A statistical mechanical theory of forest hardening is developed in which yielding arises as a phase transition. For simplicity, we consider the case of a single dislocation loop moving on a slip plane through randomly distributed forest dislocations, which we treat as point obstacles. The occurrence of slip at the sites occupied by these obstacles is assumed to require the expenditure of a certain amount of work commensurate with the strength of the obstacle. The case of obstacles of infinite strength is treated in detail. We show that the behavior of the dislocation loop as it sweeps the slip plane under the action of a resolved shear stress is identical to that of a lattice gas, or, equivalently, to that of the two-dimensional spin-1/2 Ising model. In particular, there exists a critical temperature Tc below which the system exhibits a yield point, i.e., the slip strain increases sharply when the applied resolved shear stress attains a critical value. Above the critical temperature the yield point disappears and the slip strain depends continuously on the applied stress. The critical exponents, which describe the behavior of the system near the critical temperature, coincide with those of the two-dimensional spin-1/2 Ising model.

1.
Abeyaratne
R
Knowles
JK
1988
On the dissipative response due to discontinuous strains in bars of unstable elastic material
International Journal of Solids and Structures
, vol. 
66
 
2
(pg. 
1021
-
1044
)
2.
Baskes
MI
Hoagland
RG
Tsuji
T
1997
An atomistic study of the strength of an extended dislocation barrier
3.
Baxter
RJ
Exactly Solved Models in Statistical Mechanics
Academic Press
London
4.
Binder
K
1986
Monte Carlo Methods in Statistical Physics
2
Springer-Verlag
New York
5.
Binder
K
1987
Applications of the Monte Carlo Method in Statistical Physics
2
Springer-Verlag
New York
6.
Binney
JJ
Dowrick
NJ
Fisher
AJ
Newman
MEJ
1992
The Modern Theory of Critical Phenomena
Clarendon Press
Oxford
7.
Chaikin
PM
Lubensky
TC
1995
Principles of condensed matter physics
Cambridge University Press
Cambridge, UK
8.
Chrzan
DC
Mills
MJ
1993
Collective behavior and superdislocation motion in l12 alloys
Materials Science and Engineering A
, vol. 
66
 
2
(pg. 
82
-
92
)
9.
Chrzan
DC
Mills
MJ
1994
Criticality in the plastic deformation of l12 intermetallic compounds
Physical Review B
, vol. 
66
 
2
(pg. 
30
-
42
)
10.
Cuitin˜o
AM
1996
Effect of temperature and stacking fault energy on the hardening of fcc crystals
Materials Science and Engineering
, vol. 
66
 
2
(pg. 
104
-
116
)
11.
Cuitin˜o
AM
Ortiz
M
1992
Computational Modelling of Single Crystals
Modelling and Simulation in Materials Science and Engineering
, vol. 
66
 
2
(pg. 
255
-
263
)
12.
Cuitin˜o
AM
Ortiz
M
1993
Constitutive Modeling of L12 Intermetallic Crystals
Materials Science and Engineering
, vol. 
66
 
2
(pg. 
111
-
123
)
13.
Ericksen
JL
1975
Equilibrium of bars
Journal of Elasticity
, vol. 
66
 
2
(pg. 
191
-
201
)
14.
Feynman
RP
1972
Statistical Mechanics
Addison-Wesley
Reading, MA
15.
Foreman
AJ
1995
Dislocation energies in anisotropic crystals
Acta Metallurgica
, vol. 
66
 
2
(pg. 
322
-
330
)
16.
Foreman
AJE
Makin
MJ
1966
Dislocation Movement through Random Arrays of Obstacles
Philosophical Magazine
, vol. 
66
 
2
pg. 
911
 
17.
Foreman
AJE
Makin
MJ
1967
Dislocation movement through random arrays of obstacles
Canadian J. Phys
, vol. 
66
 
2
pg. 
273
 
18.
Grosskreutz
JC
Mughrabi
H
1975
Description of the work-hardened structure at low temperature in cyclic deformation
Constitutive Equations in Plasticity
Argon
AS
M.I.T. Press
Cambridge, MA
(pg. 
251
-
326
)
19.
Guggenheim
EA
1945
The Principle of Corresponding States
Journal of Chemical Physics
, vol. 
66
 
2
(pg. 
253
-
261
)
20.
Hansen
N
Kuhlmann-Wilsdorff
D
1986
Low Energy Dislocation Structures due to Unidirectional Deformation at Low Temperatures
Materials Science and Engineering
, vol. 
66
 
2
(pg. 
141
-
161
)
21.
Hirth
JP
Lothe
J
1968
Theory of Dislocations
McGraw-Hill
New York
22.
James
RD
1979
Co-exitent phases in the one-dimensional theory of elastic bars
Archive for Rational Mechanics and Analysis
, vol. 
66
 
2
(pg. 
99
-
140
)
23.
Kocks
UF
1964
Latent hardening and secondary slip in aluminum and silver
Transactions of the Metallurgical Society of the AIME
, vol. 
66
 
2
pg. 
1160
 
24.
Kocks
UF
Franciosi
P
Kawai
M
1991
A forest model of latent hardening and its applications to polycrystal deformation
Textures and Microstructares
, vol. 
66
 
2
(pg. 
1103
-
1114
)
25.
Koonin
SE
Meredith
DC
1990
Computational Physics
Addison-Wesley
Reading, MA
26.
Kosterlitz
JM
Thouless
DJ
1972
Long Range Order and Metastability in Two Dimensional Solids and Superfluids
Journal of Physics C
, vol. 
66
 
2
(pg. 
L124
-
L126
)
27.
Kosterlitz
JM
Thouless
DJ
1973
Ordering, Metastability and Phase Transitions in Two-Dimensional Systems
Journal of Physics C
, vol. 
66
 
2
(pg. 
1181
-
1203
)
28.
Kovacs
I
1967
The Mechanism of the Work-Harderning in F.C.C. Metals
Acta Metallurgica
(pg. 
1731
-
1736
)
29.
Kova'cs
I
Zsoldos
L
1973
Dislocation and plastic deformation
Pergamon Press
Oxford, UK
30.
Kuhlmann-Wilsdorf
D
1979
Dislocation behavior in fatigue. IV. Quantitative interpretation of friction stress and back stress derived from hysteresis loops
Materials Science and Engineering
, vol. 
66
 
2
(pg. 
231
-
245
)
31.
Kuhlmann-Wilsdorf
D
1989
Theory of plastic deformation: properties of low energy dislocation structures
Materials Science and Engineering
, vol. 
66
 
2
pg. 
1
 
32.
Mughrabi
H
1975
Description of the Dislocation Structure after Unidirectional Deformation at Low Temperatures
Constitutive Equations in Plasticity
Argon
AS
M.I.T. Press
Cambridge, MA
(pg. 
199
-
250
)
33.
Nabarro
FRN
1967
Theory of crystal dislocations
Oxford University Press
Oxford, UK
34.
Nelson
DR
Halperin
BI
1979
Dislocation-Mediated Melting in Two Dimensions
Physical Review B
, vol. 
66
 
2
(pg. 
2457
-
2484
)
35.
Ortiz
M
Popov
EP
1962
A Statistical Theory of Polycrystalline Plasticity
Proceedings of the Royal Society of London
, vol. 
66
 
2
(pg. 
439
-
458
)
36.
Ortiz
M
Repetto
EA
1998
Nonconvex energy minimization and dislocation structures in ductile single crystals
Journal of the Mechanics and Physics of Solids
 
in press
37.
Phillips
R
Shenoy
V
1998
 
manuscript in preparation
38.
Stanley
HE
1971
Introduction to Phase Transitions and Critical Phenomena
Oxford University Press
Oxford, UK
39.
Yeomans
JM
1992
Statistical Mechanics of Phase Transitions
Clarendon Press
Oxford, UK
40.
Young
AP
1979
Melting and the Vector Coulomb Gas in two Directions
Physical Review B
, vol. 
66
 
2
(pg. 
1855
-
1866
)
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