Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 45, 2, pp. 337-348, Warsaw 2007 EXPERIMENTAL VERIFICATION OF FATIGUE LOADING NONPROPORTIONALITY MODEL Dariusz Skibicki Faculty of Mechanical Engineering, University of Technology and Agriculture in Bydgoszcz e-mail: dariusz.skibicki@utp.edu.pl The paper deals with the experimental verification of amodel of the fa- tigue loadingnonproportionality.The assumption of the proposedmodel is that the loading nonproportionalitydegree depends on: (a)modules of vectors of rotating stresses, (b) angular distances of these stress vectors measured in relation to the critical plane. The performed experiment involved modeling of the nonproportional fatigue loading through pro- grammed changes of principal axes positions. Different positions of the principal axes were obtained by using two loading blocks: fully rever- sed torsion and complex loading, that is to say, tension-compression and torsion. During the tests, the influence of the range of the orientation angle of principle axes on fatigue lifewas examined.The obtained results allowed one to confirm the thesis that the nonproportionality degree de- pends also on the angular distance in which the stress vector acts in relation to the critical plane. Key words: multiaxial fatigue criteria, nonproportional loading, loading nonproportionalitymeasure 1. Introduction Nowadays, one of the most important fatigue issues consist in the assessment of fatigue behavior and damage phenomena occurring in conditions of the so called nonproportional loading.We can talk about such loading if during a fatigue cycle theprincipal axes change their location.Compared to thepropor- tional loading, nonproportionality of loading can cause intensification of the fatigue damage cumulation process. In effect, fatigue life and fatigue strength are significantly impaired. Because of common character of this phenomenon and its heavy influence on fatigue behavior, this kind of loading should be taken into consideration in calculation models. 338 D. Skibicki In the authors previous works (Skibicki and Sempruch, 2001, 2002, 2004; Skibicki, 2004a,b), a fatigue strength criterion was proposed and presented, whereas fatigue life criterion can be found in Skibicki (2004b, 2006). Both criteria respect the phenomenon of fatigue loading nonproportionality and are used for biaxial sinusoidal nonproportioanl loadings in situations of the nonzero mean values cycle. These criteria are based on the critical plane approach, since it is assumed that also in conditions of nonproportional loading the fatigue damage cumu- lation process resultsmainly from shear and normal stresses acting on a given plane. Hence, in the formula for the equivalent amplitude of nonproportional stresses τ np eq(a) = τeq(a) ( 1+ t −1 b −1 H3 ) ¬ t −1 (1.1) there occurs equivalent stress amplitude τeq(a) connectedwith the critical pla- ne τeq(a) =(τa+ c1σa+ c2σm) (1.2) where t −1 – fatigue limit in torsion, b−1 – fatigue limit in bending, and c1 =1.9 t −1 b −1 −1 c2 =0.5 b −1 Rm (1.3) Rm – tensile strength, τa – shear stress amplitude on the critical plane, σa,σm – amplitude and mean value of normal stress on the critical plane consistent with the shear stress amplitude τa action moment, t−1/b−1 measure of the material sensitivity with respect to loading nonproportionality (in the form of a quotient of fatigue limit in torsion to fatigue limit in bending), H – loading nonproportionality measure. In the accepted solutions, the critical plane is determined by the action course of the maximum shear stress in the cycle. In nonproportional loading conditions, the prediction of fatigue life and strength on the basis of equivalent stress based only on the critical plane approach could be burdened with an error. The mathematical model has to account also for nonproportionality influence. The description of the loading nonproportionality effect in equation (1.1), is based on a nonproportionality term containing two functions: measure of the material sensitivity to loading nonproportionality t −1/b−1, and a function defining the loading nonpropor- tionality degree called the loading nonproportionality measure H. An commonly accepted assumption is that the bigger is the part of stresses acting beyond the critical plane in the damage cumulation process the higher is the nonproportionality degree. Two further assumptions characterizing this part have been formulated, i.e.: (a) it is directlyproportional tomodulesof stressesactingbeyondthecritical plane, Experimental verification of fatigue... 339 (b) it dependson their angular distance in suchaway that thevectors acting in a larger angular distance in relation to the critical plane increase the loading nonproportionality degree more than the vectors of the same module acting within a smaller angular distance. The nonproportionality measure H is considered here by both criteria proposed by the author and respects these theses. Assumption (a) was accounted for through application of a filling factor defined as the ratio of the loading path field of reduced stress to the field of a circle circumscribed about the loading path (Fig.1). For a proportional loading, the value of such a defined measure is equal to zero, whereas for a nonproportional loading of the highest degree its value is one. In the second case, the rotating vector of equivalent stress does not change its module, so the loading path is a circle. Fig. 1. Loading nonproportionality One of the first ideas of quantification of the degree of nonproportionality was a rotation factor proposed by Kanazawa et al. (1979). This factor is ba- sed on the interaction of slip on different planes and is defined as the ratio of the shear strain at 45◦ in the maximum shear strain range direction to the maximum shear strain range. At present, there exist many solutions which describe the nonproportionality degree through the loading path of rotating vectors (Itoh et al., 1997; Chen, 1996;Duprat, 1997;Morel et al., 1997;Borodii and Strizhalo, 2000). Some of them account for it by means of some charac- teristic dimensions of loading paths (Duprat, 1997; Morel et al., 1997). There are criteria for which the created hodograph field is the basis of the nonpro- portionality measure formulation similiarly to the hereby proposed solution (Chen, 1996). In all these cases, the nonproportionality measure is charac- terized only by modules of vectors acting beyond the critical plane. Thus, a simplification is usually accepted that all the rotating stress vectors have the same fatigue effect (they influence the fatigue damage cumulation process in the samemanner) regardless of their location. 340 D. Skibicki However, it is worth noting that there exist criteria for a general nonpro- portional loading which, for different reasons, e.g. application for insensitive materials, do not take into account the influence of nonproportionality, like for example Łagoda et al. (1999), Papadopoulos (1995), Macha (1989), Dang Van et al. (1989). There also exist criteria predicting fatigue properties in nonproportional loading conditions by means of introducing nominal loading quantities to the criterion, like a phase shift ϕ. That ideawas used in the case of Lee (1985) and Lee andChiang (1991) criteria. However, such solutions are restricted to a particular loading case. In the solution proposed by the author, the mathematical description of thesis (b) assumes a form of the weight function W . Its value in the critical plane is zero, and for the course most distant from the critical plane (course turned by 45◦) the value is one (Fig.2). Multiplying the modules of rotating vectors by W their parts are being differentiated. The stresses more distant from the critical plane correspond to a greater weight, thus their effect in the process of fatigue damage cumulation is bigger. Fig. 2.Weight function Thanks to theweight function, the accuracy of obtained calculation results increases. The influence of the vector position in the conditions of loadingnonpropor- tionality on fatigue properties has also been noticed by other authors. Expe- rimental works in this field were carried out by Sonsino et al. (2004) and Yousefi-Hashtyani (2004). These researchers claim that the nonproportionali- ty degree depends on the changing angle of the principal axes position. For small principal axes rotation angles, the influence of nonproportionality on fatigue properties is smaller than in the case when the pricipal axes rotation angle is bigger. An attempt of accounting for the influence of vector position can be found in Itohs criterion (Itoh et al., 1997). For the description of nonproportionality, the author proposes an integral of the main strain vector projection onto the direction perpendicular to the critical one. Thus, it assigns to vectors different weights depending on their particular positions. Experimental verification of fatigue... 341 Contrary to the commonly accepted concept of nonproportinality descrip- tion by means of stress modules (or indirectly by description of loading path characteristics), attempts to introduce a quantity depending on their positions into the description are very rare. Therefore, it is not difficult to notice that thesis (b) needs to be verified. In the paper, an experimental work whose aim was verification of the as- sumption that the loading proportionality degree depends both on modules of stresses acting beyond the critical plane and on angular distances of these stresses vectors measured in relation to the critical plane is discussed. 2. Testing methodology The tests require creation of nonproportional loading conditions so that there would be a possibility of controlling the rotation angle value of the principal axes position. In these tests, the controlled changes of the position of principal axes were realized bymeans of a two-block loading program.Block I consisted of fully reversed torsion and block II consisted of fully reversed cycles with biaxal torsion and compression. In literature, experimentalworksbasedona similar approach canbe found. In many papers, the effect of the main direction changes was examined (Lee and Lee, 1997; Wheelhouse et al., 2001; Morel, 2000; Bonacuse and Kalluri, 2001) sometimes referring to the character of those examinations directly as loading nonproportionality modeling (Wheelhouse et al., 2001). Those expe- riments involved realization of loading blocks of different types: torsion with push-pull (Lee and Lee, 1997;Wheelhouse et al., 2001; Bonacuse andKalluri, 2001), torsion-bending (Morel, 2000). In order to prove the influence of the vectors of angular distances on fati- gue properties, the principal axes rotation range was controlled in successive fatigue tests. This effect was obtained through the change of the principal axes position in block II and with unchanged position of the axes in block I (torsion). The assumed quantity of the principal axes position in block II was reached by establishing proper values of the shear-to-normal stress amplitude ratio. Inorder to examineonly theeffect of theprincipal axesposition change, the equivalent stress values for both blocks remained unchanged. Their concrete values resulted from the stress level accepted for a given test. Because of the applicability range of the proposed criteria those were the levels of stresses corresponding to the high cycle regime. The analysed result of carried out tests was the fatigue life. 342 D. Skibicki In connection with the fact that equivalent stresses used in the authors proposed criteria are of shear character, the achieved results were compared with the fatigue life curve in torsion. Itwas accepted that the verified assump- tion would be confirmed if along with the increase of the principal axes angle between blocks, the degree ofmodelled nonproportionality would rise, loading would bemore damaging, and the obtained fatigue lives decrease. 3. Testing conditions The tests were carried out on the Instron 8874 biaxial hydraulic mechanical testingmachine realising tension-compression in the range ±25kNand torsion in the range ±100Nmwith the possibility of phase shift. For the tests, steel X5CrNi18-10 (Rm = 740MPa, Re = 620MPa, A5 =52%)was used.As proved in the analysis carried out by Skibicki (2006), this material is very sensitive to loading nonproportionality. For this type of steel, a nonproportional loading ismuchmore destructive than a proportional loading.Thus, the use of this steel for examining the effect ofmodelled loading nonproportionality was expected to bring the best results. Geometrical features of the test sample are presented in Fig.3. The choice of the sample was made according to the standard ASTME2207-02 defining fatigue test conditions for axial loading and torsion. Deviations from the stan- dard recommendations resulted from technical parameters of the test stand. Fig. 3. Geometrical characteristics of the tested sample 4. Experiment and analysis of achieved results In the first stage of tests, the fatigue life curve in torsion was determined (Fig.4). In relation to that curve, further results of fatigue tests were compa- red. Experimental verification of fatigue... 343 Fig. 4. Fatigue life curve in torsion Due to the fact that the realized research program concerned biaxial lo- adings it was necessary to use the equivalent stress formula. For this purpose, equation (1.2) was used. Further, experimental verification of this stress was carried out. For this purpose, two fatigue tests were performed: torsion with tensionwith push-pull. The value of the stress amplitudeswas selected so that the tests could be carried out for two levels of the equivalent stress amplitude (370 and 390MPa) and for 3 shear-to-normal stress amplitude ratios (thereby for three different positions of the maximum shear stress vector). Ratios of shear-to-normal stress amplitudeswere established in such away that the angles describing the maximum shear vector position in relation to the direction of the maximal shear stress vector in the block of pure torsion had the following values: 7.5◦, 15◦ and 22.5◦ (Fig.5). In order to make the fatiguedamage cumulationandcrackgrowthprocess course for all fatigue tests similar to the basic torsion, cases of biaxial loading with torsion prevailing were chosen. For the extreme case of the angle equal 22.5◦, the amplitude ratio τ(a)xy/σ(a)x was 0.5. For both levels of the equivalent stress and all three levels of maximum tangent vector, the obtained fatigue lives are consistent with the prediction made on the basis of the torsion curve (Fig.6). Furthermore, there was chosen a level of stress for carrying out the main tests. Because of the applicability range of the proposed criteria it was a stress level of 370MPa corresponding to the high cycle range of fatigue life. The main tests included 3 variants of programmed fatigue tests. Each variant consisted of a block of 5·103 cycles of reversed torsion and a successive block of biaxial loading of the same length. These variants were different in respect of the maximum shear stress position in relation to reversed torsion. 344 D. Skibicki Fig. 5. Positions of maximum shear stress vectors for biaxial tests Fig. 6. Fatigue lives obtained in biaxial tests for three different positions of maximum shear vector: triangle – 7.5◦ (3 tests), square – 15◦ (3 tests) and circle – 22.5◦ (5 tests) Values of the angles were the same as those established for equivalent stress verification tests, i.e. 7.5◦, 15◦ and 22.5◦. On a diagram of fatigue life (Fig.7), the obtained numbers of cycles have beenmarked. For each variant, mean values have been calculated from tests. The results are presented in Fig.8. Black color is used to mark the base fatigue life obtained for the case of torsion. Gray color marks the mean value of fatigue life obtained under biaxial loading for three different maximum shear stress vector positions. They cor- respond to the value of reversed torsion whichmeans that the formula for the equivalent stress properly determines its value. Experimental verification of fatigue... 345 Fig. 7. Fatigue lives obtained in block loading tests for three different positions of maximum shear vector: triangle – 7.5◦ (4 tests), square – 15◦ (4 tests) and circle – 22.5◦ (4 tests) Fig. 8. Mean values set up in the whole research program White color is usedtomark themeanvalues of fatigue life obtained in result of programmed loadings, i.e. with principal axes positions changing between the blocks. No influenceof the changes ofprincipal axespositionson the fatigue life has been observed for angle 7.5◦. This proves that the block character of loading does not change the fatigue life and does not affect the results of tests. For angles 15◦ and 22.5◦, the influence of principal axes position changes between blocks is visible and rises along with increase of the angle value. 346 D. Skibicki 5. Conclusions • Fatigue lives obtained in the programmed tests depend on the principal axes rotation angle variability range between the loading blocks. It has been observed that along with the rise of principal axes rotation angles the obtained lives were getting smaller. • The obtained results can serve as a basis for formulation of a conclusion that the nonproportionality degree is determined bothby themodules of stresses acting in result of the principal axes rotation beyond the critical plane, and their positions in relation to the critical plane direction. It can be concluded that the contribution of these stresses in the process of fatigue damage cumulation under nonproportional loading depends on the angular distance from the critical plane. • The weight function is an important element of the proposed nonpro- portioanlity measure and its existence in the criterion notation, (1.1), is justified. References 1. Bonacuse P.J., Kalluri S., 2001, Axial and torsional load-type sequencing in cumulative fatigue: low amplitude followed by high amplitude loading,Proc. Sixth International Conference on Biaxial/Multiaxial Fatigue and Fracture, In- stituto Superior Tecnico, 1, 185-194 2. Borodii M.V., Strizhalo V.A., 2000, Analysis of the experimental data on a low cycle fatigue under nonproportional straining, International Journal of Fatigue, 22, 275-282 3. Chen X., 1996, Low-cycle fatigue under non-proportional loading, Fatigue Fract. Eng. Mater. Struct., 19, 7, 839-854 4. Dang Van K., Griveau B., Message O., 1989, On a new multiaxial fa- tigue limit criterion: theory and applications, Biaxial and Multiaxial Fatigue, Mechanical Engineering Publications, 479-496 5. DupratD., 1997,Amodel topredict fatigue life of aeronautical structureswith out-of-phase multiaxial stress condition, Proc. Fifth International Conference on Biaxial/Multiaxial Fatigue and Fracture, Technical University of Opole, 1, 111-123 6. Itoh T., Nakata T., Sakane M., Ohnami M., 1997, Nonproportional low cycle fatigue of 6061 aluminum alloy under 14 strain paths, Proc. Fifth In- ternational Conference on Biaxial/Multiaxial Fatigue and Fracture, Technical University of Opole, 1, 173-187 Experimental verification of fatigue... 347 7. Kanazawa K., Miller K.J., Brown M.W., 1979, cyclic deformation of 1% Cr-Mo-V steel under out-of-phase loads,Fatigue Fract. Eng. Mater. Struct., 2, 217-222 8. Lee B.H., Lee S.B., 1997, A stochastic modeling of multiaxial fatigue life and damage accumulation of sm45c steel,Proc. Fifth International Conference on Biaxial/Multiaxial Fatigue and Fracture, Technical University of Opole, 1, 593-607 9. Lee S.B., 1985,A criterion for fully reversedout-of-phase torsionand bending, Biaxial and Multiaxial Fatigue, Mechanical Engineering Publications, 553-568 10. Lee Y.L., Chiang J., 1991, Fatigue predictions for components under biaxial reversed loading, Journal of Testing and Evaluation, 19, 359-367 11. Łagoda T., Macha E., Będkowski W., 1999, A critical plane approach based on energy concepts: application to biaxial random tension-compression high-cycle fatigue regime, International Journal of Fatigue, 21, 431-443 12. MachaM., 1989,Generalizationof fatigue fracture criteria formultiaxial sinu- soidal loadings in the range of random loadings,Biaxial andMultiaxial Fatigue, Mechanical Engineering Publications, 425-436 13. Morel F., 2000, A critical plane approach for life prediction of high cycle fatigue under multiaxial variable amplitude loading, Int. J. Fatigue, 22, 101- 119 14. Morel F., Ranganathan N., Petit J., Bignonnet A., 1997, A meso- scopic approach for fatigue life prediction undermultiaxial loading,Proc. Fifth International Conference onBiaxial/Multiaxial Fatigue andFracture,Technical University of Opole, 1, 155-172 15. Papadopoulos I.V., 1995, A high cycle fatigue criterion applied in biaxial and triaxial out of phase stress condition, Fatigue Fract. Eng. Mater. Struct., 18, 1, 79-91 16. SkibickiD., 2004a,A fatigue failure criterion formultiaxial loadingwithphase shift andmeanvalue,Journal of Theoretical andAppliedMechanics, 42, 2, 295- 314 17. Skibicki D., 2004b, Fatigue life and strength criteria based on loading non-proportionality measure, Proc. Seventh International Conference on Bia- xial/Multiaxial Fatigue and Fracture, DeutscherVerband fürMaterialforschung und prufung e.V., 167-172 18. Skibicki D., 2006, Multiaxial fatigue life and strength criteria for non- proportional loading,MP Metalprufung, 48, 3, 99-102 19. Skibicki D., Sempruch J., 2001, The concept of non-proportional parame- ter in a complex fatigue load state, Proc. Sixth International Conference on Biaxial/Multiaxial Fatigue and Fracture, Instituto SuperiorTecnico,1, 305-312 20. SkibickiD., Sempruch J., 2002,Concept of a non-proportionalityparameter in a complex fatigue load state, Journal of Theoretical and Applied Mechanics, 40, 2, 389-400 348 D. Skibicki 21. SkibickiD., SempruchJ., 2004,Use of a loadnon-proportionalitymeasure in fatigue under out-of-phase combined bending and torsion, Fatigue Fract. Eng. Mater. Struct., 27, 5, 369-377 22. Sonsino C.M., Zenner H., Yousefi-Hashtyani F., Kuppers M., 2004, Present limitations in the assessment of components under multiaxial service loading,Proc. Seventh International Conference on Biaxial/Multiaxial Fatigue and Fracture, DeutscherVerband fürMaterialforschung und prufung e.V., 3-16 23. Wheelhouse K., Fernando U.S., Eaton D.E., 2001, The effects of non- proportional loading on the torsional fatigue life for a medium carbon steel, Proc. Sixth International Conference on Biaxial/Multiaxial Fatigue and Frac- ture, Instituto Superior Tecnico, 1, 277-284 24. Yousefi-Hashtyani F., 2004, Lifetime assessment under multiaxial random loading– calculationsandtests,Proc. Seventh InternationalConference onBia- xial/Multiaxial Fatigue and Fracture, DeutscherVerband furMaterialforschung und prufung e.V., 343-348 Eksperymentalna weryfikacja modelu nieproporcjonalności obciążenia zmęczeniowego Streszczenie W artykule przedstawiono weryfikację eksperymentalną modelu nieproporcjonal- ności obciążenia zmęczeniowego, który był podstawą sformułowanych przez autora wieloosiowychkryteriów zmęczeniowych.Wmodelu tym zakłada się, że wwarunkach obciążeń nieproporcjonalnychmożna wyróżnić płaszczyznę krytyczną. Jednocześnie, obrót osi głównychpowodowaćmoże intensyfikację procesukumulacji uszkodzeń zmę- czeniowych. Decyduje o tym stopień nieproporcjonalności obciążenia. W przyjętym modelu założono, że o stopniu nieproporcjonalności obciążenia decydują: (a) moduły wektorów obracających się naprężeń, (b) kątowe odległości wektorów tych naprężeń mierzonewstosunkudopłaszczyznykrytycznej.Zrealizowanyeksperymentpolegałna modelowaniu zmęczeniowego obciążenia nieproporcjonalnego poprzez programowane zmiany położenia osi głównych. Różne położenia osi głównych uzyskiwano realizując dwublokowyprogramobciążeń:wahadłowegoskręcania i obciążenia złożonego,w tym przypadku rozciągania ze skręcaniem. Badano wpływ zmian położenia osi głównych w trakcie próby uzyskiwaną na trwałość zmęczeniową. Otrzymane wyniki dają pod- stawę do przyjęcia tezy, że stopień nieproporcjonalności zależy nie tylko odmodułów obracającychsięnaprężeń, ale jest zależny takżeododległościkątowej,w jakiejwektor naprężenia działa w stosunku do płaszczyzny krytycznej. Manuscript received October 16, 2006; accepted for print December 1, 2006