Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 45, 4, pp. 773-784, Warsaw 2007 METAL TESTS IN CONDITIONS OF CONTROLLED STRAIN ENERGY DENSITY Stanisław Mroziński Dariusz Boroński University of Technology and Life Sciences, Faculty of Mechanical Engineering, Bydgoszcz, Poland e-mail: stmpkm@utp.edu.pl; daborpkm@utp.edu.pl A new concept of the determination of fatigue properties was presented in the paper. Experimental procedures based on strain-life fatigue tests were developed. The total strain energy density parameter calculated during a loading cycle was applied as the control signal for fatigue te- sting of the specimens. The developed procedure extends the range of fatigue testing methods in the low-cycle fatigue, which should bring a new quality in the case of its use in fatigue life calculations. Keywords: low-cycleproperties, fatigue life, plastic strain energydensity, elastic strain energy density Notations b,c – elastic and plastic exponent, respectively bw,cw – elastic andplasti strain energyexponent, respectively E – modulus of elasticity, MPa Kp,Ke – fatigue coefficient in plastic and elastic strain energy criterion, respectively, MJ/m3 Wt,Wp,We – total, plastic andelastic strain energydensity, respec- tively, MJ/m3 W+e ,W − e – elastic tensile and compression strain energy density, respectively, MJ/m3 2Nf – reversals to failure εac – amplitude of total strain ∆εac,∆εae,∆εap – total, elastic and plastic strain range, respectively ε′f – fatigue ductility coefficient σ′f – fatigue strength coefficient, MPa 774 S. Mroziński, D. Boroński σa,δσ – stress amplitude and stress increment, respectively, MPa δε – strain increment 1. Introduction With the acceptance of new quantities for fatigue life calculation such as total strain, plastic strain or plastic strain energy density Wp, there appeared the necessity of determination of new fatigue characteristics. In the case of fatigue damage assessment using a local strain-life approach, thebasic characteristic applied in fatigue life calculations is the strain rangevs. reversal of loading to the failure curve approximated with theManson-Coffin relation (Coffin, 1954; Feltner andMorrow, 1961) ∆εac 2 = ∆εae 2 + ∆εap 2 = σ′f E (2Nf) b+ε′f(2Nf) c (1.1) The data needed for determination of the characteristic described by equ- ation (1.1) (total strainor its components) aredirectlymeasuredduring fatigue tests. Determination of the fatigue characteristic in the energy-life approach de- mands determination the energy parameter and acceptance of a description model of the fatigue process in the energy approach. Basing on the litera- ture data, it can be stated that presently there are two basic approaches to description of the fatigue process in the energy approach. In the first case, the total energy cumulated in the fatigue process is taken into account. This energy is comparable with the energy dissipated in static tension tests. Suggestions of such an approach can be found for example in the papers by Feltner andMorrow (1961), Lin (1993), Lin and Haicheng (1998). In the second approach, the cumulation of dissipated energy in particular cycles of variable loading is taken into account. Three groups of suggestions can be observed in this approach. These take into account: a) plastic strain energy densityWp solely; Suggestions of this kind were presented in the papers by Gołoś (1988), Kaleta (1998),Mroziński andTopoliński (1999). Basing on the literature information, it canbe stated that thismodel of fatiguedescriptionproves to be correct in the low-cycle fatigue range. b) sum of plastic strain energy density Wp and elastic strain energy den- sity We; Metal tests in conditions... 775 Suggestions of such an approach can be found in the papers by Ellyin (1989), Gołoś (1989), Gołoś and Ellyin (1988). It proves to be correct for the high-cycle fatigue range in the loading programme. c) plastic-elastic strain energy density Wt; In the above approach separation of plastic-elastic strain energy into components is omitted. Such descriptionswere suggested in the papers by Łagoda (2001), Smith et al. (1970). Fatigue graphs in the energy-life approach with the use of Wp,We, or Wt parameters are mostly obtained indirectly basing on tests performed under controlled stress or strain, and adequate energy parameters are the resulting values calculated after realisation of a fatigue test. In the case of cyclically unstable materials it causes difficulties in a unique determination of a fatigue graph. This problemwas presented byMroziński (2006). Effective realisation of fatigue life calculations with the use of energy pa- rameters demands elaboration of new procedures for finding fatigue charac- teristics. Their determination should include research work performed with preservation of constant parameters of criterial values in following loading cycles. In the literature, there can be found examinations under controlled ener- gy parameters (plastic strain energy density or total strain energy density) during tests for metal specimens (Kasprzyczak and Macha, 2006; Mroziński and Boroński, 2006; Słowik et al., 2006), and specimens made of composite materials (Boroński et al., 2006). The basic aim of this paper is experimental verification of determination of fatigue properties for metal materials under controlled total strain energy density. An additional aim of the paper is comparative analysis of the results ob- tained using the presented concept with the results obtained with the use of the classical method (under controlled strain). 2. Assumptions to the testing method in conditions of energy control Development ofmodern strengthmachines, associatedmainlywith the impro- vement of digital control systems, allows for applying non-standard methods 776 S. Mroziński, D. Boroński of research in cyclic loading conditions. The rate of the control process in the PID feedback loop scheme and high quality ofmeasuring sensors and transdu- cers gives the possibility of controlling fatigue machines with the use of new control signal values including total strain energy density or its components. Conducting a research in conditions of controlled values of strain energy density demands accepting a suitable model of its division into individual components. In the presently used energy descriptions of the fatigue process, it is accepted that the main role in fatigue plays plastic strain energy and (to a lesser degree) elastic strain energy (Gołoś, 1988, 1989; Gołoś and Ellyin, 1988). One can write down the following equation Wt =We+Wp (2.1) where: Wt is the total strain energydensity, Wp –plastic strain energydensity, We – elastic strain energy density (We =W + e +W − e ). Graphical interpretation of the individual components of the energy is presented in Fig.1a. Basing on the assumptions, to the description of the fati- gue graph in the strain-life approach describedwith equation (1.1), analogous equations were formulated for the energy description. A relation between the energy of plastic strain Wp and the number of loading reversals to failure is as follows Wp =Kp(2Nf) cw (2.2) where: Kp denotes the regression line constant, cw – exponentof the regression line. Fig. 1. Components of the total strain energy (a), fatigue chart in the energy approach (b) Metal tests in conditions... 777 In a similar way, the relation between the energy We and the loading reversals to failure 2Nf is described with an equation We =Ke(2Nf) bw (2.3) where: Ke is the regression line constant, bw – exponent of the regression line. The sum of equations (2.2) and (2.3) enables one to define the total ener- gy Wt. Resultant equation (2.4) takes form similar to equation (1.1) Wt =We+Wp =Ke(2Nf) bw +Kp(2Nf) cw (2.4) In a bilogarythmic co-ordinate system, the Wp and We components of the total strain energy Wt described with equations (2.2) and (2.3) are straight lines. The fatigue graph described in the above way is schematically presented in Fig.1b. The values of constant coefficients and exponents occurring in the equ- ations are determined in tests conducted in conditions of a constant total strain energy Wt or plastic strain energy Wp. The testing conditions in this case are similar to the fatigue tests performed in conditions of controlled total or plastic strain, described in the ASTME606-04 [1] standard. 3. Description of verification tests 3.1. Fatigue testing procedures To realize fatigue tests in conditions of Wt = const, an original software (developed using Borland C++ Builder) for controlling the fatigue testing machine Instron (8000 series) through the digital controller Instron 8500 (8500 plus or 8800 also) was designed. The algorithm of the computer code, schematically presented in Fig.2, enables automatic change of loading parameters of the tested specimen in such away, that the ”measured” (Fig.1a) value Wt would remain during thewhole test on the same level, accepted by the operator. The level of strain energy determines the first loading cycle performed for the set value of the total strain amplitude or nominal stress amplitude. The change of loading parameters of the specimen occurs at any number of cycles, set by the operator. The change of the control signal is decided through the comparison of the base value Wtz with the current energy Wtm. During verifcation tests, as the control signal of the fatigue machine, the total strain measured with the use of an external extensometer was applied. 778 S. Mroziński, D. Boroński Fig. 2. The algorithm of a computer code used for fatigue tests in conditions of Wt = const The strain amplitude was corrected in most of the cases every second during the loading cycle by δε =0.005%. The value δε was selected in such a way to make changes of the energy due to variability of the control parameter Wt not higher than 1%. After starting the test and performing the first loading cycle with the amplitude εac (or σa in the case of selecting a force as the control signal of the machine), there follows determination of the current value of the total strain energy Wt according to the diagram shown in Fig.1a basing on relation (2.1). The strain energy Wt is calculated as a sum of areas of trapezoids described by the adjacent measurement points of the hysteresis loop (200 points per single loop), similar to numerical procedures described in Szala (1998). This value is assumed as the basis for the further part of the fatigue test. After obtaining results of measurements and calculations performed in the succeeding loading cycles and after checking the end of the test condition, a comparison of the ”base energy” Wtz and the current energy Wtm takes place. According to the comparison results, there follows either increase or decrease of εac (or σa) by addition or subtraction of the declared value of the strain increment δε (or stress increment δσ), with possible change of their value during the test. After realization of the succeeding cycle, the whole sequence of operations is repeated until one of the test criteria is fulfilled. Tests in conditions of a controlled value of the total strain energy were performed at five levels of Wt presented in Table 1. Their values at particular Metal tests in conditions... 779 levels were chosen in such a way that the obtained lives would include the whole low-cycle fatigue range. The loading frequency was 0.2Hz. Table 1. Levels of loading applied in fatigue tests No. of loading level Wt [MJ/m 3] 1 1.92 2 3.65 3 5.9 4 9.4 5 18.8 3.2. Specimen and material properties Specimens for the verification test were made of the aluminium alloy PA7 according to the standard (ASTME606-04, [1]). Geometry of the specimen is shown in Fig.3. A chemical constitution of the applied material is presented in Table 2 andmechanical properties – in Table 3. Fig. 3. Dimensions of the PA7 specimen Table 2.Chemical constitution of the aluminium alloy PA7 Cu [%] Mg [%] Mn [%] Al [%] Remaining [%] 3.95 1.476 0.54 93.690 0.342 Table 3.Mechanical parameters of the aluminium alloy PA7 σy (ReH) [MPa] σu (Rm) [MPa] Ru [MPa] E [MPa] A5 [%] Z [%] 321.7 514.7 632.7 72000 16 21 780 S. Mroziński, D. Boroński 4. Test results 4.1. The course of the stabilization process The course of the stabilization process of thePA7 aluminium in conditions of controlled total strain energy density was evaluated basing on analysis of chosen parameters of the hysteresis loop. Examples of changes of someof these parameters for the level Wt =5.8MJ/m 3 are presented in Fig.4. Fig. 4. Chosen parameters of the hysteresis loop in function of the number of loading cycles (Wt =5.8MJ/m 3): (a) hysteresis loop, (b) stress σa, σamax, σamin, (c) strain energy density Wt,Wp,We, (d) strain amplitude εac, εap, εae As it was expected, the course of changes of the energy Wt and Wp shows little pulsations resulting from the correction of the control signal by ±δε. Analysis of theobtainedgraphsallows one tonotice that for thepresented level of the total strain energy density, the testedmaterial showedminor hardening. The confirmation of this statement are the courses of changes of Wp (Fig.4c), total strain amplitudes εac and stress amplitudes εa. Metal tests in conditions... 781 4.2. Fatigue life graph The results obtained during fatigue tests performed for a constant value of total strain energydensity in the loading cycle allows one to determine a graph of fatigue life according to the description presented in Section 2. According to the accepted assumptions, changes of elastic and plastic strain energy were approximated with straight lines, whereas the total strain amplitude was cal- culated by summing up its two components. The test results were elaborated with the use of the least squares method and presented graphically in form of fatigue graphs in the W-2Nf co-ordinate system. Values of all determined coefficients and exponents of equation (2.5) were specified in Fig.5. Fig. 5. A fatigue graph of the PA7 alloy in the energy based approach Comparison analysis of the fatigue life graphs obtained by two described methods (i.e. Wt-2Nf for εac = const and Wt-2Nf for Wt = const) allows one to notice their differentiation dependent on the loading amplitude level (e.g. total strain amplitude level).Differences between the Wt energy are small for higher levels and grow together with level decreasing. Higher values of the fatigue life were obtained for the Wt = const approach in the whole range of Wt values. Differences between the fatigue life resulting from the chart presented in Fig.5 reach very high values for low levels of Wt. It will take a significant effect in the case of fatigue life calculations realised by methods based on the energy approach. Particular analysis of the influence of selected 782 S. Mroziński, D. Boroński methods for the fatigue properties determination on calculation results of the fatigue life will be carried out in further investigations. 5. Summary The results of verification tests confirmed the possibility of realization of low- cycle fatigue tests in conditions of controlled total strain energy density. The developedmethod for determination of fatigue properties in the energy appro- ach, with the procedures described inMroziński and Boroński (2006) extends the area of presently performed fatigue research in the range of low-cycle fa- tigue. The results from fatigue tests carried out in conditions of Wp = const (Mroziński and Boroński, 2006) and Wt = const enabled among others, de- termination of fatigue graphs in the energy approach, which, in consequence, may contribute to the improvement of effectiveness of fatigue life calculations of structural parts. References 1. ASTME606-04: Standard Practice for Strain-Controlled Fatigue Testing 2. BorońskiD.,CieszyńskiT., TopolińskiT.,AraszkiewiczM., 2006,Ba- dania zmęczenioweprzykontrolowanympoziomie energii dyssypacji.Koncepcja badań i wyniki badań,Proceedings of XXI Sympozjum Zmęczenia i Mechaniki Pękania, Bydgoszcz-Pieczyska,Wyd. Uczelniane ATR, 59-66 3. CoffinL.F., 1954,A studyof cyclic-thermal stresses in aductilemetal,ASME Transaction, 76, 931-950 4. Ellyin F., 1989, A criterion for fatigue under multiaxial fatigue failure, In: Biaxial and Multiaxial Fatigue, EGF3, K.J. Miller and M.W. Brown, Edit., MEP, London, 571-583 5. Feltner C.E., Morrow J.D., 1961,Microplastic strain hysteresis energy as a criterion for fatigue fracture, J. Basic Engineering ASSME, March, 15-22 6. Gołoś K., 1988, Plastic strain energy under cyclic multiaxial states of stress, Mechanika Teoretyczna i Stosowana, 26, 1, 171-177 7. Gołoś K., 1989, Trwałość zmęczeniowa stali w ujęciu energetycznym, Prace Naukowe,Mechanika, z.123, PolitechnikaWarszawska,Warszawa Metal tests in conditions... 783 8. GołośK.,EllyinF., 1988,Atotal strain energytheory for cumulative fatigue damage,TransactionASME, Journal of PressureVessel Technology,110, 35-41 9. Kaleta J., 1998,Doświadczalne podstawy formułowania energetycznych hipo- tez zmęczeniowych, OficynaWydawnicza PolitechnikiWrocławskiej 10. Kasprzyczak L., Macha E., 2006, Selection of the PID controller structure for control of stress, strain and energy parameter at the hydraulic fatigue test stand, Proceedings of the 2nd International Conference ”Mechatronic Systems and Materials MSM 2006”, Kraków,Opole University of Technology 11. Lin H., 1993, Multiaxial Plasticity and Fatigue Life Predictions of Anisotro- pic Al-6061-T6, PhDThesis,Mechanical EngineeringDepartmentNorthestern University, Boston,MA 12. Lin X., Haicheng G., 1998, Plastic energy dissipation model for lifetime prediction of zirconiumandzircaloy-4 fatiguedatRTand400C,J. Eng.Mater. Technol. ASME, 120, 114-118 13. Łagoda T., 2001, Energetyczne modele trwałości zmęczeniowej materiałów konstrukcyjnychwwarunkach jednoosiowych iwieloosiowych obciążeń losowych, Studia i monografie Politechniki Opolskiej, z.121, Opole 14. Mroziński S., 2006, Uwagi o wyznaczaniu charakterystyk zmęczeniowych w ujęciu energetycznym, Proceedings of XXI Sympozjum Zmęczenia i Mechaniki Pękania, Bydgoszcz-Pieczyska,Wyd. Uczelniane ATR, 281-286 15. Mroziński S., Boroński D., 2006, Badania niskocyklowe stali 45 w wa- runkach kontrolowanej energii odkształcenia, Proceedings of XXI Sympozjum Zmęczenia i Mechaniki Pękania, Bydgoszcz-Pieczyska,Wyd. Uczelniane ATR, 287-292 16. Mroziński S. Topoliński T., 1999, New energy model of fatigue damage accumulationand its verification for 45-steel,Journal ofTheoretical andApplied Mechanics, 37, 2, 223-240 17. Słowik J., Kasprzyczak L., Macha E., 2006, Układ sterowania maszy- ny wytrzymałościowejUFP 400 do wyznaczania energetycznej charakterystyki zmęczeniowejmateriałów,Proceedings of XXI Sympozjum Zmęczenia i Mecha- niki Pękania, Bydgoszcz-Pieczyska,Wyd. Uczelniane ATR, 383-390 18. Smith K.N., Watson P., Topper T.H., 1970, A stress-strain function for the fatigue of metals, J. Materials, 5, 767-776 19. Szala J., 1998,Hipotezy sumowania uszkodzeń zmęczeniowych, Wydawnictwa Uczelniane ATR 784 S. Mroziński, D. Boroński Badania metali w warunkach kontrolowanej gęstości energii odkształcenia Streszczenie W pracy przedstawiono nową koncepcję określania własności zmęczeniowych. Opracowanametoda badańbazuje na założeniachprowadzeniabadań zmęczeniowych sformułowanychdla opisu odkształceniowego.Parametr gęstości energii odkształcenia całkowitegoobliczanypodczas cykluobciążeniapróbki został zastosowanyjako sygnał sterujący próbą zmęczeniową. Opracowanametoda badań poszerza badania w zakre- sie zmęczenia niskocyklowego, a w przypadku jej zastosowania może przyczynić się do poprawy wyników obliczeń trwałości. Manuscript received November 13, 2006; accepted for print May 10, 2007