Models of initiation fatigue crack paths proposed by Macha

Professor E. Macha devoted his academic life to solving the problems connected with random multiaxial fatigue in components of machines and structures. In his studies he formulated stress, strain and energy criteria related to critical plane concept. He also proposed several methods to determine critical plane position. In particular, he formulated and verified weight functions applied in order to determine critical plane position. The variance method constituted another significant contribution to the development of methods for determining critical plane position. Apart from these criteria, Macha was exploring energy approach in fatigue of materials and the development of fatigue cracks. He has also observed that strain characteristics multiplied by stress amplitude determined at specimen half-life are applied to estimate fatigue life using energy criteria. However, for cyclically instable materials, stress amplitude value may differ a lot; therefore he proposed the method to determine energy fatigue characteristics directly from experimental research.


INTRODUCTION
redicting service life of different objects is a very important issue for modern engineering.Wrong service life estimation may result in accidents and disasters.Therefore, studies aimed to understand and control this phenomenon, started already in the 19 th century, are continued today.The multitude of problems connected with it suggests that scientists still have plenty of work ahead.Initially, the scope of studies was limited to uniaxial, constantamplitude issues only.With increasing knowledge on the phenomenon, the interest in multiaxial fatigue (most frequent in engineering practice) was growing.At the same time, many stress assessment criteria were proposed.Another step in the development involved attempts to assess life for random loads.This problem was explored by Professor Macha as well [1][2][3][4].He started his work from proposing mathematical models to assess fatigue life for materials in the conditions of random complex stress state, where besides stress criteria he demonstrated the method for determining critical plane position using weight functions.Further studies were connected with strain and energy criteria, and methods used to P determine critical plane position.Many scientists became interested in these studies, which resulted in numerous contacts and team work, e.g. with Carpinteri [5,6], Sakane [7], Sonsino [8], Dragon [9], Petit [9], and others.The purpose of this study is to present academic achievements of Professor Macha.

MATHEMATICAL MODELS AND THEIR EXPERIMENTAL VERIFICATION
ne of the first criteria proposed by E. Macha for multiaxial random loads [1,2] has the following form: where ( ) ns t  and ( ) n t  are: shear stress and normal stress in fracture plane, respectively; and B, K, F -constants for the selection of a given criterion version.Initially, in this criterion fracture plane was regarded as the critical plane.However subsequent analyses make it possible to observe that this plane changes especially for elastic-plastic materials.Detailed criterion guidelines are: (i) fatigue crack is generated (caused) by the activity of normal stresses σ n (t) and shear stresses τ ns (t) in the direction s  in plane with normal n  , (ii) direction s  is concurrent with average direction of shear stresses.
In the criteria related to critical plane it is very important to determine critical plane position.In order to determine its position, it was proposed to apply the weight function method.The weight function method involves finding averaged positions of main axes directions through properly selected weight functions Wk. where: L -number of averages,  1 , β 2 ,  3 -angles between main stresses and axes in the Cartesian coordinates, ( 1 , x), ( 2 , y), ( 3 , z), respectively Then, critical plane position is being determined relative to these averaged directions.6 weight functions are demonstrated in the study [1]: -Weight I -W k = 1 -it is assumed that each position of the main axes has the same effect on the critical plane position, for k = 1, 2,…,N -this weight reduces the impact of maximum main stress 1(t) value on the critical plane position, -Weight III - for k = 1, 2,…,N -according to this weight, only those positions of main axes are averaged, for which maximum stress value is  1 (t)  a• f , where  f is fatigue limit, -Weight IV - this weight was developed as a result of combining weights II and III, -Weight VI - in this proposal only those positions of main axes are taken for averaging, in which  1 (t) is higher than fatigue limit fraction, while their share is in exponential function dependent on Wöhler curve inclination.However, the selection of proper angles for averaging creates problems, and there are no physical guidelines, which angles should be averaged.The issue of averaging proper angles is discussed in the study [10], where direction cosines were made dependent on Euler angles.Matrix of direction cosines defined in this way is expressed in the following form cos cos cos sin sin cos cos sin sin cos cos sin sin cos cos cos sin sin cos sin cos cos sin sin sin cos sin sin cos Nevertheless, some transformations are required in order to obtain values of Euler angles.The first step involves calculation of the quantity: Then, Euler -Rodriguez parameters are used: to determine values of angles Euler angles calculated in this way are averaged using the following relations: Then, Macha and Będkowski [11] developed variance method to determine critical plane position.In this method, the critical plane is considered to be the plane, for which the variance of equivalent stress reduced by selected criterion reaches maximum.
The study [12] contains comparison of lives of steel specimens using variance method with damage accumulation method for criterion of maximum shear stress in the critical plane.According to this criterion, the equivalent stress eq (t) takes the following form ( ) ( ) sin(2 ) 2 ( ) cos( 2) where:  x (t) -normal stress along the specimen axis, xy (t) -shear stress in the specimen cross section,  -angle determining the critical plane position.From Eq. ( 8) it appears that the equivalent stress  eq (t) is linearly dependent on the stress state components  x (t) and  xy (t), so it can be expressed as where: a1 = sin (2), a2 = 2cos (2), x1 = x, x2 = xy.
From theory of probability [13] it results that the variance of random variable being a linear function of some random variables is expressed by the following formula where:  eq -variance of equivalent stress  eq ,  x1 -variance of normal stress  x , x2 -variance of shear stress xy,  x1x2 -covariance of normal  x and shear stress  xy stresses.Under biaxial random stationary and ergodic stress state, the variances x1, x2 and the covariance x1x2 in Eq. ( 10) are constant.In the method of variance for determination of the critical plane position the maximum function of Eq. ( 10) is searched in relation of the angle  occurring in coefficients a 1 and a 2 .After reduction, the variance of equivalent stress  eq versus the angle  can be written as sin 2 4 cos 2 2sin 4 An exemplary assessment of the critical plane position for loading combination K01 [12] obtained using the variance and damage accumulation methods is shown in Fig. 1.Strain fatigue criterion [3] expressed as is another proposal to formulate multiaxial random fatigue in the field of strains, where ε ns (t) and ε n (t) are shear and normal strain in critical plane, respectively; and a, b, k, q -constants for the selection of a given criterion version.
In 1991, Macha, Grzelak and Łagoda [14] attempted to apply spectral method to determine fatigue life.Studies on these issues were continued further in cooperation with Niesłony [15].In these studies, assuming linear effort criteria, a generalised spectral method was formulated for determining fatigue life of materials put to multiaxial loading, using the function of power spectral density in the field of frequency.Multiaxial state of stress is reduced to uniaxial state, and accumulation of damage is carried out using standard material characteristics.The study proves that the results for lives assessed using spectral method in the field of frequency and cycle counting method in the field of time are much the same.Whereas, determination of expected critical plane position using variance method for time histories gives results equivalent to the function of power spectral density.Then, Professor Macha focused his attention on stress distribution in notch root.Like in Neuber [16] and Molski-Glinka [17] criteria, Łagoda-Macha [18] proposed an energy equation for determining the state of stresses in notch bottom as where: n-exponent of cyclic strain curve, K -coefficient of cyclic strain curve.Experimental verification proved that the values obtained through this relation are between the results obtained using Neuber and Molski-Glinka relations.
Tab. 1 contains sample calculation results for the above three models [19].Professor Macha was interested most in energy criteria of multiaxial random fatigue.In this field, in cooperation with Łagoda he proposed a generalised criterion of energy density parameter for normal and shear strains in critical plane, shown as [18,20]   max ( ) ( ) where , , Q -constants for the selection of a given criterion version.Guidelines of the proposed criterion are as follows [21]: "a) this portion of strain energy density is responsible for fatigue crack, which matches the work of normal stress  n (t) in normal strain n(t), that is Wn(t) and work of shear stress ns(t) in a shear strain ns(t) in the direction s in plane with normal n, that is W ns (t), b) direction s in the critical plane matches average direction, in which density of shear strain energy is maximal, c) in boundary state, material effort is determined by the maximum value of linear combination of energy parameters Wn(t) and Wns(t)."For uniaxial stress state, strain energy density parameter is expressed as sgn ( ) sgn ( ) ( ) 0.5 ( ) ( )sgn ( ), ( ) 0.5 ( ) ( ) 2 For multiaxial stress state, the course of equivalent strain energy density parameter is calculated in the critical plane with normal n and shear direction s as The proposed energy criterion in the critical plane is applicable for cyclic and random loads for small and large number of cycles.Depending on the coefficients chosen, different criteria are obtained and thus, for: - = 0,  = 1 we have the criterion of maximum energy density for normal strain in the critical plane, - = 1,  = 0 we have the criterion of maximum energy density for shear strain in the critical plane, - = 1,  = 1 we have the criterion of maximum energy density for normal and shear strain in the critical plane.
When applying energy fatigue criteria to assess life, energy characteristics are used, developed as a result of the Coffin-Manson-Basquin characteristic multiplication [22][23][24] by stress amplitude determined for specimen half-life.However, this characteristic not fully illustrates the behaviour of cyclically unstable materials.Being aware of these differences, Professor Macha and Słowik proposed a new model to determine energy fatigue characteristics directly from experimental research.This model is described as [25]       0.5 where  i pl = (t i ) for (t i ) = 0 and i = 1, 2, 3,.... are successive numbers of the hysteresis loop (σ-ε).
In Eq. ( 17), W(t), σ(t), ε(t) are continuous functions of time t, and εi pl and εi+1 pl are constant values in time t in the hysteresis loop with the number i, while  i pl is the plastic strain registered in the moment t i , when the stress (t i ) is equal to zero, and remains constant to the moment ti+1 when the stress reaches zero again, i.e. (ti+1) = 0. Then the new registered value of plastic strain  i+1 pl replaces the previous one  i pl .This procedure is repeated for each cycle of loading.Fig. 3 shows sample hysteresis loops and energy parameter course calculated on the basis of variable-amplitude history of stresses and strains.Energy parameter course calculation procedure for variable-amplitude loads: Step 1.In point 0, individual values of stresses, strains and energy parameter are: σ(t0) = 0, ε(t0) = 0, ε0 pl = 0, W(t0) = 0.
Step 5. Point D: Fig. 3d presents energy parameter course calculated based on above procedure.1 where J Ic , J IIc , J IIIc are critical values for modes I, II and III.The criterion (18) was successfully verified while tests of aluminium alloy and steels.Different bending (cracking mode I) to torsion (cracking mode III) ratio in steel 18G2A is shown in Fig. 5 [28].It provides grounds to observe shift of experimental points towards increasing the value of parameter J I , except of the angle  = 60, where decrease in these values was confirmed.Considerable increase of parameter JI and JIII values was observed with rising fatigue crack growth rate (Fig. 5, curves 1 to 2).Diagrams 1 and 2 shown in Fig. 5 concern fatigue crack growth rates: da/dN = 1.6810 -8 m/cycle and da/dN = 4.2310 -8 m/cycle, respectively.18) for 18G2A steel [27].
Different bending (cracking mode I) to torsion (cracking mode III) ratio in AlCuMg1 alloy is shown in Fig. 6.Fig. 6 [28] provides grounds to observe shift of experimental points towards increasing the values of parameter JI -these increment values were lower than for steel 18G2A.Experimental results of interdependences between cracking mode I and III, for constant da/dN ratio value were defined by Eq. ( 18).Diagrams 1 and 2 shown in Fig. 6 concern fatigue crack growth rates: da/dN = 7.6410 -8 m/cycle and da/dN = 1.4110 -7 m/cycle, respectively.

SUMMARY
his brief description of professor Macha activity irrefutably proves his wide interests and great influence on progress regarding the issues of fatigue life assessment for components of machines and structures.During his academic career, Macha with colleagues proposed many fatigue criteria concerning the parameters of stress, strain and strain energy density both in the field of time and frequency.Macha's interests covered initiation range and propagation of fatigue cracks.Many times these criteria were verified in various load conditions for different materials, and were presented during various scientific conferences.
1, 2,…,N -only those positions of main axes are averaged, for which maximum stress value 1(t) is higher than product of Poisson's ratio and yield point,

Figure 1 :
Figure 1: Dependence of the normalized value of: a) variance, b) damage accumulation on the angle  of critical plane position for loading combination K01 (λ = 0.189) [14].

Fig. 2 Figure 2 :
Fig. 2 compares calculated life values and experimental values for variance and damage accumulation [12].

Fig. 4
Fig.4shows energy fatigue characteristic for steel C45 according to the formula(17).The models and methods proposed above were used to assess fatigue life until crack initiation.Whereas, as regards development of fatigue cracks, Rozumek and Macha proposed an energy criterion based on parameter J for three crack modes[26].This criterion was verified for mode I and mode III[27].

Figure 5 :
Figure 5: Comparison of experimental results for different bending to torsion ratios with those calculated according to the Eq.(18) for 18G2A steel[27].

Figure 6 :
Figure 6: Comparison of experimental results for different bending to torsion ratios with those calculated according to the Eq.(18) for AlCuMg1 alloy.

Table 1 :
The presentation  max depending on K t and strain energy density models.