A semi-analytic solution for plastic collapse of a thin annular disk subject to thermomechanical loading is presented. It is assumed that the yield criterion depends on the hydrostatic stress. A distinguished feature of the boundary value problem considered is that there are two loading parameters. One of these parameters is temperature and the other is pressure over the inner radius of the disk. The general qualitative structure of the solution at plastic collapse is discussed in detail. It is shown that two different plastic collapse mechanisms are possible. One of these mechanisms is characterized by strain localization at the inner radius of the disk. The entire disk becomes plastic according to the other plastic collapse mechanism. In addition, two special regimes of plastic collapse are identified. According to one of these regimes, plastic collapse occurs when the entire disk is elastic, except its inner radius. According to the other regime, the entire disk becomes plastic at the same values of the loading parameters at which plastic yielding starts to develop.

References

1.
Reid
,
L.
,
1997
, “
Incorporating Hole Cold Expansion to Meet Durability and Damage Tolerance Airworthiness Objectives
,”
SAE
Technical Paper No. 972624.10.4271/972624
2.
Ghorashi
,
M.
, and
Daneshpazhooh
,
M.
,
2001
, “
Limit Analysis of Variable Thickness Circular Plates
,”
Comp. Struct.
,
70
, pp.
461
468
.10.1016/S0045-7949(00)00140-1
3.
Lenard
,
J.
, and
Haddow
,
J. B.
,
1972
, “
Plastic Collapse Speeds for Rotating Cylinders
,”
Int. J. Mech. Sci.
,
14
, pp.
285
292
.10.1016/0020-7403(72)90084-7
4.
Hein
,
K.
, and
Heinloo
,
M.
,
1990
, “
The Design of Nonhomogeneous Equi-Strength Annular Discs of Variable Thickness Under Internal and External Pressures
,”
Int. J. Solids Struct.
,
26
, pp.
617
630
.10.1016/0020-7683(90)90033-R
5.
Gao
,
X.-L.
, and
Atluri
,
S. N.
,
1997
, “
An Elastic-Plastic Analytical Solution for the Shrink-Fit Problem With a Thin Strain-Hardening Hub and an Elastic Solid Shaft
,”
Math. Mech. Solids
,
2
, pp.
335
349
.10.1177/108128659700200307
6.
Ma
,
G.
,
Hong
,
H.
, and
Miyamoto
,
Y.
,
2001
, “
Limit Angular Velocity of Rotating Disc With Unified Yield Criterion
,”
Int. J. Mech. Sci.
,
43
, pp.
1137
1153
.10.1016/S0020-7403(00)00065-5
7.
Eraslan
,
A. N.
, and
Argeso
,
H.
,
2002
, “
Limit Angular Velocity of Variable Thickness Rotating Discs
,”
Int. J. Solids Struct.
,
39
, pp.
3109
3130
.10.1016/S0020-7683(02)00249-4
8.
Bhowmick
,
S.
,
Misra
,
D.
, and
Saha
,
K. N.
,
2008
, “
Approximate Solution of Limit Angular Speed for Externally Loaded Rotating Solid Disc
,”
Int. J. Mech. Sci.
,
50
, pp.
163
174
.10.1016/j.ijmecsci.2007.07.004
9.
Debski
,
R.
, and
Zyczkowski
,
M.
,
2002
, “
On Decohesive Carrying Capacity of Variable-Thickness Annular Perfectly Plastic Discs
,”
Z. Angew. Math. Mech.
,
82
, pp.
655
669
.10.1002/1521-4001(200210)82:10<655::AID-ZAMM655>3.0.CO;2-V
10.
Masri
,
R.
,
Cohen
,
T.
, and
Durban
,
D.
,
2010
, “
Enlargement of a Circular Hole in a Thin Plastic Sheet: Taylor-Bethe Controversy in Retrospect
,”
Q. Mech. Appl. Math.
,
63
, pp.
589
616
.10.1093/qjmam/hbq013
11.
Alexandrov
,
S.
, and
Alexandrova
,
N.
,
2001
, “
Thermal Effects on the Development of Plastic Zones in Thin Axisymmetric Plates
,”
J. Strain Anal. Eng. Des.
,
36
, pp.
169
176
.10.1243/0309324011512720
12.
Lippmann
,
H.
,
1992
, “
The Effect of a Temperature Cycle on the Stress Distribution in a Shrink Fit
,”
Int. J. Plast.
,
8
, pp.
567
582
.10.1016/0749-6419(92)90031-7
13.
Mack
,
W.
,
1993
, “
Thermal Assembly of an Elastic-Plastic Hub and a Solid Shaft
,”
Arch. Appl. Mech.
,
63
, pp.
42
50
.10.1007/BF00787908
14.
Bengeri
,
M.
, and
Mack
,
W.
,
1994
, “
The Influence of the Temperature Dependence of the Yield Stress on the Stress Distribution in a Thermally Assembled Elastic-Plastic Shrink Fit
,”
Acta Mech.
,
103
, pp.
243
257
.10.1007/BF01180229
15.
Mack
,
W.
, and
Bengeri
,
M.
,
1994
, “
Thermal Assembly of an Elastic-Plastic Shrink Fit With Solid Inclusion
,”
Int. J. Mech. Sci.
,
36
, pp.
669
705
.10.1016/0020-7403(94)90086-8
16.
Tarn
,
J.-Q.
,
2001
, “
Exact Solutions for Functionally Graded Anisotropic Cylinders Subjected to Thermal and Mechanical Loads
,”
Int. J. Solids Struct.
,
38
, pp.
8189
8206
.10.1016/S0020-7683(01)00182-2
17.
Eraslan
,
A. N.
, and
Akis
,
T.
,
2003
, “
On the Elastic-Plastic Deformation of a Rotating Disc Subjected to a Radial Temperature Gradient
,”
Mech. Based Des. Struct. Mach.
,
31
, pp.
529
561
.10.1081/SME-120023170
18.
Hodge
,
P. G.
, Jr.
, and
Sun
,
C.-K.
,
1968
, “
General Properties of Yield-Point Load Surfaces
,”
ASME J. Appl. Mech.
,
35
, pp.
107
110
.10.1115/1.3601121
19.
Spitzig
,
W. A.
,
Sober
,
R. J.
, and
Richmond
,
O.
,
1976
, “
The Effect of Hydrostatic Pressure on the Deformation Behavior of Maraging and HY-80 Steels and Its Implications for Plasticity Theory
,”
Metallurg. Trans.
,
7A
, pp.
1703
1710
.10.1007/BF02817888
20.
Kao
,
A. S.
,
Kuhn
,
H. A.
,
Spitzig
,
W. A.
, and
Richmond
,
O.
,
1990
, “
Influence of Superimposed Hydrostatic Pressure on Bending Fracture and Formability of a Low Carbon Steel Containing Globular Sulfides
,”
ASME J. Eng. Mater. Technol.
,
112
, pp.
26
30
.10.1115/1.2903182
21.
Wilson
,
C. D.
,
2002
, “
A Critical Reexamination of Classical Metal Plasticity
,”
ASME J. Appl. Mech.
,
69
, pp.
63
68
.10.1115/1.1412239
22.
Liu
,
P. S.
,
2006
, “
Mechanical Behaviors of Porous Metals Under Biaxial Tensile Loads
,”
Mater. Sci. Eng.
,
A422
, pp.
176
183
.10.1016/j.bbr.2011.03.031
23.
Drucker
,
D. C.
, and
Prager
,
W.
,
1952
, “
Soil Mechanics and Plastic Analysis for Limit Design
,”
Q. Appl. Math.
,
10
, pp.
157
165
.
24.
Alexandrov
,
S. E.
,
Lomakin
,
E. V.
, and
Jeng
,
Y.-R.
,
2010
, “
Effect of the Pressure Dependency of the Yield Condition on the Stress Distribution in a Rotating Disc
,”
Dokl.-Phys.
,
55
, pp.
606
608
.10.1134/S1028335810120050
25.
Alexandrov
,
S.
,
Jeng
Y.-R.
, and
Lomakin
E.
,
2011
, “
Effect of Pressure Dependency of the Yield Criterion on the Development of Plastic Zones and the Distribution of Residual Stresses in Thin Annular Discs
,”
ASME J. Appl. Mech.
,
78
(
3
), 031012.10.1115/1.4003361
26.
Houghton
,
S. J.
, and
Campbell
,
S. K.
,
2012
, “
Identifying the Residual Stress Field Developed by Hole Cold Expansion Using Finite Element Analysis
,”
Fat. Fract. Eng. Mater. Struct.
,
35
, pp.
74
83
.10.1111/j.1460-2695.2011.01616.x
27.
Drucker
,
D. C.
,
Prager
,
W.
, and
Greenberg
,
N. J.
,
1952
, “
Extended Limit Design Theorems for Continuous Media
,”
Q. Appl. Math.
,
9
, pp.
381
389
.
28.
Kleiber
,
M.
, and
Kowalczyk
,
P.
,
1996
, “
Sensitivity Analysis in Plane Stress Elasto-Plasticity and Elasto-Viscoplasticity
,”
Comput. Meth. Appl. Mech. Eng.
,
137
, pp.
395
409
.10.1016/S0045-7825(96)01072-9
29.
Ball
,
D. L.
,
1995
, “
Elastic-Plastic Stress Analysis of Cold Expanded Fastener Holes
,”
Fat. Fract. Eng. Mater. Struct.
,
18
, pp.
47
63
.10.1111/j.1460-2695.1995.tb00141.x
30.
Avril
,
S.
,
Pierron
,
F.
,
Pannier
,
Y.
, and
Rotinat
,
R.
,
2008
, “
Stress Reconstruction and Constitutive Parameter Identification in Plane-Stress Elasto-Plastic Problems Using Surface Measurements of Deformation Fields
,”
Exp. Mech.
,
48
, pp.
403
419
.10.1007/s11340-007-9084-2
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