JOURNAL OF THEORETICAL AND APPLIED MECHANICS 42, 2, pp. 295-314, Warsaw 2004 A FATIGUE FAILURE CRITERION FOR MULTIAXIAL LOADING WITH PHASE SHIFT AND MEAN VALUE Dariusz Skibicki Faculty of Mechanical Engineering, University of Technology and Agriculture in Bydgoszcz email: darski@atr.bydgoszcz.pl Anew criterion based on the critical plane approachhas been developed for multiaxial non-proportional fatigue failure. The criterion correctly takes into account the influence of phase shift and mean values under combined bending and torsion loading. From a certain point of view, the criterion with such a defined non-proportionality measure can be understood as a combination of the two approaches: critical plane and integral approach. The criterion has the following form τα∗(eqnp) =(τα∗(a)+ c1σα∗(a)+ c2σα∗(m)) ( 1+ t−1 b−1 Hn ) ¬ c3 where the multiplicand of the equivalent shear stress τα∗(eqnp) contains the amplitude of the shear stress τα∗(a), the amplitude σα∗(a) and me- an value σα∗(m) of the normal stress acting in the critical plane. The multiplier contains the loading non-proportionalitymeasure H. Taking into account the fact of different sensitivity of various materials to lo- ading non-proportionality, the equation also includes the material data: t−1 – fatigue limit in torsion, b−1 – fatigue limit in bending. The predictive capability of the criterionwas demonstratedby analyzing 67 experimental results from the literature. The predicted results are generally in good agreement with the experimental ones. Key words: high cycle fatigue, multiaxial fatigue, non-proportional loading, out-of-phase loading, mean value Notations b −1 – fatigue limit under bending for R=−1 b0 – fatigue limit under bending for R=0 296 D.Skibicki t −1 – fatigue limit under torsion for R=−1 σf – ultimate tensile strength σx(a) – bending stress amplitude τxy(a) – torsional shear stress amplitude σx(m) – bending stress mean value τxy(m) – torsional shear stress mean value α – plane at an angle measured from the bending stress plane α∗ – critical plane τα – shear stress on the plane α σα – normal stress on the plane α τα(a) – shear stress amplitude on the plane α τα(m) – shear stress mean value on the plane α σα(a) – normal stress amplitude on the plane α σα(m) – normal stress mean value on the plane α σα(max) – maximum value of normal stress on the plane α τt(max) – maximum value of shear stress for a given moment of time of the loading cycle on an arbitrary plane α τ̂t(max) – maximum value of shear stress in the loading cycle on an arbitrary plane α, the maximum of τt(max) τα(eq) – proportional equivalent shear stress amplitude on an arbi- trary plane α τα∗(eq) – proportional equivalent shear stress amplitude calculated for the plane α∗ τα∗(eqnp) – non-proportional equivalent shear stress amplitude calcula- ted for the plane α∗ H – loading non-proportionality measure W – weight function R – stress ratio σx(min)/σx(max) or τxy(min)/τxy(max) λ – stress amplitudes ratio σx(a)/τxy(a) ϕ – phase shift angle between σx and τxy β – phase shift angle between σα and τα 1. Effect of loading non-proportionality on fatigue properties In many cases of operational conditions one deals with non-proportional loadings. Themost significant feature of that state is the turn of the principal stress/strain axes due to fatigue. For many structural materials, such loading conditions have major influence on their fatigue life and strength. A fatigue failure criterion for multiaxial loading... 297 A group of materials for which, under such conditions, a considerable de- crease of the fatigue life is observed is ductile structural steels. The fatigue strength decrease is usually strongly correlated with extra hardening. For example, the results of Fatemi’s (from Socie, 1987) research showed that in- crease in the extra hardening by 10-15% caused reduction of the life by half. When the extra hardening was larger by 100%, the life was ten times shor- ter. Also Socie (1987), on the basis of his study, estimated that the life in non-proportional loading conditions may be even ten times smaller. There also exists substantial influence of the loading non-proportionality on the fatigue limit value. Although, the smaller the loading, the smaller the decrease of life, and the life curves for proportional and non-proportional lo- adings converge asymptotically on the level of the fatigue limit (Ellyin et al., 1991), in the range of unlimited life, a fall of the fatigue limit is still visi- ble. Extensive research concerning the topic was conducted by Nisihara and Kawamoto (1945). The causes for such fatigue scenarios are seen in the behaviour of thema- ximum shear stress vector, which changes its location in time. According to Sakane et al. (1997), when the biggest shear stress vector comprises all α planes, as a result of the non-proportional loading,many slip systems are acti- vated. As a consequence, dislocationsmoving on the planes initiate additional interactions. Fatigue damage accumulation intensifies. The effect of this are characteristic changes in the dislocation structure picture. Even on the basis of only Jiao’s et al. (1996), Rios’s et al. (1989) and Sakane’s et al. (1997) research, itmay be shown that in comparison to structu- res obtained in proportional loading conditions these are: greater dislocation density, larger wall misorientation andperfection, smaller cell sizes, remaining of the dislocation within the cells and thus, more homogeneous dislocation distribution. Non-proportional dislocation structures remain in the same kind of relation to proportional dislocation structures as, under the conditions of proportional loadings, the structures corresponding to a greater number of cycles to the structures that come into being at a smaller number of cycles. The only exception is the remaining of the dislocation in the cell inside. It can be remarked that the dislocation structures obtained under the conditions of non-proportional loading exhibit higher values of cumulated fatigue damage. What results fromthis is that thenon-proportional loading ismoredestructive in relation to the proportional one. What should also be noticed is that for other types of materials, the non- proportional loading influence is different. An increase in the fatigue life is observed under a non-proportional loading for low-ductile (so-called brittle) 298 D.Skibicki materials like cast aluminium, cast iron, sintered steels. There are also semi- ductile materials, which reveal no difference between the in- and out-of-phase multiaxial loading, e.g. cast steels andwrought aluminiumalloys, Sonsino and Maddox (2001). 2. Existing fatigue criteria Since 1935, whenGough (1950) suggested ellipse quadrant and ellipse arc equations, there has come into existence a great number of multiaxial fatigue criteria. At the same time, many propositions of their division have been sug- gested. According to Weber et al. (2001), fatigue criteria can be divided into three groups: • Empirical criteria • Critical plane approach criteria • Global approach criteria. Although, as the author emphasizes, this classification is not precise as some criteria could belong to one category from certain aspects and to another one considered from the point of view of other aspects, this division makes the analysis of some interesting criteria characteristics easier. Belonging to parti- cular groups it decides to some extent, about possible manners of including the loading non-proportionality influence in the criterion record. One of the first multiaxial fatigue criteria, like the equations of Gough (1950) and Nishihara and Kawamoto (1945), as well as some later ones, li- ke S.B. Lee’s (1989) and Y.L. Lee’s (1985), fall into the group of the Em- pirical criteria. Apart from their undoubted advantages, such as simplicity and engineer-friendliness, these criteria have significant weaknesses. Themost obvious drawback is that they are dedicated tomodel the fatigue behaviour of material under particular types of conditions only, and are generally restricted to these applications. A very important group of criteria are those based on the idea of the Cri- tical Plane. The concept of these criteria is established on the statement that the fatigue behaviour of the material is a result of action of stresses that are acting on the so-called critical plane. Application of this criterion requires, firstly, defining the orientation of the critical plane and next, determining the criterion stress quantities relative to this plane for determining the severity of the multiaxial cycle for the given material. The choices both of the critical plane definition and the stresses components that are involved in the crite- ria formulation depend on authors. To illustrate, Findley (1959), McDiarmid (1981) and Dang Van et al. (1989) proposed some criteria of that category. A fatigue failure criterion for multiaxial loading... 299 According toWeber’s et al. (2001) division, in the group of theGlobal Ap- proach Criteria three sub-categories may be distinguished. Some criteria use invariants of the stress or deviatoric stress tensors, and these are the criteria by Sines (Sines and Ohgi, 1981) and Crossland (from Dietrich et al., 1972). The next subgroup constitute the Energetic criteria as those proposed by Fro- ustey (fromWeber et al. 2001). Finally, the criteria which utilize contributions of all possible material planes passing through the material point where the fatigue assessment is realised for predicting the material fatigue behaviour. This concept termed as the Integral Approach, requires calculating of a da- mage indicator to take into account the quantities connected with all possible planes through the pointwhere the fatigue, e.g. life, is assessed. Papadopoulos (1995) proposed such a criterion. Depending on the group, the criteria differently deal with the problem of the influence of the loading non-proportionality on the fatigue properties. The easiest way to make the results of forecast fatigue properties in non- proportional loadings conditions real is to introduce the phase shift ϕ to the criterion records. The ideawas used in S.B. Lee’s (1989) andY.L. Lee’s (1985) criteria, where the non-proportional parts were functions of the out-of-phase angle. This solution is of course restricted to a particular loading case and sometimes is deprived of physical interpretation. Moreover, the phase shift angle is not an appropriate non-proportionality parameter because the loading non-proportionality degree depends also on other nominal stress parameters, like for example the amplitude relation λ or average stress values. The criterion group based on the critical plane idea possesses a strong physical interpretation. As Kanazawa et al. (1979) emphasise on the basis of physical observation, fatigue processes (crack development) and properties (fatigue limit) depend on quantities connected with the selected plane. To achieve conformity with experimental results in the non-proportional range, speciallydefinednon-proportionality functionsareused for this groupof criteria. The non-proportional factor of the non-proportional strain parameter defined by Itoh et al. (1997), the Duprat (1997) model for predicting fatigue life, Morel’s et al. (1997) phase-difference coefficient or the rotation factor introduced by Kanazawa et al. (1979) may serve as examples. The question about correct, possessing physical interpretation and in agre- ement with experimental results considering the loading non-proportionality, concerns also the criteria from the IntegralApproach group. It is believed (e.g. Witt et al., 2001) that this approach appears particularly useful in the case of a non-proportional loading, as the volume element is uniformly loaded in all planes with the direction of principal stress changing with time. However, ac- 300 D.Skibicki cording to the author, this approach does not always allow one to describe the stress state correctly either. For example, it does not make it possible to tell the difference when the stresses acting on the α planes result from the prin- ciple axis rotation, and when they are the result of turning transformation. From this model perspective, no difference is seen between the proportional and non-proportional loading state. Taking into account both group properties, Sonsino and Maddox (2001) made a division of the criteria usefulness. For materials not showing the non- proportional loading sensitiveness (semi-ductile), they suggest theCriticalPla- ne, while for materials sensitive to principle axis turns (ductile), they suggest using a criterion form the Integral Approach group. There arises a question if the imperfectness of the critical plane concept really forces us in the case of ductile materials to give up this idea for the be- nefit of the Integral Approach. Irrespectively of thematerial, fatigue processes are characterised by directivity. Together with the material change, only the critical plane discriminant changes, but the idea seems to remain always right. On the other hand, the idea of taking into consideration the influence of other planes in the state of a non-proportional loading, proposed by the Integral Approach, is also very promising. 3. Assumptions concerning the new criterion formulation It is assumed that the influence of the loading non-proportionality on the fatigue strength and life may be described bymeans of the critical plane mo- del completed with the loading non-proportionality measure being a quantity described on the basis of stresses acting outside the critical plane. The cor- rectness of the critical plane idea is assumed even in non-proportional loading conditions; however, for executing the correct results it is necessary to take into account additional interactive influence of the stresses on other planes, the planes which are embraced by the turning vector of the maximum shear stress. 4. Criterion formulation Formulation of the newly proposed criterion requires deciding about the following questions: A fatigue failure criterion for multiaxial loading... 301 • Selecting the critical plane position • Indicating other planes (outside the critical one), on which the acting stresses should be taken into account in the proposed description • Deciding on the stress state components participating in the fatigue pro- cess and deciding on the criterion quality, i.e. a mathematical form of the relation between the indicated components • Formulating the loading non-proportionality function. As amodel of the non-proportional loading, bendingwith torsionwith the phase shift and the nominal stress mean value was accepted. Fig. 1. Non-proportional stress histories; (a) definition of the directions x and α, (b) stress histories in the direction x, (c) stress histories in the directionα For the abovementionedmodel loadings, due to themaximumshear stress vector behaviour, the following possible states were distinguished: • If themeanvalues andthephase shift angle equal zero, during the fatigue cycle only the maximum shear stress value changes while the direction does not – situation (a) 302 D.Skibicki • If the phase shift angle differs form zero and the mean values equal zero, the maximum shear stress vector is fully rotated, and this stress hodograph is a closed curve – situation (b) • If themean values differ form zero and the phase shift angle equals zero, the principle axis turn is executed only to a certain extent of the angle, and the stress hodograph is an open curve – situation (c) • If both the phase shift angle and the mean values differ form zero, the maximum shear stress vector behaviour depends on these values interre- lation, i.e. at the appropriately low value of themean stresses it executes the full turn – situation (b), while when these values are appropriately high the turn is partial – situation (c). Fig. 2. The distinguished states of the maximum shear stress vector; (a) ϕ=0, σx(m) = τxy(m) =0, (b) ϕ 6=0, σx(m) = τxy(m) =0, (c) σx(m) 6=0, τxy(m) 6=0 On the basis of this analysis, the questions necessary to describe the cri- terion shape were concluded. 4.1. Critical plane selection In each of the above mentioned nominal stress states, there exists the critical plane. In Fig.2 this is the plane described by the cycle maximum shear stress vector τ̂t(max). In the case of the proposed criterion, a solution identical with Findley’s (1959) is accepted. It is assumed that, taking into account the analysed material type (ductile), what should be taken for the critical plane α∗ is the plain with the critical combination of the shear and normal stresses. 4.2. Selection of other planes considered in the proposed description During the non-proportional loading, the direction of action of the maxi- mum shear stress vector changes. The turning maximum shear stress vector A fatigue failure criterion for multiaxial loading... 303 embraces with its action many planes, potentially initiating many slide sys- tems. Therefore, these plane sets should be considered which are embraced with themaximum shear stress τt(max) action during a cycle. In situation (b) these are all planes, while in situation (c) only those which are located within a certain range of the angle of rotation. 4.3. Defining the stress state components and the relation between them For the considered fatigue quantity, i.e. the fatigue limit, the fatigue crack initiation life is a large portion of the total fatigue life (Susmel and Lazzarin, 2002). Under such conditions, themaximum fatigue damage is produced along the direction of and governed by shear stress amplitude τα(a). However, the fatigue crack initiation as well as fatigue crack growth are conditioned by the stress normal to the initiation plane σα(a). The compression component inhibits the persistent slip band laminar flow,whereas the traction component favours their flow, Susmel and Lazzarin (2002). The considered loading case takes also into account the occurrence of the mean value. Papadopoulos (1995) shows that for a very high fatigue life of the order of one million cycles or more, the limiting shear stress amplitude is independent of themean shear stress. But when it comes to the normal stress influence, the fatigue limit in bending strongly depends on the superimposed mean (static) normal stress σα(m) in such a way that the tensile mean normal stress reduces the fatigue limit, whereas the compressive mean stress leads to a net increase. On the basis of the above mentioned, the general dependence of the equ- ivalent shear stress amplitude may be formulated τα∗(eq) = τα∗(a)+ c1σα∗(a)+ c2σα∗(m) (4.1) When the equivalent stresses are calculated on an arbitrary plane α, different from the critical plane α∗, the symbol τα∗(eq) should be replaced with τα(eq). Dependance of the shape (4.1) is used in the case of e.g. McDiarmid’s criterion (1985). Sometimes, the influence of the amplitude and the normal stress mean value is combined through the highest stress value σα(max), as in the case of Findley (1959) or Fatemi and Socie (1988). In this paper, the solution with normal stress components separation was accepted. Afterwards, the coefficients c1 and c2 were defined. From Gough’s criteria (1950) it results that the coefficient c1, with the assumption that the critical planeposition is describedwith the vector τ̂t(max), has the form of 2(t −1/b−1)−1. The normal stress influence depends therefore 304 D.Skibicki on the t −1/b−1 relation andchanges from1 (which is equivalentwithaccepting themaximumprinciple stresshypothesis) to0.5 (as in thecase of themaximum shear stress hypothesis 0.5). The lower the relation, the lower the influence of the stress σα(a). In the proposed solution, the best agreement between the calculation and experimental results was obtained when the following c1 = 1.9(t−1/b−1)−1 was accepted. In the case of criteria describing similar loading conditions, the coefficient c2 is usually a function of t−1 and b−1 or b0, see e.g. Sines and Ohgi (1981) andKakuno andKawada (fromPapadopoulos, 1997). Due to the fact that b0 is seldom available, b0 may be expressed bymeans of theGoogman line in the function of σf. In both cases the coefficient c2 is expressedwith the following dependence (3t −1/b0)− √ 3 or √ 3b −1/σf. In McDiarmid’a (1985) the value c2 is expressed with the dependance 0.5b −1/(0.5σf) 2, however the normal stress appears in the 1.5 power, and in the later criterion of his (1990), c2 equals t−1/(2σf). In this paper, the best conformity of the calculationswith the experimental data was obtained when c2 =0.5b−1/σf. 4.4. Formulation of the loading non-proportionality function The moment-maximum shear stress vector embraces with its action a number of planes. Depending on the plane initiated by the maximum she- ar stress vector and on the stress conditions on the plane, the loading non- proportionality extent is different. In this paper, the non-proportionality range is described with a non-proportionality function. Similarly as in the Integral Approach, the influence of the stress acting on the selected planes on the fatigue damage cumulating process is summed. The measure of non-proportionality is defined on the basis of observation of geometry behaviour of stress hodographsdrawn for cases of different degrees of the loading non-proportionality. Themeasure is described in a detailedway in the works of Skibicki and Sempruch (2001, 2002a,b). Below, the eventual notation of the function is given H = ∫ α (τα(eq)W) π 2 (τα∗(eq)) 2 (4.2) Theparticipation of stresses acting outside the critical planewas described with a filling coefficient, definedas a quotient of the areawithin the hodograph A fatigue failure criterion for multiaxial loading... 305 and the area of the circle described on the hodograph. The quantity W appe- aring in the formula is a weight function, and takes the following form W =sin[2(α−α∗)]k (4.3) The function W becomes zero on the critical plane, and for the direction most remote fromthe critical plane (similarly as inKanazawa’s rotation factor (Kanazawa et al., 1979), this is the direction rotated by 45◦), it assumes the value of one. The taking into account the loading non-proportionality function criterion in the notation requires also considering thematerial sensitiveness (answer) to the loading non-proportionality, which is a function of the quotient t −1/b−1. As a result, the following is obtained τα∗(eqnp) = τα∗(eq) ( 1+ t −1 b −1 Hn ) ¬ c3 (4.4) In this way, for ϕ=0 and τxy(m) = σx(m) = 0, which hold in the case of proportional loading. The value H equals zero, and the equation describes the proportional loadings state τα∗(eqnp) = τα∗(eq). For ϕ 6=0 or/and τxy(m) 6=0, σx(m) 6= 0, the proportional part (multiplicand) value decreases but the non- proportionalitymeasurevalue H increases and themultiplier increases.Owing to this, the calculation value of the fatigue limit τα∗(eqnp) is constantly close to the experimental t −1. The detailed form of the criterion has been obtained bymeans of approxi- mation of the experimental data with equation (4.4) τα∗(eqnp) = τα∗(eq) ( 1+ t −1 b −1 H(W)3 ) ¬ t −1 (4.5) where τα∗(eq) = [ τα∗(a)+ ( 1.9 t −1 b −1 −1 ) σα∗(a)+ b −1 2σf σα∗(m) ] W =sin[2(α−α∗)]5 5. Calculation results Bymeans of the criterion, the fatigue limit valueswere calculated for 67 ca- ses of fatigue loadings, taken from literature (Table 1).The literature data that 306 D.Skibicki was used concern examining the influence on the fatigue limit value: firstly, of the loading nominal parameters (like phase shift angle ϕ, amplitude λ ratio and themean values) and secondly, of a wide range of materials characterized by the coefficient t −1/b−1. The data was divided into 10 groups (Column 1). A group consists of the results obtained by one author during the same type of research, concerning one type ofmaterial. The number of the data is put in the next column.The datawith numbers from1 to 18 are taken fromNisihara and Kawamoto (1945), from 19 to 40 – from Lemmp (1997), from 41 to 46 – from Sonsino (1983), from 47 to 50 – Neugebauer (from McDiarmid, 1987), from 51 to 58 – from Lemmp (from Weber et al., 2001), and from 59 to 67 – from Froustey and Lasserre (1989). The calculation results are presented in Columns 13 and 14, where Co- lumn 13 presents the equivalent stress value, and Column 14 – the relative error, i.e. τα∗(eqnp)/t−1. Table 1. Literature data and calculation results g ro u p d a ta in d ex t − 1 / b − 1 t − 1 b − 1 σ f λ σ x (a ) τ x y (a ) σ x (m ) τ x y (m ) ϕ τ α ∗ (e q n p ) τ α ∗ (e q n p ) / t − 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 1 Nisihara and 1.21 99.9 120.9 0 0 0 136.2 0.99 2 Kawamoto (1945) 1.21 103.6 125.4 0 0 60 139.1 1.01 3 0.583 137.3 235.4 – 1.21 108.9 131.8 0 0 90 144.7 1.05 4 0.5 180.3 90.2 0 0 0 137.3 1.00 5 0.5 191.4 95.7 0 0 60 135.4 0.99 6 0.5 201.1 100.6 0 0 90 138.3 1.01 7 0.21 213.2 44.8 0 0 0 127.2 0.93 8 0.21 230.2 48.3 0 0 90 130.2 0.95 2 9 Nisihara and 1.21 138.1 167.1 0 0 0 193.8 0.99 10 Kawamoto (1945) 1.21 140.4 169.9 0 0 30 200.5 1.02 11 0.625 196.2 313.9 – 1.21 145.7 176.3 0 0 60 206.8 1.05 12 1.21 150.2 181.7 0 0 90 212.4 1.08 13 0.5 245.3 122.7 0 0 0 196.5 1.00 14 0.5 249.7 124.9 0 0 30 201.0 1.02 15 0.5 252.4 126.2 0 0 60 195.9 1.00 16 0.5 258.0 129.0 0 0 90 198.0 1.01 17 0.21 299.1 62.8 0 0 0 190.3 0.97 18 0.21 304.5 63.9 0 0 90 188.0 0.96 3 19 Lemmp (1997) 0.48 140 67.2 0 0 0 102.6 1.03 20 0.568 100 176 – 0.48 168 80.6 0 0 90 107.1 1.07 A fatigue failure criterion for multiaxial loading... 307 4 21 Lemmp (1997) 0.48 201 96.5 0 0 0 148.6 1.02 22 0.575 146 254 – 0.48 234 112.3 0 0 90 152.4 1.04 5 23 Lemmp (1997) 0.21 226 47.5 0 0 0 133.8 0.98 24 0.58 136 235 – 0.21 233 48.9 0 0 90 130.6 0.96 25 0.5 180 90 0 0 0 136.2 1.00 26 0.5 187 93.5 0 0 60 131.0 0.96 27 0.5 201 100.5 0 0 90 136.5 1.00 28 1.21 98 118.6 0 0 0 133.2 0.98 29 1.21 101 122.2 0 0 60 134.7 0.99 30 1.21 109 131.9 0 0 90 143.8 1.06 6 31 Lemmp (1997) 0.21 299 62.8 0 0 0 191.8 0.97 32 0.63 198 314 – 0.21 299 62.8 0 0 90 186.8 0.94 33 0.5 245 122.5 0 0 0 197.5 1.00 34 0.5 245 122.5 0 0 60 192.4 0.97 35 0.5 245 122.5 0 0 90 190.7 0.96 36 0.5 255 127.5 0 0 90 198.5 1.00 37 1.21 137 165.8 0 0 0 193.0 0.97 38 1.21 137 165.8 0 0 0 193.0 0.97 39 1.21 142 171.8 0 0 60 203.1 1.03 40 1.21 147 177.9 0 0 90 209.6 1.06 7 41 Sonsino (1983) 0.58 135 78.3 0 0 0 112.8 0.94 42 0.6 120 200 – 0.58 152 88.2 0 0 90 117.6 0.98 43 0.58 160 92.8 0 0 0 132.3 1.02 44 0.58 168 97.4 0 0 90 126.8 0.98 45 0.58 185 107.3 0 0 0 155.3 0.97 46 0.58 207 120.1 0 0 90 161.7 1.01 8 47 Neugebauer 0.57 183 104.3 0 0 0 168.9 0.97 48 (fromMcDiarmid, 1987) 0.57 195 111.2 0 0 90 189.0 1.08 49 0.7 175 250 – 1 135 135 0 0 0 173.2 0.99 50 1 150 150 0 0 90 205.1 1.17 9 51 Lemmp 0.48 328 157 0 0 0 266.6 1.03 52 (fromWeber, 2001) 0.48 286 137 0 0 90 231.4 0.89 53 0.65 260 398 1025 0.96 233 224 0 0 0 280.6 1.08 54 0.96 213 205 0 0 90 262.2 1.01 55 0.48 266 128 0 128 0 204.2 0.79 56 0.48 283 136 0 136 90 215.6 0.83 57 0.48 280 134 280 0 0 268.6 1.03 58 0.48 271 130 271 0 90 237.6 0.91 10 59 Froustey and 0.58 485 280 0 0 0 414.1 1.01 60 Lasserre (1989) 0.58 480 277 0 0 90 390.8 0.95 61 0.62 410 660 1880 0.58 480 277 300 0 0 463.2 1.13 62 0.58 480 277 300 0 45 454.4 1.11 63 0.58 470 270 300 0 60 437.9 1.07 64 0.58 473 273 300 0 90 433.2 1.06 308 D.Skibicki 65 0.25 590 148 300 0 0 435.8 1.06 66 0.25 565 141 300 0 45 405.2 0.99 67 0.25 540 135 300 0 90 370.6 0.90 Fig. 3. Histogram of calculation results For all 67 result groups, the mean value and the relative error standard deviationwere calculated.Thesequantities valueswere respectively 1 and0.06. Apart fromfigure parameters, the results were characterised with a frequency histogram, Fig.3. What is worth noticing is the fact that more than 90% of the results falls into the range between ±10%. The results of all calculations are also presented in Fig.4. What can be seen in this diagram is the dependence between the obtained results and the non-proportionality degree expressedwith the phase shift angle (designation – �, Fig.4). To illustrate the non-proportionality function influence, the results of calculations for the criterion without the non-proportionality functionwere presented in the same diagram (designation – 4, Fig.4). The greater the phase shift angle is, theworse the results are.However, thenon-proportionality function improves the conformity of the obtained resultswith the experimental results, see e.g. points 6 and 16. What isworthmentioning is the fact that the non-proportionality function doesnot causean increase in the equivalent stressvalue in the situationswhere, despite a big phase shift angle, the loading non-proportionality, due to the other loading parameters value (e.g. λ=1.21), is small and does not have any influence on the obtained fatigue limit value (points 11 and 12). Criteria based on the phase shift angle, as the only non-proportionalitymeasure, are in these cases burdenedwith a considerable error. The non-proportionality also results fromthe existence of thenominal loadingmeanvalues. In these cases, however, the principle axis turn is limited to a minor range (Fig.2c), therefore, the A fatigue failure criterion for multiaxial loading... 309 Fig. 4. Calculation results, � – τα∗(eqnp), 4 – τα∗(eq) non-proportionality influence is in these cases insignificant (Fig.4, blackened symbols N, �). It is most clearly seen in points 61 and 65, for which ϕ=0, and the calculation error equals 2% of the result value. In general, the calculation results in the case where the mean value influ- ence is taken into account (points 55-58 and 61-67), although relatively the worst of all obtained, are still satisfactory – they fall into the scope of −21% to +13%. What is also worth carrying out is the comparison of the above results with the results obtained from other criteria for the same literature data. The comparative analysis of different criteria (Crossland, Sines, Matake, McDiarmid, Dietman, Papadopoulos) was conducted by Papaudoulos (1997). Anumberof calculationsweredone for the same literaturedata as in thispaper (groups2, 9 and10).Theresults obtainedon thebasis of theproposedcriterion as well as Papadopoulos’s analysis results are contrasted in Table 2 and in Fig.4, where the mean values and index standard deviations are compared. As a rule, the proposed criterion gives more correct results, i.e. the mean error values are closer to one, and the standard deviations in comparisonwith the first five criteria (indexes: 2, 3, 4, 5 and 6) are smaller. Only the results obtained on the basis of Papadopoulos’s criterion (1997), (index 7) are more conforming with the experimental data than the proposed criterion results. 310 D.Skibicki Table 2. Compatative analysis of the results obtained with different criteria g ro u p d a ta in d ex τ α ∗ (e q n p ) / t − 1 C ro ss la n d S in es M a ta k e M cD ia rm id D ie tm a n P a p a d o p o u lo s 1 2 3 4 5 6 7 2 9-18 mean 1.01 0.95 0.90 1.03 0.96 0.99 1.03 std. dev. 0.04 0.06 0.06 0.04 0.06 0.05 0.03 9 51-58 mean 0.95 0.87 0.83 1.00 0.83 0.91 0.96 std. dev. 0.11 0.14 0.16 0.13 0.14 0.14 0.07 10 59-67 mean 1.03 0.90 0.94 1.10 0.92 1.02 1.01 std. dev. 0.07 0.11 0.14 0.08 0.07 0.13 0.03 Fig. 5. Comparative analysis of the applied criteria 6. Conclusions The new fatigue failure criterion has been developed for the multiaxial non-proportional loading. The criterion is an attempt to combine two known but alternatively used models. The main assumptions on which the criterion is based result from the Critical Plane idea, while the non-proportionality function used in the criterion applies the Integral Approach postulates. The criterion correctly describes the influence of themeannominal loading and the influence of the non-proportionality of loading (caused by non-zero values of the phase shift and mean values) on the fatigue limit value. The predictive capability of the criterion was demonstrated by analysing 67 expe- rimental results from the literature. The predicted results were generally in good agreement with the experimental ones. A fatigue failure criterion for multiaxial loading... 311 References 1. DangVanK., GriveauB.,Message O., 1989,On a newmultiaxial fatigue limit criterion: theory and application,Biaxial and Multiaxial Fatigue, EGF 3, Edited by M.W. Brown and K.J. Miller, Mechanical Engineering Publications Limited, London, 479-496 2. Dietrich L., Turski K., 1972, Badania zmęczeniowe w złożonym stanie na- prężenia,Mechanika Teoretyczna i Stosowana, 10, 1, 9-28 3. 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Witt M., Zenner H., Yousefi F., 2001, Fatigue strength of welded com- ponents under multiaxial random loading comparison of different lifetime pre- diction concepts, Proceedings of the Sixth international Conference on Bia- xial/Multiaxial Fatigue and Fracture, Editor M. Moreira de Freitas, Instituto Superior Tecnico, Lisboa, 1, 29-40 314 D.Skibicki Kryterium wytrzymałości zmęczeniowej w warunkach wieloosiowych obciążeń z udziałem przesunięcia fazowego i wartości średnich Streszczenie Opracowano nowe, oparte o koncepcje płaszczyzny krytycznej kryterium dla wie- loosiowej wytrzymałości zmęczeniowej. Kryterium poprawnie ujmuje wpływ przesu- nięcia fazowego i wartości średnich w warunkach kombinacji zginania i skręcania. Z pewnego punktu widzenia, tak zdefiniowane kryterium może być rozumiane jako kombinacja dwóchmodeli: płaszczyzny krytycznej i podejścia całkowego (nielokalne- go). Kryteriumma następującą postać τα∗(eqnp) =(τα∗(a)+ c1σα∗(a)+ c2σα∗(m)) ( 1+ t−1 b−1 Hn ) ¬ c3 gdzie mnożna naprężenia zredukowanego τα∗(eqnp) zawiera amplitudę naprężenia stycznego τα∗(a), amplitudę σα∗(a) i wartość średnią σα∗(m) naprężenia normalnego działającychw płaszczyźnie krytycznej. Mnożnik zawieramiarę nieproporcjonalności obciążenia H. Biorąc pod uwagę fakt różnej wrażliwości materiałów na niepropor- cjonalność obciążenia, równanie zawiera również dane materiałowe: t−1 – granicę zmęczenia na skręcanie, b−1 – granicę zmęczenia na zginanie. Zgodność wyników obliczeń z wynikami uzyskanymi eksperymentalnie została zweryfikowana na 67 da- nych zaczerpniętych z literatury. Zgodność ta wwiększości przypadków jest satysfak- cjonująca. Manuscript received September 8, 2003; accepted for print January 8, 2004