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
)
This content is only available via PDF.
You do not currently have access to this content.