A Multiaxial Low Cycle Fatigue Life Prediction Model for Both Proportional and Non-proportional Loading Conditions

A Multiaxial Low Cycle Fatigue Life Prediction Modelfor Both Proportional and Non-proportional

Loading ConditionsSurajit Kumar Paul

(Submitted March 5, 2014; in revised form May 19, 2014)

This paper has presented a life prediction model in the field of multiaxial low-cycle fatigue. The proposedmodel is generally applied for constant amplitude multiaxial proportional and non-proportional loading.Depending upon applied strain path the equivalent strain varies within a cycle. Equivalent average strainamplitude is considered as fatigue damage parameter in the proposed model. The model has requirement ofonly two material constants and no other tuning parameters. The model is examined by the proportionaland non-proportional low-cycle fatigue life experimental data for eight different types of materials. Themodel is successfully correlated with multiaxial fatigue lives of eight different materials.

Keywords equivalent strain amplitude, low-cycle fatigue life,multiaxial fatigue life, non-proportional loading

1. Introduction

The components of engineering structures such as automo-biles, locomotive, aircraft, power plant, and pressure vesselusually undergo multiaxial loading. The cyclic stress-strainresponses under multiaxial loading are very complex in natureand depended upon the loading-path. As a consequence themultiaxial fatigue behavior of materials and structures is noteasy to describe. Multiaxial fatigue criteria are extremelyimportant in the field of fatigue and their aim is to reduce thecomplex multiaxial loading to an equivalent uniaxial loading.Still now, many researchers have proposed multiaxial fatiguecriteria suitable to different materials and different loadingconditions. There is not yet a unanimously accepted model inspite of a large number of criteria. There are three mainapproaches to evaluate the fatigue life; they are: stress-life,strain-life, and energy-based methods. In 1940s, the stress-lifemethod (also referred to as the Basquin model) was initiallyused to understand and quantify metals� high-cycle fatigue life(Ref 1). Then, Coffin and Mason (Ref 2, 3) proposed the strain-life method to describe the low-cycle and high-cycle fatiguefailures. Later, the energy-based method was used to predict thelow-cycle fatigue life of some metals (Ref 4, 5). Most of thecurrent models for multiaxial loading have been constructed ondepending upon the concept of the critical plane, according towhich the fatigue failure develops in the plane with themaximum shearing or normal strains that come to be known asa critical plane. A physical background for this approach isbased on the observations that the fatigue crack nucleation

occurs in persistent slip bands formed on some grain (crystal)faces of a material. The difference in the approaches consists inthat the intensity of fatigue damage which depends on thestrains or strain energy for different failure modes, such as shear(Ref 6, 7), normal (Ref 8, 9), or their combination taking intoaccount other strain and stress components on the critical planeas well (Ref 10-12). Despite the fact, a representative check ofthe accuracy of at least several approaches is missing in therecent reviews of Karolczuk and Macha (Ref 13). They pointedout in one of their summarizing comments that the informationabout the fatigue plane orientation was poor. In the presentwork, a simple multiaxial low-cycle fatigue model has beenproposed based on the equivalent average strain amplitude.

During proportional strain-controlled multiaxial low-cyclefatigue, the ratio of strain components and the direction of themaximum shear stress are maintained as constant throughoutthe cyclic deformation. As a consequence, the main mechanismfor plastic deformation slip occurs on the same planesthroughout the loading. Uniaxial tension-compression, puretorsion, and proportional tension-torsion (phase differencebetween tension and torsion loading is 0�) are the examplesof this type of loading. Whereas, during non-proportionalstrain-controlled multiaxial fatigue, the ratio of strain compo-nents and the direction of the maximum shear stress arechanged during cyclic deformation and slip occurs in differentplanes at different times. Therefore, the dislocation density andthe number of dislocation-dislocation interaction increases, andas a consequence extra hardening is observed (Ref 14).Shamsaei et al. (Ref 15) experimentally examined that in-phase cycles with a random sequence of axial-torsion cycles onan equivalent strain circle is found to cause cyclic hardeninglevels similar to 90� out-of-phase loading of 304L stainlesssteel. The biaxial loading with phase difference such as 90� outof phase, is an example of non-proportional loading. Theamount of non-proportional hardening is dependent on material(Ref 16), however, irrespective of material is non-proportion-ally hardened or not the multiaxial fatigue life is significantlyshorter for non-proportional loading condition (Ref 17).Surajit Kumar Paul, R&D, Tata Steel Limited, Jamshedpur 831007,

India. Contact e-mail: [email protected].

JMEPEG �ASM InternationalDOI: 10.1007/s11665-014-1107-4 1059-9495/$19.00

Journal of Materials Engineering and Performance

Shamsaei et al. (Ref 17) for pure titanium and titanium alloyBT9, Gladskyi and Shukaev (Ref 18) for titanium alloy BT1-0,and Norban et al. (Ref 19) for 30CrNiMo8HH alloy steelexperimentally observed that non-proportional loading resultsin significantly shorter fatigue lives whereas no non-propor-tional hardening is noticed. Therefore, it is extremely essentialto understand the cause of considerably shorter fatigue livesduring non-proportional loading. In the present investigation,multiaxial fatigue is tried to correlate with equivalent averagestrain amplitude. In view of the fact that the investigations onfatigue will remain important in the future, the goal of thepresent work is to develop an equivalent average strainamplitude based life assessment model that is based on a localapproach involving the application of the strain criterion formultiaxial fatigue.

2. Proposed Model for Multiaxial Low CycleFatigue Life

Dong et al. (Ref 20) showed that, with the increasingnumber of loading cycles, the dislocation patterns evolvefrom dislocation lines and networks to dislocation tangles,walls, and cells. After certain cycles, sub-grains are formedbecause of the re-arrangement of dislocations in the walls ofcells and inside the cells since the cross slip of dislocationscan be easily activated for the 20 carbon steel, a kind ofbody-centered cubic metal. The dislocation evolution of thenon-proportional multiaxial loading is much quicker than thatof the uniaxial one. Kida et al. (Ref 21) reported in theirinvestigation on 304 stainless steel that the stress responseand microstructure evolution are intimately related to thedegree of non-proportionality and applied strain range. Theyobserved dislocation cells, dislocation bundles, twins, andstacking faults depending upon loading conditions. Theexperimental multiaxial fatigue life investigation showed thatthe fatigue life in non-proportional multiaxial loading issignificantly lower than the uniaxial/proportional loading (Ref15-20). Therefore, it can be said that the fatigue damage innon-proportional multiaxial loading is faster than that of theuniaxial/proportional loading.

Stress-strain hysteresis loop shape during non-proportionalmultiaxial loading for different materials was reported in theliterature (Ref 16, 22). For 90� out of phase axial-shear loadingcondition (circular strain path), the unusual shape of the loopscan be attributed to the fact that at the elastic deformation of thematerial does not take place at any time. It should be noted thatthere are still linear regions in the hysteresis loops under out ofphase loading condition (other than circular strain path or 90�phase difference). These cannot be considered as elasticregions, because their slopes are not equal to the elasticmodulus (or the shear modulus) of the material (Ref 23).Therefore, it is extremely difficult to separate out elastic andplastic strain amplitudes from the hysteresis loop. Conse-quently, total equivalent average strain amplitude is calculatedin the current work and considered as a fatigue damageparameter.

For different axial-shear proportional and non-proportionalstrain paths, the equivalent stress req and equivalent strain eeqcan be defined for any instance based on the von Mises yieldtheory as follows:

req ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2 þ 3s2p

; ðEq 1Þ

eeq ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

e2 þ c2=3p

; ðEq 2Þ

where r and e are axial stress and strain, s and c are shearstress and strain. Equivalent shear strain can also be calcu-lated from Eq 2 for pure torsion case.

For proportional loading condition, the equivalent averagestrain amplitude (eaveq) is shown in Fig. 1(a). For uniaxialtension-compression, twist-untwist, and proportional axial-shear loading conditions, the equivalent average strain ampli-tude will be the half of equivalent strain amplitude. Irrespectiveof wave form shape, in the current investigation equivalentaverage strain amplitude is assumed the half of equivalent strainamplitude in uniaxial tension-compression, twist-untwist, andproportional axial-shear loading conditions. The radii of dotedcircles in Fig. 1(b)-(d) are the equivalent average strainamplitude (eaveq) for axial, shear, and axial-shear proportionalloading conditions, respectively. For 90� out of phase axial-shear strain cycling with sinusoidal waveform results constantequivalent strain throughout the cycle, shown in Fig. 2(a). Thestrain path is appeared as a circle in axial versus shear strainplot and the radius of the circle is equal to eaveq, shown inFig. 2(b). As equivalent strain remains constant throughout thecycle so no elastic unloading is occurred in the 90� out of phaseaxial-shear strain cycling with sinusoidal waveform (Ref 11,23). For 45� out of phase axial-shear strain cycling withsinusoidal waveform consequences variation of equivalentstrain within a cycle (Fig. 2c) and eaveq is shown as a radius ofdotted circle in Fig. 2(d). For triangular waveform, 90� out ofphase axial-shear strain cycling results variation of equivalentstrain within a cycle (Fig. 3a) and strain path looks rhombic inshape in axial versus shear strain plot. eaveq is revealed as aradius of dotted circle in Fig. 3(b) and (d) for 90� and 45� outof phase axial-shear strain cycling with triangular waveform,respectively. Similarly the calculations of eaveq are shown inFig. 4 for box and two-box shaped axial-shear strain paths.

A power law equation has been selected for the currentinvestigation to correlate equivalent average strain amplitude(eaveq) with multiaxial fatigue life (Nf).

eaveq ¼ B Nfð Þh; ðEq 3Þ

where B and h are the material constants. The equivalentaverage strain amplitude (eaveq) can be calculated from Eq 4.

eaveq ¼Z

DT

0

eeqdt

0

@

1

A=DT ; ðEq 4Þ

where DT is the period of the cyclic straining waveform. Thematerial constants in the Eq 3 are obtained by best fitting theexperimental data under tension-compression and torsionloading. Tension-compression and in-phase loading data canbe used if torsion loading data are absent. Only tension-com-pression data can be used for determination of material con-stants, but it follows that there is a chance of slight error.

The predictive capability of proposed equivalent averagestrain amplitude based multiaxial fatigue life prediction modelis proper for low-cycle fatigue region. That means this model issuitable for fatigue loading in the elasto-plastic region.Therefore, this multiaxial fatigue life prediction model ispertinent for semi-ductile and ductile material.


3. Application of the Proposed Model to EightDifferent Materials

3.1 Material Descriptions

Eight metallic engineering materials are used to evaluate theproposed fatigue model: pure titanium (Ref 17), titanium alloyBT9 (Ref 17), 45 steel (Ref 11), 30CrNiMo8HH steel alloy(Ref 22), BT1-0 titanium alloy (Ref 18), GH4169 nickel-basedsuperalloy (Ref 24), S460 steel (Ref 25), and SA 333 Steel (Ref26). All the experimental data have been taken from literatures.The material constants of the proposed multiaxial fatigue lifeprediction model are tabulated in Table 1 for all eight materials.

Pure and alloyed titanium are used in wide range ofapplications. Higher strength and stiffness to weight ratio oftitanium as compared to steels, excellent corrosion and creepresistance, high toughness, good temperature resistance, andlow value of thermal expansion coefficient make it desirable touse in military, aerospace, racing automotives, chemical andpetrochemical applications (Ref 17). Due to the combination ofstrength and biocompatibility of titanium, it is also widely usedin body and dental implants (Ref 17). Shamsaei et al. (Ref 17)

performed fatigue tests on pure titanium and titanium alloy BT9under various uniaxial tension-compression, torsion, in-phase,45� and 90� out-of-phase strain cycling with sinusoidalwaveform.

Chen et al. (Ref 11) conducted multiaxial fatigue tests on 45steel using thin-walled tubes with outside and inside diametersof 25 and 21 mm. They carried out multiaxial low-cycle fatigueexperiments with various axial-shear strain paths, like, circular,elliptic, square, rectangular, rhombic, 45� line. They defined thefailure life as the number of cycles at which a 25% drop fromthe maximum value occurs in either the tensile or shear stressrange. They noticed that for the same equivalent strain range,fatigue life is the shortest under a circular path, and is thelongest under a proportional path.

30NiCrMo8HH a nickel-chromium-molybdenum steel alloyis used in components such as power train and drivelinesystems that generate power and deliver it to the road surface,water, or air (Ref 22). This nickel-chromium-molybdenum steelalloy provides excellent tensile and yield strength combine withhigh hardness, stiffness, good formability, and high fatigue liferesistance. Noban et al. (Ref 22) carried out axial fatigue testson solid bar specimens and tubular specimens which are used

Fig. 1 The equivalent average strain amplitude of proportional loading conditions (a) axial equivalent strain and equivalent average strain varia-tion with time for triangular waveform; (b) equivalent average strain for uniaxial tension-compression; (c) equivalent average strain for uniaxialtorsion; and (d) equivalent average strain for Multiaxial proportional axial-shear


for pure torsion and biaxial (tension-torsion) tests. Fatiguefailure of 30CrNiMo8HH steel alloy was declared at 20% loaddrop. All tests were performed at standard laboratory temper-ature and humidity on an Instron-8874 servo hydraulic tension-torsion test machine. They conducted uniaxial tension-com-pression, torsion, in-phase and various biaxial out of phase withdifferent strain paths (30�, 45�, 60�, and 90� out-of-phase straincycling with sinusoidal waveform; square and two square axial-shear strain paths) experiments.

Uniaxial tension-compression, torsion and multiaxial (in-phase, 45� elliptic and circular axial-shear strain paths) fatigueexperiments were performed by Gladskyi and Shukaev (Ref 18)on BT1-0 titanium alloy. A tubular geometry with an outerdiameter of 11 mm and an inner diameter of 10 mm at the gagesection specimen were used for multiaxial fatigue experiments.

Fatigue tests under axial and torsional loading wereperformed on GH4169 alloy (Ref 24) loaded at 650 �C.Fatigue test results include in-phase loading and 45� out-of-phase, 90� out-of-phase loading conditions. Failure life wasdefined as the number of cycles that resulted in a 25% dropfrom the maximum value in either tensile stress or shear stress.

Typical applications of the fine grain structural steel S460Nare highly stressed steel constructions and steel-concrete

composite constructions (Ref 25). Jiang et al. (Ref 25)performed fatigue tests on S460N Steel components undervarious in-phase and out-of-phase loading conditions with arange of axial-shear strain paths.

SA-333 Gr.6. steel is used in Indian Pressurized HeavyWater Reactor�s (PHWR) primary heat transport (PHT) piping(Ref 26). The pure torsion and axial-torsion fatigue experimentswere conducted on tubular specimens and uniaxial tension-compression experiments were on solid cylindrical specimensby Arora et al. (Ref 26). All specimens were fabricated from thepipe made of SA333 Gr.6 (low carbon manganese steel). Thetests were grouped into four categories, namely axial, shear, in-phase axial-shear, and out of phase axial-shear with phase shiftsof 90�, 45�, and 180�. All these tests were conducted at roomtemperature and in air environment.

3.2 Multiaxial Fatigue Life Prediction by Proposed Model

Figure 5 has summarized the comparisons of the experi-mental fatigue lives with the predictions by the proposedequivalent average strain amplitude based fatigue model for theeight materials. The vertical axes in these figures represent theequivalent average strain amplitude (eaveq) and the horizontal

Fig. 2 The equivalent average strain amplitude for non-proportional loading conditions with sinusoidal waveform (a) axial and shear equivalentstrain; and equivalent average strain variation with time for 90� out of phase axial-shear loading; (b) equivalent average strain for 90� out ofphase axial-shear loading; (c) axial and shear equivalent strain; and equivalent average strain variation with time for 45� out of phase axial-shearloading; and (d) equivalent average strain for 45� out of phase axial-shear loading


axes are the multiaxial fatigue life (Nf). The thick red soliddiagonal lines in the figures signify a perfect prediction. Thetwo black thin solid lines in each figure represent the factor-of-two boundaries.

Predicted life obtained by employing proposed equivalentaverage strain amplitude based fatigue model is compared toexperimental life data in Fig. 5(a) for pure titanium, Fig. 5(b)for TITANIUM alloy BT9, Fig. 5(c) for 45 steel, Fig. 5(d) for30CrNiMo8HH steel alloy, Fig. 5(e) for BT1-0 titanium alloy,Fig. 5(f) for GH4169 nickel-based super alloy, Fig. 5(g) forS460N steel and Fig. 5(h) for SA333 steel. It can be seen fromthose figures that most of the experimental results are within thefactor of ±2 of proposed model�s prediction. As it can be seenfrom Fig. 5(g), most of the results which were predicted by theproposed model lie in a factor of ±2, only few results lie out ofa factor of ±2. Many experimental fatigue lives lie within104-106 ranges, plastic deformation is limited for thoseconditions. This thought can be one of the primary reasonsfor positioning few experimental fatigue data points outside thefactor of ±2.

The above analysis illustrates the new proposed equivalentaverage strain amplitude based approach which is an effective

way to predict low-cycle fatigue under both proportional andnon-proportional loading conditions. The determination ofequivalent average strain amplitude is very simple and straightforward. The current approach makes use of the macroscopicstrain quantities.

The proposed equivalent average strain amplitude basedfatigue model treats the fatigue damage as a scalar. This is amajor difference from the critical plane approaches where thefatigue damage on material plane is considered. The scalartreatment of the fatigue damage does not allow for the modelto identify the fatigue cracking direction. Therefore, theproposed model is applicable to the cases where the crackingdirection is not required to predict. Jiang et al. (Ref 25)reported that while fatigue lives can be reasonably predictedby using critical plane multiaxial fatigue approaches when thestress-strain response is known for the material, the crackingdirection is much more difficult to predict correctly. Withoutconsidering individual material planes, the application of theproposed model does not require the rotation of the coordi-nates system that is required by a critical plane approach inthe determination of the stresses and strains associated withindividual material planes.

Fig. 3 The equivalent average strain amplitude for non-proportional loading conditions with triangular waveform (a) axial and shear equivalentstrain; and equivalent average strain variation with time for 90� out of phase axial-shear loading; (b) equivalent average strain for 90� out ofphase axial-shear loading; (c) axial and shear equivalent strain; and equivalent average strain variation with time for 45� out of phase axial-shearloading; and (d) equivalent average strain for 45� out of phase axial-shear loading


Fig. 4 The equivalent average strain amplitude for non-proportional loading conditions with square waveform (a) axial and shear equivalentstrain; and equivalent average strain variation with time for 90� out of phase axial-shear loading; (b) equivalent average strain for 90� out ofphase axial-shear loading; (c) axial and shear equivalent strain; and equivalent average strain variation with time for 45� out of phase intermittentaxial-shear loading; and (d) equivalent average strain for 45� out of phase intermittent axial-shear loading

Table 1 Material constants for different materials

Sl. no. Material Ref B h

1 Pure titanium 17 0.548 �0.6462 Titanium alloy BT9 17 0.04 �0.283 45 steel 11 0.33 �0.654 30CrNiMo8HH steel alloy 22 0.14 �0.4595 BT1-0 titanium alloy 18 0.2 �0.566 GH4169 nickel-based superalloy 24 0.03 �0.317 S460 steel 25 0.012 �0.28 SA 333 steel 26 0.07 �0.39


Fig. 5 Comparison of experimental multiaxial fatigue lives and predictions by proposed equivalent average strain amplitude based model for(a) Pure Titanium (Ref 17); (b) Titanium alloy BT9 (Ref 17); (c) 45 steel (Ref 11); (d) 30CrNiMo8HH steel alloy (Ref 22); (e) BT1-0Titaniumalloy (Ref 18); (f) GH4169 nicle-based super alloy (Ref 24); (g) S460N steel (Ref 25); and (h) SA333 steel (Ref 26)


4. Conclusions

A low-cycle fatigue life prediction methodology for constantamplitude multiaxial proportional and non-proportional loadingconditions is presented in the current work. It is noticed that theequivalent average strain amplitude is different for various non-proportional strain paths. An excellent correlation is observedbetween equivalent average strain amplitude and multiaxialfatigue life. The proposed model relates equivalent averagestrain amplitude with multiaxial fatigue life through a powerlaw equation. The material constants in the model are directlyand explicitly related to the basic fatigue experiments underfully reversed tension-compression and torsion. The proposedmodel greatly facilitates its application to general multiaxialfatigue loading. Comparison of the proposed criterion predic-tions against experimental fatigue limit data which are collectedfrom literatures for eight different materials under multiaxial in-phase and out-of-phase low-cycle fatigue with triangular,sinusoidal, and square strain paths has showed good agreement.

Acknowledgments

The authors would like to acknowledge Dr. S. Shivaprasad, Dr.S. Tarafder, National Metallurgical Laboratory (CSIR), Jamshed-pur, India and Dr. Saurabh Kundu, R&D, Tata Steel Limited,Jamshedpur, India for their valuable suggestions.

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A Multiaxial Low Cycle Fatigue Life Prediction Model for Both Proportional and Non-proportional Loading Conditions