Abstract

The requirement of a non-negative dissipation rate for all possible deformation histories is generally imposed on plastic constitutive relations. This is a constraint analogous to the Coleman–Noll [Coleman, B. D., and Noll, W., 1964, “The Thermodynamics of Elastic Materials With Heat Conduction and Viscosity,” Arch. Ration. Mech. Anal., 13, pp. 167–178. 10.1007/BF01262690] postulate that the Clausius–Duhem inequality needs to be satisfied for all possible deformation histories. The physical basis for the Clausius–Duhem inequality is as a statistical limit for a large number of discrete events for a long time and is not a fundamental physical requirement for small systems for a short time. The relation between the requirement of a non-negative dissipation rate and the Clausius–Duhem inequality is considered. The consequences of imposing a non-negative dissipation rate for all possible deformation histories are illustrated for: (i) a single crystal plasticity framework that accounts for elastic lattice curvature changes as well as elastic lattice straining and (ii) for discrete defect theories of plasticity, with attention specifically on discrete dislocation plasticity for crystalline solids and discrete shear transformation zone (STZ) plasticity for amorphous solids. Possible less restrictive conditions on the evolution of dissipation in plasticity formulations are considered as are implications for stability. The focus is on open questions and issues.

References

1.
Evans
,
D. J.
, and
Searle
,
D. J.
,
2002
, “
The Fluctuation Theorem
,”
Adv. Phys.
,
51
, pp.
1529
1585
.
2.
Jarzynski
,
C.
,
2010
, “
Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale
,”
Séminaire Poincaré XV Temps
, 51, pp.
77
102
.
3.
Evans
,
D. J.
,
Cohen
,
E. G. D.
, and
Morriss
,
G. P.
,
1993
, “
Probability of Second Law Violations in Shearing Steady States
,”
Phys. Rev. Lett
,
71
, pp.
2401
2404
.
Errata Phys. Rev. Lett. 71, 3616, 1993
.
4.
Wang
,
G. M.
,
Sevick
,
E. M.
,
Mittag
,
E.
,
Searles
,
D. J.
, and
Evans
,
D. J.
,
2010
, “
Experimental Demonstration of Violations of the Second Law of Thermodynamics for Small Systems and Short Time Scales
,”
Phys. Rev. Lett.
,
89
, p.
050601
.
5.
Ostoja-Starzewski
,
M.
, and
Laudani
,
R.
,
2020
, “
Violations of the Clausius–Duhem Inequality in Couette Flows of Granular Media
,”
Proc. R. Soc.
,
A476
, p.
20200207
.
6.
Coleman
,
B. D.
, and
Noll
,
W.
,
1964
, “
The Thermodynamics of Elastic Materials With Heat Conduction and Viscosity
,”
Arch. Ration. Mech. Anal.
,
13
, pp.
167
178
.
7.
Coleman
,
B.
, and
Mizel
,
V. J.
,
1967
, “
Existence of Entropy as a Consequence of Asymptotic Stability
,”
Arch. Ration. Mech. Anal.
,
25
, pp.
243
270
.
8.
Dafermos
,
C. M.
,
1979
, “
The Second Law of Thermodynamics and Stability
,”
Arch. Ration. Mech. Anal.
,
70
, pp.
167
179
.
9.
Needleman
,
A.
,
2023
, “
Discrete Defect Plasticity and Implications for Dissipation
,”
Eur. J. Mech. A Solids
,
100
, p.
105002
.
10.
Rivlin
,
R. S.
,
1970
, “Red Herrings and Sundry Unidentified Fish in Nonlinear Continuum Mechanics,”
Inelastic Behaviour of Solids
,
M. G.
Kanninen
,
D.
Adler
,
A. R.
Rosenfield
, and
R. I.
Jaffee
eds.,
McGraw Hill
,
New York
, pp.
117
134
.
11.
Bouchbinder
,
E.
, and
Langer
,
J. S.
,
2009
, “
Nonequilibrium Thermodynamics of Amorphous Materials II: Effective-Temperature Theory
,”
Phys. Rev. E
,
80
, p.
031132
.
12.
Langer
,
J. S.
,
Bouchbinder
,
E.
, and
Lookman
,
T.
,
2010
, “
Thermodynamic Theory of Dislocation-Mediated Plasticity
,”
Acta Mater.
,
58
, pp.
3718
373
.
13.
Falk
,
M. L.
, and
Langer
,
J. S.
,
2011
, “
Deformation and Failure of Amorphous, Solidlike Materials
,”
Ann. Rev. Condens. Matter Phys.
,
2
, pp.
353
373
.
14.
McDowell
,
D. L.
,
2023
, “
Nonequilibrium Statistical Thermodynamics of Thermally Activated Dislocation Ensembles: Part 2—Ensemble Evolution Toward Correlation of Enthalpy Barriers
,”
J. Mater. Sci.
(in press)
.
15.
McDowell
,
D. L.
,
2023
, “
Nonequilibrium Statistical Thermodynamics of Thermally Activated Dislocation Ensembles: Part 3—Taylor–Quinney Coefficient, Size Effects and Generalized Normality
,”
J. Mater. Sci.
(in press)
.
16.
Nye
,
J. F.
,
1953
, “
Some Geometrical Relations in Dislocated Crystals
,”
Acta Metall.
,
1
, pp.
153
162
.
17.
Gurtin
,
M. E.
,
2002
, “
A Gradient Theory of Single-Crystal Viscoplasticity That Accounts for Geometrically Necessary Dislocations
,”
J. Mech. Phys. Solids
,
50
, pp.
5
32
.
18.
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
, pp.
281
310
.
19.
Fleck
,
N. A.
,
Hutchinson
,
J. W.
, and
Willis
,
J. R.
,
2015
, “
Guidelines for Constructing Strain Gradient Plasticity Theories
,”
ASME J. Appl. Mech.
,
82
(
7
), p. 071002.
20.
Hussein
,
A. M.
,
Rao
,
S. I.
,
Michael Uchic
,
D.
,
Dimiduk
,
D. M.
, and
El-Awady
,
J. A.
,
2015
, “
Microstructurally Based Cross-Slip Mechanisms and Their Effects on Dislocation Microstructure Evolution fcc Crystals
,”
Acta Mater.
,
85
, pp.
180
190
.
21.
Malka-Markovitz
,
A.
,
Devincre
,
B.
, and
Mordehai
,
D.
,
2021
, “
A Molecular Dynamics-Informed Probabilistic Cross-Slip Model in Discrete Dislocation Dynamics
,”
Scripta Mater.
,
190
, pp.
7
11
.
22.
Vitek
,
V.
, and
Paidar
,
V.
,
2008
, “Non-Planar Dislocation Cores: A Ubiquitous Phenomenon Affecting Mechanical Properties of Crystalline Materials,”
Dislocations in Solids
,
J. P.
Hirth
, ed.,
Elsevier
,
Amsterdam
, pp.
439
514
.
23.
Wang
,
Z. Q.
, and
Beyerlein
,
I. J.
,
2011
, “
An Atomistically-Informed Dislocation Dynamics Model for the Plastic Anisotropy and Tension–Compression Asymmetry of BCC Metals
,”
Int. J. Plast.
,
27
, pp.
1471
1484
.
24.
Chaussidon
,
J.
,
Robertson
,
C.
,
Rodney
,
D.
, and
Fivel
,
M.
,
2008
, “
Dislocation Dynamics Simulations of Plasticity in Fe Laths at Low Temperature
,”
Acta Mater.
,
56
, pp.
5466
5476
.
25.
Benzerga
,
A. A.
,
Bréchet
,
Y.
,
Needleman
,
A.
, and
Van der Giessen
,
E.
,
2003
, “
Incorporating Three-Dimensional Mechanisms Into Two-Dimensional Dislocation Dynamics
,”
Modell. Simul. Mater. Sci. Eng.
,
12
, pp.
159
196
.
26.
Eshelby
,
J. D.
,
1957
, “
The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems
,”
Proc. R. Soc. Lond. A
,
241
, pp.
376
396
.
27.
Eshelby
,
J. D.
,
1959
, “
The Elastic Field Outside an Ellipsoidal Inclusion
,”
Proc. R. Soc. Lond. A
,
252
, pp.
561
569
.
28.
Vasoya
,
M.
,
Kondori
,
B.
,
Benzerga
,
A. A.
, and
Needleman
,
A.
,
2020
, “
Energy Dissipation Rate and Kinetic Relations for Eshelby Transformations
,”
J. Mech. Phys. Solids
,
136
, p.
103699
.
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