Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 1, pp. 161-175, Warsaw 2009 APPLICATION OF THE STRAIN ENERGY DENSITY PARAMETER FOR ESTIMATION OF MULTIAXIAL FATIGUE LIFE OF SINTERED STEELS WITH STRESS CONCENTRATORS Tadeusz Łagoda Opole University of Technology, Opole, Poland e-mail: t.lagoda@po.opole.pl Cetin M. Sonsino LBF Darmstadt, Germany Paweł Ogonowski Nutricia, Opole, Poland This paper presents fatigue test results of a Fe-Cu sintered alloy (1.5%Cu) and three types of Fe-Cu-Ni sintered alloys (2% Cu, 2.5% Ni). Fe-Cu-Ni sintered alloyswere produced under different compaction pressures and sin- tering temperatures. Round specimens with fillet notches were subjected to a constant amplitude pure bending, pure torsion and a combined in-phase and out-of-phase (δ = 90◦) bending with torsion. All results have been described by the criterion of the strain energy density parameter on the critical plane. It was assumed that the fatigue life is influenced by a linear combination of normal and strain energy density parameters with the co- efficients that refer to the calculated plane. In particular, it allowed one to compute fatigue lives sufficiently comparable to the experimental ones. Key words: energymodels, cyclic loading, fatigue lifetime, sintered steel Notations bk – fatigue strength exponent for notched elements C – coefficient determining values of stresses in circumferential direction depending on stress concentration E – Young’s modulus i,j,k – unit vectors in Cartesian coordinate system 162 T. Łagoda et al. Kt – theoretical stress concentration factor l̂η,m̂η, n̂η – direction cosines of unit vector η l̂s,m̂s, n̂s – direction cosines of unit vector s Nf – number of cycles to failure Re0.2 – yield stress Rm – ultimate tensile strength sgn[i,j] – binary logical function sgnwith variables i,j t – time W – strain energy density parameter x,y,z – spatial coordinates β,κ – coefficients indicating criterion form, obtained fromuniaxial fatigue tests ε – strain ν – Poisson’s ratio γ – engineering shear strain σ – normal stress σ′fk – fatigue strength coefficient for notched elements τ – shear stress Subscripts: a – amplitude b – bending eq – equivalent s – shear t – torsion η – normal 1. Introduction Notch problems occur in the case of component section discontinuity, which causes local stress concentrations that are higher than the nominal calculated ones.Their quantity depends,amongothers, on thenotch geometry, properties of thematerial, and the loading path.The notch occurrence leads to reduction in fatigue strength of a material. Thus, an efficient method for fatigue life determination under stress concentration should be searched for. For fatigue life determination under stress concentration, it is necessary to determine local stresses and elastic-plastic strains in the notch root at first. In the case of cyclic loading, theNeubermethod (Neuber, 1961), or the strain energydensitymethod (SEDM) (Molski andGlinka, 1981) are usually applied. Application of the strain energy density paramater... 163 Both methods have been generalized to the multiaxial loading state (Mofta- khar et al., 1995). Thesemethodsdiffer only in the procedure of determination of the plastic strain energy density. The values of strains and stresses in the notch root obtained according to the Neuber method are overestimated, and those obtained according to SEDMare lowered as compared with the true re- sults. The calculated results included between those obtained according to the Neuber andMolski-Glinkamodels are presented in Łagoda andMacha (1998). In the literature, one can also found other energymodels formulated by Inoue et al. (1996) andYe et al. (2004), and the empiricalmodel proposedbySonsino (1993). Determination of local stresses and strains at the notch root requires also application of the Hencky constitutive equations and complex numerical calculations. Under a non-proportional loading, the calculations become even more complicated, because the chosen method should be expressed in an in- cremental notation, as the loading path influencemust be taken into account. Moreover, a plasticitymodel (for example, themodel proposedbyMróz (1967) or Chu (1984)) has to be applied in order to determine the relation between stresses and plastic strains. Thus, pseudoelastic stresses and strains are often used for fatigue calculations. In Sumsel (2004), a criterion assuming the ela- stic stress state near the notch was presented. Fatigue life was determined on the basis of normal and shear stresses on the plane of the maximum normal stresses. Also the criterion proposed in Grubisic and Simbürger (1976) for ductile steels was based on elastic stresses. In that case, the maximum shear stress was assumed to be as the equivalent stress amplitude. In Łagoda and Ogonowski (2005), the criterion of strain energy density parameter on the critical plane was proposed. This criterion is based on pseu- doelastic stresses and strains and it is valid under stress concentration. It was verified for 42CrMo4V steel. Next, similar criteria performed in stress and strain notations were presented (Ogonowski et al., 2004). Moreover, the effi- ciency of particular criteriawas analysed according to the test results obtained for 10HNAP steel. From the analysis, it appears that the energy notation that includes variation of both strains and stresses gives calculated fatigue lives closest to the experimental ones. Theaim of this paper is to verify the energy criterion for Fe-Cu andFe-Cu- Ni sintered steels with stress concentrators, using the test results obtained by Sonsino (1983), Sonsino and Grubisic (1989). In Sonsino (1983), Sonsino and Grubisic (1989), the criterion of themaximumnormal elastic stress amplitude, which occurs in the critical plane, is applied. 164 T. Łagoda et al. 2. Application of the strain energy density parameter in the presence of stress concentration in a component 2.1. The case of uniaxial loading Under a uniaxial loading, the strain energy density parameter (SEDP) (Grubisic and Simbürger, 1976) is defined as Wxx(t)= 1 2 σxx(t)εxx(t)sgn[σxx(t),εxx(t)] (2.1) where sgn[i,j] = sgn[i]+ sgn[j] 2 (2.2) For a constant amplitude loading, we have the following amplitude of SEDP Wxx,a = 1 2 σxx,aεxx,a (2.3) Let us note (Kluger et al., 2007) that the strain energy density parameter is of a vectorial character, whereas the strain energy density applied in mechanics is a scalar. When stress concentration occurs, stress histories in particular directions can be expressed for axial stresses (under bending) σxx(t)=Ktbσxx,η(t) (2.4) and for circumferential stresses σyy(t)=Cσxx(t) (2.5) where 0¬ C ¬ ν. Changes of the coefficient C depending on Kt factor are shown in Fig.1. Its approximate value, found from numerical calculation and extrapolated, can be determined from C = 1.84ν Kt (Kt−1) 1−ν (2.6) The relation given by Eq. (2.6) was based on calculations with the use of the finite element method for different notch geometries, Kt = Kta for axial loading and Kt =Ktb for bending. At the bottom of the notch root (surface), Application of the strain energy density paramater... 165 Fig. 1. Changes of coefficient C values against stress concentration factor Kt the radial stresses equal zero (σzz(t) = 0). Thus, assuming an elastic body model, the following strain histories are obtained εxx(t)= (1−Cν)Ktb σxx,η(t) E εyy(t)= (C−ν)Ktb σxx,η(t) E (2.7) εzz(t)=−(C+ν)Ktb σxx,η(t) E Taking into consideration Eqs (2.3), (2.4) and (2.7)1, the following formula is valid for the elastic material Wxx,a = σ2xx,aη 2E (1−νC)= σ2xx,aηK 2 tb 2E (1−νC) (2.8) Using Eq. (2.8), we can propose a new fatigue characteristic S-N that is based on the strain energy density parameter and involves the stress concen- tration field: — under control of stress or strain energy density parameter (Będkowski et al., 2004) logNf =Aw−mw logWxx,a (2.9) where AW and mW are coefficients determining the fatigue curve Wa-Nf, like for the standard Basquin curve σa-Nf —or under strain control Wa = σ ′2 fkK 2 tb 2E (1−νC)(2Nf) 2bk (2.10) 166 T. Łagoda et al. 2.2. The case of multiaxial loading As in the case of criteria for smooth specimens or elements, it is assumed that fatigue damage is described by a linear combination of the parameters of normal and shear strain energy density parameters (Wn and Wηs(t), re- spectively) on the critical plane. In Łagoda andOgonowski (2005), two forms of the criterion dependent on the assumed critical plane (Fig.2) have been presented. The calculations carried out for many materials have proved that the criterion based on the plane of maximum parameter of the normal strain energy density gives good results for cast irons only. For other materials, it is better to assume the critical plane where the shear strain energy density parameter reaches its maximum value. Thus, for calculations related to the discussed alloys, the following fatigue criterion on the plane of the maximum shear strain energy density parameter was assumed: βmax t {Wηs(t)}+κWη(t)= f(Nf) (2.11) Fig. 2. Interpretation of the critical plane The energy density parameters of normal and shear strains are determined as follows Wη(t)= 1 2 ση(t)εη(t)sgn[ση(t),εη(t)] (2.12) Wηs(t)= 1 2 τηs(t)εηs(t)sgn[τηs(t),εηs(t)] The weight coefficients β and κ define contribution of the normal and shear strain energy density parameters during fatigue life determination. For Application of the strain energy density paramater... 167 a combined bending and torsion as well as stress concentration, the normal ση(t) and shear τηs(t) stresses on the critical plane are defined as ση(t)= l̂ 2 ησxx(t)+ m̂ 2 ησyy(t)+2l̂ηm̂ησxy(t) (2.13) τηs(t)= l̂ηl̂sσxx(t)+m̂ηm̂sσyy(t)+(l̂ηm̂s+ l̂sm̂η)σxy(t) The normal εη(t) and shear strains γηs(t) being in the relation εηs(t)= 1 2 γηs(t) (2.14) can be expressed as εη(t)= l̂ 2 ηεxx(t)+ m̂ 2 ηεyy(t)+ n̂ 2 ηεzz(t)+2l̂ηm̂ηεxy(t) (2.15) εηs(t)= l̂ηl̂sεxx(t)+m̂ηm̂sεyy(t)+ n̂ηn̂sεzz(t)+(l̂ηm̂s+ l̂sm̂η)εxy(t) where σij(t), εij(t) are the components of stress and strain state tensors at the notch root, respectively. For the plane stress state, the direction cosines l̂η, m̂η, l̂s, m̂s of the vectors η̂ and ŝ referingto thenormal strain (Eqs. (2.13)1 and(2.15)1) andshear strain energy density parameter (Egs. (2.13)2 and (2.15)2 are defined by one angle α in the following relationships l̂η =cosα m̂η =sinα l̂s =−sinα m̂s =cosα n̂η = n̂s =0 (2.16) where α is the angle determining the critical plane orientation. After introducing β= k(1−νC) 1+ν (2.17) for the coefficient β and κ, according Łagoda and Ogonowski (2005), as in the following κ= [4−k(1−C)2](1−νC) (1−ν)(1+C)2 (2.18) for a given moment t, Eq. (2.10) can be written as k(1−νc) 1+ν (Wa,ηs)max+ [4−k(1−C)2](1−νC) (1−ν)(1+C)2 Wa,η = (σ′fk) 2 2E (2Nf) 2bk (2.19) 168 T. Łagoda et al. In a general case, the weight coefficient k is defined with respect to a number of cycles to failure as k= k(Nf)= σ2xx,a(Nf) τ2xy,a(Nf) (2.20) where σxx,a(Nf) – amplitude of local normal stress on the S-N curve for bending τxy,a(Nf) – amplitude of local shear stress on the S-N curve for tor- sion. Relationships (2.19) and (2.20) are the final form of the failure criterion under stress concentration on the plane of maximum shear strain energy den- sity. The right-hand side of relationship (2.19) expresses the equation of the S-N curve for tension (alternating bending) rescaled to energy notation and written in an exponential form. Thus, themultiaxial stress state is reduced to an equivalent uniaxial one. When the S-N fatigue curves for bending and torsion are parallel, thenEq. (2.20) can be written as k= σaf τaf = const (2.21) 3. Experimental verification 3.1. The experiments The criterion of strain energy density parameterwas verified forFe-Cu and three Fe-Cu-Ni sintered steels; the verification was based on the tests made by Sonsino (1983), Sonsino andGrubisic (1989). Mechanical properties of the considered materials are presented in Table 1. Various pressing pressures and temperatures of sintering as well as three types of gas mixtures were applied during production of the sintered alloys. Water atomized iron powder WPL 200 (base) was mixed with copper and nickel powders, and the whole mixture was then sintered (Table 2). Roundspecimenswithfillet notches as stress concentratorswere tested (see Fig.3). The theoretical notch factor for the given geometry was Ktb = 1.49 for bending, and Ktt =1.24 for torsion. The assumed coefficient determining the values of circumferential stress according to Eq. (2.6) was C = 0.215 (Sonsino andGrubisic, 1989). A graphical interpretation and amethod for the Application of the strain energy density paramater... 169 Table 1.Mechanical properties of the considered sinters Material ρ [g/cm3] E [GPa] Rp0.2 [MPa] Rm [MPa] A5 [%] Z [%]Tension/Compression Fe-Cu 7.4 167 290/312 371 13 16 Fe-Cu-Ni(1) 7.1 149 270/312 358 10 11 Fe-Cu-Ni(2) 7.4 155 247/293 401 12 14 Fe-Cu-Ni(3) 7.4 165 299/325 399 10 14 (1) T1 =1280 ◦C (2) T1 =900 ◦C, T2 =1120 ◦C (3) T1 =900 ◦C, T2 =1280 ◦C A5 – elongation Z – reduction of area Table 2.Composition andmanufacturingparameters of the considered sinters Material Compositions p1 [MPa] T1 [ ◦C] p2 [MPa] T2 [ ◦C] Fe-Cu WPL 200 550 900 500 1280 +1.5%Cu atmosphere 1 atmosphere 3 Fe-Cu- Ni(1) WPL 200 +2%Cu +2.5%Ni 460 1280 – – atmosphere 3 Fe-Cu- Ni(2) 500 900 550 1120 atmosphere 1 atmosphere 2 Fe-Cu- Ni(3) 500 900 550 1280 atmosphere 1 atmosphere 3 Atmosphere 1: 36% H2, 19% CO, 0.4% CO2, 45% N2 Atmosphere 2: 75% H2, 25% N2 Atmosphere 3: 70% N2, 30% atmosphere 2 Fig. 3. Geometry of the round specimen determination of the approximate value of the coefficient C were presented in Ogonowski et al. (2004). 170 T. Łagoda et al. Fatigue tests were conducted under a constant-amplitude pure bending, pure torsion, and combinations of proportional (in-phase, δ = 0◦) and non- proportional (out-of-phase, δ = 90◦) bending with torsion. For a combined loading, the selected ratio of the nominal shear stress amplitude and the no- minal bending amplitude τan/σan was 0.58. The tests were performed under control of bending and torsional moments at the frequency 18.5Hz. For the consideredmaterials, themultiaxial stress statewas substituted by the equivalent stress state corresponding to a pure alternating bending. The S-N curves for the alternating equivalent bending and the obtained fatigue strengths at the knee point of the curves σak for the tested alloys are shown in Fig.4. Fromcomparison of the data inTable 1 andTable 2, it appears that the sintering parameters strongly influence mechanical properties of the obtained materials. As the yield strength Rp0.2 and the ultimate tensile strength Rm increase, the fatigue life increases, too.Thus, the sintering parameters strongly influence the fatigue life (seealsoFig.4).Under loadswithout-of-phase loading (90◦), the obtained amplitudes for a given life level were greater than those obtained under the in-phase loading, i.e. the out-of-phase loading increases the fatigue life of the investigated alloys in contrast to the in-phase loading. This remark concerns all the sintered alloys considered in the paper. Fig. 4. S-N curves for alternating bending 3.2. Steps of the fatigue lifetime calculation procedure The determination of fatigue life for known theoretical notch coefficients Ktb and Ktt as well as coefficient C according to the proposed criterion inc- ludes: Application of the strain energy density paramater... 171 • determination of the histories of components of the stress tensor σij(t) and the strain tensor εij(t), including the influence of stress concentra- tion, • determination of the angle of critical plane position α, • determination of histories of stresses, strains andnormal and shear strain energy density parameters on the critical plane, • determination of the equivalent history of the strain energy density pa- rameter on the assumed plane, • iterative computations of the fatigue life Ncal =Nf calculated according to Eq. (2.17) including variations of the coefficient k (2.18). The calculated fatigue lives Ncal obtained according to fatigue criterion (2.17) were compared with the experimental lives Nexp (see Figs. 5-8). Fig. 5. Comparison of fatigue life Ncal calculated according to Eq. (2.17) and experimental fatigue life Nexp for Fe-Cu sintered steel Correlation between the calculated and experimental fatigue lives seems to be satisfactory. They are included into a scatter band with the factor 3, like for the alternating bending. Only some results for the non-proportional combination of the Fe-Cu-Ni(3) alloy are located outside that band, but at the so-called safe side. The taking into account of elastic-plastic stresses and strains at the notch root could give better correlation between Ncal and Nexp. However, in such a case, calculations of the fatigue life would bemuchmore complicated. 172 T. Łagoda et al. Fig. 6. Comparison of fatigue life Ncal calculated according to Eq. (2.17) and experimental fatigue life Nexp for Fe-Cu-Ni (1) sintered steel Fig. 7. Comparison of fatigue life Ncal calculated according to Eq. (2.17) and experimental fatigue life Nexp for Fe-Cu-Ni (2) sintered steel 4. Conclusions The calculated fatigue lives correlate well with the experimental data. The results were found to be included within a scatter band with the factor 3 (Ncal/Nexp = 3 or Ncal/Nexp = 1/3), as in the case of alternating bending. Only someresults for thenon-proportional combinationofFe-Cu-Ni(3) sintered steels are located outside that band, but at the so-called safe side. Application of the strain energy density paramater... 173 Fig. 8. Comparison of fatigue life Ncal calculated according to Eq. (2.17) and experimental fatigue life Nexp for Fe-Cu-Ni (3) sintered steel A better correlation between the results should be expected if elastic- plastic stresses and strains at the notch root were taken into account. Ho- wever, in such a case, the determination of the fatigue life might be much more complicated. References 1. 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Fatigue, 26, 447-455 Zastosowanie parametru gęstości energii odkształceń do oceny wieloosiowego zmęczenia stali spiekanych z koncentratorami naprężeń Streszczenie W pracy przedstawiono rezultaty badań zmęczeniowych spieku Fe-Cu o zawarto- ści 1.5%miedzi oraz trzech rodzajów spieku Fe-Cu-Ni o zawartości 2%miedzi i 2.5% niklu. Do badań zastosowano próbki okrągłe z koncentratorami naprężeń w posta- ci odsadzenia. Zakres badań obejmował stałoampitudowe zginanie, skręcanie oraz kombinacje proporcjonalnego i nieproporcjonalnego zginania ze skręcaniem. Wyniki badań eksperymentalnych zostały opisane przy zastosowaniu kryterium parametru energetycznego, opartego na koncepcji płaszczyzny krytycznej. Trwałość zmęczenio- wa została wyznaczona na podstawie liniowej kombinacji parametru gęstości ener- gii odkształceń normalnych oraz stycznych z pewnymi współczynnikami w przyjętej płaszczyźnie krytycznej.Wwiększości przypadkówotrzymano zadowalającązgodność trwałości obliczeniowych z eksperymentalnymi. Manuscript received April 11, 2008; accepted for print October 13, 2008