Colloque National MECAMAT- Aussois Janvier 2007

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,

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

Introduction Fatigue failure
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.

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.

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 alternating stresses”, by H. Gough and H. Pollard. Proc. Institution of Mechanical Engineers 131, (1935)

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.

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 Arbitrary plane orientation

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 determined in some cases .

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 T after which crack initiation occurs, can be expressed as follows: Stress based - k – array of material coefficients Strain based - Energy based -

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 In general, the critical plane is the plane for which fatigue life will be determined. The problem is: which plane orientation is critical?

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

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 and Low Cycle Fatigue regimes, HCF-LCF energy criteria, HCF and LCF

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

Stress criteria

Generalised hypothesis of material strength
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 (1) (2) (3)

Findley criterion (1956) Critical planes in criteria based on stresses
where: f and k are material coefficients, (for ductile materials k  0.2, 0.3) (4) or Form:

The critical plane (5) The critical plane is a plane with the maximum value of linear relation of shear and normal stress 50 60 70 80 90 100 Angle , deg maximum value critical plane

Findley did not defined a mathematical formula for coefficient f . Some researchers (Park and Nelson, 2000; Backstrom and Marquis, 2001) assume that it can be determined from the shear-mode cracking (6)

Experimental verification (7) 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

McDiarmid criterion (1972)
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 (Park and Nelson, 2000) as follows (9)

McDiarmid criterion (1972)
Critical planes in criteria based on stresses McDiarmid criterion (1972) The critical plane The critical plane is a plane with the maximum shear stress amplitude (10) Proposed criterion correlated experimental data for proportional and non-proportional bending 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. Experimental verification

Dietman et al. criterion (1974)
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 interaction between changes of principal stress directions and fatigue life. They proposed to modify the criterion of octahedral shear stress to take into account the changes of direction 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)

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

Simbürger and Grubisic criterion (1976)
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 principal stress directions. In plane stress state, all possible orientations of the considered plane can be described by the angle α. The proposed fatigue parameter S is formulated as follows (12) where: Material coefficients a1 and a2 are functions of fatigue limits af and af , whereas a,c represents the critical stress amplitude for a given number of cycles to failure. Coefficients m allows to take into account the mean stress value eq,m.

Simbürger and Grubisic criterion (1976)
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 because the 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 of Sn parameter. Experimental verification Simbürger and Grubisic did not define a fatigue characteristic (Nf −S) which should be used to calculate fatigue life.

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

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

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.

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-stresses by macro-stresses.

Robert et al. criterion (1992)
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 normal stress 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 tensile test (R=0). The fatigue criterion is defined by (17) where a3 is the third criterion parameter.

Robert et al. criterion (1992)
Critical planes in criteria based on stresses Robert et al. criterion (1992) The number of cycles to failure Nf is the solution of the Equation (17) and it is obtained from an 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 equivalent stress 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.

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

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

Carpinteri and Spagnoli criterion (2001)
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

Carpinteri and Spagnoli proposed to compute the critical plane orientation n with respect to the averaged maximum principal stress direction 1 in the plane of 1,3 by 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). This criterion was analyze under cyclic proportional and non-proportional loading and under proportional random loading. Experimental verification

Summary: Among stress criteria based on the critical plane approach we can distinguish criteria which 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 of shear 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 critical plane.

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.

Strain criteria

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 (24) (25) (26)

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

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

Critical planes in criteria based on strains Brown-Miller criterion (1973) The critical plane Brown-Miller criterion: Maximum shear strain plane Wang-Brown criterion: The critical plane of maximum damage 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. Experimental verification

Lohr and Ellison criterion (1980)
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 assessed by a linear combination of shear ns,a and normal strain n,a amplitudes in the critical plane (32) where k is material coefficient (k = 0.2 for 1Cr–Mo–V steel).

Lohr and Ellison criterion (1980)
Critical planes in criteria based on strains Lohr and Ellison criterion (1980) The critical plane The critical plane is the plane inclined by 45◦ to the free surface of material. 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). Experimental verification

Socie-Fatemi et al. criterion (1985)
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. crack (33)

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

Socie-Fatemi et al. criterion (1985, 1987)
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 This criterion was analyze under torsion, tension-compression and their combination for proportional, non-proportional constant-amplitude loading. Experimental verification

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

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.

Energy criteria

Critical planes in criteria based on energy Generalised hypothesis of material strength The first criteria basing on strain energy came from static hypotheses of material strength. The most often applied criterion from this group is Huber-Mises-Hencky’s hypothesis. In case of elastic strain state, Huber-Mises-Hencky’s hypotheses can be describe be means of shear stress or shear strain in octahedral plane.

Smith-Watson-Topper criterion (1970)
Critical planes in criteria based on energy Smith-Watson-Topper criterion (1970) Smith and co-authors proposed a simple form of a damage parameter, SWT, described as stress and strain product σmaxεa for fatigue life determination under uniaxial tension-compression. Form: (37)

Smith-Watson-Topper criterion (1970)
Critical planes in criteria based on energy Smith-Watson-Topper criterion (1970) Socie modification (1987) (38) The critical plane The plane of maximum normal strain range 1 The criterion is applied under multiaxial cyclic proportional and non-proportional loading for materials with crack Mode I. Experimental verification

Nitta-Ogatta-Kuwabara criterion (1988)
Critical planes in criteria based on energy Nitta-Ogatta-Kuwabara criterion (1988) Form: for crack Mode I for crack Mode II (39) (40) Under non-proportional loading, cracks with mixed Modes I and II were noticed. So, they proposed to calculate fatigue life Nou for non-proportional loading in the following form (41)

Nitta-Ogatta-Kuwabara criterion (1988)
Critical planes in criteria based on energy Nitta-Ogatta-Kuwabara criterion (1988) The critical plane - the plane of maximum range of normal strain energy density WI - the plane of maximum range of shear strain energy density WII The criterion is applied under multiaxial cyclic proportional and non-proportional loading (under low cycle fatigue regime). Experimental verification where A1, A2 and β1, β1 are material constants.

Liu criterion,VSE (1993) Critical planes in criteria based on energy
Liu proposed an energy method to estimate the fatigue life, based on virtual strain energy (VSE). The parameters of virtual strain energy are associated with two different Modes of fatigue cracks. Form: (42) (43) Proportional loading

Despite formal similarity of presented formulas for WII,A and WII,B, shear and normal stresses and strain ranges are calculated according to Mohr’s cycle in a different way: for Case A, from maximum and minimum normal stresses and strains σ1, σ3 and ε1, ε3; for Case B, from σ1, σ2 and ε1, ε2. In uniaxial stress state, the parameter VSE can be written as follows: (44) The parameter VSE contains the elastic We and plastic strain Wp energy density. It can be described by the area of the rectangles defined by the hysteresis loop ranges.  σ

In uniaxial stress state, the energy density W is equivalent to the Smith–Watson–Topper parameter and, as a function of fatigue life Nf , it is written as follows: (45) where B1, B2, a1, a2 are materials constants.

Non-proportional loading (45) (47) Non-proportional loading Virtual paths

Comments: - The criterion proposed by Liu is limited only to some special kind of loading. - We cannot agree that all non-proportional paths of loading are represented by a rotated ellipse. - Liu did not define the fatigue parameter for random loading or even for triaxial state of stress.

The critical plane - the plane of maximum range of normal strain energy density WI - the plane of maximum range of shear strain energy density WII The criterion is applied under multiaxial cyclic proportional and non-proportional loading. Experimental verification

Critical planes in criteria based on energy
Glinka and et al. (1994, 1999) Glinka et al. proposed an energy parameter, being a part of total strain energy density, expressed by stress and strain in the critical plane Form: (48) In order to take into account the mean value, authors have modified the above parameter, (49)

Glinka and et al. (1994, 1999) In 1999 (Pan et al., 1999) noticed that the influence of strain energy in shear direction, (ns/2)(ns/2), on fatigue life is different than the influence of strain energy calculated in normal direction, (εn/2)(σn/2). For this reason, they proposed to modify Glinka criterion by applying two coefficients determined by experiments Pan and et. al. modification (1999) (50) where the coefficients are equal to

Glinka and et al. (1994, 1999) The critical plane The plane of maximum range of shear strain ns The criterion was applied under cyclic torsion, tension, bending and combined proportional bending and torsion loading with zero and non-zero mean stress. The fatigue parameters (48)– (50) were used to correlate experimental fatigue life Nf . However, any formula to determine relation between fatigue life Nf and the proposed strain energy parameters was not proposed. Experimental verification

Rolovic and Tipton (1999) Critical planes in criteria based on energy
Rolovic and Tipton proposed a criterion for multiaxial cyclic proportional and non-proportional fatigue loading including the mean value of normal stress. In general, the criterion is written as follows (51) The specific form of Equation (51) was proposed as (52)

Rolovic and Tipton (1999) Critical planes in criteria based on energy
The critical plane The critical plane is determined a plane with the highest calculated damage level. Experimental verification Uniaxial and biaxial in phase and out-of-phase fatigue data from three materials were used to verify the proposed model. The proposed model can be used under multiaxial random loading. Unfortunately, this very interesting model was very little verified (only by authors). The problem is to determine the formula of functions: f1(σn,max) and f2(σn,max) for other materials.

Chen et al. (1999) Critical planes in criteria based on energy (53)
Chen et al. proposed two criteria: the first one for materials characterised by Mode I crack, and the second one for materials characterised by Mode II crack. For materials characterised by Mode I crack, the critical plane is the plane of maximum normal strain range εn (53) For materials with Mode II crack, the critical plane is the plane of maximum shear strain range γns (54)

Chen et al. (1999) Critical planes in criteria based on energy
The proposed damage parameters correspond to the parameters proposed by Glinka and co-authors. In this criterion the different influence of the normal and the strain energy density on fatigue life is not included. The critical plane The critical plane orientations depends on the observed dominating crack orientation. Experimental verification The criterion was verified under cyclic, non-proportional loading.

Farvani – Farahani criterion (2000)
Critical planes in criteria based on energy Farvani – Farahani criterion (2000) Varvani-Farahani proposed a fatigue parameter as the summation of the normal σnεn and shear ns,maxns,max/2 strain energy density ranges calculated in the critical plane of maximum shear strain. (55) The fatigue parameter (55) was used to correlate experimental fatigue life Nf (cyclic, non-proportional), but the function f(Nf ) was not formulated.

Critical plane criteria based on
strain energy density Conclusions The fatigue failure criteria based on energy are able to take into account the effect of cyclic hardening or softening since they use both strain and stress histories.

Summary Multiaxial fatigue criteria can be divided according to the critical plane orientation Crack Mode I Crack Mode II Crack Mode III

Summary Crack Mode I Stress criteria 1. Maximum normal stress
2. Macha (1979) 3. Carpinteri-Spagnoli (2001) Strain criteria 1. Maximum normal strain 2. Macha (1988) Energy criteria 1. Smith - Watson – Topper (1970) 2. Liu (1993) 3. Lagoda-Macha (1998) 4. Chen (1999)

Summary Crack Mode II and III Stress criteria 1. Maximum shear stress
2. McDiarmid (1972) 3. Dietman-Issler (1974) 4. Matake (1977) 5. Macha (1979) 6. Dang Van (1982) 7. Papadopoulos (1993) 8. Carpinteri - Spagnoli (2001) Strain criteria 1. Maximum shear strain 2. Brown-Miller (1973) 3. Lohr-Ellison (1980) 4. Fatemi-Socie (1987) 5. Macha (1988) Energy criteria 1. Liu (1993) 2. Glinka - Shen – Plumtree (1994) 3. Lagoda-Macha (1998) 4. Chen (1999) 5. Varvani-Farahani (2002)

Summary Mixed crack Mode Stress criteria 1. Octahedral shear stress
2. Findley (1956) 3. Simburger (1974) 4. Macha (1979) 5. Sempruch (1992) 6. Vidal (1996) 7. Zolochevski et al. (2000) 8. Carpinteri – Spagnoli (2001) 9. Kuppers-Sonsino (KoNoS) (2003) Strain criteria 1. Octahedral shear strain 2. Macha (1988) Energy criteria 1. Nitta - Ogata - Kuwabara (1988) 2. Lagoda-Macha (1998) 3. Rolovic-Tipton (1999) 4. Hoffmeyera et al. (2001) 5. Lee et al. (2003)

Generalized criteria Summary Form: Macha criterion (1979)
(56) Lagoda-Macha criterion (1998) where: B, K, F, b, k, q, , , Q are constants to select a particular form of criterion Main assumption: fatigue crack is controlled by normal strain n(t), stress n(t) or energy Wn(t) and shear strain ns(t), stress ns(t) or energy in Wns(t) the critical plane

Generalized criteria Summary Experimental verification
The criterion is a general form of many criteria and the particular form of the criterion depends on many conditions: material, temperature, loading etc. For example: for ductile materials the influence of shear strain, stress or energy is greater than the influence of normal strain, stress or energy, thus, K,k,  0 (aluminum) for brittle materials the influence of normal strain, stress or energy is greater than the influence of shear strain energy, thus, B,b,  0 (cast iron) for elastic-plastic materials, B,b,  0 and K,k,  0 (alloys)

Summary Through the literature review of the critical plane approach to multiaxial fatigue the following comments can be drawn: 1. The terminology applied in a case of static or proportional loading turned out to be false for non-proportional loading. 2. The rotation of principal stress and strain directions is not always taken into account, there are treated as scalar variables. 3. In a few criteria of multiaxial fatigue failure it is assumed that the critical plane position is defined in relation to the averaged directions of principal stress and strain. 4. There is not enough information concerning the influence of mean stress and strain on the critical plane and fracture plane orientation. 5. Information about fatigue fracture plane positions under multiaxial random loading is very pure. 6. One part of the multiaxial criteria applies a few critical plane orientations for calculation of a single damage parameter. 7. There are attempts to include different mechanism of fatigue process (Mode I, II, III, stage I, stage II etc.) to fatigue life calculation through:

Summary the application of several criteria based on different fracture mechanism and the fatigue life is established by criterion with highest damage level, (Socie, Das), summation of damage levels calculated according to several multiaxial criteria, (Socie, Nitta).

Algorithm of the fatigue life calculation

Algorithms of fatigue life calculation based on the critical plane approach have in general the same structure. A generalised scheme of this algorithm applicable under multiaxial random loading is as follows: 1) Recording, generation, computing ij(t), ij(t), (i, j = x, y, z) 2) Determination of the critical plane orientation: n, s 3) Computation of the equivalent damage parameter course Based on the fatigue criterion 4) Cycle counting 5) Damage degree computation S(T0) Fatigue damage accumulation hypothesis Tcal = T0/S(T0) 6) Fatigue life calculation Tcal

2) Determination of the critical plane orientation: n, s
Algorithm of the fatigue life calculation 2) Determination of the critical plane orientation: n, s The critical plane The position of the critical plane is determined by normal vector n and shear components are determined by shear vector s on that plane. The direction n and s may be computed by three methods: damage accumulation method, variance method, weight function method.

Damage accumulation method The damage accumulation method is the most often used for determination of the critical plane orientation. This comes from the fact that this method refers directly to the fatigue life. The critical plane orientation is this method is the plane experiencing the highest damage degree, this means the plane of the theoretical fracture and the lowest fatigue life. Unfortunately, the damage accumulation method is the most time consuming. In the damage accumulation method the algorithm of fatigue life calculation is repeated once per each considered plane orientation.

Damage accumulation method - simulations The aim of the simulation is the determination of the critical plane orientation for different plane stress states. Normal and shear stresses in an arbitrary plane orientation for plane stress state can be presented as a function of the angle α

Damage accumulation method - simulations Cyclic loading 1) a = 200 MPa; a = 200 MPa  = a / a = 1  = 0

Damage accumulation method - simulations 1) a = 200 MPa; a = 200 MPa;  = 0

Damage accumulation method - simulations eq(t) = n(t) + k ns(t) k=0.2

Damage accumulation method - simulations 2) a = 200 MPa; a = 200 MPa;  = /2

Damage accumulation method - simulations eq(t) = n(t) + k ns(t) k=0.2

Variance method The variances of stress, strain or strain energy density are important parameters in the fatigue process. The variance method assumes that the plane experiencing the maximum variance of equivalent damage parameter (e.g. stress, strain or energy) is critical. In the case of linear fatigue failure criteria the equivalent e.g. stress may be expressed as a function of: tensor components (in general as Yij(t))), the critical plane orientation and material constants, k)

Variance method Variance of the equivalent course is nonlinear function of the direction cosines of n, s, but it is linear function of the covariance matrix components of Yij(t) tensor. Thus, the variance of equivalent course may be expressed by the sum of the products of the covariance matrix components μykl and nonlinear function of material constants and direction cosines of n, s vectors: ak, al

Variance method

Variance method - simulations For a given plane stress state with zero mean value of stresses the covariance matrix is as follows where E[x] – mean the expected value of x

Variance method - simulations where

Variance method - simulations 1) a = 200 MPa; a = 200 MPa;  = 0 2) a = 200 MPa; a = 200 MPa;  = /2

Weight function method Maximum principal stress direction 1 Averaged direction of maximum principal stress 1 Averaged directions of principal stresses 1, 2, 3 Are they related to fatigue fracture plane position? The weight function method consist in weighted through a suitable function average process of instantaneous principal axis directions. The critical plane orientation is determined in relation to the averaged principal axes directions.

Weight function method Principal axis directions 1(t), 2(t), 3(t) could be described by Euler angles. Principal axes 123 described by Euler angles ϕ, θ, ψ

Weight function method Weight functions, examples: According to such a weight function, each position of the principal axes influences the mean position of the principal axes in the same degree, irrespective of the stress values. Application of this weight function leads to the arithmetic mean of Euler angles This weight function includes only those positions of the principal stress axes for which the maximum principal stress σ1(t) is greater than the product of c and the fatigue limit σaf . Averaging process exponentially depends on the parameter mσ of the Wöhler curve

Weight function method Algorithm of fatigue life calculation with the critical plane orientation determined by the weight function method

Comparison of methods The critical plane orientations αcal according to the criterion of maximum normal stress The critical plane orientation αcal according to the maximum variance of normal stress σn(t) The critical plane orientation according to weight function W2

Exemplary application of simple energy based criteria in fatigue life calculation

Exemplary application of simple energy based criterion in fatigue life calculation
Lagoda - Macha criterion (1998) Lagoda and Macha formulated a generalised criterion of normal Wn(t) and shear strain energy density Wns(t) in the critical plane. Main assumption: Fatigue crack is formed by the part of strain energy density which corresponds to work of normal stress σn(t) on normal strain n(t) - Wn(t) and work of shear stress ns(t) on shear strain occurring in s direction in the critical plane with normal n - Wns(t) (1)

Exemplary application of simple energy based criterion in fatigue life calculation
where the strain energy density parameter is (2) The main aims of the example are: Verification of the energy criteria of multiaxial fatigue proposed by Lagoda and Macha for low-cycle non-proportional loading The analyze of the history of energy parameters in the critical plane

The energy parameters (3) or (4) THE CRITICAL PLANE
Four particular versions of the generalized criterion of normal and shear strain energy density are verified: (3) or (4) THE CRITICAL PLANE Maximum normal strain energy density Maximum shear strain The criterion of normal strain energy density (C1) The criterion of shear strain energy density (C2) The criterion of normal and shear strain energy density (C3) The criterion of normal and shear strain energy density (C4)

The energy parameters THE CRITERION OF MAXIMUM NORMAL STRAIN ENERGY DENSITY IN THE CRITICAL PLANE (C1) generally: for  =0,  = 1 (5) in the plane of in uniaxial tension-compression tests right side of equation (5) is (6) Fatigue effort under multiaxial state Eq. (5) must be equivalent to fatigue effort under uniaxial state Eq. (6), thus (7) Fatigue life is computed from energy characteristic obtained from uniaxial tests Eq. (6) (8)

The energy parameters THE CRITERION OF MAXIMUM SHEAR STRAIN ENERGY DENSITY IN THE CRITICAL PLANE (C2) generally: for  =1,  = 0 (9) in the plane of in uniaxial tension-compression tests right side of equation (9) is (10)

(11) Tension-compression fatigue tests
Mohr’s circles for stress and strain state (11)

The energy parameters (12) (13)
Fatigue effort under multiaxial state Eq. (9) must be equivalent to fatigue effort under uniaxial state Eq. (11), thus, (12) Fatigue life is computed from energy characteristic obtained from uniaxial tests Eq. (8) (13)

The energy parameters THE CRITERION OF MAXIMUM SHEAR AND NORMAL STRAIN ENERGY DENSITY IN THE CRITICAL PLANE (C3) generally: for  1,  = 1 (14) in the plane of in uniaxial tension-compression tests right side of equation (14) is (15) Fatigue effort under multiaxial state Eq. (14) must be equivalent to fatigue effort under uniaxial state Eq. (15), thus (16)

The energy parameters (17) (18) (19)
The coefficient  should be chosen to obtain the best correlation between uniaxial and multiaxial fatigue tests. For simplicity (criterion C2) the coefficient  is chosen as: (17) Thus, the equivalent parameter is: (18) Fatigue life is computed from energy characteristic obtained from uniaxial tests (19)

The energy parameters THE CRITERION OF MAXIMUM SHEAR AND NORMAL STRAIN ENERGY DENSITY IN THE CRITICAL PLANE (C4) generally: for  1,  = 1 (20) in the plane of in uniaxial tension-compression tests right side of equation (20) is (21)

(22) Tension-compression fatigue tests
Mohr’s circles for stress and strain state (22)

The energy parameters (23)
Fatigue effort under multiaxial state Eq. (20) must be equivalent to fatigue effort under uniaxial state Eq. (22), thus, (23)

The energy parameters (24) (25) (26)
The coefficient  should be chosen to obtain the best correlation between uniaxial and multiaxial fatigue tests. For simplicity of Eq. 29 the coefficient  is chosen as: (24) Thus, the equivalent parameter is: (25) Fatigue life is computed from energy characteristic obtained from uniaxial tests (26)

The energy parameters, summary
The equivalent parameters are: C1: in the plane of C2: in the plane of C3: in the plane of C4: in the plane of Fatigue life:

Fatigue tests (M. Ohnami and N. Hamada)
Materials: SUS 304 steel Al 6061 aluminum alloy Specimens: cylindrical thin-walled Loading: combined tension-compression and torsion under controlled strain Tests were performed under 14 different strain paths Strain paths

Fatigue tests Histories of energy parameters SUS 304 Case 13
Wn, MJ/m3 Not taken into account Wns, MJ/m3 Not taken into account Case 13 Strain path

Fatigue tests Histories of energy parameters Al 6061 Case 12
Not taken into account Wns, MJ/m3 Case 12 Strain path

Calculated and experimental results
3(2Nf) (2Nf)/3 Amplitudes of strain energy density Weq,q according to criterion C1 against the energy fatigue characteristic for SUS 304

Amplitudes of strain energy density Weq,q according to criterion C2 against the energy fatigue characteristic for SUS 304

Amplitudes of strain energy density Weq,q according to criteria C1-C4 against the energy fatigue characteristic for SUS 304

Amplitudes of strain energy density Weq,q according to criterion C1 against the energy fatigue characteristic for Al 6061

Amplitudes of strain energy density Weq,q according to criteria C1-C4 against the energy fatigue characteristic for Al 6061

in the plane of Comparison of the calculated Ncal and experimental lives Nexp

Histograms of scatter coefficients

Conclusions For 6061 Al aluminum alloy, the best relation between the experimental life and the energy parameter was obtained according to the criterion C2 of shear strain energy density in the critical plane. For SUS 304 steel, the best relation between the experimental life and the energy parameter was obtained according to the criterion C3 of normal and shear strain energy density in the critical plane In general, the criterion C2 of shear strain energy density in the critical plane can be applied for both materials For unstable materials and regimes the uniaxial fatigue tests should be carried out under energy control system to obtained the fatigue energy characteristic (Wa-2Nf) used in criteria based on energy parameters

Colloque National MECAMAT- Aussois Janvier 2007

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Colloque National MECAMAT- Aussois Janvier 2007

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