L., 2010, “Size Effects in Phenomenological Strain Gradient Plasticity Constitutively Involving the Plastic Spin,” Int. J. Eng. Sci., 48(5), pp. 550–568. [CrossRef]
Nye, J. F., 1953, “Some Geometrical Relations in Dislocated Crystals,” Acta Metall., 1(2), pp. 153–162. [CrossRef]
Fleck, N. A., and Hutchinson, J. W., 1997, “Strain Gradient Plasticity,” Adv. Appl. Mech., 33, pp. 295–361. [CrossRef]
Bassani, J. L., 2001, “Incompatibility and a Simple Gradient Theory of Plasticity,” J. Mech. Phys. Solids, 49(9), pp. 1983–1996. [CrossRef]
Forest, S., and Amestoy, M., 2008, “Hypertemperature in Thermoelastic Solids,” Comptes Rendus Mécanique., 336(4), pp. 347–353. [CrossRef]
Gurtin, M. E., 1996, “Generalized Ginzburg–Landau and Cahn–Hilliard Equations Based on a Microforce Balance,” Physica D, 92(3–4), pp. 178–192. [CrossRef]
Voyiadjis, G. Z., and Faghihi, D., 2012, “The Effect of Temperature on Interfacial Gradient Plasticity in Metallic Thin Films,” Advanced Materials Modelling for Structures, Vol. 19, Springer-Verlag, Berlin, pp. 337–349. [CrossRef]
Xiang, Y., Chen, X., and Vlassak, J. J., 2005, “Plane-Strain Bulge Test for Thin Films,” J. Mater. Res., 20(9), pp. 2360–2370. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Taylor, G. I., and Quinney, H., 1932, “The Plastic Distortion of Metals,” Philos. Trans. R. Soc. London Ser. A, 230, pp. 323–362. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Callen, H. B., 1960, Thermodynamics, Wiley, New York.
Kuhlmann-Wilsdorf, D., 1989, “Theory of Plastic-Deformation—Properties of Low-Energy Dislocation-Structures,” Mater. Sci. Eng. A, 113, pp. 1–41. [CrossRef]
Kuhlmann-Wilsdorf, D., 1999, “The Theory of Dislocation-Based Crystal Plasticity,” Philos. Mag. A, 79(4), pp. 955–1008. [CrossRef]
Dascalu, C., and Maugin, G. A., 1993, “Material Forces and Energy-Release Rates in Homogeneous Elastic Bodies With Defects,” C. R. Acad. Sci. Ser. Vie Sci., 317(9), pp. 1135–1140.
Maugin, G. A., and Trimarco, C., 1995, “On Material and Physical Forces in Liquid-Crystals,” Int. J. Eng. Sci., 33(11), pp. 1663–1678. [CrossRef]
Zaiser, M., and Aifantis, E. C., 2006, “Randomness and Slip Avalanches in Gradient Plasticity,” Int. J. Plast., 22(8), pp. 1432–1455. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Caillard, D., and Martin, J. L., 2003, Thermally Activated Mechanisms in Crystal Plasticity (Pergamon Materials Series, No. 8), D. Caillard and J. L. Martin, eds., Pergamon, Oxford, UK.
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. [CrossRef]
Coleman, B. D., and Gurtin, M. E., 1967, “Thermodynamics With Internal State Variables,” J. Chem. Phys., 47(2), pp. 597–613. [CrossRef]
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., 51(5), pp. 641–659. [CrossRef]
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. [CrossRef]
Kröner, E., 2001, “Benefits and Shortcomings of the Continuous Theory of Dislocations,” Int. J. Solids Struct., 38(6–7), pp. 1115–1134. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Soer, W. A., and De Hosson, J. T. M., 2005, “Detection of Grain-Boundary Resistance to Slip Transfer Using Nanoindentation,” Mater. Lett., 59(24–25), pp. 3192–3195. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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, pp. 189–211. [CrossRef]
Wo, P. C., and Ngan, A. H. W., 2004, “Investigation of Slip Transmission Behavior Across Grain Boundaries in Polycrystalline Ni3Al Using Nanoindentation,” ASME J. Mater. Res., 19(1), pp. 189–201. [CrossRef]
Shen, Z., Wagoner, R. H., and Clark, W. A. T., 1986, “Dislocation Pile Up and Grain-Boundary Interactions in 304 Stainless-Steel,” Scr. Metall., 20(6), pp. 921–926. [CrossRef]
Soer, W. A., Aifantis, K. E., and De Hosson, J. T. M., 2005, “Incipient Plasticity During Nanoindentation at Grain Boundaries in Body-Centered Cubic Metals,” Acta Mater., 53(17), pp. 4665–4676. [CrossRef]
Eliash, T., Kazakevich, M., Semenov, V. N., and Rabkin, E., 2008, “Nanohardness of Molybdenum in the Vicinity of Grain Boundaries and Triple Junctions,” Acta Mater., 56(19), pp. 5640–5652. [CrossRef]
Gurtin, M. E., and Murdoch, A. I., 1976, “Effect of Surface Stress on Wave-Propagation in Solids,” J. Appl. Phys., 47(10), pp. 4414–4421. [CrossRef]
Gurtin, M. E., and Murdoch, A. I., 1978, “Surface Stress in Solids,” Int. J. Solids Struct., 14(6), pp. 431–440. [CrossRef]
Fredriksson, P., and Gudmundson, P., 2005, “Size-Dependent Yield Strength and Surface Energies of Thin Films,” Mater. Sci. Eng. A, 400, pp. 448–450. [CrossRef]
Shockley, W., and Read, W. T., 1949, “Quantitative Predictions From Dislocation Models of Crystal Grain Boundaries,” Phys. Rev., 75(4), pp. 692–692. [CrossRef]
Read, W. T., and Shockley, W., 1950, “Dislocation Models of Crystal Grain Boundaries,” Phys. Rev., 78(3), pp. 275–289. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Abu Al-Rub, R. K., and Faruk, A. N. M., 2010, “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. [CrossRef]
Cahn, J. W., and Hilliard, J. E., 1959, “Free Energy of a Nonuniform System. III. Nucleation in a Two-Component Incompressible Fluid,” J. Chem. Phys., 31(3), pp. 688–699. [CrossRef]
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. [CrossRef]
Meyers, M. A., and Chawla, K. K., 2009, Mechanical Behavior of Materials, 2nd ed., Cambridge University Press, Cambridge, UK, pp. xxii, 856.
Chung, Y., 2007, Introduction to Materials Science and Engineering, CRC/Taylor & Francis, Boca Raton, FL, p. 287.
Borg, U., and Fleck, N. A., 2007, “Strain Gradient Effects in Surface Roughening,” Modell. Simul. Mater. Sci. Eng., 15(1), pp. 1–12. [CrossRef]
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. [CrossRef]
Aifantis, K. E., and Konstantinidis, A. A., 2009, “Hall-Petch Revisited at the Nanoscale,” Mater. Sci. Eng. B, 163(3), pp. 139–144. [CrossRef]
Nieh, T. G., and Wang, J. G., 2005, “Hall-Petch Relationship in Nanocrystalline Ni and Be–B Alloys,” Intermetallics, 13(3-4), pp. 377–385. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Faghihi, D., and Voyiadjis, G. Z., 2010, “Size Effects and Length Scales in Nanoindentation for Body-Centred Cubic Materials With Application to Iron,” Proc. Inst. Mech. Eng., Part N, 224(1–2), pp. 5–18. [CrossRef]
Anisimov, S. I., Kapeliovich, B. L., and Perel'Man, T. L., 1974, “Electron Emission From Metal Surfaces Exposed to Ultrashort Laser Pulses,” Sov. JETP, 39, pp. 375–377.
Brorson, S. D., Fujimoto, J. G., and Ippen, E. P., 1987, “Femtosecond Electronic Heat-Transport Dynamics in Thin Gold-Films,” Phys. Rev. Lett.59(17), pp. 1962–1965. [CrossRef] [PubMed]
Brorson, S. D., Kazeroonian, A., Moodera, J. S., Face, D. W., Cheng, T. K., Ippen, E. P., 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. [CrossRef] [PubMed]
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. [CrossRef] [PubMed]
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. [CrossRef]
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. [CrossRef]
Voyiadjis, G. Z., and Faghihi, D., 2013, “Localization in Stainless Steel Using Microstructural Based Viscoplastic Model,” Int. J. Impact Eng., 54, pp. 114–129. [CrossRef]
Gurtin, M. E., Fried, E., and Anand, L., 2010, The Mechanics and Thermodynamics of Continua, Cambridge University Press, Cambridge, UK.
Gerberich, W. W., Kramer, D. E., Tymiak, N. I., Volinsky, A. A., Bahr, D. F., and Kriese, M. D., 1999, “Nanoindentation-Induced Defect-Interface Interactions: Phenomena, Methods and Limitations,” Acta Mater., 47(15–16), pp. 4115–4123. [CrossRef]
Gouldstone, A., Koh, H. J., Zeng, K. Y., Giannakopoulos, A. E., and Suresh, S., 2000, “Discrete and Continuous Deformation During Nanoindentation of Thin Films,” Acta Mater., 48(9), pp. 2277–2295. [CrossRef]
Giannakopoulos, A. E., and Suresh, S., 1999, “Determination of Elastoplastic Properties by Instrumented Sharp Indentation,” Scr. Mater., 40(10), pp. 1191–1198. [CrossRef]
Larsson, P. L., Giannakopoulos, A. E., Soderlund, E., Rowcliffe, D. J., and Vestergaard, R., 1996, “Analysis of Berkovich Indentation,” Int. J. Solids Struct., 33(2), pp. 221–248. [CrossRef]
Hirth, J. P., and Lothe, J., 1982, Theory of Dislocations, 2nd ed., Wiley, New York, p. 857.
Lasalmonie, A., and Strudel, J. L., 1986, “Influence of Grain-Size on the Mechanical-Behavior of Some High-Strength Materials,” J. Mater. Sci., 21(6), pp. 1837–1852. [CrossRef]
Wan, L., and Wang, S., 2009, “Shear Response of the Σ11, 〈1 1 0〉 {1 3 1} Symmetric Tilt Grain Boundary Studied by Molecular Dynamics,” Modell. Simul. Mater. Sci. Eng., 17(4), p. 045008. [CrossRef]
Kumar, R., Nicola, L., and Van der Giessen, E., 2009, “Density of Grain Boundaries and Plasticity Size Effects: A Discrete Dislocation Dynamics Study,” Mater. Sci. Eng. A, 527(1–2), pp. 7–15. [CrossRef]
Voyiadjis, G. Z., Faghihi, D., and Zhang, C., 2011, “Analytical and Experimental Detemination of Rate- and Temperature-Dependent Length Scales Using Nanoindentation Experiments,” J. Nanomech. Micromech., 1(1), pp. 24–40. [CrossRef]
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. [CrossRef]
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. Phy., 93(2), pp. 793–818. [CrossRef]
Swartz, E. T., and Pohl, R. O., 1989, “Thermal Boundary Resistance,” Rev. Mod. Phy., 61(3), pp. 605–668. [CrossRef]
ABAQUS, 2008, “Users' Manual,” Habbit, Karlsson and Sorensen, Inc., Providence, RI.
Nemat-Nasser, S., and Guo, W. G., 2000, “High Strain-Rate Response of Commercially Pure Vanadium,” Mech. Mater., 32(4), pp. 243–260. [CrossRef]
Estrin, Y. Z., Isaev, N. V., Grigorova, T. V., Pustovalov, V. V., Fomenko, V. S., Shumilin, S. E., Braude, I. S., Malykhin, S. V., Reshetnyak, M. V., and Janecek, M., 2008, “Low-Temperature Plastic Strain of Ultrafine-Grain Aluminum,” Low Temp. Phys., 34(8), pp. 665–671. [CrossRef]
Ahmed, N., and Hartmaier, A., 2011, “Mechanisms of Grain Boundary Softening and Strain-Rate Sensitivity in Deformation of Ultrafine-Grained Metals at High Temperatures,” Acta Mater., 59(11), pp. 4323–4334. [CrossRef]
Haque, M. A., and Saif, M. T. A., 2003, “Strain Gradient Effect in Nanoscale Thin Films,” Acta Mater., 51(11), pp. 3053–3061. [CrossRef]
Han, S., Kim, T., Lee, H., and Lee, H., 2008, “Temperature-Dependent Behavior of Thin Film by Microtensile Testing,” 2nd Electronics System-Integration Technology Conference (ESTC 2008), Greenwich, UK, Sept. 1–4, pp. 477–480. [CrossRef]
Xiang, Y., Tsui, T. Y., and Vlassak, J. J., 2006, “The Mechanical Properties of Freestanding Electroplated Cu Thin Films,” ASME J. Mater. Res., 21(6), pp. 1607–1618. [CrossRef]
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. [CrossRef]
Bauschinger, J., 1881, “Tension Prisma Stabe,” Civilingenieur, 27, pp. 289–301.
Yefimov, S., Groma, I., and van der Giessen, E., 2004, “A Comparison of a Statistical-Mechanics Based Plasticity Model With Discrete Dislocation Plasticity Calculations,” J. Mech. Phys. Solids, 52(2), pp. 279–300. [CrossRef]
Taylor, G. I., 1938, “Plastic Strain in Metals,” J. Inst. Metals, 62, pp. 307–324.
Mandel, J., 1973, “Thermodynamics and Plasticity,” Proceedings of the International Symposium on Foundations of Continuum Thermodynamics, Bussaco, Portugal, July 22–26, J. J. Delgado Domingas, M. N. R. Nina, and J. H. Whitelaw, eds., Halsted Press, New York, pp. 283–304.
Hill, R., 1965, “Continuum Micro-Mechanics of Elastoplastic Polycrystals,” J. Mech. Phys. Solids, 13(2), pp. 89–101. [CrossRef]
Teodosiu, C., 1970, “A Dynamic Theory of Dislocations and Its Applications to the Theory of the Elastic-Plastic Continuum,” Conference on Fundamental Aspects of Dislocation Theory, Gaithersburg, MD, Apr. 21–25, 1969, U.S. National Bureau of Standards, pp. 837–876.
Rice, J. R., 1971, “Inelastic Constitutive Relations for Solids: An Internal-Variable Theory and Its Applications to Metal Plasticity,” J. Mech. Phys. Solids, 19(6), pp. 443–455. [CrossRef]
Evers, L. P., Brekelmans, W. A. M., and Geers, M. G. D., 2004, “Scale Dependent Crystal Plasticity Framework With Dislocation Density and Grain Boundary Effects,” Int. J. Solids Struct., 41(18–19), pp. 5209–5230. [CrossRef]
Shi, J., and Zikry, M. A., 2009, “Grain–Boundary Interactions and Orientation Effects on Crack Behavior in Polycrystalline Aggregates,” Int. J. Solids Struct., 46(21), pp. 3914–3925. [CrossRef]
Ertürk, İ., van Dommelen, J. A. W., and Geers, M. G. D., 2009, “Energetic Dislocation Interactions and Thermodynamical Aspects of Strain Gradient Crystal Plasticity Theories,” J. Mech. Phys. Solids, 57(11), pp. 1801–1814. [CrossRef]
Han, C.-S., Gao, H., Huang, Y., and Nix, W. D., 2005, “Mechanism-Based Strain Gradient Crystal Plasticity—I. Theory,” J. Mech. Phys. Solids, 53(5), pp. 1188–1203. [CrossRef]
Borg, U., 2007, “A Strain Gradient Crystal Plasticity Analysis of Grain Size Effects in Polycrystals,” Eur. J. Mech. A, 26(2), pp. 313–324. [CrossRef]
Cordero, N. M., Gaubert, A., Forest, S., Busso, E. P., Gallerneau, F., and Kruch, S., 2010, “Size Effects in Generalised Continuum Crystal Plasticity for Two-Phase Laminates,” J. Mech. Phys. Solids, 58(11), pp. 1963–1994. [CrossRef]
Shu, J. Y., and Fleck, N. A., 1999, “Strain Gradient Crystal Plasticity: Size-Dependent Deformation of Bicrystals,” J. Mech. Phys. Solids, 47(2), pp. 297–324. [CrossRef]
Kubin, L. P., Canova, G., Condat, M., Devincre, B., Pontikis, V., and Bréchet, Y., 1992, “Dislocation Microstructures and Plastic Flow: A 3D Simulation,” Solid State Phenom., 23-24, pp. 455–472. [CrossRef]
Vandergiessen, E., and Needleman, A., 1995, “Discrete Dislocation Plasticity—A Simple Planar Model,” Modell. Simul. Mater. Sci. Eng., 3(5), pp. 689–735. [CrossRef]
Ghoniem, N. M., Tong, S. H., and Sun, L. Z., 2000, “Parametric Dislocation Dynamics: A Thermodynamics-Based Approach to Investigations of Mesoscopic Plastic Deformation,” Phys. Rev. B, 61(2), pp. 913–927. [CrossRef]
Weygand, D., Friedman, L. H., Van der Giessen, E., and Needleman, A., 2002, “Aspects of Boundary-Value Problem Solutions With Three- Dimensional Dislocation Dynamics,” Modell. Simul. Mater. Sci. Eng., 10(4), pp. 437–468. [CrossRef]
Deshpande, V. S., Needleman, A., and Van der Giessen, E., 2005, “Plasticity Size Effects in Tension and Compression of Single Crystals,” J. Mech. Phys. Solids, 53(12), pp. 2661–2691. [CrossRef]
Yasin, H., Zbib, H. M., and Khaleel, M. A., 2001, “Size and Boundary Effects in Discrete Dislocation Dynamics: Coupling With Continuum Finite Element,” Mater. Sci. Eng. A, 309–310, pp. 294–299. [CrossRef]
Khraishi, T. A., and Zbib, H. M., 2002, “Free-Surface Effects in 3D Dislocation Dynamics: Formulation and Modeling,” ASME J. Eng. Mater. Technol., 124(3), pp. 342–351. [CrossRef]
Yan, L., Khraishi, T. A., Shen, Y.-L., and Horstemeyer, M. F., 2004, “A Distributed-Dislocation Method for Treating Free-Surface Image Stresses in Three-Dimensional Dislocation Dynamics Simulations,” Modell. Simul. Mater. Sci. Eng., 12(4), p. S289. [CrossRef]
Nakano, A., Bachlechner, M. E., Kalia, R. K., Lidorikis, E., Vashishta, P., Voyiadjis, G. Z., Campbell, T. J., Ogata, S., and Shimojo, F., 2001, “Multiscale Simulation of Nanosystems,” Comput. Sci. Eng., 3(4), pp. 56–66. [CrossRef]
Voyiadjis, G. Z., Aifantis, E. C., and Weber, G., 2003, “Constitutive Modeling of Plasticity in Nanostructured Materials,” Trends in Nanoscale Mechanics: Analysis of Nanostructured Materials and Multi-Scale Modeling,V. M.Harik, and M. D.Salas, eds., Kluwer Academic Publishers, Amsterdam, The Netherlands, Chap. 5.
Nair, A. K., Parker, E., Gaudreau, P., Farkas, D., and Kriz, R. D., 2008, “Size Effects in Indentation Response of Thin Films at the Nanoscale: A Molecular Dynamics Study,” Int. J. Plast., 24(11), pp. 2016–2031. [CrossRef]
Chandra, N., and Dang, P., 1999, “Atomistic Simulation of Grain Boundary Sliding and Migration,” J. Mater. Sci., 34(4), pp. 655–666. [CrossRef]
Salahshoor, H., and Rahbar, N., 2012, “Nano-Scale Fracture Toughness and Behavior of Graphene/Epoxy Interface,” J. Appl. Phys., 112(2), p. 023510. [CrossRef]
Tan, T., Meng, J., Rahbar, N., Li, H., Papandreou, G., Maryanoff, C. A., and Soboyejo, W. O., 2012, “Effects of Silane on the Interfacial Fracture of a Parylene Film Over a Stainless Steel Substrate,” Mater. Sci. Eng. C, 32(3), pp. 550–557. [CrossRef]
Samvedi, V., and Tomar, V., 2009, “Role of Interface Thermal Boundary Resistance in Overall Thermal Conductivity of Si-Ge Multi-Layered Structures,” Nanotechnology, 20(36), p. 365701. [CrossRef] [PubMed]
Che, J., Çağın, T., Deng, W., and Goddard, W. A., 2000, “Thermal Conductivity of Diamond and Related Materials From Molecular Dynamics Simulations,” J. Chem. Phys., 113(16), pp. 6888–6900. [CrossRef]
Li, J., Porter, L., and Yip, S., 1998, “Atomistic Modeling of Finite-Temperature Properties of Crystalline β-SiC: II. Thermal Conductivity and Effects of Point Defects,” J. Nucl. Mater., 255(2–3), pp. 139–152. [CrossRef]
Volz, S., Saulnier, J. B., Chen, G., and Beauchamp, P., 2000, “Computation of Thermal Conductivity of Si/Ge Superlattices by Molecular Dynamics Techniques,” Microelectron. J., 31(9–10), pp. 815–819. [CrossRef]
Daly, B. C., Maris, J. H., Imamura, K., and Tamura, S., 2002, Molecular Dynamics Calculation of the Thermal Conductivity of Superlattices, Vol. 66, American Physical Society, Ridge, NY.
Abramson, A. R., Tien, C.-L., and Majumdar, A., 2002, “Interface and Strain Effects on the Thermal Conductivity of Heterostructures: A Molecular Dynamics Study,” ASME J. Heat Transfer, 124(5), pp. 963–970. [CrossRef]
Lee, Y. H., Biswas, R., Soukoulis, C. M., Wang, C. Z., Chan, C. T., and Ho, K. M., 1991, “Molecular-Dynamics Simulation of Thermal Conductivity in Amorphous Silicon,” Phys. Rev. B, 43(8), pp. 6573–6580. [CrossRef]
Ding, K., and Andersen, H. C., 1986, “Molecular-Dynamics Simulation of Amorphous Germanium,” Phys. Rev. B, 34(10), pp. 6987–6991. [CrossRef]
Tomar, V., and Gan, M., 2011, “Temperature Dependent Nanomechanics of Si-C-N Nanocomposites With an Account of Particle Clustering and Grain Boundaries,” Int. J. Hydrogen Energy, 36(7), pp. 4605–4616. [CrossRef]
Izvekov, S., Chung, P. W., and Rice, B. M., 2010, “The Multiscale Coarse-Graining Method: Assessing Its Accuracy and Introducing Density Dependent Coarse-Grain Potentials,” J. Chem. Phys., 133(6), p. 064109. [CrossRef] [PubMed]
Farrell, K., and Oden, J. T., 2014, “Calibration and Validation Methods of Coarse-Grained Models of Atomic Systems: Application to Semiconductor Manufacturing,” Comput. Mech., 54(1), pp. 3–19. [CrossRef]
Curtin, W. A., and Miller, R., 2003, “Atomistic/Continuum Coupling in Computational Materials Science,” Modell. Simul. Mater. Sci. Eng., 11(3), p. R33. [CrossRef]
Weinan, E., Li, X., and Vanden-Eijnden, E., 2004, “Some Recent Progress in Multiscale Modeling,” Multiscale Modelling and Simulation, S.Attinger and P.Koumoutsakos, eds., Springer, Berlin, pp. 3–21.
Liu, W. K., Karpov, E. G., Zhang, S., and Park, H. S., 2004, “An Introduction to Computational Nanomechanics and Materials,” Comput. Methods Appl. Mech. Eng., 193(17–20), pp. 1529–1578. [CrossRef]
Bauman, P. T., Dhia, H. B., Elkhodja, N., Oden, J. T., and Prudhomme, S., 2008, “On the Application of the Arlequin Method to the Coupling of Particle and Continuum Models,” Comput. Mech., 42(4), pp. 511–530. [CrossRef]
Bauman, P. T., Oden, J. T., and Prudhomme, S., 2009, “Adaptive Multiscale Modeling of Polymeric Materials With Arlequin Coupling and Goals Algorithms,” Comput. Methods Appl. Mech. Eng., 198(5–8), pp. 799–818. [CrossRef]
Prudhomme, S., Bauman, P. T., and Oden, J. T., 2006, “Error Control for Molecular Statics Problems,” Int. J. Multiscale Comput. Eng., 4(5-6), pp. 647–662. [CrossRef]
Oden, J. T., Prudhomme, S., Romkes, A., and Bauman, P. T., 2005, “Multiscale Modeling of Physical Phenomena: Adaptive Control of Models,” SIAM J. Sci. Comput., 28(6), pp. 2359–2389. [CrossRef]

Figures

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Fig. 1

Illustration of phonon interactions and scattering mechanisms of free electrons within metal lattice

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Fig. 2

Schematic illustration of grain boundary separating grains A and B along with single slip system at each grain and the misalignment angles [122,127]

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Fig. 4

Expansion of the plastic zone under the indenter and the effect of grain boundary. According to the figure: (a) nucleation of dislocations (point E); (b) expansion of plastic zone with no grain boundary influence (line EA); (c) dislocation pile-up at the interface (line AB); and (d) dislocation transmission to the adjacent grain (line BC) [127].

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Fig. 5

(a) Qualitative stress–strain response from 1D solution of the Faghihi [127] model and (b) one-dimensional model of grain and grain boundary

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Fig. 3

A schematic nano-indentation response near the grain boundary. The solid line represents the test far from the grain boundary (grain interior response) and the dashed line denotes the response when indentation is conducted near the grain boundary (L < 1 μm) [122,127].

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Fig. 6

One-dimensional model for a single crystal bounded by two grain boundaries

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Fig. 10

Effect of temperature and strain rate power (n and m) on ((a) and (c)) stress–strain responses; ((b) and (d)) evolution of temperature with plastic strain at the midpoint of the grain (ℓen/L = 0.1 and ℓdis/L = 1)

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Fig. 7

Size effect due to the dissipative length scale when the energetic backstress vanishes (i.e., ℓen→0): (a) plastic strain distribution across the grain; (b) stress–strain; (c) temperature distribution across the grain; and (d) evolution of temperature with plastic strain at the midpoint of the grain [122]

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Fig. 8

Effect of isotropic hardening constants on the stress–strain responses when the energetic backstress vanishes (i.e., ℓen→0): (a) hardening parameter, h and (b) hardening power, r

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Fig. 14

Grain size effect in case of nonvanishing energetic and dissipative interfacial length scales: (a) plastic strain distribution across the grain and (b) stress–strain behavior [122]

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Fig. 11

Effect of thermal parameter (TY) and initial temperature (T0) on ((a) and (c)) stress–strain responses; ((b) and (d)) evolution of temperature with plastic strain at the midpoint of the grain (ℓen/L = 0.1, ℓdis/L = 1 for all curves, m = 0.05 and strain rate of 2000/s are considered for (c) and (d)).

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Fig. 12

Size effect due to the energetic interfacial length scale when energy dissipation vanishes (i.e., ℓdisI,ℓdis→0). The interface is considered as compliant (ℓenI/ℓen = 0.1) in the left column and stiff (ℓenI/ℓen = 10) in the right column: ((a) and (b)) stress–strain; ((c) and (d)) plastic strain distribution across the grain; ((e) and (f)) temperature distribution across the grain (127).

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Fig. 13

Size effect due to the energetic interfacial length scale when energy storage vanishes (i.e., ℓenI,ℓen→0). The interface is considered as compliant (ℓdisI/ℓdis = 0.01), intermediate (ℓdisI/ℓdis = 0.2), and stiff (ℓdisI/ℓdis = 1.5): ((a) and (b)) stress–strain; ((c) and (d)) plastic strain distribution across the grain; ((e) and (f)) temperature distribution across the grain.

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Fig. 15

Effect of interfacial thermal parameters (nI and TYI) on stress–strain responses and plastic strain distribution across the grain (ℓen/L = ℓenI/L = 0.1 and ℓdis/L = ℓdisI/L = 1)

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Fig. 16

Effect of initial temperature T0 on: (a) plastic strain distribution across the grain and (b) stress–strain responses (ℓen/L = ℓenI/L = 0.1, ℓdis/L = ℓdisI/L = 1, nI = 0.1)

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Fig. 9

Size effect due to the energetic length scale when the dissipative strengthening vanishes (i.e., ℓdis→0): (a) plastic strain distribution across the grain; (b) stress–strain; (c) temperature distribution across the grain; and (d) effect of backstress power, a, on the stress–strain responses (r = 1) [122,127]

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Fig. 17

Model predictions and experimental measurements of (a) the film thickness effect on the stress–strain curves of sputter-deposited Al films (experimental data taken from Ref. [255]) and (b) the temperature effect on the stress–strain curves of sputter-deposited Ni films (experimental data taken from Refs. [256,122])

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Fig. 18

Model predictions and experimental measurements of the film thickness effect on the stress–strain curves of electroplated Cu films with both surface passivated by 20 nm of Ti (experimental data taken from Refs. [21,156])

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Fig. 19

Model predictions and experimental measurements of the actual yield stress of the electroplated Cu films as a function of inverse film thickness (experimental data taken from Refs. [21,156])

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Fig. 20

Distribution of (a) plastic strain and (b) plastic strain gradient, across the film thickness [156]

Grahic Jump Location
Fig. 21

Model predictions and experimental measurements of the surface conditions (passivated-freestanding) on the stress–strain curves of sputter-deposited Al films (experimental data taken from Refs. [20,156])

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