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Research Papers

# Guidelines for Constructing Strain Gradient Plasticity Theories

[+] Author and Article Information
N. A. Fleck

Department of Engineering,
Cambridge University,
Trumpington Street,
Cambridge CB2 1PZ, UK

J. W. Hutchinson

School of Engineering
and Applied Sciences,
Harvard University,
Cambridge, MA 02138

J. R. Willis

Centre for Mathematical Sciences,
Cambridge CB3 0WA, UK

The forward difference approximation would take $γ=λ=0$. For the backward difference approximation, $γ=λ=1$. Both have an error of order $Δt$. The central difference approximation is defined by $γ=λ=1/2$ and has an error of order $(Δt)2$.

For the first increment, $ɛ0p=0$. Retention of $ɛ0p$ allows the general formula to apply, with re-numbering, to any increment and also to the case of uniform straining with surfaces unpassivated up to a uniform plastic strain $ɛ0p$, as considered in Ref. [1].

The integral to follow is obtained via the variable transformation transformation $cosθ=z¯/[1-R∧(1-z¯)]$, as in Ref. [1].

Strictly, it is necessary to demonstrate that there exist fields $qUR$ and $τUR$ that do not exceed the yield criterion, when $R∧. This demonstration was made in a slightly different context in Ref. [1]; it is omitted here.

We refrain from recording the analysis, in the interest of conciseness.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received December 21, 2014; final manuscript received March 12, 2015; published online June 3, 2015. Assoc. Editor: Erik van der Giessen.

J. Appl. Mech 82(7), 071002 (Jul 01, 2015) (10 pages) Paper No: JAM-14-1595; doi: 10.1115/1.4030323 History: Received December 21, 2014; Revised March 12, 2015; Online June 03, 2015

## Abstract

Issues related to the construction of continuum theories of strain gradient plasticity which have emerged in recent years are reviewed and brought to bear on the formulation of the most basic theories. Elastic loading gaps which can arise at initial yield or under imposition of nonproportional incremental boundary conditions are documented and analytical methods for dealing with them are illustrated. The distinction between unrecoverable (dissipative) and recoverable (energetic) stress quantities is highlighted with respect to elastic loading gaps, and guidelines for eliminating the gaps are presented. An attractive gap-free formulation that generalizes the classical $J2$ flow theory is identified and illustrated.

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## References

Fleck, N. A., Hutchinson, J. W., and Willis, J. R., 2014, “Strain Gradient Plasticity Under Non-Proportional Loading,” Proc. R. Soc., A, 470(2170), p. 20140267.
Bardella, L., and Panteghini, A., 2015, “Modelling the Torsion of Thin Metal Wires by Distortion Gradient Plasticity,” J. Mech. Phys. Solids, 78, pp. 467–492.
Gurtin, M. E., 2003, “On a Framework for Small-Deformation Viscoplasticity: Free Energy, Microforces, Strain Gradients,” Int. J. Plast., 19(1), pp. 47–90.
Gudmundson, P. A., 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 Hutchinson, J. W., 1997, “Strain Gradient Plasticity,” Adv. Appl. Mech., 33, pp. 295–361.
Muhlhaus, H. B., and Aifantis, E. C., 1991, “A Variational Principle for Gradient Plasticity,” Int. J. Solids Struct., 28(7), pp. 845–857.
Fleck, N. A., and Willis, J. R., 2009,“A Mathematical Basis for Strain Gradient Plasticity Theory. Part I: Scalar Plastic Multiplier. Part II: Tensorial Plastic Multiplier,” J. Mech. Phys. Solids, 57(1), pp. 161–177.
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Danas, K., Deshpande, V. S., and Fleck, N. A., 2010, “Compliant Interfaces: A Mechanism for Relaxation of Dislocation Pile-Ups in a Sheared Single Crystal,” Int. J. Plast., 26(12), pp. 1792–1805.
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.
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Niordson, C. N., and Legarth, B. N., 2010, “Strain Gradient Effects in Cyclic Plasticity,” J. Mech. Phys. Solids, 58(4), pp. 542–557.
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.
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.
Evans, A. G., and Hutchinson, J. W., 2009, “A Critical Assessment of Theories of Strain Gradient Plasticity,” Acta Mater., 57(5), pp. 1675–1688.

## Figures

Fig. 1

Comparison overall stress–strain response based of deformation theory for a layer of thickness 2h with surfaces passivated from the start and stretched in plane strain. The material is incompressible. The lower curve applies to an unpassivated layer or, equivalently, a layer with ℓR/h=0. The upper two curves have ℓR/h=1. The top curve is based on formulation (2.4), and it has an elastic loading gap on the vertical axis from 1 to 1.825. The middle curve is based on Eq. (2.5) and it has no elastic loading gap.

Fig. 2

Elastic loading gap at the onset of yield for a passivated layer in plane strain for the deformation theory based on formulation (2.4) with ℓ=ℓR. This same gap arose for the nonincremental theory considered in Ref. [1] for a layer stretched into the plastic range and then passivated followed by further stretch with ℓ=ℓUR.

Fig. 3

Pure bending in plane strain with no passivation followed by continued bending with passivation. The constitutive law is specified by a dissipation potential ϕ given by Eq. (3.2) with no recoverable contributions. The material is taken to be incompressible and the computation in (a) is carried out using the rate-dependent version with a strain-rate exponent m=0.1, as in Ref. [1]. The elastic loading gap as specified by the curvature increase Δκ after passivation without plastic flow is plotted as a function of the curvature κ at passivation in (b). The predictions in (b) are based on the rate-independent formulation and minimum principle I (3.9).

Fig. 4

Average stress versus stretching strain for a passivated layer specified by the gap-free incremental theory defined in Sec. 5. The material is incompressible and the deformation is plane strain.

Fig. 5

(a) The plastic strain at the center of the passivated layer as a function of the strain imposed on the layer—a comparison between asymptotic and exact results. (b) The distribution of the normalized plastic strain across the layer at 2ɛ11/3=3ɛY. In both parts, for the incompressible, incremental material in Sec. 5 with N=0.2 and p=0.5.

Fig. 6

An unpassivated layer of thickness 2h stretched into the plastic range and then passivated followed by additional stretch, as predicted by the incremental theory in Sec. 5 for an incompressible layer in plane strain

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