Investigation of low cycle asymmetric torsion/Mazaciklio asimetrinio sukimo tyrimas.
Daunys, M. ; Cesnavicius, R.
1. Introduction
During exploitation the materials of constructions start gradually
accumulate the damage, which finally causes fracture of the
construction. The accumulation of damage occurs as a result of the high
cycle and low cycle damage due to the cyclic overloads, which cause the
elastic-plastic deformation. Especially dangerous is the overload as
cyclically varying stresses exceed the proportional limit of the
material and plastic strain starts, which results the hysteresis loop of
the plastic strain and causes the reduction in material fatigue life up
to thousands or hundreds cycles. It is defined that 75% of fracture of
the mechanical systems' constructions occurs due to material
fatigue [1 6].
The problems of metal fracture remain actual despite years of
long-lasting investigation of the cyclic loading of metals. While
selecting the material, it is necessary to know properties of the
material and the laws change of their characteristics under different
type loading in the areas of periodically varying elastic-plastic
strain. In majority of the modern mechanisms and devices under loading
the elastic-plastic deformation takes place in the stress concentration
areas, near sudden change of the shape, e.g. in key seats, near shafts
diameter changing places, as a result of incorrectly chosen fillet
radius, in welded joints, because of the various welding defects and
etc. [1, 7].
The considerable part of experiments related with the low cycle
fatigue damage were carried out under the axial loading, i.e. under
tension-compression and less of them at pure bending and the smallest
part of scientific publications consider the torsion. The considerable
amounts of the parts, operating in real exploitation conditions, are
exactly under cyclically varying torsion loading (shafts of the
mechanisms, springs and etc.).
Though the cyclically loaded parts work following the symmetrical
cycle, however, different transitional forms also frequently occur. In
real constructions the most common is asymmetrical loading (e.g. in the
stress concentration areas, the crack areas), which results that both
the hysteresis loop's width and also the fatigue life are highly
dependent on the stress ratio [1, 7, 8].
2. Experimental setup and used specimens
The experimental analysis of the monotonic and also the low cycle
torsion, considered in this paper, were carried out at room temperature.
The specimens were tested under symmetric and asymmetric loading and
strain data was recorded up to crack initiation. For the experimental
fatigue analysis under monotonic and symmetric and asymmetric low cycle
torsion the experimental low cycle setup, designed and made at
Engineering design department of Kaunas University of Technology, was
used.
Experimental setup consists of the testing machine with maximum
possible T = 5 kNm moment of torsion and the electronic part, which
records the stress strain diagrams, semicycles and controls the motor
reversal.
[FIGURE 1 OMITTED]
For the monotonic and low cycle symmetric torsion experiments the
tubular shape specimens with t/d = 1/20 working part were used. The
specimens were made of grade 45 steel rod, following the dimensions
shown in Fig. 1, a. During the cyclic torsion in the wall of the tubular
specimen is uniform stress state, i.e. there is no influence of the
stress gradient. The working part of the specimen (l = 30 mm) was chosen
taking into account the previously used torsion specimens. The fillet
radius while passing into the working part of the specimen, was R = 25
mm, aiming to decrease the stress concentration to minimum (the
theoretical stress ratio [a.sub.[sigma]] [approximately equal to] 1.03).
For the asymmetric low cycle torsion experiments the solid circular
cross-section specimens have been used. All specimens were made of the
same grade 45 steel rod following the dimensions presented in the Fig.
1, b.
To determine the torgue T, the resistance wire gauges were glued on
the surface of the tenzometer device with cylindrical working part d =
18.0 mm. The tenzometer device was made of thermal treatment grade 60S2A
spring steel (HRC 42-45). The working strain gauges were glued to the
cylinder's surface along the main strain directions e1 and
[e.sub.3] (at 45[degrees] angle, in opposite sides).
[FIGURE 2 OMITTED]
The torsion strain is measured by the attachment, which identifies
torsion angle [phi] in the working part of the specimen. The device for
torsion angle measurements, presented in Fig. 2, consists of two rings 1
and 2, each of them has bolt fastened half rings, that are attached to
the specimen by means of the 4 conical tip bolts, locating them at
identical angles. Two spring steel plates 3 and 4 are fastened to the
top ring. Working gauges (R = 100 [ohm]) are glued along
tension-compression sides of the plates. Free end of each plate rests on
bolt-adjusted bottom retainer ring. During torsion of the specimen, the
rings turn relative to each other and sprung steel plates act as
cantilever rods during bending [9].
3. Experimental analysis
3.1 Monotonic loading
During the monotonic torsion experiments the monotonic torsion
curve was defined. The monotonic torsion curve in
[[tau].sub.max]--[[gamma].sub.max] coordinates is shown in Fig. 3. The
defined mechanical characteristics of grade 45 steel under torsion are
presented in Table 1.
[FIGURE 3 OMITTED]
3.1 Asymmetric low cycle loading
It was mentioned earlier, that in real constructions most common is
the asymmetric loading, which results that hysteresis loop's width
[[bar.[delta]].sub.k] is highly dependent on the stress ratio
[r.sub.[sigma]]. Fig. 4 shows the stress amplitude dependence both on
the mean stress and the stress ratio [7].
[FIGURE 4 OMITTED]
The hysteresis loop's width [[bar.[delta]].sub.k] of the
semicycles depends both on the stress amplitude [[bar.[sigma]].sub.a]
and mean stress [[bar.[sigma]].sub.m]. These equations and the method
for their determination (Fig. 4) have been used in research works of M.
Daunys, H. Medeksas and R. Sneiderovic to calculate the results of
tension-compression experiments [1, 7, 8]. Thus this dependence may be
written
[bar.[sigma]] = -tg(90 - [phi]][[bar.[sigma]].sub.m] +
[[bar.[sigma]].sub.a] (1)
or
[bar.[sigma]] = ctg[phi][[bar.[sigma]].sub.m] +
[[bar.[sigma]].sub.a] (2)
and introducing the notations ctg[phi] = k and [bar.[sigma]] =
[bar.[sigma]].sub.con], the following is obtained
[[bar.[sigma]].sub.con] = [[bar.[sigma]].sub.a] +
[KAPPA][[bar.[sigma]].sub.m] (3)
where [[bar.[sigma]].sub.con] is conditional stress.
For symmetric cycle, [[bar.[sigma]].sub.m] = 0 and results
[[bar.[sigma]].sub.con] = [[bar.[sigma]].sub.a] + [sigma] (4)
While using the earlier mentioned equations, the experimental
results, despite the stress ratio in coordinates [[bar.e].sub.con] -
[[bar.[delta]].sub.1] are coincident with the results of the symmetric
cycle and the following may be written
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
To simplify the calculations, it was taken, that cyclic
proportional limit [[bar.s]sub.T] is independent on [[bar.e].sub.con]
[1], the number of semicycles k and on stress ratio [r.sub.[sigma]].
Therefore, for the asymmetrical cycle, in all the equations used for the
symmetrical cycle, the initial strain [[bar.e].sub.0] is replaced by
[[bar.e].sub.con], which is defined from the monotonic diagram by
[[bar.[sigma]].sub.con].
For anisotropic materials, two parameters [KAPPA] : [[KAPPA].sub.1]
are for uneven and [[KAPPA].sub.2] for even semicycles and consequently
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for uneven and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for even semicycles.
These parameters characterize the dependence of the strain diagrams on
the stress ratio [r.sub.[sigma]] [8].
The experiments were carried out under negative asymmetric loading
cycles. The cycles were corresponding to loading of the real
constructions and the following stress ratio were chosen: [r.sub.[tau]]
= -0.75 and [r.sub.[tau]] = -0.5.
The experiments of the asymmetric low cycle torsion were carried
out under constant hysteresis loop's width of the first semicycle
([[bar.[delta]].sub.1] = const). The calculated amplitude stress
dependence both on the mean stress and the stress ratio [r.sub.[sigma]]
is shown in Fig. 5.
[FIGURE 5 OMITTED]
From the Eq. (2), as [r.sub.[sigma]] =
[[sigma].sub.min]/[[sigma].sub.max], we can express
[P.sub.1;2] = 1 + [[KAPPA].sub.1;2] 1 + [r.sub.[sigma]]/1 -
[r.sub.[sigma]] (6)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
During the symmetric ([r.sub.[tau]] = -1.0) and asymmetric
stress-limited torsion experiments under stress ratio ([r.sub.[tau]] =
-0.75 and [r.sub.[tau]] = -0.5), the hysteresis loop's width
dependence on the number of loading semicycles k was determined. Fig. 6
shows, that as the number of the loading semicycles k increases, for
grade 45 steel, the loop width [[bar.[delta]].sub.k] remains constant,
thus we have a cyclic stable material.
Where
[[bar.[tau]].sub.0] = [[bar.[tau]].sub.0]/[[bar.[tau]].sub.pl];
[[bar.[delta]].sub.k] = [[delta].sub.k]/[[gamma].sub.pl] (8)
here [[tau].sub.0] is shear stress at the initial semicycle,
[[tau].sub.pl] and [[gamma].sub.pl] are the stress and strain of
proportional limit under torsion.
[FIGURE 6 OMITTED]
During the experiments of the low cycle asymmetrical torsion,
differently than under symmetrical torsion, it was determined, that
grade 45 steel accumulates plastic strain along the initial torsion
direction (Fig. 7, a, b). Then the accumulated plastic strain in initial
torsion direction after k loading semicycles is calculated as follows
[[bar.e].sub.pk] = [[bar.e].sub.0] - [[bar.[sigma]].sub.0] +
[k.summation over (1)] [(-1).sup.k] [[bar.[delta]].sub.k] (9)
where [[bar.[sigma]].sub.0] is stress of the initial semicycle.
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
Loop width dependences of the first and second semicycles on the
conditional initial strain are presented in Fig. 8. The defined cyclic
characteristics of the material are given in Table 2.
4. Accumulated damage under stress-limited loading
Thus, the specimen under stress limited torsion fractures due to
quasistatic damage [d.sub.K], caused by the accumulated plastic strain
[[bar.e].sub.pk], and due to fatigue damage [d.sub.N, resulted by the
cyclic plastic strain, which is characterized by the hysteresis
loop's width [[bar.[delta]].sub.k]. Therefore total damage d may be
written
d = [d.sup.g.sub.K] + [d.sup.1.sub.N] (10)
Fatigue damage is calculated using the following equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
where [k.summation over (1)] [[bar.[delta]].sub.k] is fatigue
damage accumulated during k loading semicycles, [[k.sub.c].summation
over (1)] [[bar.[delta]].sub.k] is fatigue damage accumulated till the
crack initiation.
Quasistatic damage
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
where [e.sub.pk] is accumulated plastic strain after k loading
semicycles, whereas [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
is uniform strain under monotonic loading.
The analytical curves of the cyclically stable grade 45 steel, as
only fatigue damage is taken into account, were calculated applying the
following equation [7, 9, 10]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
Table 3 presents characteristics of the Coffin's curves
[C.sub.2], [C.sub.3], [m.sub.1], [m.sub.2], [m.sub.3], which were
obtained from the experimental data [9] for the grade 45 steel under low
cycle strain limited torsion loading.
Fig. 9 shows graphical relationship between the quasi-static and
fatigue damage for the analysed grade 45 steel under different stress
ratio. The mentioned figure indicates, that Eq. (10) constants are q = l
= 1, whereas damage accumulation by using Eqs. (10)-(12) is presented in
Table 4.
It is seen from Fig. 9 and Table 4, that under asymmetric loading
the quasistatic damage depends on the stress level and is increased at
highest loading levels ([r.sub.[tau]] = -0.75, when [bar.[tau]] = 1.80 -
70.6%; [r.sub.[tau]] = -0.5, when [bar.[tau]] = 1.95 - 82.2%). When
stress level decreases, the part of quasi-static damage also decreases
([r.sub.[tau]] = -0.75, when [bar.[tau]] = = 1.33 - 27.8%; [r.sub.[tau]]
= -0.5, when [bar.[tau]] = 1.41 - 48.1%).
[FIGURE 9 OMITTED]
Fig. 10 shows the experimental low cycle fatigue curves for
symmetric and asymmetric cycles and also the fatigue curves, as only
fatigue damage is taken into account. It is seen from this figure, the
same as from the Table 4, that quasistatic damage more significant
decrease in the fatigue life under higher strain level. At medium
loading levels, when [tau] = 1.60 and asymmetry [r.sub.[tau]] = -0.75,
the experimental curves show, that fatigue life, if compared to the
theoretical fatigue curves, diminishes from [N.sub.c] =2840 to [N.sub.c]
=1830 cycles. Under the same loading level [tau] = 1.60 and asymmetry
[r.sub.[tau]] = -0.5, results decrease in fatigue life from [N.sub.c] =
9050 to [N.sub.c] = 5010 cycles.
[FIGURE 10 OMITTED]
5. Conclusions
Grade 45 steel under stress-limited monotonic torsion, low cycle
symmetric and asymmetric torsion loading was analysed, using hollow and
solid specimens of the circular cross-section.
1. It was determined that for the analysed grade 45 steel the
hysteresis loop width is independent on the number of the loading
semicycles k under symmetric ([r.sub.[tau]] = -1.0) and asymmetric
([r.sub.[tau]] = -0.75, -0.5) torsion, i.e. this steel is cyclically
stable and parameter [alpha] = 0.
2. Under the asymmetric loading, when stress ratio are
[r.sub.[tau]] = -0.75 and [r.sub.[tau]] = -0.5, the accumulation of the
plastic strain in the direction of the initial loading, which does
not occur during the symmetric loading cycle was determined.
3. The analysed case of the symmetric loading showed smaller
fatigue life if compared to that of the asymmetric loading, whereas
under asymmetric loading, due to the accumulation of plastic strain, the
quasistatic damage occurs, which reduces the fatigue life at the highest
levels of loading ([r.sub.[tau]] = -0.75, as [bar.[tau]] = 1.80 - 70.6%;
[r.sub.[tau]] = -0.5, when [bar.[tau]] = 1.95 - 82.2%) and at low levels
of loading ([r.sub.[tau]] = -0.75, as [bar.[tau]] = 1.33 - 27,8%;
[r.sub.[tau]] = -0.5, when [bar.[tau]] = 1.41 - 48.1%).
Received August 30, 2010
Accepted December 07, 2010
References
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M. Daunys, Kaunas University of Technology, Kestucio 27, Kaunas,
44312, Lithuania, E-mail: mykolas.daunys@ktu.lt
R. Cesnavicius, Kaunas University of Technology, Kestucio 27,
Kaunas, 44312, Lithuania, E-mail: ramunas.cesnavicius@ktu.It
Table 1
Mechanical characteristics of grade 45 steel
Series [[tau].sub.pr], MPa [[tau].sub.0.3], MPa
1 174 226
2 224 209
3 188 211
Mean
[bar.x] 195 215
Series [[tau].sub.u], MPa [[gamma].sub.u], %
1 4245 23.4
2 435 25.2
3 420 19.7
Mean
[bar.x] 426 22.7
Table 2
Cyclic characteristics of the grade 45 steel
Hollow specimens, Solid specimens,
[r.sub.[[tau]] = -1.0 [r.sub.[[tau]] = -1.0
A [[bar.S].sub.T] [alpha] A [[bar.S].sub.T] [alpha]
1.1 1.40 0 1.1 1.40 0
Solid specimens, r = -0.75
[[bar.S]. [[kappa]. [[kappa].
[A.sub.1] [A.sub.2] sub.T] [alpha] sub.1] sub.2]
0.51 0.55 1.45 0 -0.25 -0.26
Solid specimens, [r.sub.[tau]] = -0.75
[[bar.S]. [[kappa]. [[kappa].
[A.sub.1] [A.sub.2] sub.T] [alpha] sub.1] sub.2]
0.23 0.29 1.40 0 -0.25 -0.26
Table 3
Values of Coffin's constants C and m
[C.sub.2] [C.sub.3] [m.sub.1] [m.sub.2] [m.sub.3]
727 440 0.49 0.58 0.88
Table 4
Accumulated damage under asymmetric loading
Solid specimens, [r.sub.[tau]] = -0.75
[bar.[tau]] 1.80 1.64 1.56 1.41 1.33
[d.sub.N] 0.303 0.406 0.416 0.428 0.715
[d.sub.K] 0.728 0.559 0.432 0.395 0.275
d 1.031 0.966 0.847 0.823 0.990
Solid specimens, [r.sub.[[tau] = -0.5
[bar.[tau]] 1.95 1.80 1.72 1.64 1.41
[d.sub.N] 0.182 0.314 0.362 0.400 0.544
[d.sub.K] 0.839 0.678 0.656 0.622 0.503
d 1.021 0.992 1.018 1.022 1.047
