JOURNAL OF THEORETICAL AND APPLIED MECHANICS 43, 1, pp. 65-80, Warsaw 2005 EXPERIMENTAL METHOD OF DEFINING BIAXIAL FATIGUE PROPERTIES OF ELASTIC-PLASTIC CONSTRUCTION MATERIALS. PART 2 – EXPERIMENTAL RESEARCH Artur Cichański Janusz Sempruch Faculty of Mechanical Engineering, University of Technology and Agriculture in Bydgoszcz e-mail: artur.cichanski@atr.bydgoszcz.pl The paper is concernedwith the authors’ ownmethod of defining biaxial fatigue properties of elastic-plastic construction materials, for which a theoretical basiswas formulatedbefore (Cichański andSempruch, 2005). In the present paper, results of experimental investigations (divided in- to preliminary andmain research) appropriate for the proposedmethod are described. In the preliminary research phase, an experimental verifi- cation of major assumptions and numerical verification of conditions on preparationof test specimenswere carried out.During themain research phase, model parameters of plastic and fatigue properties were descri- bed and the evaluation of the proposedmodel (Cichański andSempruch, 2005) was done. Key words: biaxial fatigue, yield surface, testingmethods 1. Introduction Experimental determinatiion of biaxial fatigue properties requires choosing a research method appropriate for the specified quantity. Numerous research works carried out in the field have not yet led to formulation of adequate normalization acts. A review of research methods of biaxial fatigue in bi- bliography indicates three commonly-used methods (Cichański and Świtała, 2002) and a number of rarely used ones. Analysis of biaxial fatigue research methods carried out towards tests specimen shaping shows that in the group ofmost often usedmethods, the research is done with the use of a cylindrical, 66 A.Cichański, J.Sempruch thin-walled tubular and cruciform specimen (Cichański and Świtała, 2002). In the group of more rarely used methods, specimens with uniaxial loading can bemet, in which the stress state biaxiality results from the specimen geo- metry. The group includes oval (Schewchuk et al., 1968) and rhombic plate specimens (Zamrik et al., 1997) exposed to cyclic bending, and thick-walled tubular specimens (McDiarmid, 1991) cyclically loaded with altering pressu- re. Also specimens in the shape of a rotating disc (Findley et al., 1961) can bemet as well as specimens whose geometrical shape result from researcher’s own experience (Sawert, 1943). In the case of shape specimens, the change of realized stress proportions is obtained by the change of specimen geometrical dimensions. The advantage of methods using shape specimens for biaxial fati- gue research is the possibility of conducting the research at a test stand that generates cyclic uniaxial loading. Their disadvantage comes from the limit of their application to the proportional fatigue. The paper is concerned with the authors’ own method of defining biaxial fatigue properties of elastic-plastic construction materials. In the first part of thepaper, the theoretical basis of the proposedmethod is discussed (Cichański and Sempruch, 2005). In the continuing part, the results of experimental re- search, divided into the preliminary andmain research, are presented. During the preliminary research phase, the most important assumptions were veri- fied experimentally, while the conditions for preparing the specimens for tests were verified numerically. At the main research phase, the model parameters describing plastic and fatigue properties are given, and the evaluation of the proposedmodel (Cichański and Sempruch, 2005) is done. 2. Experimental procedure The idea of themethod for conducting an experimental researchwas outli- ned in Cichański and Sempruch (2005). The characteristic feature of the me- thod is the possibility of obtaining controlled plastic straining of the tested material, as it is schematically shown in Fig.1. Higher quality constructional steel 45 in the shape of plates of size 450× 200×4mmwas used for the research. The chemical composition of the tested steel is presented in Table 1. The plates from which the specimens were cut out were subject to normalising in order to remove the texture resulting from rolling. Experimental method.... Part 2 – Experimental research 67 Fig. 1. Themethod of obtaining specimens for tests (Cichański and Sempruch, 2005) Table 1.Tested steel chemical composition Element C Mn Si P S Determined 0.48% 0.74% 0.42% 0.02% 0.01% Polish Standard 0.42-0.5% 0.5-0.8% 0.1-0.4% 0-0.04% 0-0.04% The research on plastic properties was conducted in a tension test carried out on flat bars, Fig.2. The specimen geometric shape and other research conditions were in accordance with the Polish Standard [8]. Fig. 2. Geometric shape of the specimen for determination of plastic properties The fatigue research was carried out on a universal INSTRON 8501 ma- chine in conditions of uniaxial sinusoid symmetrical (R = −1) tension- compression loading. For the preliminary research of fatigue properties flat dumbbell specimens, made of the virgin material, were used (Fig.3) in accor- dance with the Polish Standard [9]. 68 A.Cichański, J.Sempruch Fig. 3. Geometric shape of the specimen for the preliminary fatigue research During the preliminary research, it was found that the specimen presented in Fig.3, made of prestrained material, underwent buckling. A new geome- trical specimen shape (Fig.4), not subject to stability loss, was established experimentally. Eventually, in the main research, specimens made of prestra- inedmaterial were used with geometry shown in Fig.4. Fig. 4. Geometric shape of the specimen used in the main fatigue research 3. Verification of test conditions 3.1. Verification of the method of specimen preparation for tests Due to specificity of themethod of preparing specimens for tests inducing inhomogeneity of plastic strain distribution in restrained material plates, a numerical verification of the appointed method was carried out. The aim of the analysis was to define the plate area, where plastic strain gradients have negligible influence on the repeatability of properties test specimens. Nume- rical calculations were done with the use of the finite element method in the ANSYS program environment. The research was a two-dimensional nonlinear analysis, taking into account elastic-plastic material behaviour. Experimental method.... Part 2 – Experimental research 69 Making use double symmetry of both geometric shape and boundary con- ditions of the plate, one fourth of the plate was tested. Flat elements with two translatory degrees of freedom in each node, described as PLANE42 in theANSYS library, were used for automaticmeshing of the chosen plate area. The omitted plate parts were modelled by establishing adequate degrees of freedom in nodes lying on plate section edges. The elastic-plastic model with kinematic multi-linear hardening was used for description of material beha- viour. Characteristic model points were read on a diagram σ-ε, obtained in a static tensile testing of the virgin material. The force course was controlled during calculations. Its highest value 300kN was established on the basis of experimental research as a force causing a strain of about 2% in themeasure- ment base of 200mm. The calculation results in the form of a map of strains remaining in the plate after the initial plastic prestrain process are presented in Fig.5. Fig. 5. The distribution of reduced plastic strains in the plate The reduced plastic strains εpl take the highest values near the holes, throughwhich the load is transmitted onto the plate.Moreover, the strains εpl in plate areas surrounding the holes are characterised by significant gradients. The strains are distributed more and more uniformly together with moving away fromthehole line in thedirection towards theplate centre.The relatively lower uniformity can be observed at the edge of the plate area, where the grip parts of specimens are to be shaped. 70 A.Cichański, J.Sempruch On the basis of carried out analysis, it can be stated that plastic strain gradients in the distance of approx. 100mm from the plate centre have a negligible value from the point of view of the research purpose. Therefore, little variability of othermechanical properties in this area canbeassumed.For further needs of the experimental research, the testing specimens cut out from the area of prestrained plates were assumed to be characterised by repeatable properties. 3.2. Research on plastic properties Theaimof this researchphasewas experimental verification of the assump- tion about isotropy of plastic properties of the virgin material. Also identifi- cation of the influence of plastic prestrain on the course of the yield stress defined on the basis of agreed methodology was carried out. During preliminarymetallographic tests, the preferential structural orien- tation of the virginmaterial was observed, which was the remains of the tech- nological process of sheet rolling. Structures occurring near the sheet surface and in the middle of sheet intersection were chosen for metallographic obse- rvations. What results from the observations is that the heat treatment of annealing caused full recrystallisation only near the sheet surface (Fig.6). Fig. 6. Sheet metallographic structures: (a) near the surface; (b) in the middle of the intersection (multiplied by 80×) To evaluate the observed banding influence on mechanical properties of the tested virgin material sheet, a number of tests on specimens cut out at the angles α∈ {0◦,15◦, . . . ,90◦} were carried out. Values Re and Rm, obta- ined during the tests, are presented in Fig.7. Apart from the experimentally determined values, the recommended Re for tested steels according to Polish Standards is shown in Fig.7 (dashed line). Experimental method.... Part 2 – Experimental research 71 To evaluate the method of preparing specimens for tests, a number of tests on specimens cut out from the prestrained material at the angles α ∈ {0◦,15◦, . . . ,90◦}, for which εpl =2%. For this material, not showing a clear yield point, values of R0.2, whose course is presented in Fig.7, were found. Fig. 7. Research results on chosenmonotonic properties The yield point Re found for the virgin material reaches the predicted by PS value of 355MPa with the accuracy of ±3%, not showing a clear relation to the angle α. Also the course of Rm does not indicate the correlation of the tested properties with the sheet banding direction. For the needs of testing, it was assumed that for the virgin material, the assumption about the isotropy of plastic properties was fulfilled. Introduction of material initial strains of εpl = 2% caused a distinct mo- dification of plastic properties, which found its representation in the course of R0.2, see Fig.12. The largest growth of R0.2 by +14.9% occurred in the direction of the initial plastic prestrain α=0◦. In the direction perpendicular to the initial plastic prestrain α = 90◦ one observes the highest fall of R0.2 by −16.3%. For needs of the tests, the course of changes of R0.2 was assumed to well represent the anisotropy of mechanical properties of the prestrained material 3.3. Estimation of fatigue properties Thefirst stage of preliminary researchon fatiguepropertieswas to estimate the influence of the preferential orientation observed in the metallographic structure on the fatigue strength. Due to their time consumption, the tests were carried out for three directions of cut-out specimens at α=0◦, 45◦ and 90◦. The tests were conducted on flat dumbbell specimensmade of the virgin 72 A.Cichański, J.Sempruch material, see Fig.3. During the tests the amplitude of sinusoid-cycle-variable stress with the frequency of 10Hz was controlled. The test results in the form of Wöhler’s lines are presented in Fig.8. Fig. 8.Wöhler’s lines for the virginmaterial A slight aberration of Wöhler’s lines observed for three angles α indicate that the fatigue properties of the tested material do not show a significant dependence on the test direction. Because of the test aim, an assumption was made that the virgin material was characterised by the isotropy of fatigue properties. 4. The main research 4.1. The aim and scope of research The main goal of the research was to verify experimentally the allowable stress amplitude presented in themodel approach in Cichański and Sempruch (2005). Due to specificity of the model formulated in the mentioned work, which consists in combining the plastic and fatigue properties altogether, the realization of the research goal required realisation of three detailed aims. The detailed aims were as follows: carrying out experimental research on plastic properties, carrying out experimental research on fatigue properties and ana- lyzing the allowable stress amplitude on the basis of test results in both areas of the research. Experimental method.... Part 2 – Experimental research 73 The first group of the main research concerned the determination of the parameters describing the plastic part of the model. At that phase, the yield stress for seven directions described by angles α∈ {0◦,15◦, . . . ,90◦} was de- termined. The tests were conducted for the virgin material as well as for the prestrainedmaterial. In the second phase of research, the parameters describing the fatigue part of the model were determined. Due to a minor dependence of the fati- gue properties of the virgin material on the investigated angle, Wöhler’s line was determined for one direction described with the angle α = 90◦. For the prestrained material, Wöhler’s lines were determined for three characteristic directions: α=0◦, 45◦ and 90◦. 4.2. Results of the investigation on plastic properties The results of experimental research on plastic properties for both types of materials are presented in the form of yield lines distinguished on the yield surface. For the virginmaterial, characterised by adistinct yield point, Re was assumed as the yield stress value. For the prestrained material, which shows no clear yield point, values of yield stresses corresponding to three different plastic strains: R0.05,R0.1,R0.2 were determined. The obtained results served to establish, on the basis of methodology described in Cichański and Sem- pruch (2005), experimental shapes of the BEC ellipse (Fig.10) for the virgin material, and B′E′C′ for the prestrainedmaterial. Figure 9 presents both el- lipses seen in the direction D0 (accordingly to designation in Fig.10), while Fig.11, shows them in the direction A0. In both figures, the BEC ellipse is markedwith bold lines, while the remaining lines represent different shapes of the B′E′C′ ellipse, depending on plastic strains assumed for the definition of yield stresses. For the virgin material, the yield line seen in the direction D0 (Fig.9) is a line segment. For the angle α = 60◦, the line curves. For the prestrained material, theyield line shapedependson the strain εpl acceptedas thematerial yield measure. In the case of assuming the yield stress as the quantity R0.05, the yield line curving can be noticed both angles α < 15◦ and α > 45◦; therefore, the condition was omitted in further considerations. In the case of assuming the yield stress as R0.1 or R0.2, the yield line for angles α ∈< 0◦,45◦ > is a straight line segment. The smallest value of εpl for which yield line does not curve was taken for further consideration. Yield stresses in the formof R0.1 correspond to this case. The yield line curving for angles α> 45◦ can be observed; however, the effect deepens with increasing angle α. The strongest effect is observed for the angle α=90◦. 74 A.Cichański, J.Sempruch Fig. 9. Ellipses BEC and B′E′C′ – view along D0 direction Fig. 10. Huber-Mises-Hencky’s ellipsoid For the virgin material, the yield line seen in the direction A0 (Fig.11) is an ellipse. In the case of the prestrainedmaterial, a shift of this ellipse occurs in the direction σx – the same direction in which the plastic prestrain was made. All of this agrees with the hypothesis of kinematic hardening. 4.3. Results of the investigation on fatigue properties The results of experimental tests on fatigue properties in the form of Wöhler’s lines are presented in Fig.12. The lines were determined with the use of specimens (Fig.4) made of the virgin material and with the use of Experimental method.... Part 2 – Experimental research 75 Fig. 11. Ellipses BEC and B′E′C′ – view along A0 direction Fig. 12.Wöhler’s line for the virgin and prestrainedmaterials specimens (Fig.4) made of the prestrained material for three characteristic directions α=0◦, 45◦ and 90◦. During the research conducted for the prestrained material, the highest fatigue lives were achieved for the angle α = 45◦, lower fatigue lives were determined for the angle α = 90◦, while the lowest values were obtained for the angle α = 0◦. Fatigue lives determined for the virgin material are found slightly below the values determined for the prestrainedmaterial for the angle α=45◦ 76 A.Cichański, J.Sempruch σvirgina (N)= 382.28−13.078lnN (4.1) σprestr.0 ◦ a (N)= 369.71−13.643lnN (4.2) σprestr.45 ◦ a (N)= 394.15−13.832lnN (4.3) σ prestr.90◦ a (N)= 425.97−17.315lnN (4.4) Equations (4.1)-(4.3) characterising the behaviour of the virgin material and theprestrainedmaterial tested for the angle α=0◦ and 45◦ are described with parameters slightly different from each other. Due to only minor diffe- rences in the direction coefficient, the lines describedwith these equations are situated in a slightly non-parallel way. The decisive factor for the position of Wöhler’s lines in a particular direction is the absolute term in the equations taking the highest value for the prestrained material and angle α = 45◦, see (4.3), and slightly lower for the virgin material, see (4.1). The lowest value of the absolute term, corresponding to the lowest fatigue lives, was determined for the prestrained material and α = 0◦, (4.2). The equation describing the behaviour of the prestrainedmaterial tested at the angle α=90◦, (4.4), both for the directional coefficient and the absolute term, considerably differs from the other three equations. TheWöhler line describedwith equation (4.4) runs obliquely with respect to Wöhler’s lines described with equations (4.1)-(4.3). 5. Achieved results and comparison of proposed models The results achieved during the experimental research on plastic and fa- tigue properties served to verify the allowable stress amplitude described in Cichański and Sempruch (2005). In the plastic part, themodel parameter was the plastic strain accepted for definition of the yield stress. To verify nota- tion used in the model R0.1 was accepted as a definition of yield stress. In the fatigue part, the model parameter was the number of the limiting cycle established for considerations. The verification of notation was made for the considered fatigue life of Nlim =200000 cycles. The main model parameter is the coefficient aN. Its value decides about the relationship between the B′E′C′ plastic ellipse and b′e′c′ fatigue one, determined for theprestrainedmaterial. The coefficient aN wasdeterminedon the basis of experimental results for the virginmaterial. The yield stress value εizopl equalled 346MPa.The stress value σ izo A definedon thebasis of dependence (4.1) for the considered fatigue life Nlim was 222.6MPa. Eventually, the aN parameter obeys a following formula Experimental method.... Part 2 – Experimental research 77 a200000 = σizoA σizo pl = 222.6MPa 346MPa =0.64 (5.1) According to the accepted assumption about the effect of allowable am- plitude of the fatigue stress on yield stresses, and using the established aN coefficient as well as plastic properties found from experiments, the model el- lipse b′e′c′ was determined. Its course is presented in Fig.14 with lines having trianglemarkers. For comparison, in the samefigure, theplastic ellipse B ′E′C′ determined for the prestrainedmaterials indicated with square markers. Fig. 13. Results of investigatiions and comparison of proposedmodels The correctness of the model ellipse b′e′c′ was verified in three points corresponding to three characteristic directions α = 0◦, 45◦ and 90◦, where the fatigue research was conducted. According to the accepted assumption on the isotropy of fatigue properties, the stress σizoA for these directions was established on the basis of dependence (4.1), see Table 2. The σanizoA stress, being a modified quantity for particular angles α, and determined for the considered fatigue life Nlim on the basis of (4.2)-(4.4), is also presented in Table 2. Table 2. Fatigue properties α [◦] σizoA [MPa] σ anizo A [MPa] 0 222.6 203.2 45 222.6 225.3 90 222.6 214.6 78 A.Cichański, J.Sempruch The stress σmodelA , determined for three characteristic directions on the basis of plastic properties and formula (5.1), the stress σekspM , determined on the basis of fatigue properties and the model approach to the allowable am- plitude for these directions, are presented in Table 3. The table also presents the relative error resulting from the comparison of σekspM and σ model A . Table 3.Numerical results for the proposedmodel α [◦] σekspM [MPa] σ model A [MPa] Error [%] 0 252.9 254.5 −0.63 45 218.5 222.8 −1.99 90 235.1 166.3 29.29 The stresses σeksp M , calculated for appropriate angles α, after some trans- formations, are presented in Fig.14 with round markers. For angles α = 0◦ and 45◦, the points determined on the basis of the model course are situated on the model ellipse b′e′c′. The numerical error determined for these angles is smaller than 2%. For the angle α = 90◦, the point determined on the ba- sis of the model course lies outside the model ellipse b′e′c′. In this case, the numerical error equals approximately 30%. 6. Conclusions Experimental verification of the research conditions allows one to accept an assumption on the isotropy of plastic and fatigue properties of the virgin material. On the basis of the conducted numerical analysis, it can be assumed that, despite noticeable influence of the manner in which plates are loaded on the plastic strain distribution, there exists such an area on the prestrained material plate fromwhich specimens for tests with repeatablemechanical pro- perties can be cut out. The agreement between the model and research results for angles α=0◦ and 45◦ as well as lack of such agreement for α = 90◦ were observed. The determineddifference is causedby two factors.Theyield line, found fromexpe- riments on plastic properties, shows curving for the discussed angle.Wöhler’s line, resulting from research on fatigue properties in the prestrained material (α=90◦) runs obliquely towards Wöhler’s line determined for the remaining angles. On this basis, it can be assumed that the allowable stress amplitude Experimental method.... Part 2 – Experimental research 79 in the model approach sufficiently describes the material behaviour as for the non-deformed yield line. The research method proposed in the paper successfully underwent the experimental verification of most important assumptions and the numerical verification of specimen preparation for testing conditions. The paper indica- ted the possibility of application of thatmethod for determination of proposed characteristic that describe biaxial fatigue properties of an elastic-plastic ma- terial. References 1. Cichański A., Świtała A., 2002, Przegląd doświadczalnychmetod badania dwuosiowego zmęczenia, Zeszyty Naukowe ATR, Bydgoszcz, 5-16 2. Cichański A., Sempruch J., 2005, Experimental method of defining biaxial fatigue properties of elastic-plastic constructionmaterials. Part 1 –Model for- mulation, Journal of Theoretical and Appleid Mechanics, 43, 1 3. FindleyW.N.,MathurP.N., Szczepanski E.,TemelA.O., 1961,Energy versus stress theories for combined stress – a fatigue experimentusinga rotating disk,Trans. ASME, J. Basic Engng. D, 83, 2, 10-14 4. McDiarmid D.L., 1991, A general fatigue criterion for high cycle multiaxial fatigue failure, Fatigue Fract. Engng. Mater. Struct., 14, 4, 429-453 5. Sawert W., 1943, Verhalten der Baustähle bei wechselndermehrachsiger Be- anspruchung, Z. Ver. Deut. Ing., 87, 39/40, 609-615 6. Schewchuk J., Zamrik S.Y., Marin J., 1968, Low-cycle fatigue of 7075- T651 aluminium alloy in biaxial bending,Experimental Mechanics, 8, 11, 504- 512 7. Zamrik S.Y., Ledger D.J., Date C., 1997, Fatigue characteristics of thin titaniumplatesdue tobiaxial stress cycling,Proceedings of the 5th International Conference onBiaxial/Multiaxial Fatigue and Fracture, Opole-Cracow, II, 167- 187 8. PN-EN10002-1+AC1:1998,Metale.Próbarozciągania.Metodabadaniawtem- peraturze otoczenia 9. PN-74/H-04327, Badanie metali na zmęczenie. Próba osiowego rozciągania – ściskania przy stałym cyklu obciążeń zewnętrznych 80 A.Cichański, J.Sempruch Doświadczalna metoda wyznaczania dwuosiowych własności zmęczeniowych sprężysto-plastycznych materiałów konstrukcyjnych. Część 2 – Badania eksprerymentalne Streszczenie Niniejsza praca jest zorientowana na własną metodę wyznaczania dwuosiowych własności zmęczeniowych sprężysto-plastycznych materiałów konstrukcyjnych, dla której podstawy teoretyczne zostały sformułowane wcześniej (Cichański i Sempruch, 2005).W prezentowanej pracy przedstawiono wyniki badań doświadczalnych właści- wych dla zaproponowanej metody, podzielonych na badania wstępne i główne. Na etapie badań wstępnych wykonano doświadczalną weryfikację najważniejszych zało- żeńmetody oraz numerycznąweryfikację warunkówprzygotowania próbek do badań. Na etapie badań zasadniczychwyznaczonomodelowe parametry opisu własności pla- stycznych i zmęczeniowych orazdokonanoocenymodelu zaproponowanegowCichań- ski i Sempruch (2005). Manuscript received June 25, 2004; accepted for print October 18, 2004