JOURNAL OF THEORETICAL AND APPLIED MECHANICS 43, 1, pp. 51-64, Warsaw 2005 EXPERIMENTAL METHOD OF DEFINING BIAXIAL FATIGUE PROPERTIES OF ELASTIC-PLASTIC CONSTRUCTION MATERIALS. PART 1 – MODEL FORMULATION Artur Cichański Janusz Sempruch Faculty of Mechanical Engineering, University of Technology and Agriculture in Bydgoszcz e-mail: artur.cichanski@atr.bydgoszcz.pl The paper presents formulation of a method of biaxial fatigue testing in conditions of cyclic uniaxial loadswith the use of specimensmade of ama- terial characterised by controlledmechanical properties. The origin for the paperwas a successful attempt to adapt the plastic potential for describing biaxial fatigue strength made by Troost and El.-Magd (1974) in relation to the ”stress anisotropy” effect, pointed out by him. On the basis of the carried-out reasoning, a model formulation of the allowable stress ampli- tude was introduced, which is a formalised record of the relations between the fatigue stress amplitudes and the yield stresses. The article suggests an experimental method of defining the appointed model parameters, which is a development of Szczepiński’s method (Szczepiński, 1963). Key words: biaxial fatigue, yield surface, testingmethods 1. Introduction The fatigue research is carried out in order to define the fatigue life or strength. Both the features may be assigned in tests treated as material or constructional ones. The material research is conducted on standardised spe- cimens and in standardised conditions [13,15]. The constructional research is realised onactual construction elements, e.g. Jachimowicz et al. (2002). One of the significant elements determining the research conditions is a time-variable load defined for each particular test separately [12]. The basic type of the time-variable load, defined as standard for the use of fatigue tests is a uniaxial 52 A.Cichański, J.Sempruch load, sinusoid-cycle variable with a defined value of the realised amplitude of stress, strain or load [13]. The cycle position in relation to zero value is defined by the mean level. There existbranch legal documents,whichdefinethe loadmodel corresponding to operational loads in the form of a spectrum (Berger et al., 2002). The spectrum describes the distribution of conventional variations of amplitude values and the mean level of cycles assigned during the courses of randomly changing operational loads (Szala and Kocańda, 1997). The above-mentioned considerations relate univocally to a uniaxial cyclic load of the analysed specimenor construction element [14]. Due to geometrical complexity of actual constructional elements and relations between geometri- cal forms and appearing loads occurring in them, leaving cyclic bi- or triaxial loads arising in the area ofweak construction elements out of account is frequ- ently too far-reaching simplification of the loading model (Bogdaniuk, 1998). The phenomenological relation (fatigue criterion) of the fatigue stress or strain state constituents, occurring in such a situation, allows one to assign the reduced value and towrite down the strength conditionwith the allowable value assigned during a uniaxial fatigue test taken into account. The amount of such relations, their limited range of application and inmost cases not satis- fying the verification level cause that this approach in the engineering practice has a simplified form and is only limited to the best-known hypothesis, e.g. Garud (1981). This approach does not take into account many characteristic features of the currently verified state. The alternative approach, i.e. assigning an appropriate material or con- structional featuresdirectly to experimental testing requires specialised testing machines realisingmulti-axial cyclic load anddefining a typical testingmetho- dology. Numerous research designs connected with the problem have not led to the standardisation of the testing methods (Cichański and Świtała, 2002). This paper has undertaken such an attempt, assuming that the suggested experimental testingmethod is to result indefining the constructionalmaterial sensitivity to the occurrence of a complex fatigue stress state. To describe the sensitivity, amodel formulation of the allowable stress amplitudewas derived. Another significant assumption was that the tests should be realised with the use of a standard strength machine. On the basis of bibliography studies and the authors’ own preliminary tests, a thesis has been formulated that it is possible to carry out fatigue research in a uniaxial load state, with the use of a specimen made of a material characterised by mechanical properties purposefully modified by direction-oriented plastic strain, in such a way that the resultswill one allow to conclude about the fatigue strength corresponding to the biaxial load state (Cichański and Sempruch, 2001). Experimental method.... Part 1 – Model formulation 53 2. Theoretical background Let us consider the distribution of fatigue biaxial stresses in an elementary area (Fig.1). In this area, the co-ordinate system 0x∗y∗ has been agreed in suchaway that its axis directions overlapwith thedirections of loadoperation. Thevalues σ∗x,σ ∗ y and τ ∗ xy arenominal stresses resulting fromthe applied load. Together with a change in the angle ϑ, indicating the planewhere the stresses are being considered, the rotation of the system 0xy takes place, accompanied by the rotation of related stresses σx, σy and τxy. Fig. 1. The fatigue stress distribution in the elementary area Oneof theways for realising biaxial fatigue is such anapplication of biaxial sinusoid loads that the nominal stresses σ∗x, σ ∗ y and τ ∗ xy resulting from them may be expressed bymeans of the following dependences σ ∗ x(t)=σ ∗ xm+σ ∗ xa sin(ωxt) σ ∗ y(t)=σ ∗ ym+σ ∗ ya sin(ωyt−φy) (2.1) τ ∗ xy(t)= τ ∗ xym+ τ ∗ xya sin(ωxyt−φxy) In a general notation, the components of the stress state can be put down as σi(t), where the index i∈ (x,y,xy) describes the direction and character of a given component σi(t)=σim+σia sin(ωit−φi) (2.2) 54 A.Cichański, J.Sempruch The thus defined biaxial cyclic stresses are called proportional when for all i directions themean level σim and the phase shift φi equal zero, and the frequencies ωi are the same. The biaxial stresses are called non-proportional when the above-mentioned condition is not fulfilled. In the case when stress non-proportionality occurs, the fatigue strength decreases. The fatigue strength in the i direction, in conditions of the mean level different from zero, is described by the allowable stress amplitude σiA. If the realised amplitude σia exceeds the value σiA depending on σim, fatigue damage occurs on the plane ϑkr, called the critical plane. Bibliography gives a number of suggestions onhow towrite down the relation of σiA to σim (Troost andEl.-Magd, 1981). One of the commonly approved approaches assumes that the value of the allowable stress amplitude σxA, corresponding to Zrc, changes together with the angle ϑ as it is described below σxA(ϑ)=Zrc− 2Zrc−Zrj Zrj σxm(ϑ) (2.3) σxm(ϑ)= (1+cos2ϑ) σ∗xm 2 +(1− cos2ϑ) σ∗ym 2 − τ ∗ xym sin2ϑ A diagram presenting dependences of the allowable amplitudes σxA on the angle ϑ for different cases, where a non-zero mean cycle level occurs, is presented in Fig.2. Fig. 2. Dependence of σxA on the angle ϑ The picture of the allowable amplitudes distribution in polar coordinates for loads showing the mean level equal to zero is a circle. This indicates the Experimental method.... Part 1 – Model formulation 55 independence of σxA of the angle ϑ. In the case of introducing non-zeromean levels, the circle takes an oval-like shape (Fig.2). The analysis of this stress state leads to definition of the angles, for which σxA achieves extreme values. This indicates the dependence of the allowable amplitudes distribution on the direction described by the angle ϑ. This phenomenon was described by Troost as anisotropic fatigue behaviour of an isotropic material (Troost and El.-Magd, 1974), and the QVH hypothesis was introduced for the description of the fatigue strength in such conditions. The plasticity potential formulated to describe plasticity conditions of the anisotropic material (Troost and El.- Magd, 1981) was the basis of the QVH hypothesis. One of the ways of causing anisotropy of material mechanical properties, among other fatigue properties, is initiating plastic changes in the material. Plastic changes cause modification of material characteristics, including the yield surface. Thin-walled tubular specimens are widely used for experimental material yield surface testing in the initial state and after preliminary plastic prestrain. A unique method of such tests with the use of flat specimens was introduced by Szczepiński (1963). This method is schematically presented in Fig.3. Fig. 3. Methods for preliminary and secondary specimen loading (Szczepiński, 1963) The initial loading P , causing plastic strains of a predefined value, is ap- plied to large thin-plates made of a material in the initial state. Small spe- cimens are cut out of thus prepared large plates with a certain angle with respect to the initial loading operation direction. Under conditions of secon- dary loads F , the tested plastic properties are defined using these specimens. 3. Experimental procedure To fulfil the aim of the work it was necessary to formulate such amethod of conducting the experimental tests, whichwould take into account the possi- 56 A.Cichański, J.Sempruch bility of controlling the plastic strain of the tested material. Owing to this, a specificmethod of specimen preparation, which is a development of Szczepiń- ski’s method (Szczepiński, 1963), was proposed. Thematerial in the form of a thin-plate is prestrained by forces Pprst in a specially designed grip (Fig.4). Fig. 4. The specimen acquisitionmethod The forces Pprst have such values so as to cause plastic strains εpl in the plate central part. Flat dumbbell specimens for further research are cut out of the strained plates at the angle α (Fig.4). These specimens are tested in the conditions of uniaxialmonotonic loads Fmono with the aim of thematerial plastic properties assignment. Material fatigue properties are defined under the conditions of cyclic uniaxial loads Fcyc. The load Pprst is transmitted from the grip onto the stretched plate thro- ugh a series of five pins placed in specially prepared holes. The arrangement of holes is expected to affect the shape of plastic strain fields occurring in the plate. The stipulated area, where the gradients εpl have negligible influence on the repeatability of properties of the test specimens, is presented in Fig.4 with a dotted line. 4. Model for determination of allowable stress amplitude 4.1. Assumptions Correspondingly to the agreed experimental testing method, the material is used in the initial state and after a preliminary plastic prestrain. For the Experimental method.... Part 1 – Model formulation 57 material in the initial state, further called the virgin material, an assumption on themechanical properties isotropy wasmade. The yield surface of thema- terial is described by the Huber-Mises-Hencky (H-M-H) condition. The virgin material fatigue strength is proportional to the yield stresses. An assump- tion about kinematic hardening was made for describing the yield surface of the preliminarily plastically prestrained material, further called the prestra- ined material. For this material, an assumption that the initial plastic strain caused a decrease in the fatigue strength was made. The considerations we- re limited to plane stress, which appeared on free surfaces, where the main changes deciding about the fatigue strength occured. 4.2. Description of plastic properties The following dependence presents the yield H-M-H condition, expressed bymeans of nominal stresses in the approach typical for a plane stress σ2x+σ 2 y −σxσy +3τ 2 xy =σ 2 pl (4.1) The yield condition has its graphic representation in the form of a yield surface situated in the stress space described with the co-ordinate system 0σxσyτxy. An ellipsoid in Fig.5 depicts relationship (4.1). The ellipsoid is si- tuated in such a way that one of its axis overlaps with the axis 0τxy, and the other two constitute bisectors between the axis 0σx and 0σy. Fig. 5. Huber-Mises-Hencky ellipsoid On the H-M-H ellipsoid, a number of ellipses (Fig.5) may be seen, which results from surface intersection with planes describing different stress states 58 A.Cichański, J.Sempruch realised in the course of experimental testswith theuse of specimens of various geometrical shapes. For further consideration, the ellipse BEC was chosen, createdbyH-M-Hellipsoid intersectionwithaplanedescribedby the following dependence σx+σy =σpl (4.2) Analysis of the stress state meeting the conditions described by (4.1) and (4.2) leads to the conclusion that one of the main stresses in this state must be equal to zero. The ellipse BEC is a geometrical picture of the appointed stress state and its points correspond to yield stresses assigned to uniaxial load conditions for the virgin material. Coordinates of the ellipses BEC in the stress space are described by means of transforming dependences (4.3), where the angle ϕ defines the direction for appointing εpl in the agreed co- ordinate system σx = 1 2 σpl(1+cos2ϕ) σy = 1 2 σpl(1− cos2ϕ) (4.3) τxy = 1 2 σpl sin2ϕ Ellipses BEC, seen in the direction D0, according to the system inFig.5, are presented in Fig.6 by means of a line with the square marker. Fig. 6. Ellipses BEC and B′E′C′ – D0 direction view (assumption) The plastic strain causes modification of the yield surface. According to the assumptionabout the kinematic nature ofmaterial hardening, the ellipsoid BEC moves in the direction defined by the initial plastic load. What depicts it is the change of the ellipses BEC corresoinding to the virginmaterial. The new ellipse marked as B′E′C′ corresoinding to the prestrained material, is Experimental method.... Part 1 – Model formulation 59 presented inFig.6 bymeans of a linewith the trianglemarker.Theco-ordinate system 0σxσyτxy was selected in such a way that the ellipse B′E′C′ moved in the direction 0σx, which corresponds to the angle ϕ=0◦. Changes in ellipses BEC and B′E′C′ seen in Fig.6, indicate an increase in yield stresses in the direction of ϕ=0◦ (point B′) and a decrease in yield stresses in the direction of ϕ=90◦ (point C′). 4.3. Description of fatigue properties For most materials, the fatigue strength σiA is proportional to the yield stress σpl, as it is symbolically presented by the relation σiA def = aNσpl (4.4) The proportionality coefficient aN, appearing in dependence (4.4), is de- fined on the basis of tests on the virgin material as a relation of appropriate stresses, as follows aN = σizoA σizo pl (4.5) On the basis of results of tests regarding plastic properties of thematerial related to the angle ϕ aswell as equation (4.4) and the coefficient aN, theore- tical values of the allowable stress amplitudes σA can be assigned. The values, after transformation by means of relation (4.3), create a curve in the space 0σxaσyaτxya, called a fatigue ellipse, which is shown andmarked bec (Fig.7). Fig. 7. Ellipses bec for different fatigue life – D0 direction view (assumption) Thetheoretical values of allowable stress amplitudesdependon thenumber of cycles N, assumed in the definition of the coefficient aN. The greater is 60 A.Cichański, J.Sempruch the cycle number N, the lower remains the value of the appointed amplitude, which is depicted by lessening of the ellipses bec (Fig.7). Throughequation (4.4), the fatigue ellipse bec is connectedwith the ellipse BEC, assigned for thevirginmaterial. For theprestrainedmaterial, theplastic ellipse takes the shape of B′E′C′ (Fig.6), which has the corresponding fatigue ellipse b′e′c′ presented in Fig.8. Fig. 8. Ellipses b′e′c′ for different fatigue life – D0 direction view (assumption) 4.4. Allowable stress amplitude model approach formulation Plastic strain causes that the material is given a preferential orientation. This orientation is assumed to be described with an angle α measured with respect to direction of the initial load operation (Fig.4). The thesis about the possibility of using thematerial with controlled mechanical properties for biaxial fatiguemodelling is expressed by the postulate about the angles ϕ and α equality. According to the hypothesis of kinematic hardening in the direction of plastic load operation, yield stresses become higher. Due to the fact that the fatigue ellipse b′e′c′ is determined on the basis of the plastic ellipse B′E′C′, also quantities constituting the ellipse b′e′c′ grow in the direction of plastic load operation. One of the assumptions for the model is a claim that the initial plastic prestrain causes a decrease of the fatigue strength. The tests results indicate that this assumption is valid (Ingerma and Ranna, 1974). To assure the conformity of behaviour of the fatigue ellipses with experimental results, σM is introduced, which is amodel formulation of the allowable stress amplitude (Cichański and Sempruch, 2001). The amplitude σM is a quantity to be compared with the amplitudes creating the fatigue ellipses b′e′c′. Experimental method.... Part 1 – Model formulation 61 The model formulation of the allowable stress amplitude σM (4.6)1 con- stitutes the base part σMb (4.6)2 and the modifying part σMo (4.6)3 of the stress. The base part is independent of the direction α, while the modifying part changes together with alteration of α σM =σMb+σMo σMb =σ izo A (4.6) σMo = 1 aN (σizoA −σ anizo A ) where: aN – model scaling coefficient σizoA – allowable stress amplitude for the virgin material σanizoA – allowable stress amplitude for the prestrainedmaterial σM – allowable stress amplitude in the model formulation σMb – base model stresses σMo – modifying model stresses. The values of σizoA and σ anizo A are defined on the basis of Wöhler’s li- nes, assigned for the virgin and prestrainedmaterials. To describe the fatigue properties, three characteristic directions α = 0◦, 45◦ and 90◦ were selec- ted. Wöhler’s lines for the virgin material, according to the assumption on the isotropy of mechanical properties, should have a similar course for three characteristic directions (Fig.9a). Fig. 9.Wöhler’s lines: (a) virgin material; (b) prestrainedmaterial (assumption) There exist fatigue properties variability together with the angle α for the prestrainedmaterial. According to the assumption that the initial plastic strain causes the decrease of fatigue strength, Wöhler’s line determined in 62 A.Cichański, J.Sempruch the direction of the plastic load operation (for α = 0◦) should be placed in the lowest position (Fig.9b). The influence of the initial prestrains on the fatigue strength should descend together with deviation from the direction of plastic load operation. It should be reflected by higher and higher positions of Wöhler’s lines for rising values of the angle α (Fig.9b). On the basis of Wöhler’s diagram for the virgin material (Fig.9a), the values of σizoA are defined. On the basis of a diagram corresponding to the considered angle α for the prestrainedmaterial (Fig.9b), the values of σanizoA are defined. The amplitudes read on suitable Wöhler’s diagrams, for the ap- plied strength N, are introduced in equations (4.6). For the virgin material σanizoA = σ izo A , which causes that for this material the modifying part equals zero. The thus determined values of σM and dependent on the cycle num- ber N, after transformation by means of (4.3), lead to generation of ellipses bec and b′e′c′. In the variability interval of the N-th cycle, the limited cycle number Nlim can be defined for which the value σM, found from the fatigue research, is comparable with the value σA, found from the rescaling of the plastic properties. For this case, equation (4.4) describes the relation between the ellipses BEC and bec for the virgin material, and B′E′C′ and b′e′c′ for the prestrainedmaterial. 5. Conclusions Thepaperpresents an idea of amethodof investigating biaxialmaterial fa- tigue properties.The characteristic of thematerial, called amodel formulation of the allowable stress amplitude, corresponds to the proposed testingmethod. A graphic representation of this characteristic is a fatigue ellipse, which is a curve lying in the stress space 0σxσyτxy and describing a flat fatigue stress state. The testing method formulated in the paper and resulting material cha- racteristics are an attempt to describe the constructional material sensitivity to a complex fatigue stress state. Using the suggested testingmethod and the model, such tests can be carried out on specimensmade of the plastically pre- strained material in conditions of cyclic simple loads. The determined tests results will allow one to conclude about the biaxial material fatigue strength. The presented notation of the model formulation of the allowable stress amplitude is characterised by two parameters. The obvious parameter of the model is the relation of the allowable stress amplitude to yield stresses found Experimental method.... Part 1 – Model formulation 63 for the virginmaterial. Anon-evidentmodel parameter is thenumber of cycles suitable for defining the allowable stress amplitude. To estimate the correctness of themodel formulation of the allowable stress amplitude, basic experimental tests need to be made with the use of the sug- gested method. References 1. Berger C., Eulitz K.-G., Heuler P., Kotte K.-L., Naundorf H., Schuetz W., Sonsino C.M.,Wimmer A., Zenner H., 2002, Betriebfestig- keit in Germany – an overview, International Journal of Fatigue, 24, 603-625 2. Bogdaniuk M., 1998, Kryterium wytrzymałości zmęczeniowej we współcze- snych przepisachklasyfikacji i budowy statkówmorskich,Materiały XVII Sym- pozjum Zmęczenia Materiałów i Konstrukcji, Bydgoszcz-Pieczyska, 33-38 3. Cichański A., Sempruch J., 2001, Modelling of fatigue plane stress, Proce- edings of the 6th International Conference on Biaxial/Mulitaxial Fatigue and Fracture, Lisbona, Portugal, 1, 223-230 4. Cichański A., Świtała A., 2002, Przegląd doświadczalnychmetod badania dwuosiowego zmęczenia, Zeszyty Naukowe ATR, Bydgoszcz, 5-16 5. GarudY.S., 1981,Multiaxial fatigue: a survey of the state of the art, J. Test. Eval., 9, 165-178 6. Ingerma A., Rannat E., 1974,Wpływmałych odkształceń plastycznych na granicę wytrzymałości zmęczeniowej metali, Czas. Techn. Mechanika, Wyd. Politechniki Krakowskiej, Zeszyt 10 (181), Kraków, 1-6 7. Jachimowicz J., Kajka R., Karliński W., Kowalski W., 2002, Analiza przyczyn zmęczeniowego zniszczenia elementu podwozia samolotu, Materiały XIX Sympozjum Zmęczenia Materiałów i Konstrukcji, Bydgoszcz-Pieczyska, 141-148 8. SzczepińskiW., 1963,On the effect of plastic deformation on yield condition, Archiwum Mechaniki Stosowanej, 15, 2, 275-296 9. Szala J., Kocańda S., 1997,Podstawy obliczeń zmęczeniowych, PWN,War- szawa 10. Troost A., El.-Magd E., 1974, Anisotropes Ermüdungsverhalten isotro- per metallischerWerk-stoffe,Metall-Internationale Zeitschrift für Technik und Wirtschaft, 28, 1, 49-55 11. Troost A., El.-Magd E., 1981, Schwingfestigkeit bei mehrachsiger Bean- spruchung ohne undmit Phasenverschiebung,Konstruktion, 33, 297-304 64 A.Cichański, J.Sempruch 12. PN-74/H-04327, Badanie metali na zmęczenie. Próba osiowego rozciągania – ściskania przy stałym cyklu obciążeń zewnętrznych 13. PN-76/H-04325, Badanie metali na zmęczenie. Pojęcia podstawowe i ogólne wytyczne przygotowania próbek oraz przeprowadzenia prób 14. PN-76/H-04326, Badanie metali na zmęczenie. Próba zginania 15. PN-EN10002-1+AC1:1998,Metale.Próbarozciągania.Metodabadaniawtem- peraturze otoczenia Doświadczalna metoda wyznaczania dwuosiowych własności zmęczeniowych sprężysto-plastycznych materiałów konstrukcyjnych. Część 1 – Sformułowanie modelu Streszczenie W artykule przedstawiono sformułowanie metody badania dwuosiowego zmęcze- niawwarunkachcyklicznychobciążeń jednoosiowychzwykorzystaniempróbekwyko- nanychzmateriałuokontrolowanychwłasnościachmechanicznych. Jakogenezę pracy wskazanoudaną próbę adaptacji potencjału plastycznego do opisu dwuosiowego zmę- czenia wykonaną przez Troosta Troost i El.-Magd, 1974) w odniesieniu do wskaza- nego przez niego efektu „anizotropii naprężeniowej”. Na podstawie przeprowadzone- go rozumowania wyprowadzono modelowe ujęcie dopuszczalnej amplitudy naprężeń będące sformalizowanym zapisem zależności amplitudy naprężeń zmęczeniowych od naprężeń uplastyczniających.Wpracy zaproponowanodoświadczalnąmetodęwyzna- czania parametrówwskazanegomodelu, będącą rozwinięciemmetody Szczepińskiego (Szczepiński, 1963). Manuscript received June 25, 2004; accepted for print October 18, 2004