COMPONENT FATIGUE STRENGTH

Transcription

1 COMPONENT FATIGUE STRENGTH M. Hajeck, S. Keusemann, C. Broeckmann, P. Beiss, FAPMI Institute for Materials Applications in Mechanical Engineering, RWTH Aachen University, Germany ABSTRACT A method, which seems suitable to predict the fatigue strength of components under constant amplitude loading at 50 % survival probability, has been developed. First, under unit static load, the highly stressed zones of the component are identified by linear elastic finite element analysis (FEA). These zones are remeshed and investigated in submodels by high resolution FEA to determine the highest local first principal stresses Imax and the associated highly loaded volumes V 90 which are exposed to at least 90 % of Imax in each zone. The ratios of highest local first principal stress Imax and the unit load are the load transfer coefficients Q for each spot considered. From the material fatigue characteristics, expressed by a density modified three parameter Weibull distribution at a given survival probability, the locally endurable fully reversed fatigue strength amplitudes are calculated for all highly stressed zones of the component. The lowest quotient of local fatigue strength and associated load transfer coefficient defines the crack critical location of the part and delivers the endurable load amplitude under fully reversed constant amplitude. From an analytical expression for the Haigh diagram the endurable mean stress and stress amplitude are predicted for the crack critical zone at stress ratios R 1. Various components were investigated whose material behavior is known. These parts were fatigue tested to compare the calculated strength with the experimental performance. The agreement is satisfactory. INTRODUCTION During a research program, which is still going on, the fatigue strength of two medium strength sintered steels was investigated. One major objective was to measure the fully reversed nominal fatigue strength amplitude A at 10 7 cycles and 50 % survival probability for a variety of notched and unnotched specimens under axial, torsional and plane bending loading. All geometries were analyzed with the help of FEA to determine the highest first principal stresses, Imax, under unit load, the associated stress concentration factors K t and the highly loaded material volumes V 90, exposed to at least 90 % of Imax, surrounding the peak stress locations. The product of nominal stress amplitude A and stress concentration factor K t is the locally endurable stress, e. g. in a notch. In [1, 2] it was shown, that there is a continuous relationship between the locally endurable stress amplitude K t A and the highly loaded volume V 90, 1

2 irrespective of specimen size, notch geometry or loading mode. This relationship can be expressed by a density modified three parameter Weibull distribution at a survival probability p = exp(1): K t A A0 A0ref V V 90 ref 1/ n 0 m (1) (K t A : locally endurable fatigue strength amplitude at 10 7 cycles and 36.8 % survival probability; the survival probability of 36.8 % is approximated by 50 % for practical purposes; A0 : asymptotic volume independent strength amplitude at full density 0 ; A0ref : volume dependent part of the strength amplitude at full density 0 and at a highly loaded reference volume V ref = 1 mm 3 ; n: Weibull exponent; : density; m: density exponent) Figure 1. Effect of highly loaded volume V 90 on the locally endurable fatigue strength amplitude K t A at a stress ratio of R = 1, 10 7 cycles and 50 % survival probability for a diffusion bonded iron containing 4 % Ni, 1.5 % Cu, 0.5 % Mo and 0.6 % admixed graphite 2

3 Figure 2. Effect of highly loaded volume V 90 on the locally endurable fatigue strength amplitude K t A at a stress ratio of R = 1, 10 7 cycles and 50 % survival probability for a prealloyed 1.5 % molybdenum steel containing 2 % admixed copper and 0.65 % graphite With a sufficient number of experimental results the four coefficients A0, A0ref, n and m can be obtained by multiple nonlinear regression analysis. The findings for the two steels investigated were normalized to a common density of 7.0 g/cm 3 via eq. 1 and are presented in Fig. 1 and 2. The data points marked with an asterisk were excluded from the regression evaluation for microstructural reasons. Usually real PM parts are not subjected to fully reversed cyclic stresses but rather to static mean stresses different from zero with superimposed cyclic stress amplitudes. Therefore, a second objective of the project is to develop an analytical approach to take the mean stresses into account. To this end, S-N curves at different stress ratios R are considered in double logarithmic coordinates in the finite life region, as shown in Fig. 3. At any number of cycles Basquin s equation delivers C(R 1) [K t a (R 1)] k(r 1) = C(R = 1) [K t a (R = 1)] k(r=1) (2) Since C(R 1), C(R = 1), k(r 1) and k(r = 1) are constants eq. 2 stays valid also at nominal endurance limits A if the S-N curves at R 1 and at R = 1 have similar knee points. With C R 1 C R 1 1/ k R1 KR and k(r = 1) / k(r 1) = q(r) eq. 2 becomes q(r) R 1 KR K R K (3) t A t A 1 Figure 3. Schematic illustration of the effect of mean stresses on the locally endurable stress amplitude in double logarithmic coordinates Eq. 3 predicts the locally endurable fatigue strength at any stress ratio R 1 as a function of the fully reversed fatigue strength at R = 1 if the stress ratio dependence of the factor K(R) and of the exponent 3

4 q(r) is known. In [3] an attempt was made to establish analytical expressions for these functions derived from additional fatigue tests at the stress ratios R = 5, R = 0 and R = 0.5 with both steels. Within the limits of experimental error K(R) and q(r) did not depend on the type of steel investigated. The current status of the project is presented in Fig. 4 with the expressions K(R) = expr (4) q(r) = [1 exp(expr)] (5) and with the coefficients of eq. 1 in Table 1. All these regression coefficients given here are subject to change when all results of the project are available for a final evaluation. Figure 4. Functional relationship between stress ratio R, factor K(R) and exponent q(r) in eq. 3 Table 1: Regression coefficients of eq. 1 for fully reversed fatigue strength amplitudes, R = 1 steel A0 N/mm 2 A0ref N/mm 2 1/n m Distaloy AE % C Atomet % Cu % C Eqs. 4 and 5 offer the opportunity to calculate a synthetic Haigh diagram to any fully reversed fatigue strength amplitude K t A (R = 1), Fig. 5. 4

5 Fig. 5: Synthetic Haigh diagram according to eq. 3, 4 and 5 for the two steels investigated We think that the locally endurable fatigue strength amplitude K t A (R) from the specimen results, as estimated from eq. 1 and eq. 3, can directly be compared with the locally endurable first principal stress amplitude in a component under cyclic load. If this assumption is correct, eq. 1 and 3 in combination with an FEA are suitable to predict the fatigue strength of components. The FEA is needed to determine the highly loaded volumes V 90 around positive stress peaks and the proportionality factors between outer load and local first principal stresses. COMPONENT FATIGUE STRENGTH The approach to predict the constant amplitude fatigue strength of components manufactured from the two steels considered can be separated in several steps: 1. FEA with a unit load as applied under service conditions to identify highly stressed zones of the component 2. Remesh a high resolution submodel of each potential crack critical zone 3. Calculate the maximum first principal stress Imax for each location, the highly loaded volume V 90 surrounding each stress peak and the load transfer coefficient Q as quotient of Imax and external unit load for each location under consideration 4. Measure the local density at each potential failure site of the component 5. Calculate the endurable fully reversed (R = 1) stress amplitude for each location from eq. 1 with V 90 and and the coefficients in Table 1 6. Divide each locally endurable stress amplitude by its associated load transfer coefficient Q. The lowest quotient indicates the failure critical highly loaded zone of the component. 7. Convert the stress amplitude at the lowest endurable load at R = 1 to the applicable stress ratio R via eq. 3, 4 and 5. The resulting stress amplitude is divided by the load transfer coefficient Q and delivers the predicted endurable fatigue load amplitude of the component at 50 % failure probability and 10 7 cycles. This procedure makes use only of the FEA results ( Imax, V 90, Q) and material specific fatigue testing results from laboratory specimens expressed by eq. 1 and eq. 3 without any adjustment or correction factor. The viability must be demonstrated by component tests. EXAMPLE SYNCHRONIZER HUB 5

6 One of the components investigated was a synchronizer hub of a manual transmission. The results have been published in [4] where a different mathematical treatment was applied. The component is shown in Fig. 6. It has 38 inner spline teeth, the key slots are open to one face only, and the outer perimeter with a pitch diameter of 90 mm has 11 spline teeth between the key slots. The inner end of the slots is generously rounded. The part was compacted net shape from Distaloy AE (4 % Ni, 1.5 % Cu and 0.5 % Mo diffusion bonded) % graphite UF4, sintered in an industrial belt furnace at 1120 C for 20 min in an N 2 -H 2 atmosphere and cooled with a low rate of about 0.3 C/s in the temperature range of the eutectoid transformation. In service the component is exposed to pulsating torque (R = 0) and has three highly loaded zones, the outer splines, the inner splines and the three key slots. In the global FEA model the inner splines turned out to be only lightly stressed, so they were excluded from further treatment. The zones of high stress concentration were the three key slots and each second tooth on the outer circumference behind the key slots in the direction of the applied torque. The FEA submodels of these zones gave the results in Table 2. äouter splines key slots inner splines Figure 6. Synchronizer hub Table 2: FEA submodel results of the highly stressed zones of the synchronizer hub location outer splines; 2nd tooth behind key slots; three equivalent sites key slot; inner radius; three equivalent sites highly loaded volume V 90, mm highest first principal stress Imax at a torque of 2000 Nm, N/mm load transfer coefficient Q, N/mm 2 /Nm The synchronizer hubs were manufactured with as-sintered densities of 7.0 and 7.2 g/cm 3. These values were assumed representative for the two potential failure sites without measuring local densities. For 6

7 Distaloy AE % C the pore-free density 0 = is 7.90 g/cm 3. With the constants in Table 1 eq. 1 yields the fully reversed locally endurable stress amplitudes for the outer splines and the key slots in Table 3. Table 3: Locally endurable fully reversed fatigue strengths and corresponding torques location outer splines key slot radii density, g/cm K t A (R = 1) from eq. 1, N/mm endurable torque at R = 1 K t A (R = 1) / Q, Nm It is evident that the key slots have a much higher load bearing capacity than the three second teeth behind the key slots. Thus, the outer splines represent the failure critical regions of this type of synchronizer hub, opposite to hubs with key slots that are open to both faces. Using eq. 3, 4 and 5 the fully reversed local fatigue strength K t A (R = 1) is now converted to the real load case of pulsating torque with R = 0. For R = 0 eq. 4 yields K(R = 0) = and eq. 5 delivers q(r = 0) = Table 4 compares the predicted with the measured fatigue torques from [4]. The difference is less than 10 %. Table 4: Comparison between predicted and measured torque at R = 0 density, g/cm locally endurable stress amplitude K t A (R = 0) from eq. 3, N/mm predicted endurable torque K t A (R = 0)/Q, Nm measured fatigue strength at cycles, R = 0, Nm measured/predicted EXAMPLE INJECTOR CLAMP The injector clamp in Fig. 7 is symmetric with regard to the central through hole accepting a fastening bolt submitting the component under a high static mean bending stress. In service the static bending stress is superimposed by cyclic bending stress components from periodic fuel injection. For this investigation the part was compacted from Distaloy AE % C, sintered at 1120 C and slowly cooled to match the microstructural characteristics of the laboratory specimens on which Fig. 1 is based. 7

8 Figure 7. Investigated injector clamp The global FEA model revealed a single rather large highly loaded zone around the conical opening of the central through hole on the lower face of the clamp, Fig. 8, indicating the potential failure site. 8

9 Figure 8. Distribution of first principal stresses I with Imax on the lower face around the conical hole opening with submodel The submodel of this zone was remeshed in several steps down to 40 m element size until the value of the highly loaded volume had converged to V 90 = 6.34 mm 3, and the load transfer coefficient was Q = N/mm 2 /N. The local density was determined on thin slices weighing less than 4 g cut adjacent to the fatigue fracture surfaces with a diamond wire saw. The average of five measurements was 7.11 g/cm 3. With 0 = 7.90 g /cm 3 for Distaloy AE % C eq. 1 delivers the locally endurable fully reversed stress amplitude K t A (R = 1) = N/mm 2. The component was submitted to bending fatigue testing at a stress ratio of R = 0.5 which is rather realistic in this application. From eq. 4 K(R = 0.5) = and from eq. 5 q(r = 0.5) = are obtained. With these values eq. 3 yields K t A (R = 0.5) = N/mm 2. The predicted endurable bending load amplitude is K t A (R = 0.5)/Q = 135.2/ N = 4159 N The measured fatigue strength at R = 0.5, 10 7 cycles and 50 % survival probability was 3958 N. The ratio measured/predicted is 3958/4159 = Also in this case the deviation from the testing result is below 10 %. 9

10 EXAMPLE TOOTH ROOT BENDING STRENGTH A small spur gear, shown in Fig. 9, with a nominal pitch diameter of mm, a module of mm and 10 mm thickness was compacted and sintered at 1120 C with slow cooling for tooth root bending tests in the as-sintered condition at two densities. The steels used were DistaloyAE % C and the 1.5 % Mo prealloyed steel powder Atomet % Cu % C. Evidently, from the way of loading, only the root of the two loaded teeth are potential failure sites. From the FEA submodel with 20 m element size the highly loaded volume around Imax of the two teeth was obtained as V 90 = mm 3 with a load transfer coefficient of Q = N/mm 2 /N. Fig. 9: Experimental set-up for tooth root bending at R = 0.1 To measure the local densities in the tooth root very tiny samples of about 1 g were cut from the highly loaded area with a diamond coated wire saw so as to eliminate all surface densification during the sample preparation. The density was measured according to the Archimedian principle with a high precision balance. Because of the time involved in sample preparation only two measurements per experimental condition were made. To calculate the fully reversed local fatigue strength from eq. 1, the pore free densities were assumed 0 = 7.85 g/cm 3 for the Atomet 4901 base steel and 7.90 g/cm 3 for the 4 % Ni Distaloy steel. The tests were performed at a stress ratio of R = 0.1. From eq. 4 and 5 K(R = 0.1) = and q(r = 0.1) = were obtained. These figures are valid for both steels which respond statistically the same way to static mean stresses. Knowing K(R = 0.1) and q(r = 0.1) the anticipated fatigue strength amplitude at R = 0.1 can be calculated from eq. 3 and subsequently the load bearing capacity. All results of this example are listed in Table 5. Table 5: Results of tooth root bending tests Atomet steel Distaloy steel density, g/cm locally endurable fully reversed fatigue strength K t A (R = 1), N/mm locally endurable fatigue strength amplitude at R = 0.1 K t A (R = 0.1), N/mm

11 predicted load amplitude K t A (R = 0.1)/Q, N measured fatigue strength amplitude at R = 0.1, 10 7 cycles and 50 % survival probability measured/predicted By and large, the predicted tooth root bending strengths lie within 10 % around the measured strength data. In the case of the Atomet 4901 steel the scatter is too high and the deviations are a little larger. We attribute this to experimental measuring errors due to inadvertence because the strength differences are much larger than expected from the density differences and also the comparison with the Distaloy steel yields unsystematic strength ratios. Usually, at comparable densities, the fatigue performance of slowly cooled 1.5 Mo steel should be slightly inferior to the Distaloy steel by a more or less constant factor. SUMMARY A simple analytical concept has been developed for two common medium strength porous sintered steels which seems to have the potential to predict the constant amplitude fatigue strength of real components directly from results obtained from bar shaped notched and unnotched laboratory specimens. Background is a density modified three parameter Weibull distribution at a constant survival probability of p = exp(1) to describe the dependence of the fully reversed fatigue strength on the highly loaded volume. In order to take static mean stresses into account, a synthetic Haigh diagram applicable to both steels has been established. The necessary data input comprises - the highly loaded volume V 90 defined as the volume, surrounding a positive stress peak, exposed to at least 90 % of the maximum first principal stress Imax - the load transfer coefficient Q as quotient from Imax and the corresponding service load - the local density in the zone of the stress peak and the full density 0 of the alloy - four material characterizing constants, A0, A0ref, 1/n and m, to describe the fully reversed fatigue strength as depending on density, specimen or component size, loading mode and stress concentrations - the functions K(R) and q(r), which are also material specific, to convert fully reversed fatigue strength amplitudes to stress ratios R 1 V 90 and Q must be obtained from fine meshed FEA submodels, while the material specific parameters must be obtained experimentally from specimens of simple shape. The comparison between experimental and predicted fatigue strengths is straightforward and simple if the material characteristics are available. So far only a few components have been investigated to test the method suggested here. The first results are, however, very encouraging. ACKNOWLEGEMENTS This work was supported by Fachverband Pulvermetallurgie FPM and partially financed by Stiftung Stahlanwendungsforschung under Project Numbers AVIF A253 and AVIF A270. The powders were contributed by Höganäs AB and by Rio Tinto Iron & Titanium GmbH. The injector clamps were compacted and sintered by AMES SA in Barcelona, the spur gears were compacted by Höganäs AB and sintered by GKN Sinter Metals Engineering in Radevormwald. All this help is gratefully acknowledged. REFERENCES 1. P. Beiss, A. Zafari, K. Lipp, J. Baumgartner: Fatigue Behavior of a Sintered Steel Containing 4 % Ni, 1.5 % Cu, 0.5 % Mo and 0.6 % C; Int. J. Powder Metall. 48 (2012) 1, p

12 2. A. Zafari, P. Beiss, C. Broeckmann, K. Lipp: Assessing the Fatigue Strength of a Sintered Steel as Affected by the Highly Stressed Volume; Proc. Euro PM 2011, Barcelona, Vol. 1, p ; EPMA, Shrewsbury, S. Keusemann, C. Broeckmann, P. Beiss: A Synthetic Haigh Diagram for Structural Sintered Steels; Proc. CD 2012 PM World Congr. Yokohama, paper no. 16 A - T9-5; JSPM, Tokyo, D. Amos, P. Delarbre, K. Lipp, H. Kaufmann: Reliable Component Fatigue Design Applying Appropriate Cyclic Properties; Proc PM World Congr. Florence, Vol. 3, p ; EPMA, Shrewsbury,