JOURNAL OF THEORETICAL AND APPLIED MECHANICS 42, 2, pp. 269-283, Warsaw 2004 THE EFFECT OF THE METHOD OF DETERMINATION OF YOUNG’S MODULUS ON THE ESTIMATION OF FATIGUE LIFE OF STRUCTURAL ELEMENTS Dariusz Boroński Department of Machine Design, University of Technology and Agriculture in Bydgoszcz e-mail: daborpkm@atr.bydgoszcz.pl Cyclic loading of a material entails modifications of its properties. In the paper, the problem of the influence of determination of the elasticity modulus E on calculations of the fatigue life of structural elements was presented. Different values of the modulus of elasticity obtained by dif- ferentmethodswere used formodelling of cyclic stress-strain curves and for analysis of local stresses and strains. In the calculations the strain-life and energy-life approach was applied. Key words: fatigue, modulus of elasticity, material properties 1. Introduction The divergence between fatigue lives of structural elements determined analytically and experimentally is the cause for continuous search for theways of modification of the existing methods of fatigue life calculation or creating new ones. The research conducted at the Department of Machine Design at the University of Technology andAgriculture in Bydgoszcz (UTA) (Szala and Boroński, 1995; Szala et al., 1994, 1998) has shown possible error sources in the prediction of the fatigue life, such as extrapolating life diagrams onto areas not included in fatigue tests ormaking useYoung’smodulus E determined in a monotonic tension test in further calculations. Analysis of bibliographical data (e.g. Fatigue Design Handbook, 1988; Ko- cańda and, Szala 1991; Roessle and Fatemi, 2000; Schijve, 2001; Smith et al., 1970; Topper and Lam, 1997) indicates common application of Young’s mo- dulus E with values assigned for monotonic loadings. The recently realised research has given the possibility to check the variability of the modulus E 270 D.Boroński according to the type of the material loading. This, in turn, has allowed one to define the differences in the life estimation in the strain- and energy-based calculation methods, which are caused by alterations of the value E. 2. Theoretical basis When carrying out calculations using both the strain-based method and the energy-based one (Szala, 1998), low-cycle material properties are used, which are the basis for the estimation of the fatigue life diagrams (Fig.1) and material-describing ones, e.g. stress-strain curves (Fig.2). The modulus E appears in descriptions of these diagrams as a proportionality coefficient for elastic strain components. Fig. 1. Fatigue life diagrams In the case of fatigue life diagram description in the strain approach using Manson-Coffin’s equation (Fig.1a) εac = εap+εae = ε ′ f(2Nf) c+ σ′f E (2Nf) b (2.1) themodulus E does not affect the strain or life value. This is because fatigue tests according to standards (PN 84/H04334; ASTM 606-92) make use of the quotient σ′f/E, and changes in themodulus E cause only changes in σ ′ f with a constant value of σ′f/E maintained, and thus the fatigue diagram remains unchanged. The situation becomes different in the case of stress-strain curves de- scribing a material being loaded under variable conditions, described with The effect of the method of determination... 271 Ramberg-Osgood’s dependence ε= εe+εp = σ E + ( σ K′ ) 1 n ′ (2.2) A standard research conducted to find the discussed characteristics gives only values describing the plastic element of the strain in dependence (2.2), i.e. K ′ and n′. Therefore, the assumedmodulus E may change thematerial descrip- tion quantitatively. Fig. 2. Material properties (description in text) A stress-strain diagram is used in the strain-based calculation method to determine the local strain and stress, e.g. using Neuber’s method (Neuber, 1961), see Fig.2a, while in the energy-based calculation method, it may be used to describe the hysteresis loop (Fig.2b). Observation of the first and next load cycles (Fig.3) allows one to notice that a change in the material stiffness takes place, which causes a change in the modulus E. Hence, it may be agreed that there exist at least two E values: ”static” (monotonic) and ”dynamic”. Moreover, a precise analysis of the hysteresis loop shows that it actually does not possess a linear-elastic segment, which increases the number of the possible-to-use E values by the static and linear approximation-based ones. The ambiguity connected with the determination of themodulus of elasti- citywas presented among others in thework byKandil (1999). The analysis of the methods of plastic strain determination using Young’s modulus E based on Standards BS 7270:1990; ASTM E606-92; ISO/DIS 12106; prEN 3988 for specimensmade of theNimonic 101 nickel based superalloy showeddifferences of over 30%. Moreover, different values of elasticity modulus for tension and compression semicycles and its decrement together with an increase in the 272 D.Boroński Fig. 3. Hysteresis loops for 45 steel strain were observed. The latter phenomena were described also in the work byMorestin andBoivin (1996) on investigations of specimensmade of a plain steel and alloys A33, XC 38 and AU4G with a large cross-section and upon XE280D sheet metal type with a small cross-section. The aim of the paper is a quantitative analysis of the Young modulus E that depends on loading conditions (monotonic or cyclic variable), and in the case of cyclic variable loading, depends also on the loading amplitude. Moreover, differences in the estimated fatigue lives resulting from application of different moduli E are to be determined. The tests have been carried out for three different materials. Additionally, changes in the modulus E during the fatigue tests have been observed. 3. Experimental data The determination of the longitudinal elasticity modulus requires, accor- ding to ASTME111 – 82/88 Standard, carrying out amonotonic tension test. The thus obtained stress dependence (force by cross-section) in function of strain is the basis for defining Young’s modulus. In the case of linear elastic materials, the modulus is a directional coefficient of a straight line describing the range of linear (proportional) material properties. In the case of non-linear elastic materials, the tangent and secant moduli are introduced. Themethods for their determination are shown in Fig.4. The effect of the method of determination... 273 Fig. 4. Methods of determination of the elasticity modulus: (a) linear, (b) tangent, (c) secant Three types of materials have been considered in the described research: Steel 45, Steel 30HGSA and AluminiumPA6. The investigations were carried out in the Laboratory of Department of MachineDesign at theUniversity ofTechnology andAgriculture inBydgoszcz using the INSTRON 8501 servohydraulic fatigue system with digital control system 8500. In the tests, standard specimens with round cross-sections were applied. Themain dimensions of the specimens are shown in Fig.5. Fig. 5. Specimens used in the investigations The INSTRONaxial extensometers (collaborating with the strain channel of the loading control system) with two values of gauge length were used for strainmeasurements: 50mm in the case ofmonotonic tension tests and 10mm in the case of low cycle fatigue testing. During the tests, the specimens were fastened by standard INSTRON hy- draulic grips. The monotonic tension tests for all three materials revealed in the elastic range nearly linear courses. The coefficients of the straight line correlation de- scribing the rangewere r2 =0.99993 forSteel 45, r2 ∈ 〈0.9999,1〉 for 30HGSA and r2 ∈ 〈0.9983,0.9988〉 forAluminiumPA6. The obtained courses were also described by a second degree polynomial, and tangent moduli were determi- 274 D.Boroński ned. Under those circumstances, the correlation coefficients were as follows: r2 ∈ 〈0.9999,1〉 for Steel 45, r2 =1 for 30HGSA and r2 ∈ 〈0.9997,0.9999〉 for AluminiumPA6. The obtained average moduli E are presented in Table 1. The variability coefficients determined according to ASTMand calculated on the basis of the dependence V1 =100 √ 1 r2 −1 K−2 (3.1) where: K is the number of data pairs and r2 is the correlation coefficient, are given there as well. Table 1 Young’s modulus, Tangent modulus, Material linear approxim., V1 polynomial approxim., V1 the average value the average value 45 Steel 222164MPa 0.5% 226189MPa < 1.2% 30HGSA Steel 208463MPa < 0.5% 212920MPa 0% PA6 Aluminium 74237MPa < 1.58% 83726MPa < 1% Thehysteresis loops recordedduring the fatigue tests in the low-cycle range allowed one to determine the elasticity modulus, further called the dynamic modulus Ed. Due to the fact that the hysteresis loop branches (Fig.6) do not possess the linear elastic part, the modulus of elasticity was decided to be described according to the methodology shown in Fig.4. Fig. 6. Analysis of the hysteresis loop branch The effect of the method of determination... 275 In determining the tangentmodulus, the elastic part of the hysteresis loop was approximated by a second degree polynomial with choosing such a frag- ment of the loop that the correlation coefficient was r2 =0.9999. Due to the fact that the secant value of themodulus E does not have a physical sense in the description of a cyclic strain diagram (the total strainmust be on the right side of the linear-elastic part), the directional coefficient of the straight line approximating the initial segment of the hysteresis loop was agreed to be the value of the second modulus E (the same one as in the polynomial approxi- mation), obtaining r2 > 0.9999. The thus obtained moduli E (average ones) for different strain levels and for different materials are presented in Table 2. Table 2 Dynamic modulusEd, linear approximation Material Strain εac% Average 0.35 0.5 0.8 1.0 2.0 value 45 Steel 165896 176302 180356 181431 174707 175738 30HGSA Steel 198774 190304 192406 192082 189422 192598 PA6 Aluminium 67304 73995 73330 70799 69660 71018 Dynamic modulusEd, linear approximation Material Strain εac% Average 0.35 0.5 0.8 1.0 2.0 value 45 Steel 209516 205587 209674 209369 201509 207131 30HGSA Steel 213649 208898 209842 209888 216278 211711 PA6 Aluminium 68337 76057 80365 74707 72727 74439 The obtained moduli are also presented in Fig.7. 4. Analysis of tests on the modulus E The analysis of the obtained values shows that the so-called dynamicmo- dulus is lower than its static counterpart found inmonotonic tests. Moreover, the modulus changes depending upon the strain which is applied to the spe- cimen. The changes are irregular, and the realisation of the fatigue tests on a considerably greater number of strain levels would be required to determine a possible function describing the dependence of the modulus E on loading. Themodulus Ed differswith respect to E, at themaximum, byabout 21% for Steel 45, about 8% for Steel 30HGSA and about 11% for AluminiumPA6. 276 D.Boroński Fig. 7. Values of the modulus E; 1 –E (static) linear approximation, 2 –E (static) tangent, 3 –Ed (dynamic) linear, 4 –Ed linear-mean, 5 –Ed (dynamic) tangent, 6 –Ed tangent-mean Fig. 8. Exemplary courses of the modulus E for specimens made of the three tested materials During the research, variability of themodulus E in function of the num- ber of realised load cycles were observed as well. Figure 8 presents examples of moduli E for specimensmade of the three testedmaterials, loaded in sym- metric cycles of a constant strain value εac. The comparison of the observed values of the tangentmodulus and of themodulus resulting from linear appro- ximation in the case of steel specimens indicates increasingly weaker linearity The effect of the method of determination... 277 of the elastic hysteresis loop segment (an increase in the tangent modulus with a simultaneous decrease in the ”linear” modulus), and a decrease in the material stiffness. In the case of aluminium specimens, a minor increase in the modulus values occurred, which may indicate a growth in the material stiffness. The differences between the initial and final moduli were growing together with the intensification of the load applied to the specimens. For all the materials, the differences did not exceed 10%. 5. Analysis of fatigue life calculations The determined, on the basis of tests, static and dynamic, values of the modulus E were used for calculation of the fatigue life of notched structural elements by means of the strain-based method (for the three materials) and energy-based method (for 45 Steel). In the case of the strain-based calculationmethod, thenotched elementwas modelled according toNeuber’s hypothesis (Fig.2a), assuming in thematerial model description a stress-strain curve of different values of the modulus E. In the case of the energy-based calculation method, the plastic strain energy was calculated by describing the hysteresis loop branch bymeans of a doubled stress-strain curve (Fig.2b) andNeuber’s local strain and stress analysis ∆ε 2 = ∆σ 2E + (∆σ 2K′ ) 1 n ′ (5.1) assuming different values of E. Low-cycle properties of the three tested materials, discussed in the works by Szala et al. (1998), were used in the calculations. These properties are presented in Table 3. Table 3 Material c b ε′f σ′ f K′ n′ C0 m [MPa] [MPa] 45 Steel −0.43915−0.11668 0.165836 1304 1436 0.226796 2.68604−0.5933 30HGSASteel −0.81030−0.08716 2.139042 1660 1068 0.066426 PA6Alumin. −0.84613−0.10016 0.117573 894 797 0.085848 The stress-strain curves used in both methods and defined for different values of E are shown in Fig.9. The diagram analysis indicates fairly low 278 D.Boroński Fig. 9. Stress-strain curves for different values of the modulus E influence of variability of the modulus E on the qualitative course of the material modelling curve. For 45 Steel, the greatest stress differences equal about ∆σ = 40MPa for the strain ε = 0.1%, and in the case of strain of about ∆ε = 0.08% for stresses from the range of σ∈ 〈400,440〉MPa. Thismeans that themaximum difference in the case of strain-based stress determination is about 20%, while reciprocally, i.e. determining the strain on the grounds of stress the resultmay differ by about 14.5%. For 30HGSA Steel, the greatest differences equal about ∆σ = 40MPa for the strain ε = 0.27% (8% difference), and in the case of strain of about ∆ε=0.035% for stresses σ∈ 〈620,660〉MPa (12% difference). ForPA6Aluminiumthegreatest stress differences are about ∆σ=50MPa for the strain range ε∈ 〈0.3,0.5〉% (which gives the difference of about 24%), The effect of the method of determination... 279 and in the case of strain of about ∆ε=0.1% for stresses σ∈ 〈460,520〉MPa (difference by about 14.3%). The changes of the analytically determined local strain and plastic strain energy (the hysteresis energy) for the given range of variability of the mo- dulus E in a structural element made of 45 Steel are presented in Fig.10. Three values of the nominal stress S (200, 300 and 400MPa) and two values of the stress concentration factor Kt (1.5, 2.5) were used in the calculations. The differences in the local strains (Fig.10a) were about 20%. In the case of fatigue life calculations (Fig.10b) it caused differences from 57 up to 100%. Fig. 10. Changes of the local strain (a), plastic strain energy (c), fatigue life for strain-based calculations (b) and energy based calculations (d) for 45 Steel Similarly, in the case of the energybasedapproach, themaximaldifferences of the plastic strain energy ∆W (Fig.10c) for the analysed range of variability 280 D.Boroński of the modulus E were 13-23%, which entailed differences in the fatigue life (Fig.10d) from 21 up to 38%. Fig. 11. Changes of the local strain (a), fatigue life for strain-based calculations (b) for 30HGSA Steel Fig. 12. Changes of the local strain (a), fatigue life for strain-based calculations (b) for PA6Aluminium The results of analogous strain-based calculationsmade for 30HGSASteel and PA6 Aluminium are shown in Fig.11 and Fig.12. The differences in the prediction of the local strains and fatigue life for the three analysedmaterials are collected in Table 4. The effect of the method of determination... 281 Table 4 Differences of Material ε 2Nf ∆W Nc εr/εm (2Nf)r/(2Nf)m ∆Wr/∆Wm (Nc)r/(Nc)m [%] [%] [%] [%] 45 Steel 21-23 57-100 13-23 21-38 30HGSA Steel 9-10 23-104 PA6 Aluminium 15-17 27-130 where subscript r= range, m=mean. 6. Conclusions On the basis of the conducted tests and calculations, several conclusions concerning the sensitivity of Young’smodulus to variable loads can be drawn: • cyclic loading in the range of loads producing plastic strains decreases Young’s modulus; in the conducted tests, the phenomenon was least visible in the case of Steel 30HGSA (tangentmodulus), andmost visible in the case of Steel 45 (linear approximation modulus), • the value of modulus obtained by the linear approximation decreases with the number of cycles which, with somewhat growing tendency of the tangentmodulus, indicates increasinglyweaker linearity of the elastic part of the hysteresis loop, • a too small number of the realised loading levels does not allow one to determine the influence of the loading amplitude on Young’s modulus. The subsequently made simulations, in which different moduli E were applied, allowed one to notice that the determination method did not consi- derably affect the calculated local strains and plastic strain energy. However, even a small variation of the local strainmay cause big differences in the fati- gue life when the strain-life approach is applied, especially for small values of strain. For the least advantageous case, for PA6 Aluminium, with the coefficient Kt = 1.5 and the nominal stress value S = 200MPa, the difference in the estimation of the fatigue life was ∆=130%. 282 D.Boroński In spite of this, it can be stated that making use of moduli E defined on the basis of the literature data (including handbooks) should not produce significant errors in engineering calculations on the fatigue life of structural elements. Theobtainedcalculation results indicate also thenecessity of further search on the sources of errors that appear in various methods of determination of the fatigue life. References 1. Fatigue Design Handbook, 1988, II edition, Society of Automative Engineering 2. Kandil F.A., 1999, Potential ambiguity in the determination of the plastic strain range component in LCF testing, International Journal of Fatigue, 21, 1013-1018 3. Kocańda S., Szala J., 1991,Podstawy obliczeń zmęczeniowych, PWN,War- szawa 4. Morestin F., Boivin M., 1996, On the necessity of taking into account the variation in the Young modulus with plastic strain in elastic-plastic software, Nuclear Engineering and Design, 162, 107-116 5. NeuberH., 1961,Theory of stress concentration for shear strained prismatical bodies with arbitrary non-linear stress-strain law, Journal of Applied Mecha- nics, 28, 544-550 6. Roessle M.L., Fatemi A., 2000, Strain-controlled fatigue properties of steels and some simple approximations, International Journal of Fatigue,22, 495-511 7. Schijve J., 2001, Fatigue of Structures and Materials, Kluwer Academic Pu- blishers, Dordrecht 8. Smith K.N., Watson P., Topper T.H., 1970, A stress-strain function for the fatigue of metals, Journal of Materials, JMLSA, 5, 4, 767-778 9. Szala J., 1998,Hipotezy sumowania uszkodzeń zmęczeniowych, Wydawnictwa Uczelniane ATR, Bydgoszcz 10. Szala J., Boroński D., 1995, Comparative analysis of experimental and cal- culated fatigue life of the 45 steel notched structural member, The Archive of Mechanical Engineering, 1-2, 111-123 11. Szala J., Mroziński S., Boroński D., 1994, Fatigue life of machines parts in periodical and irregular loading conditions, KBN Report (State Committee for Scientific ResearchGrant No 712829101) The effect of the method of determination... 283 12. SzalaJ.,Mroziński S.,BorońskiD., 1998, Investigationsof fatiguedamage summing process in low-cycle fatigue range,KBNReport (StateCommittee for Scientific ResearchGrant No 7T07A03508) 13. Topper T.H., Lam T.S., 1997, Effective strain-fatigue life data for variable amplitude fatigue, International Journal of Fatigue, 19, Supp. No. 1, 137-143 14. Standard ASTME606-92, Standard Practice for Strain-Controlled Fatigue Te- sting 15. StandardASTME111– 82/88:Determination ofYoung’s, Tangent, andChord Modulus 16. Standard BS 7270:1990,Method for constant amplitude strain controlled fati- gue testing, British Standards Institution, 1990 17. Standard ISO/DIS 12106,Metallic materials fatigue testing – axial strain con- trolled method, 1998 18. Standard PN-84/H-04334, Badania niskocyklowego zmęczenia metali 19. Standard PrEN 3988:1998, Aerospace series, Test methods for metallic mate- rials – constant amplitude strain-controlled low cycle fatigue testing, AECMA, draft no.2, 1998 Wpływ sposobu określania wartości modułu E na obliczenia trwałości zmęczeniowej elementów konstrukcyjnych Streszczenie W pracy przedstawiono analizę ilościowej zmienności modułu E w zależności od rodzaju obciążenia (monotonicznie i cyklicznie zmienne), a w przypadku obciążenia cyklicznie zmiennego także od wartości amplitudy obciążenia. Ponadto wyznaczono różnicę szacowanych trwałości wynikającą z przyjęcia różnych wartości modułu E. Badania przeprowadzono dla trzech różnych materiałów. Dodatkowo obserwowano zmianę wartości modułu E w różnych okresach trwałości zmęczeniowej. Manuscript received November 19, 2003; accepted for print December 22, 2003