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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.

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## 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. 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])

Fig. 3

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

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