A review of multiaxial fatigue failure criteria based on the critical plane approach

1. Colloque National MECAMAT- Aussois 2007 - 21-26 Janvier 2007 Ecole de Mécanique des Matériaux A review of multiaxial fatigue failure criteria based on the critical plane approach Aleksander KAROLCZUK Ewald MACHA Opole University of Technology, POLAND Department of Mechanics and Machine Design, e-mail: karol@po.opole.pl

2. Plan of the presentation • Introduction • Critical plane approach - definition - assumptions - range of application - general expressions • Multiaxial fatigue failure criteria based on the critical plane approach - stress based criteria - strain based criteria - energy based criteria • Algorithm of the fatigue life calculation • Determination of the critical plane orientation - damage accumulation method - variance method - weight function method • Exemplary application of simple energy based criterion in fatigue life calculation Part I Part II

3. Introduction In material science, fatigue is the progressive, localised, and permanent structural damage that occurs when a material is subjected to cyclic stresses that have maximum values less than (often much less than) the static yield strength of the material. Fatigue failure Issues: Many mechanical and structural components are subjected to uniaxial or multiaxial fatigue loading that could lead to catastrophic failures (aircrafts, ships, trains). Proper determination of fatigue life of components and structures is important issue at the designing and operating stages.

4. Introduction Real service loading often generates random and multiaxial stress/strain state, which complicates the analysis. Many researchers have attempted to reduce multiaxial stress/strain state to uniaxial one, which is used in fatigue life calculation. Such uniaxial parameter is often called ‘equivalent’ and it means that the same fatigue life is obtained under uniaxial (‘equivalent’) and multiaxial stress/strain state. The reduction is based on the multiaxial fatigue failure criterion. Numerous multiaxial fatigue failure criteria have been proposed in recent decades.

5. Critical plane approach Among these criteria, one type called the critical plane approach can be distinguished. This approach dates back to 1935 when Stanfield suggested a new criterion but without calling it „critical plane criterion” and without making any further research on this subject. Another possible criterion (...), was that in which the two components of stress acting across any plane, i.e. shear and direct stress, might be taken as each contributing a definite quota to ”disruption” combined by a simple arithmetical relation. (...). The planes on which such effect were maximum would not be the principal planes, (...) G. Stanfield, 1935 G. Stanfield. Discussion on ”The strength of metals under combined alternatingstresses”, by H.Gough and H.Pollard. Proc. Institution of Mechanical Engineers 131, (1935)

6. Critical plane approach This concept was not developed until the Fifties when Findley (1956, 1959), Stulen and Cummings (1954) introduced the phrase ”critical plane” and verified fatigue stress criteria based on the critical plane approach. The critical plane approach assumes that the fatigue failure of the material is due to some stress or/and strain components acting on the critical plane. It is based upon the experimental observation that in metallic materials fatigue cracks initiate and grow on certain planes. In this approach, the aspect of microdamage or even short crack propagation are not considered.

7. Arbitrary plane orientation Critical plane approach The critical plane approach concerns the crack initiation process that is usually (in most papers) related to fatigue failure at high cycle fatigue regime. However, it was successfully used also at low cycle fatigue regime. Transformation

8. Critical plane approach Summary: • critical plane criteria reduce the multiaxial state of stress/strain to the equivalent-uniaxial state, • this single parameter (often called damage parameter) is used to calculate fatigue life or damage degree on a plane using the standard S-N curves (a - Nf , a - Nf , a - 2Nf , a - 2Nf), • any phenomena regarding to the crack propagation are not considered, • fatigue life to crack initiation is therefore (usually) estimated, • orientation of the fatigue fracture plane (crack orientation) could be determinedin some cases .

9. Critical plane approach Significance of the critical plane approach has increased during last years, because of its effectiveness and broad application range (proportional, non-proportional, cyclic and random loading). The phrase „critical plane” is included in around 350 articles in databases of Elsevier Journals and Springer Journals during the last 5 years. • The problem is stated as follows: • construct fatigue failure criterion for multiaxial cyclic/random states of stress • determine the critical plane orientation The general form of a critical plane-type failure criterion, that determines the fatigue life Tafter which crack initiation occurs, can be expressed as follows: Stress based - k – array of material coefficients Strain based - Energy based -

10. Critical plane approach Dozens (or more) of multiaxial fatigue failure criteria based on the critical plane approach were proposed. 32 of them are described in paper: KAROLCZUK A., MACHA E.: A review of critical plane orientations in multiaxial fatigue failure criteria of metallic materials, International Journal of Fracture, vol 134, 2005, pp. 267-304 In general, the critical plane is the plane for which fatigue life will be determined. The problem is: which plane orientation is critical?

11. Critical plane approach Multiaxial fatigue criteria based on the criticalplane concept usually apply different loading parameters in the critical plane whose orientationis determined by (a) only shear loading parameters (crack Mode II or III), (b) only normal loadingparameters (crack Mode I) or sometimes (c) mixed loading parameters (mixed crack Mode). Thereare also criteria based on few critical plane orientations and criteria based on critical plane orientationsdetermined by a weighted averaging process of rotating principal stress axes.

12. Critical plane approach • Multiaxial fatigue criteria based on the critical plane approach can be divided according to the fatigue parameter into three groups: • - stress criteria, • High Cycle Fatigue regime, HCF • strain criteria, • High andLow Cycle Fatigue regimes, HCF-LCF • energy criteria, • HCF and LCF

13. Critical plane approach • The present review of the multiaxial fatigue failure criteria based on the critical plane approach is focused on • presentation of the large spectrum of the damage parameters resulting from the critical plane approach and • survey of the critical plane orientation used in the fatigue critical plane criteria

14. Stress criteria

15. (1) (2) (3) Critical planes in criteria based on stresses Generalised hypothesis of material strength Adaptation of the static hypotheses of material strength to fatigue as the replacement of stress static values in these hypotheses by amplitudes or range of fatigue loading • The criteria most frequently verified with experimental data, were the criteria of: • maximum normal stress, • maximum shear stress, • octahedral shear stress

16. Form: or (4) where: fand kare material coefficients, (for ductile materials k  0.2, 0.3) Critical planes in criteria based on stresses Findley criterion (1956)

17. The critical plane is a plane with the maximum value of linear relation of shear and normal stress maximum value (5) critical plane 50 60 70 80 90 100 Angle , deg Critical planes in criteria based on stresses Findley criterion (1956) The critical plane

18. Critical planes in criteria based on stresses Findley criterion (1956) Findley did not defined a mathematical formula for coefficient f. Some researchers (Park and Nelson, 2000; Backstrom and Marquis, 2001) assume that itcan be determined from the shear-mode cracking (6)

19. This criterion was effective for proportional bending with torsion with non-zero mean stress value under the same ratio of normal to shear stress amplitudes for variable loading and static loading (7) Critical planes in criteria based on stresses Findley criterion (1956) Experimental verification

20. Critical planes in criteria based on stresses McDiarmid criterion (1972) Form: (8) where: af – shear fatigue strength for Case A or Case B of fatigue cracks, u – ultimate tensile strength The criterion distinguished type A crack (along the surface) and type B crack (into the material) From the criterion (8) damage parameter can be deduced (Parkand Nelson, 2000) as follows (9)

21. The critical plane is a plane with the maximum shear stress amplitude (10) Experimental verification Proposed criterion correlated experimental data for proportional and non-proportionalbending with torsion loading with zero and non-zero mean value. Only for one case of loading a/a= 0.5 and phase shift /2 where all planes are planes with maximum shear stress range, the criterion was ineffective. Critical planes in criteria based on stresses McDiarmid criterion (1972) The critical plane

22. Critical planes in criteria based on stresses Dietman et al. criterion (1974) Dietman et al. were among the first researchers who paid attention to the interactionbetween changes of principal stress directions and fatigue life. They proposed tomodify the criterion of octahedral shear stress to take into account the changes ofdirection of principal stress axes. This criterion assumes that material fatigue failure occurs when the shear stress amplitude, ns,a, in the critical octahedral plane reaches the critical stress value, ns,a,c, characteristic for a given material (11)

23. Critical planes in criteria based on stresses Dietman et al. criterion (1974) The critical plane The critical plane isthe octahedral plane at time t, for which octahedral shear stress oct,max achievesthe maximum value. Experimental verification Unfortunately, this criterion was used only to determine thefatigue limit and the results were not compared to any standard fatigue characteristic.

24. Critical planes in criteria based on stresses Simbürger and Grubisic criterion (1976) Simbürger and Grubisic proposed a criterion including mean stress value and rotation of principalstress directions. In plane stress state, all possible orientations of the considered planecan be described by the angle α. The proposed fatigue parameter S is formulated asfollows (12) where: Material coefficients a1 and a2 are functions of fatigue limits af and af , whereas a,crepresents the critical stress amplitude for a given number of cycles to failure. Coefficientsm allows to take into account the mean stress value eq,m.

25. Critical planes in criteria based on stresses Simbürger and Grubisic criterion (1976) The critical plane In principle,this criterion does not belong to the critical plane approach becausethe parameter Sn is independent of a specific plane orientation. However, Simbürger and Grubisic determined the position of fatigue fracture plane as a plane with maximum value ofSn parameter. Experimental verification Simbürger and Grubisic did not define a fatigue characteristic (Nf −S) which shouldbe used to calculate fatigue life.

26. (13) where: af– fatigue limit for fully torsion loading k– material coefficient The critical plane The critical plane is one of two planes of maximum shear stress nswith a higher value of normal stress n. (14) Critical planes in criteria based on stresses Matake criterion (1977) Form:

27. Experimental verification This criterion was created to analyze cyclic torsion, bending and proportional torsion with bending. The constant position of principal stresses direction were assumed. Critical planes in criteria based on stresses Matake criterion (1977)

28. Critical planes in criteria based on stresses Dang Van criterion (1982) This criterion is based on the concept of micro-stresses in the critical volume of material. (15) where:  is the microscopic shear stress in grain area, ,h is the microscopic hydrostatic stress, a1, a2 are constants determined from cyclic uniaxial fatigue tests.

29. Critical planes in criteria based on stresses Dang Van criterion (1982) The critical plane The critical plane is a plane with the maximum microscopic shear stress  Experimental verification Many researchers have simplified the Dang Van criterion by replacement of micro-stressesby macro-stresses.

30. Critical planes in criteria based on stresses Robert et al. criterion (1992) Robert et al. proposed a criterion which takes into account the shear stress ns(t), the normalstress n(t) and the mean value of the normal stress n,m in the critical plane (16) where a1(Nf), a2(Nf) are criterion parameter depending on uniaxial fatigue characteristic:fully reversed axial and torsion loading (R=−1), and tensiletest (R=0). The fatigue criterion is defined by (17) where a3 is the third criterion parameter.

31. Critical planes in criteria based on stresses Robert et al. criterion (1992) The numberof cycles to failure Nf is the solution of the Equation (17) and it is obtained froman iterative process. If we assume that a1 =a2 =0.5k, then we will obtain the criterion similar to criteria proposed by Findley, McDiarmid and Matake. The critical plane The critical plane is the plane in which the equivalentstress eq(t) reaches the maximum value. Experimental verification Advantages: (i) the criterion parameters a1, a2, a3 were identified with the use of three uniaxial S–N curves and (ii) the criterion could be applied for random loading. This criterion was successfully used under random, proportional loading.

32. (18) where:,  are material parameters, Ta is a generalised shear stress amplitude is hydrostatic stress. (19) Critical planes in criteria based on stresses Papadopoulos criterion (1993) Form:

33. Critical planes in criteria based on stresses Papadopoulos criterion (1993) Generalised shear stress amplitude: (20) (21)

34. Experimental verification This criterion was analyze under cyclic multiaxial proportional and non-proportional loading. Critical planes in criteria based on stresses Papadopoulos criterion (1993) The critical plane The critical plane is plane where generalized shear stress amplitude Taachieves maximum value.

35. Critical planes in criteria based on stresses Carpinteri and Spagnoli criterion(2001) Form: (22) where: af– fatigue limit for fully torsion loading af – fatigue limit for fully reversed axial loading The critical plane The critical plane orientation is correlated with the averaged principal stress directions The averaged principal stress directions are computed using a weight function which depends on the maximum principal stress σ1(t) and two material parameters

36. Experimental verification This criterion was analyze under cyclic proportional and non-proportional loading and under proportional random loading. Critical planes in criteria based on stresses Carpinteri and Spagnoli proposed to compute the critical plane orientation nwithrespect to the averaged maximum principal stress direction 1 in the plane of 1,3by the following relationship (23) where the angle α is expressed in degrees. According to Eq. (23), the angle α is equal to 0◦ for af/σaf= 1 (hard metals) and α = 45◦ for af/σaf = √3/3 (between hard and mild metals).

37. Critical planes in criteria based on stresses Summary: • Among stress criteria based on the critical plane approach we can distinguish criteriawhich assume that fatigue failure is due to: • the linear combination of shearns and normal n stresses acting on the critical plane; • the linear combination ofshear parameter (ns or Ta), acting on the critical plane, with hydrostatic stress h; • the nonlinear combination of shear ns and normal n stresses acting on the criticalplane.

38. Critical plane criteria based on stresses Conclusions • The promising fatigue criteria seem to be the criteria which can be used under the most general loading, i.e. multiaxial random loading. Unfortunately, only a few stress criteria were experimentally verified (very little) under random loading. • The fatigue failure criteria based on stresses are not able to take into account the effect of cyclic hardening or softening. If the fatigue tests are carried out under stress/force controlled system, the effect of cyclic hardening or softening is visible only in strain history, which is not taken into account in the fatigue failure criteria based on stresses.

39. Strain criteria

40. (24) (25) (26) Critical planes in criteria based on strains Generalised hypothesis of material strength Adaptation of the static hypotheses of material strength to fatigue as the replacement of strain static values in these hypotheses by amplitudes or range of dynamic loading • The criteria most frequently verified with experimental data, were the criteria of: • maximum normal strain, • maximum shear strain, • octahedral shear strain

41. or (27) Kandil-Brown-Miller modification (1982): (28) where: S is a coefficient determined by experiment Wang-Brown modification (1993): (29) Critical planes in criteria based on strains Brown-Miller criterion (1973) Form:

42. Critical planes in criteria based on strains Brown-Miller criterion (1973) The difference between the two criteria above (Equations (28) and (29)) is based ondifferent definitions of the normal strain range. The normal strain εn* (called normalstrain excursion by authors) in Equation (29) is calculated in the plane of maximumshear strain range ns. (30) Fatigue life is calculated based on the following expression (31)

43. Experimental verification This criterion was analyze under torsion, tension-compression and their combination for proportional, non-proportional constant-amplitude and variable-amplitude loading. The calculated fatigue life obtained in the maximum shear strain plane and in the critical plane of maximum damage according to the Brown–Wang criterion were comparable. Critical planes in criteria based on strains Brown-Miller criterion (1973) The critical plane Brown-Miller criterion: Maximum shear strain planeWang-Brown criterion: The critical plane of maximum damage

44. Critical planes in criteria based on strains Lohr and Ellison criterion (1980) Lohr and Ellison proposed a criterion to calculate fatigue life in low-cyclic fatigue regime. This criterion assumes that fatigue life and crack growth rate can be assessedby a linear combination of shear ns,a and normal strain n,a amplitudes in the criticalplane (32) where k is material coefficient (k=0.2 for 1Cr–Mo–V steel).

45. Experimental verification Correlation of the experimental test results (fatigue life, Nf ) with the calculated equivalent strain parameter eq,a based of the proposed criterion was satisfactory (under cyclic, proportional loading). Critical planes in criteria based on strains Lohr and Ellison criterion (1980) The critical plane The critical plane is theplane inclined by 45◦ to the free surface of material.

46. crack Critical planes in criteria based on strains Socie-Fatemi et al. criterion (1985) Socie et al. observing fatigue fractures came into conclusion similar to those by Brown and Miller, that is, the normal strain n in the plane of maximum shear strain accelerates the fatigue damage process through crack opening. Crack opening (by maximum normal stress) decreases the friction force between slip planes. (33)

47. (34) where: y -yield stress nis an experimental coefficient - maximum normal stress (35) Critical planes in criteria based on strains Socie-Fatemi et al. criterion (1987) Form: (36)

48. Experimental verification This criterion was analyze under torsion, tension-compression and their combination for proportional, non-proportional constant-amplitude loading. Critical planes in criteria based on strains Socie-Fatemi et al. criterion (1985, 1987) The critical plane The plane experiencing the maximum value of shear strain amplitude ns,a

49. Critical planes in criteria based on strains Summary: • Among strain criteria based on the critical plane approach we can distinguish criteriawhich assume that fatigue failure depends on: • the linear combination of shearns and normal n strains acting on the critical plane; • the nonlinear combinationof shear strain ns and different kind of tensile parameters acting on the criticalplane.

50. Critical plane criteria based on strains Conclusions In Table is clearly visible that the critical plane of maximum shear strain dominates. The reason of this is that the fatigue failure criteria based on strains are usually applied for non-brittle materials, where crack Mode II and III dominate.