Low cycle stress strain curves and fatigue under tension-compression and torsion/Mazaciklio deformavimo ir suirimo kreives esant tempimui-gniuzdymui ir sukimui.

Daunys, M. ; Cesnavicius, R.

1. Introduction

During exploitation damage gradually appears in the constructions materials and results their fracture. The gradually accumulated damage depends on material properties, magnitude and character of the time-dependent stress and strain variation, environment conditions. It was observed, that 75 % of fracture in mechanical constructions is causes by the material fatigue. Especially dangerous are the overloads, as cyclically varying loading exceeds the proportionality limit of the material and causes plastic strain and formation of the hysteresis loop, while durability of the material decreases to thousands or hundreds of cycles [1]. In most mechanisms and devices under loading the elastic-plastic strain appears in stress concentration areas, near the 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. [2, 3]. Under cyclic elastic-plastic loading, after the cycle number of hundreds--thousands, the fatigue crack appears which commonly causes failures with hardly predictable outcome.

The problems of metal fracture remain actual despite years of long-lasting investigation of the cyclic loading of metals [4]. While selecting the material, it is necessary to know properties and change laws of their characteristics under different type loading in the areas of the periodically varying elastic-plastic strain. Most common are the following three types of loading: tension-compression, bending and torsion [5].

If compared to tension-compression and bending tests, the number of performed low cycle torsion loading tests is not so considerable. It should be noted, that a large amount of the parts in real operating conditions, i.e. shafts, springs and others parts of the mechanisms, are exactly under cyclically varying torsion loading [6, 7].

2. Experimental setup and used specimens

All performed experimental analyses: monotonous tension, monotonous torsion, low cycle tension-compression and low cycle torsion were carried out under ambient temperature. For both the mentioned cases, the specimens were under symmetric loading and experimental data was registered up to crack initiation.

For monotonous tension and low cycle tension-compression fatigue analysis the experimental low cycle setup, designed and made at Machine Design Department of the Kaunas University of Technology, was used. Experimental setup consists of 50 kN testing machine and an electronic part, which is designed to record the stress strain diagrams, semicycles and control the motor reversal.

[FIGURE 1 OMITTED]

The specimens of circular cross-section have been used for the monotonous tension and low cycle tension-compression experiments. The specimens were made of the grade 45 steel rods, following the dimensions presented in Fig. 1.

For the monotonous and low cycle torsion fatigue tests the experimental low cycle setup with T=500 Nm torgue and the same electronic equipment, as in tension-compression analysis, was used.

[FIGURE 2 OMITTED]

Tubular shape specimens with t/d=1/20 working part were used for the experiments. The specimen is shown in Fig. 2. During the cyclic torsion uniform stress state is produced within the wall of the tubular specimen, i.e. the stress gradient does not have the influence. To fulfil the working part of the test, the same fillet radius R = 25 mm was used for both the torsion and tension-compression specimens, aiming to decrease the stress concentration to minimum (the theoretical stress concentration coefficient [[alpha].sub.[sigma]] [approximately equal to] 1.03).

To determine the torque T, resistance wire gauges were glued on the surface of the device with cylindrical working part d=18.0 mm. This device is made of the thermal treatmed grade 60S2A spring steel (HRC 42-45). The working strain gauges were glued to the cylinder's surface along the main strain directions [e.sub.1] and [e.sub.3] (at 45 [degrees] angle, in opposite sides).

[FIGURE 3 OMITTED]

The torsion strain is measured by the attachment, which identifies the torsion angle [phi] in the working part of the specimen. The device for torsion angle measurements, presented in Fig. 3, 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.

3. Experimental analysis

3.1. Investigation of the monotonous loading

During the experiments of monotonous loading, the monotonous tension and monotonous torsion curves were obtained. The curves of the monotonous tension and torsion in coordinates [[sigma].sub.i] - [e.sub.i], and [[tau].sub.max] - [[gamma].sub.max] are presented in Figs. 4 and 5. The determined mechanical characteristics of the grade 45 steel under tension are given in Table 1 and under torsion--in Table 2.

The curves of monotonous tension in [[sigma].sub.i] - [e.sub.i] coordinates were obtained applying the equalities

[[sigma].sub.i] = [[sigma].sub.l]; [e.sub.i] = [e.sub.l] (1)

[FIGURE 4 OMITTED]

The curves of monotonous torsion in [[sigma].sub.i] - [e.sub.i] coordinates were obtained by the Eqs. 2.

[[sigma].sub.3] [square root 3[tau]]; [e.sub.i] = [[gamma]/[square root 3] (2)

[FIGURE 5 OMITTED]

The curves of monotonous tension in [[tau].sub.max] - [[gamma].sub.max] coordinates were obtained by

[[tau].sub.max] = [[sigma].sub.1]/2; [[gamma].sub.max] = 1.5[e.sub.l] (3)

The curves of monotonous torsion in [[tau]s.ub.max] - [[gamma].sub.max] coordinates were obtained by the Eq. 4.

[[tau].sub.max] = [tau]; [[gamma].sub.max] = [gamma] (4)

It is seen from the Figs. 4 and 5 that monotonous tension and torsion curves in [[sigma].sub.i] - [e.sub.i] coordinates are closer than the same curves in [[tau].sub.max] - [[gamma].sub.max] coordinates.

3.2. Low cycle stress strain curves

Under stress limited low cycle loading, the determined hysteresis loop width dependence both on the number of loading semicycles k and loading level a [degrees], is presented in Fig. 6. The mentioned data was obtained during the tension-compression experiments, using loading levels from [[bar.[sigma]].sub.0] = 1.08 to [[bar.[sigma]].sub.0] = 1.93, where

[[bar.[sigma]].sub.0] = [[sigma].sub.0]/[[sigma].sub.pl]; [[bar.[delta]].sub.k] = [[bar.[sigma]].sub.k]/[e.sub.pl] (5)

here [[sigma].sub.0] is loading stress amplitude, [[delta].sub.k] is width of the hysteresis loop of plastic strain for loading semicycle k, [[sigma].sub.pl] and [e.sub.pl] are stress and strain of proportionality limit [1].

[FIGURE 6 OMITTED]

Fig. 6 shows, that at increasing the number k of the loading semicycles, the hysteresis loop's width [bar.[delta]] for grade 45 steel is not changing, i.e. remains constant, consequently we have stable material.

Low cycle torsion stress limited loading experiments were carried out using the loading levels from [[bar.[tau]].sub.0] = 1.12 to [[bar.[tau]].sub.0] = 1.94 . Fig. 7 shows, that increasing the number of loading semicycles k for the grade 45 steel hysteresis loop width [bar.[delta]] is not changing as under tension-compression, i.e. it remains constant.

[FIGURE 7 OMITTED]

Width of the hysteresis loop during tension-compression for the cyclic anisotropic materials is wider at even semicycles and smaller at uneven, i.e. [[bar.[delta]].sub.even] > [[bar.[delta]].sub.uneven]. For the case of torsion, the width of the hysteresis loop remains constant. Therefore, the hysteresis loop width dependence on the number of semicycles is written as follows [1]

[[bar.[delta]].sub.k] = [A.sub.1,2] ([[bar.e].sub.0] - [[bar.s].sub.T]/2)[k.sup.[alpha]] (6)

where [A.sub.1], [A.sub.2] and a are cyclic characteristics of the material, [[bar.e].sub.0] is relative initial strain, [[bar.s].sub.T] is cyclic proportionality limit.

To determine tension-compression constants [A.sub.1] and [A.sub.2], and torsion constant A under stress limited low cycle loading, [[bar.[delta].sub.1,2] = f ([[bar.e].sub.0]) graphs of the semicycle hysteresis loop width dependence on initial strain have been used [1], i.e.

[A.sub.1,2] = [[bar.[delta].sub.1,2]/([[bar.e].sub.0] - [[bar.s].sub.T]/2)(7)

Dependences of semicycle's loop width on the initial strain are shown in Figs. 8 and 9, whereas determined cyclic characteristics of the material are given in Table 3.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Carrying out the low cycle tension-compression tests, it was obtained, that grade 45 steel is accumulating plastic strain in tension direction (Fig. 10). Thus, the accumulated plastic strain after loading semicycles k, can be expressed as follows [1]

[[bar.e].sub.pk] = [[bar.e].sub.0] - [[bar.[sigma].sub.0] + [k.summation over (l)] [(-1).sup.k] [[bar.[delta].sub.k] (8)

[FIGURE 10 OMITTED]

Carrying out the low cycle torsion tests, it was obtained, that grade 45 steel does not accumulate plastic strain.

3.2. Low cycle fatigue curves

Fig. 11 presents curves of low cycle fatigue and reduction of area [psi] for grade 45 steel under stress limited tension-compression loading.

[FIGURE 11 OMITTED]

Fig. 12 presents low cycle fatigue curves of grade 45 steel under stress limited torsion.

Under strain limited low cycle loading, the accumulation of plastic strain [[bar.e].sub.pk] is not available. Experimental analysis of low cycle strain limited loading was carried out at tension-compression levels from [[bar.e].sub.0min] = 3.42 to [[bar.e].sub.0max] = 16.15 and for the torsion levels - from [[bar.e].sub.0min] = 4.56 to [[bar.e].sub.0max] = 19.63 . Figs. 13 and 14 present low cycle fatigue curves under strain limited loading in coordinates lg [bar.[delta]] - lg [k.sub.c] and lg [bar.[epsilon] - lg [k.sub.c].

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

4. Damage under low cycle loading

Under tension-compression and stress limited loading, fracture of the specimen occurs due to quasistatic damage [d.sub.K], caused by the accumulated plastic strain [[bar.e].sub.pk], and fatigue damage [d.sub.N], caused by the cyclic plastic strain, which is caused by the hysteresis loop width [[delta].sub.k], whereas total damage may be written [1]

d = [d.sup.q.sub.K] + [d.sup.l.sub.N] (9)

where d is total damage.

Fatigue damage is calculated by the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where [k.summation over (l)] [[bar.[delta]].sub.k] is fatigue damage accumulated during the k loading semicycles, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is fatigue damage accumulated till crack initiation.

Quasistatic damage

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where [[bar.e].sub.pl] is accumulated plastic strain during k semicycles of loading, whereas [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the maximum uniform strain under monotonous loading which corresponds [[sigma].sub.u].

If stress limited loading is approached as non-stationary strain limited loading, when the damage, accumulated during one semicycle k, is expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Then condition of the crack initiation is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

The analysis of strain limited low cycle loading when strain is limited and quasistatic damage is not occurring was performed. In this case, damage of the specimen is predetermined only by the cyclic plastic strain, i.e. under strain limited loading, the fatigue curve in coordinates lg [bar.[delta]] - lg [k.sub.c] has a shape of straight line. The constants m and C have been determined by the equation of straight line

lg [bar.[delta]] = - m lg [k.sub.c] + lg C (14)

or

[bar.[delta]] [k.sup.m.sub.c] = C (15)

where [bar.[delta] is average width of the hysteresis loop.

From the fatigue curve, formed under strain limited loading, in coordinates lg [bar.[delta]]- lg [k.sub.c] and lg[[bar.[epsilon]] lg[k.ub.c] and applying the L. Coffin's equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

In expression (16), the average width of the plastic hysteresis loop was calculated by the equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Applying the coordinates lg [bar.[epsilon] - lg[k.sub.c], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

After the applied Eq. (20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

By introducing the [m.sub.3] = 1 - [m.sub.2]/[m.sub.1] and applying the Eqs. (13) - (21), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

Because of good agreement between the experimental and calculated data, Eq. (22) was used to calculate the damage in works [8, 9].

[FIGURE 15 OMITTED]

The curve 3 in Fig. 15 presents the fatigue damage under stress limited tension-compression as only fatigue damage is taken into account and is close to the fatigue curve under stress limited torsion (curve 2), because under stress limited torsion loading, strain accumulation is not observed, i.e., the quasistatic damage does not occur.

The curve 4 confirms Eq. (22), as it shows satisfactory agreement to fatigue curve under stress limited torsion, because during the torsion experiments the quasistatic damage was not observed. The curves 3 and 4 confirm that according to the results of cyclic tension-compression it is possible to calculate the durability under cyclic torsion. Besides, the durability under cyclic torsion (in coordinates [[sigma].sub.i] - [N.sub.c]) is higher than under cyclic tension-compression loading, because under cyclic torsion there is no accumulation of plastic strain, i.e. there is no quasistatic damage.

5. Conclusions

Grade 45 steel was investigated under monotonous tension, monotonous torsion, the low cycle tension-compression and low cycle torsion with stress and strain limited loading, using the circular cross-section specimens for tension-compression and thin walled specimens - for torsion.

1. It was determined, that characteristics of the cyclic stress strain curves for the analyzed grade 45 steel at cyclic tension-compression, and under cyclic torsion, are similar. For both the analyzed loading cases the parameter [alpha] = 0, i.e. the material is cyclically stable. Values of the parameters A, which characterizes the hysteresis loop width of the first semicycle, are also similar.

2. During the stress limited loading, under cyclic torsion, the accumulation of plastic strain was not observed, i.e. under stress limited torsion, there is no quasistatic damage.

3. The durability under cyclic torsion is higher (in coordinates [[sigma].sub.i] - [N.sub.c]), than that under cyclic tension - compression loading, because under cyclic torsion there is no accumulation of plastic strain, i.e. is no quasistatic damage.

Received September 29, 2009 Accepted November 23, 2009

References

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

(2.) Krenevicius, A., Leonavicius, M. Fatigue life prediction for threaded joint. -Mechanika, -Kaunas: Tech nologija, 2008, Nr.3(71), p.5-11.

(3.) Krenevicius, A., Juchnevicius, Z. Load distribution in the threaded joint subjected to bending. -Mechanika, -Kaunas: Technologija, 2009, Nr.4(78), p.12-16.

(4.) Findley, WN. Effects of extremes of hardness and mean stress on fatigue of AISI steel in bending and torsion. -ASME Journal of Engineering Materials and Technology, 1989, p.119-122.

(5.) Shigley, JE., Mischke, CR. Mechanical Engineering Design, 5th ed. -McGraw-Hill, 1989.-1018p.

(6.) Marquis, G., Socie, D. Long-life torsion fatigue with normal mean stresses. -Fatigue and Fracture of Engineering Materials and Structures, 2000, p.293-300.

(7.) McClaflin, D., Fatemi, A. Torsional deformation and fatigue of hardened steel including mean stress and stress gradient effects. -International Journal of Fatigue. -Elsevier, 2004, 26, p.773-784.

(8.) Daunys, M., Rimovskis, S. Analysis of circular crosssection element, loaded by static and cyclic elasticplastic pure bending. -International Journal of Fatigue. -Elsevier, 2006, 28, p.211-222.

(9.) Daunys, M., Sabaliauskas, A. Influence of surface hardening on low cycle tension compression and bending characteristics. -Mechanika. -Kaunas: Technologija, 2005, Nr.3(53), p.12-16.

M. Daunys *, R. Cesnavicius **

* Kaunas University of Technology, Kqstucio 27, 44312 Kaunas, Lithuania, E-mail: mykolas.daunys@ktu.lt

** Kaunas University of Technology, Kqstucio 27, 44312 Kaunas, Lithuania, E-mail: ramunas.cesnavicius@ktu.lt

Table 1

Mechanical characteristics of the grade 45 steel under
tension

[[sigma].      [[sigma].       [[sigma].     [[sigma].
sub.pl], MPa   sub.0.2], MPa   sub.u], MPa   sub.f] MPa

319            320             673           1000
325            334             686           993
324            328             688           995

                           [bar.x]

323            327             682           996

[[sigma].
sub.pl], MPa   [e.sub.u2], %   [psi], %

319            13.1            43.8
325            12.5            37.6
324            13.0            40.6

                     [bar.x]

323            12.9            40.7

Table 2

Mechanical characteristics of the grade 45 steel under
torsion

[[tau].sub.   [[tau].sub.   [[tau].sub.   [[gamma].
pl], MPa      0.3], MPa     u], MPa       sub.u2], %

174           226           425           23.4
224           209           435           25.2
188           211           420           19.7

                         [bar.x]

195           215           426           22.7

Table 3

Cyclic characteristics of the grade 45 steel

Tension-compression

[A.sub.1]   [A.sub.2]   [[bar.S].sub.t]   [alpha]

                        Grade 45 steel

0.93        1.01        1.65              0

Torsion

A      [[bar.S].sub.t]   [alpha]

1.14        1.40            0

Table 4

Values of Coffins constants C and m

                        Steel 45

[C.sub.2]   [C.sub.3]   [m.sub.1]   [m.sub.2]   [m.sub.3]

                Low cycle tension-compression

314         198         0.42        0.51        1.14

                     Low cycle torsion

727         440         0.49        0.58        0.88