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International Journal of Pressure Vessels and Piping 110 (2013) 50e56

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International Journal of Pressure Vessels and Piping

journal homepage: www.elsevier .com/locate/ i jpvp

Multiaxial low cycle fatigue life under non-proportional loading

Takamoto Itoh a,*, Masao Sakane a,1, Kazuki Ohsuga b

aDepartment of Mechanical Engineering, Faculty of Science and Engineering, Ritsumeikan University, 1-1-1, Nojihigashi, Kusatsu-shi, Shiga 525-8577, JapanbGraduate School of Mechanical Engineering, University of Fukui, Japan

Keywords:Low cycle fatigueMultiaxial loadingNon-proportional loadingLife predictionDesign criteria

* Corresponding author. Tel.: þ81 (0)77 561 4965;E-mail addresses: itohtaka@fc.ritsumei.ac.jp (T. Itoh

(M. Sakane).1 Tel.: þ81 (0)77 561 2746; fax: þ81 (0)77 561 266

0308-0161/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.ijpvp.2013.04.021

a b s t r a c t

A simple and clear method of evaluating stress and strain ranges under non-proportional multiaxialloading where principal directions of stress and strain are changed during a cycle is needed for assessingmultiaxial fatigue. This paper proposes a simple method of determining the principal stress and strainranges and the severity of non-proportional loading with defining the rotation angles of the maximumprincipal stress and strain in a three dimensional stress and strain space. This study also discussesproperties of multiaxial low cycle fatigue lives for various materials fatigued under non-proportionalloadings and shows an applicability of a parameter proposed by author for multiaxial low cyclefatigue life evaluation.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Most design codes use equivalent values to express the intensityof multiaxial stress or strain, like von Mises or Tresca equivalentstress and strain, and fatigue lives are usually estimated usingequivalent values under multiaxial stress and strain states. Theequivalent value means a scalar parameter that expresses intensityof a physical phenomenon in multiaxial stress states and should bereduced to be a uniaxial value in uniaxial stress state. Most widelyused equivalent parameters are the von Mises and the Trescaequivalent stresses and strains. The von Mises equivalent stressphysically expresses the intensity of shear strain energy and theTresca equivalent stress that of the maximum shear stress. Forexample, ASME Section III, Division 1 NH [1] uses the von Misesequivalent strain and ASME Section VIII, Division 3 [2] themaximum shear stress.

However, the von Mises equivalent stress and strain have nonegative values so that they have a difficulty of expressing stressand strain ranges. The Tresca equivalent stress and strain havenegative values but they also have a difficulty to put a sign to theshear stress and strain under multiaxial loading. Especially, in non-proportional loading where the principal stress and strain changetheir directions, giving a sign to them becomes more difficult.

fax: þ81 (0)77 561 2665.), sakanem@se.ritsumei.ac.jp

5.

All rights reserved.

A simple and clear method of calculating stress and strain ranges isneeded for describing multiaxial fatigue.

Multiaxial low cycle fatigue (LCF) lives are reduced under straincontrolled non-proportional loading accompanied by additionalcyclic hardening compared with proportional loading [3e10] andan appropriate design method of evaluating the non-proportionalfatigue life is needed for a reliable design and maintenance ofstructural components. Classical models particularly applicable inmultiaxial fatigue life evaluation under proportional loadings leadto overestimate the lives under non-proportional loadings. For lifeevaluation under non-proportional loading, commonly proposedmodels are critical plane approaches that consider specific planeapplied the critical damage, such as a SimitheWatosoneTopper[11] and a FatemieSocie [12] models. The authors also proposed astrain parameter (ItoheSakane model) estimating the non-proportional LCF lives for several materials under various strainhistories [6,7,13e16]. This parameter is the strain based model withintroducing two parameters, non-proportional factor and materialconstant. The former one reflects the intensity of non-proportionalloading reducing life and the latter one is related to the materialdependence for degree of life reduction due to non-proportionalloading.

The SimitheWatosoneTopper, the FatemieSocie and the ItoheSakane models have been demonstrated to be applicable to lifeevaluation under non-proportional loading using hollow cylinderspecimens in a laboratory level. However, these models can beapplicable to the life evaluation under limited non-proportionalloadings such as the loadings in the plane stress state. Therefore,there is a limit of application of the models to the design of actual

Fig. 2. Definition of principal stress and strain directions in XYZ coordinates.

T. Itoh et al. / International Journal of Pressure Vessels and Piping 110 (2013) 50e56 51

components where variation in principal directions of stress andstrain vs. time is changed 3-dimensionally.

This study proposes a method of evaluating the principal stressand strain ranges and the mean stress and strain, and also proposesa method of calculating the non-proportional factor which ex-presses the severity of non-proportional loading in 3-dimantional(3D) stress and strain space. This study also discusses the materialconstant, a, used in the strain parameter proposed by author for lifeestimation under non-proportional multiaxial LCF and presents asimple method to reevaluate a in relation to material constantsobtained in a static tension test [16].

2. Definition of stress and strain ranges under non-proportional loading

2.1. Definition of stress and strain

Fig. 1 illustrates three principal vectors, Si(t), applied to a smallcube in material at time t in xyz-coordinates (spatial coordinates),where “S” is the symbol denoting either stress “s” or strain “ε”.Thus, Si(t) are the principal stress vectors for the case of stress andthe principal strain vectors for the case of strain. The subscript, i,takes 1, 2 or 3 in descending order of principal stress or strain. Themaximum principal vector, SI(t), is defined as Si(t) whose absolutevalue takes maximum one, i.e., SI(t) ¼ S1(t) when S1(t) takesmaximum magnitude among Si(t). The maximum principal value,SI(t), is defined as the maximum absolute value of Si(t) as,

SIðtÞ ¼ jSIðtÞj ¼ Max½jS1ðtÞj; jS2ðtÞj; jS3ðtÞj� (1)

The “Max” denotes taking the larger value from the three in thebracket. The maximum value of SI(t) during a cycle is defined as themaximum principal value, SImax, at t ¼ t0 as follows,

SImax ¼ jSI t0ð Þj ¼ Max jS1 t0ð Þj; jS2 t0ð Þj; jS3 t0ð Þj½ � (2)

2.2. Definition of principal stress and strain directions

Fig. 2 illustrates two angles, x(t)/2 and z(t), to express the rota-tion or direction change of the maximum principal vector, SI(t), inthe new coordinate system of XYZ, where XYZ-coordinates are thematerial coordinates taking X-axis in the direction of SI(t0) with theother two axes in arbitrary directions. The two angles of x(t)/2 andz(t) are given by

xðtÞ2

¼ cos�1�

Siðt0Þ$SiðtÞjSiðt0ÞjjSiðtÞj

��0 � xðtÞ

2� p

2

�(3)

z�t� ¼ tan�1

�SiðtÞ$eZSiðtÞ$eY

�ð0 � zðtÞ � 2pÞ (4)

where dots in Eqs. (3) and (4) denote the inner product and eY andeZ are unit vectors in Y and Z directions, respectively. Si(t) are the

Fig. 1. Principal stress and strain in xyz coordinates.

principal vectors of stress or strain used in Eq. (1) and the subscripti takes 1 or 3.

The rotation angle of x(t)/2 expresses the angle between theSI(t0) and SI(t) directions and the deviation angle of z(t) is the angleof SI(t) direction from the Y-axis in the X-plane.

2.3. Definitions of stress and strain in polar figure

Fig. 3 shows the trajectory of SI(t) in 3D polar figure for a cyclewhere the radius is taken as the value of SI(t), and the angles of x(t)and z(t) are the angles shown in the figure. A new coordinate sys-tem is used in Fig. 3 with the three axes of S1I , S

2I and S3I , where

S1I -axis directs to the direction of SI(t0). The rotation angle of x(t) hasdouble magnitude compared with that in the specimen shown inFig. 2 considering the consistency of the angle between the polarfigure and the physical plane presentation. The principal range, DSI,is determined as the maximum projection length of SI(t) on theS1I -axis. The mean value, SImean, is given as the center of the range.DSI and SImean are equated as,

DSI ¼ Max½SImax � cos xðtÞSIðtÞ� ¼ SImax � SImin (5)

SImean ¼ 12ðSImax þ SIminÞ (6)

SImin is the SI(t) to maximize the value of the bracket in Eq. (5). Thesign of SImin in the figure is set to be positive if it does not cross theS2I eS

3I plane and the sign negative if it crosses the plane.

The advantage of the definitions of the maximum principalrange and mean value above mentioned is that the two are deter-minable without human judgments for any loading case in 3Dstress and strain space. The range and mean value are consistentused in simple loading cases which are discussed in the case studiesin the followings. SI(t) can be replaced by equivalent values of stressor strains, such as the von Mises and the Tresca, in case of necessityfrom user’s requirement.

Fig. 3. Definition of principal range and mean principal value.

Table 1List of materials tested and mechanical properties.

Test material Mechanical property in static tension test

Types CS Young’s modulusE (GPa)

Yield/Proof stresssY (MPa)

StrengthsB (MPa)

SUS316 FCC 197 260 575SUS304 197 290 750SUS304 (923K) 150 130 480SUS310S 196 215 520OFHC (Cu) 117 182 2406061A1 77 253 3901070Al 70 112 116SGV410 BCC 216 275 470SUS430 200 263 480S25C 200 354 493S45C 205 445 630S55C 203 485 695

T. Itoh et al. / International Journal of Pressure Vessels and Piping 110 (2013) 50e5652

3. Definition of non-proportionality

The authors proposed the non-proportional strain rangeexpressed in Eq. (6) for correlating LCF lives under non-proportionalloading [6,7,13e16].

DεNP ¼ ð1þ afNPÞDεI (6)

In the equation, DεI is the principal strain range discussed previ-ously. a is a material constant related to the amount of additionalhardening by non-proportional loading, which will be mentionedmore detail in Sections 4.2 and 4.3.

fNP is the non-proportional factor that expresses the severity ofnon-proportional loading in the form as,

fNP ¼ bTεImax

ZT

0

ðjsinðxðtÞÞjεIðtÞÞdt (7)

where T is the time for a cycle. b is a constant for making fNP ¼ 1 inthe circular loading on ε� g=

ffiffiffi3

pplot and b ¼ p/2 [6,7].

This paper presents f 0NP in Eq. (8) in 3D expression as anextended form from fNP in 2D shown in Eq. (7).

f0NP ¼ p

2SImaxLpath

ZC

e1 � eRSI tð Þds; Lpath ¼ZC

ds (8)

where eR is a unit vector directing to SI(t), ds the infinitesimal tra-jectory of the loading path shown in Fig. 3. Lpath the whole loadingpath length during a cycle and “�” denotes vector product. Thescalars, SImax and Lpath, before the integration in Eq. (8) is set tomake f 0NP unity in the circler loading in 3D polar figure.

Fig. 4 compares the values of fNP with those of f 0NP for severalloading paths for the case of strain. Small difference in the valuebetween fNP and f ’NP is found because of different definition be-tween them. However, f 0NP has the advantages applicable to 3Dstress and strain conditions.

Fig. 4. Comparing fNP and f 0N

4. Multiaxial low cycle fatigue lives under non-proportionalloading

This chapter shows multiaxial LCF life properties under non-proportional loadings for several materials and shows the appli-cability of the strain parameter for life estimation, which werestudied in authors’ previous study [14].

4.1. Materials and test procedure

Test materials employed were 12 metallic materials of whichcrystal structures (CS) are face-centered cubic structure (FCC) andbody-centered cubic structure (BCC) as listed in Table 1 with me-chanical properties obtained by static tension test. The specimenused was a hollow cylinder specimen with 12 mm outer diameter,9 mm inner diameter and 7 mm gauge length as shown in Fig. 5.

Total strain controlled multiaxial LCF tests were conducted un-der 2 types of strain paths. Fig. 6 (a) and (b) show the strain pathson ε� g=

ffiffiffi3

pplot and the strain waveforms of ε and g, respectively,

where ε and g are total axial and total shear strains. Case 1 is thepushepull test and Case 2 the 90� sinusoidal out-of-phase (circle)

P under several loadings.

Fig. 5. Shape and dimensions of specimen (mm).

T. Itoh et al. / International Journal of Pressure Vessels and Piping 110 (2013) 50e56 53

loading test. The former is the proportional loading test and thelatter the non-proportional loading test. Total axial strain ranges(Dε) were set to the same ranges in Case 1 and Case 2 and total axialstrain and total shear strain ranges were the same ranges based onvon Mises, Dε ¼ Dg=

ffiffiffi3

p, in Case 2. Strain rate was 0.1%/sec based

on von Mises basis.

4.2. Multiaxial LCF life and additional hardening

To evaluate the material dependency of failure life and cyclichardening behaviors under non-proportional loading, this sectionshows the multiaxial LCF test results for SUS316 and SGV410fatigued in the pushepull and the circle tests using the hollowcylinder specimen (Case 1 and Case 2).

ASME code case [1] defines a strain parameter to express thenon-proportional fatigue damage. The strain parameter is origi-nated from the equivalent strain range based on von Mises but it

Fig. 6. Strain paths and

(a) SUS316

102 103 104 1050.2

0.5

1

2

5

Case 1 (Push-pull) Case 2 (Circle)

SUS316

Number of cycles to failure Nf

Stra

in r

ange

Δ ε

, %

Fig. 7. Relationship be

was modified to have a maximum value taking any times C and Dalong strain paths as shown in Eq. (9).

DεASME ¼ Max εC � εDð Þ2 þ 13gC � gDð Þ2

� �12

(9)

where εC and gC are the axial and shear strains at time C and εD andgD those at time D tomaximize the strain in the bracket. In the testsof Case 1 and Case 2, the values of DεASME correspond with thosegiven by total axial strain range, Dε.

Fig. 7 (a) and (b) show failure lives (Nf) of SUS316 and SGV410correlated by Dε (¼DεASME). In the figure, the bold solid line wasdrawn based on the data of Case 1 and the two thin lines show afactor of 2 band. For SUS316, Nf in Case 2 is about 1/5 of that in Case1. The similar trend can be seen for SGV410, too.Nf in Case 2 is about1/5 of that in Case 1. Therefore, the degrees of reduction in failurelife due to non-proportional loading between these two steels arealmost equivalent.

The overestimation of Nf in Case 2 by the life curve in Case 1 alsohas been reported and it is known that the reduction in failure lifeunder non-proportional loading is related to the additional hard-ening due to non-proportional loading depending on material[5,16e19].

Fig. 8 (a) and (b) show cyclic stressestrain relations for SUS316and SGV410 respectively obtained by a multiple step-up test undertwo strain paths using the hollow cylinder specimen. The strainpaths employed were the pushepull straining (Case 1) and thecircular straining (Case 2) where von Mises’ equivalent strainamplitude was increased by 0.05% at each 10 cycles. In the figures,DεI and DsI are the maximum principal strain and stress rangesunder non-proportional loading which can be calculated by ε, g ands, s. The obtained result shows clearly that behaviors of the

strain waveforms.

(b) SGV410

102 103 104 1050.1

0.4

2

Case 1 (Push-pull) Case 2 (Circle)

SGV410

Number of cycles to failure Nf

Stra

in r

ange

Δε

, %

tween Dε and Nf.

(a) SUS316 (b) SGV410

0 0.5 10

200

400

600

800

1000

1/2Δε I, %

Case 1 (Push-pull) Case 2 (Circle)

1/2

Δσ I,

MPa

SUS316α = 0.75

0 0.5 1 1.50

200

400

600

800

1/2

Δσ I,

MPa

1/2Δε I, %

Case 1 (Push-pull) Case 2 (Circle)

SGV410α = 0.39

Fig. 8. Cyclic stress and strain relation obtained by multiple step-up test under push-pull and circular strainings.

0.4

2 SGV410α* = 0.85

ran

ge

Δε N

P,

%

T. Itoh et al. / International Journal of Pressure Vessels and Piping 110 (2013) 50e5654

additional hardening due to non-proportional loading are differentbetween SUS316 and SGV410. The degree of additional hardening ofSUS316 was approximately twice than that of SGV410, whereas LCFlife in Case 2 was decreased down to 1/5 in comparisonwith that inCase 1 for both steels as shown in Fig. 7. Therefore, the additionalhardening and the reduction in failure life are closely related, whichdepends on material resulted from the difference in the deforma-tion behavior due to crystal structural dependency [16,18,19].

Fig. 9 (a) and (b) show Nf correlated by non-proportional strainrange, DεNP. a employed here is the material constant evaluatedfrom the degree of additional hardening. For SUS316 (a ¼ 0.75) inFig. 9 (a), Nf in Case 2 is almost the same as that in Case 1. On theother hand, Nf for SGV410 (a ¼ 0.39) in Fig. 9 (b) is correlatedunconservatively in Case 2. The similar trend also can be observedin other FCC and BCC materials which will be shown in thefollowing section.

Fig. 10 shows the re-plot of relationship between DεNP and Nf forSGV410 by using a* as material constant for evaluating the degreeof reduction in failure life. The correlation in this figure shows Nf inCase 2 is plotted within the factor of 2 band with a* ¼ 0.85. Thevalue of a* is slightly larger than that for SUS316 (a* ¼ a ¼ 0.75).

102 103 104 1050.1

Case 1 (Push-pull) Case 2 (Circle)

Number of cycles to failure Nf

Stra

in

Fig. 10. Relationship between DεNP and Nf for SGV410 with a* ¼ 0.85.

4.3. Evaluation of material constant a

In order to investigate the relationship between multiaxial LCFlife and cyclic hardening under non-proportional loading for 12kinds of test materials. The relationship between a and a* is dis-cussed based on the experimental results. The universal slope

(a) SUS316

102 103 104 1050.2

0.5

1

2

5

Case 1 (Push-pull) Case 2 (Circle)

SUS316α = 0.75

Number of cycles to failure Nf

Stra

in r

ange

Δε

NP,

%

Fig. 9. Relationship bet

method equated in Eq. (10) [20] was employed to obtain the lifecurves with a small number of data in Case 1 and Case 2 for eachmaterial. The equation is shown by,

DεNP ¼ ð1þ a*fNPÞDεI ¼ N�0:12f þ BN�0:6

f (10)

where the coefficients A and B are equated as 3.5sB/E and ε0:6f

respectively, according to the definition of the universal slopemethod. Here, E, sB and εf are Yong’s modulus, tensile strength andelongation. In this study, A is put to 3.5sB/E but B and a* aredetermined as life curves in Case 1 and Case 2 are corresponding atsame DεI for each material.

(b) SGV410

102 103 104 1050.1

0.4

2

Case 1 (Push-pull) Case 2 (Circle)

SGV410α = 0.39

Number of cycles to failure Nf

Stra

in r

ange

Δ ε

NP,

%

ween DεNP and Nf.

Fig. 11. Relationship between a and a*.Fig. 13. Relationship between (sB�sY)/sB and a.

T. Itoh et al. / International Journal of Pressure Vessels and Piping 110 (2013) 50e56 55

Fig. 11 shows the relationship between a and a* for each ma-terial. The open mark shows the data for FCC materials, the solidmark the data for BCC materials. The relationship is shown by twostraight lines separately in FCC and BCC materials although a fewdata are scattered on the both sides of the band. The result in Fig. 11shows that reduction in failure life has close relationship withadditional hardening in non-proportional loading, which dependson crystal structure of tested materials. The relationship between a

and a* can be expressed experimentally as,

a* ¼�0:8aþ 0:1 for FCC2ð0:8aþ 0:1Þ for BCC

(11)

In order to verify the application of life evaluation under non-proportional loading, the comparison of Nf in Case 2 obtained fromexperiment evaluated by Eq. (10) based on life curve in pushepulltest (Case 1) is shown in Fig. 12. In Eq. (10), a* was used for materialconstant. In the figure, Nexp

f is the failure life in experiment and Ncalf

the failure life estimated by Eq. (10). All the data are correlatedwithin the factor of 2 band. Consequently, the good correlation oflives in Fig. 12 suggests that failure life under non-proportionalloading for various materials can be estimated by DεNP if the in-tensity of additional hardening is obtained from experiment.

4.4. A simple method for evaluation of a and life estimation

As discussed above,multiaxial LCF life shows the large reductionin failure life under non-proportional loading in comparison withthat under proportional loading. By using non-proportional strainparameter, DεNP in Eqs (6) and (10), multiaxial LCF lives can be

Fig. 12. Comparison of Nf in Case 2 between calculation and experiment.

estimated from the data in pushepull loading test. However, toobtain the value of material constants, a and a*, multiaxial fatiguetests under non-proportional loading are necessary, but it is usuallydifficult procedure. If a and a* can be obtained without conductingthe multiaxial fatigue test, it will be very convenient for engineersto estimate LCF life under non-proportional loading. This sectiondiscusses the reevaluation of a by focusing on the relationshipbetween a and material constants obtained by the static tensiontest. Cyclic hardening and additional hardening behaviors shouldhave close relationship with static deformation behavior, then arelationship between (sB�sY)/sB and a is shown in Fig.13. Althoughsome scatter of data is shown, the relationship can be equatedapproximately as,

0:8aþ 0:1 ¼ ðsB � sYÞsB

(12)

where sB is tensile strength and sY yielding or 0.2% proof stress,According to Eq. (10)�(12), non-proportional strain range, Dε0NP,can be rewritten as,

Dε0NP ¼�1þ S

sB � sYsB

fNP

�DεI ¼ AN�0:12

f þ BN�0:6f (13)

where coefficient S takes S ¼ 1 for FCC materials and S ¼ 2 for BCCmaterials.

Fig.14 shows the comparison ofNf in Case 2 between experimentand calculation. In the figure, Nexp

f is the failure life in experimentand Ncal

f the failure life estimated based on life curve in pushepulltest (Case 1 test) by using Dε0NP in Eq. (13). Consequently, all the data

Fig. 14. Comparison of Nf in Case 2 between experiment and calculation by Dε 0NP.

T. Itoh et al. / International Journal of Pressure Vessels and Piping 110 (2013) 50e5656

are correlated within a factor of 3 band andmost of them correlatedwithin the factor of 2 band. The good correlation in Fig. 14 suggeststhat multiaxial LCF life under non-proportional can be estimated byEq. (13) with material constants obtained by the static tension test.

5. Conclusions

1. This paper showed a simple method of determining the prin-cipal stress and strain ranges together with themean stress andstrain under proportional and non-proportional loading in 3Dstress and strain space. It also presented themethod of definingthe rotation and deviation angles of the maximum principalstress and strain.

2. The paper extend the non-proportional factor, fNP, from 2D to3D stress and strain space with the consistency with the pre-vious definition of it in the 2D space.

3. Reduction in failure life has close relationship with additionalhardening under non-proportional loading, which depends oncrystal structure of tested materials.

4. The parameter a* which relates to the degree of reduction infailure life is effective in life evaluation for various kinds ofmaterials.

5. The relationship between a* and a which relates to the addi-tional hardening due to non-proportional loading is equated bylinear relationships separately in BCC and FCC materials.

6. a is closely related with the behavior of static tensile test andcan be equated by 0.8a þ 0.1 ¼ (sB�sY)/sB.

7. Failure life under non-proportional loading for various mate-rials can be evaluated by DεNP if the intensity of additionalhardening is obtained as equated by DεNP and Dε0NP.

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