82782133 Evaluation and Comparison of Several Multi Axial Fatigue Criteria

International Journal of Fatigue 26 (2004) 17–25www.elsevier.com/locate/ijfatigue

Evaluation and comparison of several multiaxial fatigue criteria

Ying-Yu Wang, Wei-Xing Yao∗

Department of Aircraft Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 16 January 2003; received in revised form 29 April 2003; accepted 30 May 2003

Abstract

In this paper several multiaxial fatigue criteria are reviewed. The criteria are divided into three groups, according to the parametersused to describe the fatigue life or fatigue strength of materials. They are stress criteria, strain criteria and energy criteria. Theirpredictive capabilities are checked against the experimental data of six materials under proportional and nonproportional loading.Among the stress criteria, the criterion of Lee is in the best agreement with the test data. Among the strain criteria, the Kandil,Brown and Miller’s criterion has the best correlation with the experimental data of the materials employed. The Farahani’s criterionyields the most satisfactory result among the energy criteria. Its fatigue life correlation for 1045HR steel and 304 stainless steelfell within factors of 2 and 3, respectively. 2003 Elsevier Ltd. All rights reserved.

Keywords: Multiaxial fatigue; Nonproportional loading; Stress criteria; Strain criteria; Energy criteria

1. Introduction

The components of engineering structures such as air-craft and automobiles usually undergo multiaxial load-ing. The cyclic stress–strain responses under multiaxialloading, which depend on the loading-path, are verycomplex and the fatigue behavior of materials and struc-tures is very difficult to be described. Multiaxial fatiguecriteria, whose aim is to reduce the complex multiaxialloading to an equivalent uniaxial loading, are veryimportant in the study of multiaxial fatigue. Up to now,many researchers have proposed multiaxial fatigue cri-teria suitable to different materials and different loadingconditions. There is not yet a universally accepted modelin spite of a great number of criteria. Even though thereare extensive reviews of multiaxial fatigue criteria,which were presented by Garud[1], Brown and Miller[2], You and Lee[3], Papadopoulos[4], Macha and Son-sino [5] and others, few critiques on multiaxial fatiguecriteria with a lot of experimental data yet exist.

In the present paper, the multiaxial fatigue criteria arereviewed and the most criteria published in literaturesare classified into three categories, namely stress criteria,

∗ Corresponding author. Fax:+86-25-4891422.E-mail address: [email protected] (W.-X. Yao).

0142-1123/$ - see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0142-1123(03)00110-5

strain criteria and energy criteria. Furthermore, in orderto evaluate the capabilities of some popular criteria tocorrelate multiaxial fatigue experiments for both pro-portional and nonproportional loading conditions,fatigue experimental data of six different materials arequoted. Some various criteria are critically examined andthe comparisons are made.

2. Stress criteria

2.1. Brief review of stress criteria

Gough and Pollard[6,7] proposed two equations formetals under combined in-phase bending and torsion,namely the ellipse quadrant for ductile metals,� Sb

f�1�2 � � St

t�1�2 � 1 (1)

and the ellipse arc for brittle metals,� St

t�1�2 � � Sb

f�1�2�f�1

t�1�1� � � Sb

f�1��2�f�1

t�1� � 1 (2)

Considering the phase difference between loadings, Lee[8] modified the ellipse quadrant of Gough as follows

18 Y.-Y. Wang, W.-X. Yao / International Journal of Fatigue 26 (2004) 17–25

Nomenclature

b, c axial fatigue strength exponent and axial fatigue ductility exponent, respectivelyb�, c� torsional fatigue strength exponent and axial fatigue ductility exponent, respectivelyE, G Young’s modulus and shear modulus, respectivelyf�1, t�1 fatigue limits in reversed bending and torsion, respectivelyJ2,a amplitude of the second invariant of the stress deviatorNP, NE predicted life and experimental life, respectivelySb, St bending and torsional stress amplitudes, respectivelytA,B shear fatigue strengtha material constant relating to the additional hardeningb, k, l, S, C material constants�e equivalent strain range between time A and time B�e22, �g21 the normal and shear strain ranges on the critical plane, respectively�emax range of the maximum strain�en normal strain range on the critical plane�gmax range of the maximum shear strain�s22, �t21 the normal and shear stress ranges on the critical plane, respectively�sn normal stress range on the critical plane�t shear stress range on the critical plane�tmax maximum shear stress range on the critical planeen normal strain on the plane of gmax

e∗n strain normal to plane of g∗e�n the normal strain excursion between adjacent turning pointsgmax maximum shear straing∗ maximum shear strain on the plane p/4 to surfacesmax

12 , smax22 maximum absolute values of shear and normal stresses on the critical plane, respectively

sa,eq amplitude of equivalent stresss�f, e�f axial fatigue strength coefficient and axial fatigue ductility coefficient, respectivelysH,m mean value of the hydrostatic stresssH,max maximum value of the hydrostatic stresssn,cr maximum normal stress on the critical planesn,m mean stress on the critical planesm

n mean stress normal to the critical planesmax

n maximum normal stress on the critical planesu ultimate tensile strengthsy yield stressta,cr shear stress amplitude on the critical planet�f, g�f torsional fatigue strength coefficient and axial fatigue ductility coefficient, respectivelyq angle of the cycle path orientation with respect to the principle axis� coefficient of nonproportionalityj phase difference between applied torsion and bending

sa,eq � Sb�1 � �f�1K2t�1

�x�1/x(3)

where x = 2(1 + bsinj), K = 2St /Sb. Carpinteri and Spag-noli [9] used the maximum normal stress and the shearstress amplitude on the critical plane as parameters tomodify Gough’s quadrant equation.

Sines [10] proposed a popular high-cycle fatigue cri-terion�J2,a � ksH,m�l (4)

Kakuno and Kawada [11] suggested separating theeffects of the amplitude and mean value of the hydro-static stress. Crossland [12] proposed that the influenceof the hydrostatic stress must appear in the fatigue for-mula by its maximum value.

Findley [13], Matake [14] and McDiarmid [15] usedthe shear stress amplitude and the maximum value ofthe normal stress on the critical plane as parameters andrespectively proposed a similar fatigue criterion as fol-lows

ta,cr � ksn,cr�l (5)

19Y.-Y. Wang, W.-X. Yao / International Journal of Fatigue 26 (2004) 17–25

Findley [13] determined the critical plane by maximizinga linear combination of shear stress amplitude andmaximum value of the normal stress. However, Matake[14] and McDiarmid [15] proposed that the critical planeis the plane on which the amplitude of the shear stressattains its maximum. McDiarmid [15] considered differ-ence of crack propagation and defined k and l as follows

k � tA,B

2su, l � tA,B (6)

Papadopoulos [16] proposed a fatigue limit criterionwhich could be applied in the case of constant amplitudemultiaxial proportional and nonproportional loading inthe field of high-cycle fatigue. The fatigue limit criterionwere written as

ta,cr � ksH,max � l (7)

2.2. Critique on stress criteria

In the present section, four stress criteria, namely thecriteria of Gough, Lee, Sines and McDiarmid are selec-ted for application against some experimental data. Theexperimental data related to in-phase and 90°out-of-phase loading are retrieved from the following publi-cations: Papadopoulos [4] and Carpinteri and Spagnoli[9]. Correlations of experimental data with predictedparameter are shown in Figs. 1–4. Involved materialparameters are reported in Table 1.

Fig. 1 illustrates that the Gough’s criterion does notfit in with the test data under nonproportional loading.The prediction of this criterion under proportional load-

Fig. 1. The Gough’s criterion.

Fig. 2. The Lee’s criterion.

Fig. 3. The Sines’ criterion.

ing showed a deviation less than ±10%, but the predic-tion of this criterion under nonproportional loadingshowed a deviation greater than ±10%. From Fig. 2, onecan observe that the Lee’s criterion has a good life pre-diction capability under both proportional loading andnonproportional loading. The prediction of the Sines’criterion under proportional loading showed a deviationless than ±10%, see Fig. 3. The errors between experi-


Fig. 4. The McDiarmid’s criterion.

Table 1Material properties

Material f�1 t�1 f�1/t�1 su

(MPa) (MPa) (MPa)

Hard steel [9] 313.9 196.2 1.6 704.1Mild steel [9] 235.4 137.3 1.71 518.842CrMo4 [4] 398 260 1.53 1025

mental data and the predictions of the Sines’ criterionunder nonproportional loading are bigger. The deviationdegrees are different depending on the material, namelythe error of the mild steel is the smallest, the error ofthe hard steel is bigger and the error of the 42CrMo4 isthe biggest. Fig. 4 shows that the McDiarmid’s criterionhas a good correlation within ±5% for hard steel andmild steel, but has a nonconservative correlation for42CrMo4. This coincides with the result in the paper[15]. The error index I is defined to measure the relativedifference between the equivalent stress (sa,eq) and thefatigue limits (f�1 or t�1), that is

I � sa,eq�xx (%) (8)

I � 1n�n

i � 1

|Ii| (9)

where sa,eq is the amplitude of the equivalent stress got

from the multiaxial fatigue criteria, and x equal f�1 ort�1 depending on different criterion. The average absol-ute errors of the four stress criteria are presented in Table2. Considering all sides, the criterion of Lee is in thebest agreement with the test data. Its average absoluteerror is less than 5% for all the materials and all kindof loading conditions employed. However, in the Lee’scriterion there are material constants, which can bedetermined by a lot of experimental data, so the appli-cation of the Lee’s criterion is limited. The Gough’s cri-terion is valid for a lot of materials under proportionalloading and its calculation is relatively simple.

3. Strain criteria

3.1. Review of strain criteria

Brown and Miller [17] proposed a theory based on aphysical interpretation of mechanisms of fatigue crackgrowth. Kandil, Brown and Miller (KBM) [18] then pro-posed a specific form for it

gmax � Sen � C (10)

Lohr and Ellison [19] defined the critical plane as a planep/4 to surface. The proposed fatigue parameter is

g∗ � ke∗n � C (11)

Socie et al. [20] modified the formulation of Eq. (10) byincluding the effect of mean stress as follows

gmax � en � sn,m

E� C (12)

Fatemi and Socie [21] changed the normal strain termin Eq. (10) to a normal stress and gave the following cri-terion

gmax�1 � ksmax

n

sy� � C (13)

Table 2The average absolute error of four stress criteria

Criterion The average absolute error I (%)

Proportional loading Nonproportional loading

Hard Mild 42CrMo4 Hard Mild 42CrMo4steel steel steel steel

Gough 2.4 4.0 9.0 4.7 10.9 10.7Lee 2.4 4.0 9.0 1.5 5.0 4.2Sines 2.7 3.5 8.5 12.5 4.8 27.8McDiarmid 2.6 3.2 8.7 9.4 5.3 30.2


Additional cyclic hardening developed during out-of-phase loading is included in the normal stress term in theabove parameter. Shang and Wang (SW) [22] proposed apath-independent parameter, which is based on the criti-cal plane approach. This parameter is made with themaximum shear strain range �gmax and the normal strainexcursion e�n between adjacent turning points. It isgiven as�eeq / 2 � �e�2

n � 13��gmax

2 �2�1/2

(14)

Under proportional loading the parameter agrees withVon Mises effective strain. Borodii and Strizhalo (BS)[23] take into account the influence of the different cyc-lic path and proposed the following reduced strain range�enp � (1 � ksinq)(1 � a�)�e (15)

3.2. Critique on strain criteria

The results predicted by existing strain criteria arecompared with experimental data found in literature forSAE-1045 steel reported in Kurath et al. [24], 304 stain-less steel reported in Socie et al. [25] and 6061 alumi-num alloy tested by Itoh et al. [26]. The correspondingstrain paths are in-phase and 90° out-of-phase strainingfor three materials. Results are shown in Figs. 5–7. Table3 lists the material properties used.

Correlation for three strain criteria is within a factorof three in life except for the data under out-of-phase

Fig. 5. Predicted versus experimental lives for 1045HR steel: straincriteria.

Fig. 6. Predicted versus experimental lives for 304 stainless steel:strain criteria.

Fig. 7. Predicted versus experimental lives for 6061Al alloy: straincriteria.

loading. The error index E is defined to measure thedeviation between predicted lives and experimentallives, that is

E � log(NP /NE) (16)

22 Y.-Y. Wang, W.-X. Yao / International Journal of Fatigue 26 (2004) 17–25T

able

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E � 1n�n

i � 1

|Ei| (17)

The mean absolute errors of the three strain criteria ana-lyzed are presented in Table 4. It is shown that theKBM’s criterion has a universal good correlation overlow to high lives for 1045HR steel, 304 stainless steeland 6061Al alloy. The SW’s criterion has a good corre-lation within a factor of 3 except for 1045HR steel. Onthe contrary, the BS’s criterion has a good correlationfor 1045HR steel.

4. Energy criteria

4.1. Review of energy criteria

Smith et al. (SWT) [27] proposed an experimentaldamage parameter as follows

W � smaxn

�emax

2(18)

For tensile mode failures, Socie [28] modified the SWTparameter by considering that the parameters, whichcontrol damage, were the maximum principal strainrange and the maximum principal stress on themaximum principal strain plane. Chen et al. [29] pro-posed different damage parameters for the tensile modefailures and the shear mode failures. For the tensile modefailures, Socie’s model were modified by including theshear components for the tensile mode failures

W � �emax1 ·�s1 � �g1·�t1 (19)

For the shear mode failures, the following criterionwere given

W � �gmax·�t � �en·�sn (20)

There are other developments of the SWT approach pro-posed by Chu et al. [30], Liu [31] and others.

Glinka et al. [32] proposed a fatigue parameter byusing the sum of elastic and plastic energy densities inthe critical plane to be

Table 4The average absolute errors of three strain criteria

Material The average absolute error E (%)


KBM SW BS KBM SW BS

1045HR steel 20 23 23 42 72 29304 Stainless 31 29 38 16 32 646061Al alloy 8 8 8 18 7 45


W � �g12

2·�s12

2� �e22

2·�s22

2(21)

By considering the mean stress effect, Glinka et al.(GWP) [33] modified Eq. (21) in the following form

W∗ � �g12

2·�s12

2 � 11�smax

12 /t�f

� 11�smax

22 /s�f� (22)

Farahani (F) [34] proposed a new parameter that doesnot use an empirical fitting factor. The proposed para-meter is the sum of the normal energy range and theshear energy range calculated for the critical plane

W � 1s�fe�f

(�sn�en) (23)� (1 � smn /s�f)

t�fg�f(�tmax�gmax)

Pan Wen-Fang et al. (PHC) [35] proposed that the influ-ence of strain energy on the fatigue life in shear directionshould be different from its influence in normal direc-tion. They modified the fatigue strain energy densityparameter proposed by Glinka et al. [32] in Eq. (21)

W� � �g21

2�t21

2� k1k2

�e22

2�s22

2(24)

where k1 = g�f /e�f and k2 = s�f /t�f are two weight con-stants for strain and stress amplitudes, respectively.

4.2. Critique on energy criteria

The test data related to in-phase and 90° out-of-phasesinusoidal loading in [24,25] are used to evaluate thecriteria of Glinka et al. (Eq. (21)), Farahani and PanWen-Fang et al. (Pan). Predicted lives versus experi-mental lives are shown in Fig. 8 and the average absoluteerrors of the three energy criteria analyzed are presentedin Table 5.

The F’s criterion yields the most satisfactory result.Its correlation for 1045HR steel and 304 stainless steelare within factors of 2 and 3 in life, respectively. ThePHC’s criterion takes second place. The GWP’s criterionhas a good correlation for 1045HR steel. In addition,Fig. 8(a) shows that scatter of data is noticeably greaterat longer lives.

5. Conclusion

1. Among the stress criteria, the Lee’s criterion is mostpromising. Its average absolute error is less than 5%for the materials and loading conditions employed.However, in the Lee’s criterion there are material con-stants, which can be determined by a lot of experi-

Fig. 8. Predicted versus experimental lives: the criteria of Glinka,Farahani and Pan.

mental data, so the application of the Lee’s criterionis limited. The Gough’s criterion is relatively simpleto use but it can only be used under proportional load-ing.

2. Among the strain criteria, the KBM’s criterion has agood correlation over low to high lives for 1045HRsteel, 304 stainless steel and 6061Al alloy.


Table 5The average absolute errors of three energy criteria

Material The average absolute error E (%)


GWP F PHC GWP F PHC

1045HR steel 12 14 23 28 28 54304 Stainless 54 29 39 39 24 41

3. The F’s criterion leads to the most satisfactory resultamong the energy criteria. Its fatigue life correlationfor 1045HR steel and 304 stainless steel fell withinfactors of 2 and 3, respectively. However, the scatterof data estimated by using the energy criteria isnoticeably greater at longer lives.

4. Critique and comparison on multiaxial fatigue criteriarely on multiaxial fatigue experimental test, but dueto the complexity and expense, especially the tests ofhigh-cycle fatigue are very few. It is difficult to do acomprehensive and systemic comparison on the exist-ing multiaxial fatigue criteria. Therefore, enhancingthe study on the multiaxial fatigue test is importantto multiaxial fatigue research.

Acknowledgements

The authors gratefully acknowledge the financial sup-port of National Doctoral Foundation of China undergrant 20020287022.

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82782133 Evaluation and Comparison of Several Multi Axial Fatigue Criteria

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