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

[1.] Daunys, M. Strength and Fatigue Life under Low Cycle Non-Stationary Loading. -Vilnius: Mokslas, 1989. -256p. (in Russian).

[2.] Krenevicius, A., Juchnevicius, Z. Load distribution in the threaded joint subjected to bending. -Mechanika. -Kaunas: Technologija, 2009, No.4(78), p.12-16.

[3.] Djakov, I., Prentkovskis, O. Optimization problems in designing automobiles. -Transport. -Vilnius: Technika, 2008, vol.23, No.4, p.316-322.

[4.] Kala, Z., Melcher, J., Puklicky, L. Material and geometrical characteristics of structural steels based on statistical analysis of metallurgical products. -Journal of civil engineering and management. -International re search and Achievements. -Vilnius: Technika, 2009, vol.15, No.3, p.299-307.

[5.] Gasiak, G., Pawliczek, R. Application of an energy model for fatigue life prediction of construction steles under bending, torsion and synchronous bending and torsion. -International Journal of Fatigue 25. -Elsevier, 2003; p.1339=1346.

[6.] Brazenas, A., Vaiciulis, D. Determination of stresses and strains in two-layer mechanically inhomogeneous pipe subjected to internal pressure at elastic plastic loading. -Mechanika. -Kaunas: Technologija, 2009, No.6(80), p.12-17.

[7.] Daunys, M. Cycle Strength and Durability of Structures. -Kaunas: Technologija, 2005. -286p. (in Lithuanian).

[8.] Medeksas, H. Strength and fracture under low cycle loading at elevated temperatures: summary: mechanics thesis for degree of Doctors of habilitation. -Kaunas: Technologija, 1994.-25p.

[9.] Daunys, M., Cesnavicius, R. Low cycle stress strain curves and fatigue under tension-compression and torsion. -Mechanika, -Kaunas: Technologija, 2009, No.6(80), p.5-11.

[10.] Daunys, M., Rimovskis, S. Analysis of circular cross-section element, loaded by static and cyclic elastic-plastic pure bending. -International Journal of Fatigue. -Elsevier, 2006, 28, p.211-222.

M. Daunys, Kaunas University of Technology, Kestucio 27, Kaunas, 44312, Lithuania, E-mail: [email protected]

R. Cesnavicius, Kaunas University of Technology, Kestucio 27, Kaunas, 44312, Lithuania, E-mail: [email protected]

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