0
Research Papers

A Thermodynamic Consistent Model for Coupled Strain-Gradient Plasticity With Temperature

[+] Author and Article Information
Danial Faghihi

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

Boyd Professor
Computational Solid Mechanics Laboratory,
Department of Civil and
Environmental Engineering,
Louisiana State University,
Baton Rouge, LA 70803
Department of Civil and
Environmental Engineering,
World Class University,
Hanyang University,
Seoul, South Korea

3 Such continuum theories of plasticity break down at scales when the numbers of dislocations are too small for them to be treated collectively. By increasing the resolution of the theory (e.g. Discrete Dislocation models), individual dislocations can be modeled incorporating other length scales than continuum models.

4Referred to quasi-conservative by Ziegler and Wehrli [26].

5The convexity of the rate dissipation potential as described by Ziegler and Wehrli [26] is only regard to strain. As it is shown by the same authors, such a function is concave with respect to temperature in the case of classical heat transfer. Faghihi and Voyiadjis [29] showed the concavity of dissipation potential with respect to gradient of temperature in driving the microscale heat transfer to address the phenomena is small spatial and temporal scales.

6While 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.

7According to Gurtin [57] “the plastic spin vanishes identically when the free energy is independent of curl of plastic strain, but not generally otherwise.” A formal discussion based on experience with other gradient theories suggests 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.

8Referred as Burgers tensor in the works of Gurtin [57].

9Using direct notation, Eq. (7) reads as α = (curl εp)T. It should be note that the transpose of α is referred as Burgers tensor, G, in the works of Gurtin [57], therefore: G = curl εp.

10In 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 SSD and GND associated with plastic strain and gradient of plastic train) and entanglements.

11Gurtin and Reddy [30] 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.

12Specifically, densities of edge and screw GNDs according to Gurtin [57,78]. It should be noted that the kinematic hardening in the current theory does not account for certain interval of plastic strain during the loading reversal where dislocation density remains constant [79]. 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 [75].

13Such correlation between GNDs will quickly decrease to zero within a few dislocation distances [80].

14Starting 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 $∂2Ψ/∂T2(δT2)=-∂2e/∂s2(δs)2$. 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 $∂2e/∂s2>0=>∂2Ψ/∂T2<0$ [84]. Moreover, according to Callen [85] one can assume that that entropy is a monotonically increasing function of temperature, thus: $∂s/∂T=-∂2Ψ/∂T2$.

15Conventionally, dislocations impinging transversely on a slip plane are termed forest dislocations and are thought to be responsible for the second stage of hardening [86,87].

16For exmple, Dascalu and Maugin [89] and Maugin and Trimarco [90] for an inclusive treatment of the configurational (material) forces.

17Assuming the interaction between only two GNDs (or pile ups), this stress acts on them in the opposite direction of slip.

18As it is discussed by Bardella [58], 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 and the moving elastic–plastic boundaries (i.e., imposing any higher-order boundary condition at the internal surfaces between elastic and plastic domains).

19For further details regarding the effect of the interaction coefficient value, see Dahlberg and Faleskog [99].

20According to the second paragraph after Eq. (40 )

21Both Kuroda and Tvergaard [112] and Gurtin and Ohno [71] discussed the crystal plasticity case, which can also be applied to the continuum plasticity.

22Due to the high strain rates and short duration of the loading, heat loss through conduction, convection, or radiation is neglected in comparison to the thermoplastic heating, and therefore qi,i = 0.

23According to Fleck and Hutchinson [67] the aforementioned simplification has insignificant influence for the results in the plastic regime.

24The average grain sizes of the films with 1.00, 1.90, and 4.20 μm are 1.5 ± 0.05, 1.51 ± 0.04, and 1.5 ± 0.05 μm respectively.

25A 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.

26Taylor law gives a simple description of the dislocation interaction processes at the microscale and relates the shear strength to the dislocation density.

27He suggested that the deformation resistance in the presence of plastic strain gradient depends on additive coupling of the SSDs and GNDs densities.

28Generalized based on the pioneer works of, e.g., Refs. [1,5,60,116,125,126,127].

29The dislocation density, ρT, used here to derive the gradient independent stored energy of cold work is equivalent to the forest dislocation density introduced by Gurtin and Ohno [71]. In other words, it is assumed here that the energy of the cold work results in an extra strain (i.e., latent)-hardening which is recoverable and temperature independent but it does not affect the yield stress. Indeed in the current developed theory the forest dislocation density only affects the isotropic hardening (see the paragraph after Eq. (36)).

1Present address: Postdoctoral Fellow, Institute for Computational Engineering and Science, The University of Texas at Austin, Austin, TX 78712.

2Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received May 9, 2013; final manuscript received September 18, 2013; published online November 7, 2013. Assoc. Editor: Mohammed Zikry.

J. Eng. Mater. Technol 136(1), 011002 (Nov 07, 2013) (14 pages) Paper No: MATS-13-1080; doi: 10.1115/1.4025508 History: Received May 09, 2013; Revised September 18, 2013

Abstract

The mechanical responses of small volume metallic compounds are addressed in this work through developing a nonlocal continuum theory. In this regard, a thermodynamic-based higher-order strain-gradient plasticity framework for coupled thermoviscoplasticity modeling is presented. The concept of thermal activation energy and the dislocations interaction mechanisms are taken into consideration to describe the choice of thermodynamic potentials such as Helmholtz free energy and rate of dissipation. The theory is developed based on the decomposition of the thermodynamic conjugate forces into energetic and dissipative counterparts, which provides the constitutive equations to have both energetic and dissipative gradient length scales. The derived constitutive model is calibrated against the experimental data of bulge test conducted on thin films.

<>

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.
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.
Vlassak, J. J., Xiang, Y., and Chen, X., 2005, “Plane-Strain Bulge Test for Thin Films,” J. Mater. Res., 20(9), pp. 2360–2370.
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.
Ma, Q., and Clarke, D. R., 1995, “Size-Dependent Hardness of Silver Single-Crystals,” J. Mater. Res., 10(4), pp. 853–863.
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.
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.
Arsenlis, A., and Parks, D. M., 1999, “Crystallographic Aspects of Geometrically-Necessary and Statistically-Stored Dislocation Density,” Acta Mater., 47(5), pp. 1597–1611.
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.
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.
Aifantis, E. C., 1999, “Gradient Deformation Models at Nano, Micro, and Macro Scales,” ASME J. Eng. Mater. Technol., 121(2), pp. 189–202.
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.
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.
Xiang, Y., and Vlassak, J. J., 2005, “Bauschinger Effect in Thin Metal Films,” Scr. Mater., 53(2), pp. 177–182.
Xiang, Y., and Vlassak, J. J., 2006, “Bauschinger and Size Effects in Thin-Film Plasticity,” Acta Mater., 54(20), pp. 5449–5460.
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.
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.
Muhlhaus, H. B., and Aifantis, E. C., 1991, “A Variational Principle for Gradient Plasticity,” Int. J. Solids Struct., 28(7), pp. 845–857.
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.
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.
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.
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.
Schiotz, J., Di Tolla, F. D., and Jacobsen, K. W., 1998, “Softening of Nanocrystalline Metals at Very Small Grains,” Nature, 391, pp. 561–563.
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.
Aifantis, E. C., 1984, “On the Microstructural Origin of Certain Inelastic Models,” ASME J. Eng. Mater. Technol.106(4), pp. 326–330.
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.
Ziegler, H., 1962, Some Extremum Principles in Irreversible Thermodynamics, With Application to Continuum Mechanics, Swiss Federal Institute of Technology, Zurich, Switzerland.
Ziegler, H., 1958, “An Attempt to Generalize Onsager's Principle, and Its Significance for Rheological Problems,” Z. Angew. Math. Phys., 9(5), pp. 748–763.
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.
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.
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.
Reddy, B. D., 2012, “Erratum to: The Role of Dissipation and Defect Energy in Variational Formulations of Problems in Strain-Gradient Plasticity, Part 1, Polycrystalline Plasticity,” Continuum Mech. Thermodyn., 24(1), p. 79.
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.
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.
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.
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.
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.
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.
Lemaitre, J., 1985, “Coupled Elasto-Plasticity and Damage Constitutive-Equations,” Comput. Methods Appl. Mech. Eng., 51(1–3), pp. 31–49.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Mroz, Z., and Oliferuk, W., 2002, “Energy Balance and Identification of Hardening Moduli in Plastic Deformation Processes,” Int. J. Plast., 18(3), pp. 379–397.
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.
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.
Zehnder, A. T., 1991, “A Model for the Heating Due to Plastic Work,” Mech. Res. Commun., 18(1), pp. 23–28.
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.
Saanouni, K., Forster, C. H., and Hatira, F. B., 1994, “On the Anelastic Flow With Damage,” Int. J. Damage Mech., 3(2), pp. 140–169.
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.
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.
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.
Bardella, L., 2010, “Size Effects in Phenomenological Strain Gradient Plasticity Constitutively Involving the Plastic Spin,” Int. J. Eng. Sci., 48(5), pp. 550–568.
Nye, J. F., 1953, “Some Geometrical Relations in Dislocated Crystals,” Acta Metall., 1(2), pp. 153–162.
Fleck, N. A., and Hutchinson, J. W., 1997, “Strain Gradient Plasticity,” Adv. Appl. Mech., 33, pp. 295–361.
Bassani, J. L., 2001, “Incompatibility and a Simple Gradient Theory of Plasticity,” J. Mech. Phys. Solids, 49(9), pp. 1983–1996.
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.
Gudmundson, P., 2004, “A Unified Treatment of Strain Gradient Plasticity,” J. Mech. Phys. Solids, 52(6), pp. 1379–1406.
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.
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.
Voyiadjis, G. Z., and Faghihi, D., 2012, “Thermo-Mechanical Strain Gradient Plasticity With Energetic and Dissipative Length Scales,” Int. J. Plast., 30–31(0), pp. 218–247.
Fleck, N. A., and Hutchinson, J. W., 2001, “A Reformulation of Strain Gradient Plasticity,” J. Mech. Phys. Solids, 49(10), pp. 2245–2271.
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.
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.
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.
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.
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.
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.
Taylor, G. I., and Quinney, H., 1932, “The Plastic Distortion of Metals,” Philos. Trans. R. Soc. London, Ser. A, 230, pp. 323–362.
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.
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.
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.
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.
Hasegawa, T., Yakou, T., and Karashima, S., 1975, “Deformation Behavior and Dislocation-Structures Upon Stress Reversal in Polycrystalline Aluminum,” Mater. Sci. Eng., 20(3), pp. 267–276.
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.
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.
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.
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.
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.
Callen, H. B., 1960, Thermodynamics, John 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.
Kuhlmann-Wilsdorf, D., 1999, “The Theory of Dislocation-Based Crystal Plasticity,” Philos. Mag. A, 79(4), pp. 955–1008.
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.
Dascalu, C., and Maugin, G. A., 1993, “Material Forces and Energy-Release Rates in Homogeneous Elastic Bodies With Defects,” C. R. Acad. Sci. Ser. II, 317(9), pp. 1135–1140.
Maugin, G. A., and Trimarco, C., 1995, “On Material and Physical Forces in Liquid-Crystals,” Int. J. Eng. Sci., 33(11), pp. 1663–1678.
Zaiser, M., and Aifantis, E. C., 2006, “Randomness and Slip Avalanches in Gradient Plasticity,” Int. J. Plast., 22(8), pp. 1432–1455.
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.
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.
Caillard, D., and Martin, J. L., 2003, Thermally Activated Mechanisms in Crystal Plasticity (Pergamon Materials Series), Elsevier, 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.
Coleman, B. D., and Gurtin, M. E., 1967, “Thermodynamics With Internal State Variables,” J. Chem. Phys., 47(2), pp. 597–613.
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.
Dahlberg, C., and Faleskog, J., 2012, “An Improved Strain Gradient Plasticity Formulation With Energetic Interfaces: Theory and A Fully Implicit Finite Element Formulation,” Comput. Mech., pp. 1–19.
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.
Kröner, E., 2001, “Benefits and Shortcomings of the Continuous Theory of Dislocations,” Int. J. Solids Struct., 38(6–7), pp. 1115–1134.
Forest, S., and Amestoy, M., 2008, “Hypertemperature in Thermoelastic Solids,” C. R. Mec., 336(4), pp. 347–353.
Forest, S., 2009, “Micromorphic Approach for Gradient Elasticity, Viscoplasticity, and Damage,” ASCE J. Eng. Mech.135(3), pp. 117–131.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Vlassak, J. J., and Nix, W. D., 1992, “A New Bulge Test Technique for the Determination of Young Modulus and Poisson Ratio of Thin-Films,” J. Mater. Res., 7(12), pp. 3242–3249.
Xiang, Y., Chen, X., and Vlassak, J. J., 2005, “Plane-Strain Bulge Test for Thin Films,” J. Mater. Res., 20(9), pp. 2360–2370.
Xiang, Y., Tsui, T. Y., and Vlassak, J. J., 2006, “The Mechanical Properties of Freestanding Electroplated Cu Thin Films,” J. Mater. Res., 21(6), pp. 1607–1618.
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.
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.
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.
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.
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.
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 DislocationPlasticity Calculations,” J. Mech. Phys. Solids, 52(2), pp. 279–300.
Ashby, M. F., 1970, “Deformation of Plastically Non-Homogeneous Materials,” Philos. Mag., 21(170), pp. 399–424.
Voyiadjis, G., and Abu Al-Rub, R. K., 2002, “Length Scales in Gradient Plasticity Theory,” Proceedings of the IUTAM Symposium on Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials, Morocco, October 20–25, pp. 167–174.
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.
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.
Kocks, U. F., Argon, A. S., and Ashby, M. F., 1975, “Thermodynamics and Kinetics of Slip,” Prog. Mater. Sci., 19, pp. 1–281.
Hirth, J. P., and Lothe, J., 1982, Theory of Dislocations, 2nd ed., Krieger Publishing, Malabar, FL, p. 857.
Meyers, M. A., and Chawla, K. K., 2009, Mechanical Behavior of Materials, Vol. xxii, 2nd ed., Cambridge University Press, Cambridge, New York, p. 856.

Figures

Fig. 1

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 Xiang and Vlassak [15])

Fig. 2

Model predictions and experimental measurements of the actual yield stress Cu films as a function of inverse film thickness (experimental data taken from Xiang and Vlassak [15])

Fig. 3

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

Fig. 4

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 Xiang and Vlassak [14])

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections