Microsoft Word - 001 PLENARY Marciniak Rozumek POLAND Paper FINAL Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01 1 Focussed on Crack Paths Models of initiation fatigue crack paths proposed by Macha Z. Marciniak Opole University of Technology, Poland z.marciniak@po.opole.pl D. Rozumek Opole University of Technology, Poland d.rozumek@po.opole.pl ABSTRACT. 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. KEYWORDS. Multiaxial fatigue criteria; Energy; Crack growth; Variance method; Weight function. 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 19th 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, constant- amplitude 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- 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 Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01 2 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:  max ( ) ( )ns n t B t K t F   (1) 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 σn(t) and shear stresses τns(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. 1 1 1 1ˆ cos L k k k l W W     , 2 2 1 1 ˆ cos L k k k m W W     , 3 3 1 1 ˆ cos L k k k n W W     , (2) where: 1 L k k W W    - sum of weights, L – number of averages, 1, β2, 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 – Wk = 1 – it is assumed that each position of the main axes has the same effect on the critical plane position, - Weight II – 1 1min 1max 1min k kW        for k = 1, 2,…,N – this weight reduces the impact of maximum main stress 1(t) value on the critical plane position, - Weight III – 1 1 0 0 1 1 k af k k af for a W a for a              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 – 1 1 0 1 k e k k e for R W for R             for k = 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, O Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01 3 - Weight V – 1 1 1 max 1min 10 k f k af k k k af a for a W for a                   – this weight was developed as a result of combining weights II and III, - Weight VI – 1 1 10 m k k af f k k af for a a W for a                    – 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                                          . (3) Nevertheless, some transformations are required in order to obtain values of Euler angles. The first step involves calculation of the quantity:  1 2 3 1 arccos 1 2 l m n     , 3 21 2sin m n u    , 1 32 2sin n l u    , 2 13 2 sin l m u    . (4) Then, Euler - Rodriguez parameters are used: 1 sin 2 u    , 2 sin 2 u    , 3 sin 2 u    , cos 2    (5) to determine values of angles arctg arctg                   , 3arcsin sin m          , arctg arctg                   . (6) Euler angles calculated in this way are averaged using the following relations:     1 1 ˆ L k k W k W      ,     1 1ˆ L k k W k W      ,     1 1 ˆ L k k W k W      . (7) 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 Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01 4 ( ) ( ) sin( 2 ) 2 ( ) cos( 2 )eq x xyt t t        , (8) 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 1 1 2 2 1 n eq j j j a x a x a x     , (9) 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 1 2 1 2 2 2 2 1 2 1 2 1 2 2 n eq j xj j k xjk x x x x j j k a a a a a a a              , (10) 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 a1 and a2. After reduction, the variance of equivalent stress eq versus the angle  can be written as       1 2 1 2 2 2sin 2 4 cos 2 2sin 4eq x x x x         . (11) 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. a) b) 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]. Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01 5 Fig. 2 compares calculated life values and experimental values for variance and damage accumulation [12]. a) b) Figure 2: Comparison of calculated and experimental fatigue lives with critical planes determined according to: a) variance and b) damage accumulation methods [14]. Strain fatigue criterion [3] expressed as  max ( ) ( )ns n t b t k t q   (12) is another proposal to formulate multiaxial random fatigue in the field of strains, where εns(t) and εn(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 1 2 max max max 1 2 1 n LM n W E n K             , (13) 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]. Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01 6 Kt Model Neuber max (MPa) Molski-Glinka max (MPa) Macha-Łagoda max (MPa) 9.61 421 386 405 4.30 329 302 316 3.23 294 270 282 1.85 233 215 224 Table 1: The presentation max depending on Kt and strain energy density models. 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 ( ) ( )ns n t W t W t Q   or max ( )eq t W t Q    (14) 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 Wns(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 t t W t t t t t t t            . (15) 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     ( ) ( ) ( ) 0.5 ( ) ( )sgn ( ), ( ) 0.5 ( ) ( )sgn ( ), ( ) eq ns n n n n n ns ns ns ns W t W t W t t t t t t t t t               . (16) 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 [22-24] 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 pliW t t t     , (17) where ipl = (ti) for (ti) = 0 and i = 1, 2, 3,.... are successive numbers of the hysteresis loop (σ-ε). Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01 7 In Eq. (17), W(t), σ(t), ε(t) are continuous functions of time t, and εipl and εi+1pl are constant values in time t in the hysteresis loop with the number i, while ipl is the plastic strain registered in the moment ti, when the stress (ti) 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+1pl replaces the previous one ipl. 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, ε0pl= 0, W(t0) = 0. Step 2. In point A, individual parameters have the following values: σ(tA) = σA, ε(tA) = εA, εApl= ε0pl =0, W(tA) = 0.5·σA׀εA- ε0pl0.5 = ׀·σA·εA. Step 3. Then, going to point B we obtain: σ(tB) = σB=0, ε(tB) = εB = 0, εBpl= εBpl, W(tB) = 0.5·σB· ׀εB- εBpl0 = ׀. Step 4. Point C: σ(tC) = σC, ε(tC) = εC, εCpl= εBpl, W(tC) = 0.5·σC· ׀εC- εBpl׀ . Step 5. Point D: σ(tD) = σD =0, ε(tD) = εD =0, εDpl= εDpl, W(tD) = 0.5·σD·׀εD- εDpl0 = ׀. Step 6. Point E: σ(tE) = σE, ε(tE) = εE, εEpl= εDpl, W(tE) = 0.5·σE· ׀εE- εD׀pl, etc. Fig. 3d presents energy parameter course calculated based on above procedure. Figure 3: Sample hysteresis loops a), stress courses b), strain courses c), energy parameter courses d). Fig. 4 shows 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]. 2 2 2 1I II III Ic IIc IIIc J J J J J J                       , (18) where JIc, JIIc, JIIIc 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 JI, except of the Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01 8 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. Figure 4: Energy fatigue characteristic for steel C45. Figure 5: Comparison of experimental results for different bending to torsion ratios with those calculated according to the Eq. (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. Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01 9 Figure 6: Comparison of experimental results for different bending to torsion ratios with those calculated according to the Eq. (18) for AlCuMg1 alloy. 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. REFERENCES [1] Macha, E., Mathematical models of the life to fracture for materials subject to random complex stress systems, Monographs no 13, Wrocław University of Technology, Wrocław, (1979) (in Polish). [2] Macha, E., Generalization of fatigue fracture criteria for multiaxial sinusoidal loadings in the range of random loadings. Biaxial and Multiaxial Fatigue, EGF 3, Eds M.W. Brown and K.J. Miller, Mechanical Engineering Publications, London, (1989) 425-436. [3] Macha, E., Generalization of strain criteria of multiaxial cyclic fatigue to random loadings, Studies and Monographs, no. 23, Opole University of Technology, Opole, (1988) (in Polish). [4] Macha, E., Simulations investigations of the position of fatigue plane in materials with biaxial loads, Mat.-wiss. U. Werkstofftech., 20 (1989) 132-136. [5] Carpinteri, A., Macha, E., Brighenti, R., Spagnoli, A., Expected principal stress directions for multiaxial random loading - Part I, Theoretical aspects of the weight function method, Int. J. Fatigue, 21 (1999) 83-88. [6] Carpinteri, A, Macha, E, Brighenti, R, Spagnoli, A., Expected principal stress directions under multiaxial random loading. Part II: numerical simulation and experimentally assessment through the weight function method. Int. J. of Fatigue, 21 (1999) 89-96. T Z. Marciniak et alii, Frattura ed Integrità Strutturale, 34 (2015) 1-10; DOI: 10.3221/IGF-ESIS.34.01 10 [7] Będkowski, W., Macha, E., Ohnami M., Sakane M., Fracture plane of cruciform specimen in biaxial low cycle fatigue – estimate by variance method and experimental verification, Journal of Engineering Materials and Technology, 117 (1995) 183-190. [8] Macha, E., Sonsino, C.M., Energy criteria of multiaxial fatigue failure, Fatigue Fract. Engng. Mater. Struci., 22 (1999) 1053-1070. [9] Lagoda, T, Macha, E, Dragon, A, Petit, J., Influence of correlations between stresses on calculated fatigue life of machine elements, Int. J. Fatigue, 18 (1996) 547–555. [10] Carpinteri, A, Karolczuk, A., Macha, E., Vantadori, S., Expected position of the fatigue fracture plane by using the weighted mean principal Euler angels, Int. J. of Fracture, 115 (2002) 87-99. [11] Będkowski, W., Macha, E., Fatigue fracture plane under multiaxial random loadings – prediction by variance of equivalent stress based on the maximum shear and normal stresses, Mat.-wiss. U. Werkstofftech., 23 (1992) 82-94. [12] Marciniak, Z., Rozumek, D., Macha, E., Verification of fatigue critical plane position according to variance and damage accumulation methods under multiaxial loading, Int. J. of Fatigue, 58 (2014) 84-93. [13] Korn, GA, Korn TM., Mathematical Handbook, Sec. Ed., Mc Graw-Hill Book Company, New York, (1968). [14] Grzelak, J., Lagoda, T., Macha, E., Spectral-analysis of the criteria for multiaxial random fatigue, Mat.-wiss. U. Werkstofftech., 22 (1991) 85-98. [15] Niesłony, A., Macha, E., Spectral Method in Multiaxial Random Fatigue, Springer-Verlag Berlin Heidelberg, (2007) 147. [16] Neuber, H., Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress-strain law, ASME J. Applied Mech. 28 (1961) 544-550. [17] Molski K., Glinka G. A method of elastic-plastic stress and strain calculation at a notch root, Mat. Sci. and Engng. 50 (1981) 93-100. [18] Łagoda, T., Macha, E., Multiaxial random fatigue of machine elements and structures, Cyclic energy based multiaxial fatigue criteria to random loading, Part III, Studies and Monographs 104, Opole University of Technology, Opole, (1998) (in Polish). [19] Rozumek, D., Marciniak, Z., Fatigue properties of notched specimens made of FeP04 steel, Materials Science, 47 (2012) 462-469. [20] Łagoda, T., Macha, E., Będkowski, W., A critical plane approach based on energy concepts: application to biaxial random tension–compression high-cycle fatigue regime, Int. J. Fatigue, 21 (1999) 431–443. [21] Karolczuk, A., Macha, E., Critical planes in multiaxial fatigue of materials, monograph. Fortschritt-Berichte VDI, Mechanik/Bruchmechanik, reihe 18, nr. 298. Düsseldorf: VDI Verlag, (2005) 204. [22] Manson, S.S., Behaviour of materials under conditions of thermal stress, NACA TN-2933, (1953). [23] Coffin, L.F., A study of the effects of cyclic thermal stresses on a ductile metal, Trans. ASME 76 (1954) 931-950. [24] Basquin, O.H., The experimental law of endaurance test, ASTM, 10 (1910) 625-630. [25] Macha, E., Słowik, J., Pawliczek, R., Energy based characterization of fatigue behavior of cyclically unstable materials, Solid State Phenomena, 147-149 (2009) 512-517. [26] Rozumek, D., Macha, E., A survey of failure criteria and parameters in mixed-mode fatigue crack growth, Materials Science, 45 (2009) 190-210. [27] Rozumek, D., Macha, E., J-integral in the description of fatigue crack growth rate induced by different ratios of torsion to bending loading in AlCu4Mg1, Mat.-wiss. U. Werkstofftech., 40 (2009) 743-749. [28] Rozumek, D., Mixed mode fatigue cracks of constructional materials, Studies and Monographs, no. 241, Opole University of Technology, Opole, (2009) (in Polish).