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Transient heating of a rotating elastic-plastic solid shaft
E. Arslan*,1, W. Mack2, and A. N. Eraslan3
1 Department
of Mechanical Engineering, Inonu University, Malatya, Turkey
of Mechanics and Mechatronics, Vienna University of Technology, Wiedner Hauptstraße 8-10, A-1040
Vienna, Austria
3 Department of Engineering Science, Middle East Technical University, Ankara, 06531 Turkey
2 Institute
Based on Tresca’s yield criterion and the flow rule associated with it, the transient plasticization in a rotating elastic-plastic
solid shaft with temperature dependent yield stress subject to a temperature cycle is studied. Special attention is paid to the
residual stresses.
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1
Introduction
Rotating machine parts that exhibit purely elastic behaviour may undergo plastic deformation when exposed to an
inhomogeneous temperature field. This is caused by the additional stress due to the temperature gradient, on the one hand,
and by softening of the material due to heating, on the other hand; the latter effect may cause plasticization even for
homogeneous heating (see, e.g., [1,2]). Hence, the subject of the present (analytical) investigation is a rotating solid cylinder
in a state of generalized plane strain, the free surface of which is subjected to a temperature cycle according to Fig. 1a (where
also the meaning of the symbols can be seen). The material is presumed to obey Tresca’s yield criterion and the flow rule
associated with it. Furthermore, the yield stress decreases linearly with temperature (compare [3]), i.e., σth=σ0(1-βT).
2
Elastic-plastic states
1.2
(b)
1.85
T1.0m
1.80
t
0.8
1.75
unloaded region
u
plastic
region I
1.70
0.6
Time
Temperature of the surface (r=b)
(a)
1.65
plastic
region II
r1
0.4
approximated
unloading
r2
elastic
region
1.60
0.2
1.55
0.0
tt
tt  tc
2t t  t c
0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0
1.50
0
0.0
0.2
Time
0.4
0.6
Radial coordinate
b
0.8
1.0
Fig. 1. a Sketch of the prescribed surface temperature of the shaft; b Sketch of the evolution of the plastic regions.
With the above presuppositions, a certain type of behaviour is found to be typical [4]. That is, at the axis of the shaft two
plastic regions emerge simultaneously. The image points of the inner plastic region I lie on an edge and the ones of the outer
plastic region II on an adjacent side of the Tresca prism as in related cylinder problems [5] (see Fig. 1b), i.e.,
 r   z   th ,     z   th
    z   th
for
for 0  r  r1 ,
r1  r  r2 .
(1)
(2)
____________________
*
Corresponding author: e-mail erarslan@metu.edu.tr, Phone +90 312 210 2387, Fax +90 312 210 4462
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2
Section 00
At a certain time t=tu the plastic region does not expand any more, and elastic unloading occurs at the plastic-elastic border
r=r2. Thereafter, the unloaded region spreads rapidly into the plastic zone. The exact calculation of this procedure is very
intricate but, as is shown in [4], not really necessary. Hence, sudden unloading in the entire plastic zone at t=tu is assumed
[6], yielding a very good approximation.
3
Residual stresses
As is expected, a remarkable redistribution of the stresses takes place for constant angular speed in the course of the
temperature cycle. From the engineering point of view, the stress distribution after (complete) cooling is of particular
interest. For the numerical calculations it is convenient to introduce the non-dimensional quantities
q
Tm G
,
0
r
x ,
b
 *   Tm ,
 i* 
i
,
0
(3)

t
b
2
,
2 
 2b2
,
0
(4)
where κ means the thermal diffusivity and α the coefficient of (linear) thermal expansion; the parameter q is a measure for
the maximum surface temperature during the transient heating. Figure 2a shows the stresses after cooling for q=0.6, β*=0.2,
ν=0.3 and an angular speed Ω=1.5875, which is 90% of the elastic limit angular speed at reference temperature [7], Fig. 2b
depicts the ones at subsequent stand-still. In both figures curves 1 represent the stresses for a fast temperature cycle with
τt=1.2 and τc=0.3, curves 2 the ones for very slow heating and cooling giving rise to a quasi-homogeneous temperature field
(τt=60, τc=1.0). While for the parameter values under consideration the effect of thermal softening only is small (curves 2),
the inhomogeneity of the temperature field for the fast cycle has a pronounced effect particularly on the axial residual stress
at the centre.
(a) 1.5
Non-dimensional stresses
2
0.9
 *
 r*
 z*
0.10
1
Non-dimensional stresses
1
1.2
(b) 0.12
 *
 r*
 z*
0.6
1
0.3
2
0.0
0.08
0.06
0.04
0.02
1
2
0.00
2
-0.3
-0.02
1
-0.04
-0.6
0.0
0.2
0.4
0.6
0.8
Non-dimensional radial coordinate
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Non-dimensional radial coordinate
Fig. 2. Stress distribution after cooling; a at Ω=1.5875, b at subsequent stand-still; curves 1: fast temperature cycle, curves 2: very
slow temperature cycle.
References
[1] W. Mack and M. Plöchl, Int. J. Engng. Sci. 38, 921-938 (2000).
[2] W. Mack, M. Plöchl and U. Gamer, Z. Angew. Math. Mech. (ZAMM) 81, S351-S352 (2001).
[3] H. Lippmann, Int. J. Plast. 8, 567-582 (1992).
[4] E. Arslan, W. Mack and A. N. Eraslan, Acta Mech. 195, 129-140 (2008).
[5] U. Gamer and M. Sayir, J. Appl. Math. Phys. (ZAMP) 35, 601-617 (1984).
[6] W. Mack and U. Gamer, Forsch. Ing. Wes. 54, 48-52 (1988).
[7] W. Mack, Tech. Mech. 12, 119-124 (1991).
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