An Analytical Investigation on Fatigue Behaviour Of Carbon Steel Materials Subjected To Rotating Bending Loads

Transcription

1 Seoul FISITA World Automotive Congress June -5,, Seoul, Korea FA8 An Analytical Investigation on Fatigue Behaviour Of Carbon Steel Materials Subjected To Rotating Bending Loads Harkali Setiyono UPT-Laboratorium Uji Konstruksi BPP Teknologi Kompleks PUSPIPTK Serpong Tangerang 534 -mail : harkali@luk.puspiptek.net Indonesia An analytical model used to evaluate fatigue behaviour of carbon steel materials subjected to rotating bending loads is presented in the paper. The application of the analytical model is aimed at estimating lives of initial crack formation and subsequent crack propagation so that the whole fatigue lives and fatigue crack growth behaviour of the investigated carbon steel material can be identified. The lives of initial crack formation are analytically determined according to the value of locally total strain ranges at the critical part of the material. The other lives required by the initial crack to subsequently propagate to its critical size are estimated by utilizing a relationship of stress intensity ranges in the vicinity of the propagating crack tip and crack growth rates as formulated by Paris. The material analysed is represented by a V-notched round bar sample and it is made of a carbon steel material, which is normally used for the design of transmission shafts, axles, crankshafts etc. In order to assess the accuracy of the analytical model used, fatigue tests on a number of the same sample design of specimens were carried out under the test loads of rotating bending. The test loads cause the V-notch tip of the tested specimens to be affected by completely reversed stresses with a stress ratio R = - and test results obtained are used to verify the analytical-predicted ones. In the paper, the verification of the analytical-predicted results is presented in the form of comparing analytical and experimental behaviour of fatigue life (S-N diagrams and fatigue crack growth (a-n curves. The comparison indicates that the results of the analytical model are very close to the ones of the experimental approach. Keywords: fatigue, rotating bending, strain, stress intensity and crack growth. INTRODUCTION In this paper, fatigue behaviour of a carbon steel material normally used for the design of transmission shafts, crankshafts, axles etc. is analytically assessed using a combined approach of local strain and stress intensity analyses. Since a fatigue damage process of materials is generally initiated by the formation of a crack at a critical part of the materials, which is subsequently followed by the continuous propagation of the crack to the stage of final fracture, total fatigue lives of the materials will be composed of the lives of initial crack formation (N i and crack propagation (N P. The analytical model is mainly aimed at utilizing the local strain analysis to estimate the lives of initial crack formation (N i while the subsequent lives consumed by the initial crack to propagate to its critical size (N P are obtained from analysing the correlation of the stress intensity and crack propagation rates. In the analysis, the investigated material is represented by a V-notched round bar sample made of a carbon steel material and a load applied to the sample is selected in such a way that the load is similar to the one frequently undergone by the shafts in service. This type of load is usually a rotating bending one and subject to this load, the V-notch tip that simulates a critical part of the sample will be affected by completely reversed stresses with a stress ratio R = -. Due to stress concentrations at the V-notch tip, the fatigue crack is expected to start from the notch tip so that the lives of crack formation can be obtained from analysing local strain ranges at the notch tip. Once the initial fatigue crack has been formed, the crack propagation rate (da/dn and the corresponding stress intensity range ( K in the vicinity of crack tip at each crack extension can be calculated so that the lives of crack propagation (N P is obtainable by integrating the correlation of da/dn to K from the initial crack size to the critical crack one. In order to assess the accuracy of the analytical model, its predicted results are compared to the ones obtained from a number of fatigue tests on V-notched round specimens of a carbon steel material under the same loading condition. Comparisons of analytical and experimental results cover the behaviour of fatigue life ( S- N diagrams and fatigue crack growth (a-n curves. ANALYTICAL APPROACH In the analysis, the analytical model is used to evaluate fatigue behaviour of a sample design made of a carbon steel material shown in Figure. The chemical composition of the sample material can be seen in the table and its mechanical properties are as follows: Ultimate tensile strength, UTS = 66 MPa Yield strength, y = 4 MPa lastic modulus, = 7x 3 MPa longation = 4%

2 Reduction area at fracture = 47% ndurance limit, D = 35 MPa Fracture toughness, K IC = 45 MPa m.5 Figure. Design of the carbon steel sample analysed. Table. The chemical composition. lement C Si Mn S P Fe Percentage ( % The rest When the sample shown in Figure is subjected to rotating bending loads, the stress concentrator of the V- notch tip will be affected by completely reversed stresses such as illustrated in the following figure. R = 5 35 R = o 5 φ.5 φ.4 φ. 3 R = = ; = min + mean = = Figure. Cyclic stresses undergone by the notch tip. ANALYSIS OF CRACK FORMATION LIVS Cycles of applied load required to form an initial fatigue crack (N i can be obtained from analysing totally local strain ranges in the vicinity of the notch tip ( T of the sample in Figure. On the basis of predicted-total strain ranges ( T, the value of the crack formation lives (N i can be calculated from the following equation. T a α N { } i =... ( f m t a is an endurance strain range and it is obtainable from: ( r end yp a = yp ( r + end (+ r end is an endurance strain and its value is: end = D... (3... ( D : endurance limit based on constant-amplitudecyclic loading with a mean stress, mean = : elastic modulus Because the sample in Figure is notched, the value of end is therefore determined using a notch factor K f =. [], so that the endurance strain ( end used in the analysis is: D end =.... yp is a yield strain and it is calculated from: y yp =... (4 (5 y is the yield strength and r is equal to R where the value of stress ratio R in the analysis of rotating bending effect is -, so that the equation ( changes to be: = a end... (6 m is a mean strain at the notch tip and its value is as follows: m = T max (+ r e... (7 Tmax : maximum total strain at the notch tip r e : T min T max, Tmin : minimum total strain at the notch tip In the equation (, f is a fatigue ductility coefficient and α is a fatigue ductility exponent. Both factors of the fatigue ductility can be obtained from a plastic component of a Coffin-Manson s (Low cycle fatigue diagram for the investigated material. Due to the reason of efficiency in generating the low cycle fatigue diagram, the diagram was not experimentally determined but it was empirically estimated according to the method of Four-point correlation prediction [4]. The predicted result of the Coffin-Manson s diagram can be seen in Figure 3 and the plastic component of the diagram provides the values of f =.56 and α = In using the equation (, the value of total strain range ( T at the notch tip is determined by considering the value of maximum notch stress ( in satisfying the following criteria. - If < y, the total strain range ( T at the notch tip is determined only according to the elastic strain range ( e as follows: T = e =... (8 = -, because R = - the value of = -

3 so that =, = K... (9 max t nom K t is a stress concentration factor and for the V-notch of the sample shown in Figure, the value of K t =.5 while nom is an appliedly nominal stress. - If y, the total strain range ( T at the notch tip is determined according to the sum of elastic ( e and plastic ( P strain ranges as follows: T = e + P = + P... ( APPLID STRAIN RANGS Figure 3. mpirically predicted Coffin-Manson s diagram. lastic diagram PRDICTD-LASTIC DATA PRDICTD-PLASTIC DATA Tmax Pmax emax Plastic diagram.... x 3 APPLID LOAD CYCLS The value of totally plastic strain range ( P is determined by means of an iteration method. The iteration is initially carried out at a half cycle of tensile loading and then followed by the one at a half cycle of compression loading. In the iteration process of P, the material near by the notch tip is assumed to behave as a stress (-strain ( relationship shown in Figure 4. A computer program was written to perform the iteration of P and the value of P is obtained when the value of iterated notch stresses relating to T in the equation ( has converged to the stresses actually applied to the notch tip. ANALYSIS OF CRACK PROPAGATION LIVS As the number of load cycles continuously increases, the initial crack propagates to its critical size (a cr, which causes fatigue failure and its crack propagation lives can be obtained from analysing the following well-known Paris equation. da = c K dn n... da dn : crack propagation rate K : stress intensity factor range c and n : material constants ( With reference to Figure, a circumference-fatigue crack starts to grow from the notch tip towards the longitudinal axis of the sample. The formulation of stress intensity factor range ( K can be expressed in term of the depth of the circumference crack (a from the notch tip (Figure 5. K = π a (.... ( π Kt K t is a stress concentration factor and equal to.5 for the notch as indicated in the sample, while π is a correction factor for the circumference crack as seen in the following figure. 5 a φ. (5 a A φ.5 φ.4 Tmin emin Pmin A a : crack depth measured from the notch tip 5 4 Section AA Figure 4. Material behaviour near by the notch tip. Pmax : maximum plastic strain Pmin : minimum plastic strain emax : maximum elastic strain ( e max = emin : minimum elastic strain ( e min = and are maximum and minimum notch stresses. Figure 5. Direction of circumference crack growth. The critical depth of the circumference crack (a cr is determined by considering that the value of the maximum stress intensity (K Imax ahead of the critical crack tip has approached to the value of fracture toughness (K IC of the investigated material. Accordingly, the depth of critical crack (a cr is obtained from iterating the crack depth (a using a computer program until the following condition is satisfied. KI max = π a(. KIC... (3 3

4 Substitution of the equation ( into the equation ( and then by integrating it from the initial circumference crack depth (a i up to the critical one of the circumference crack (a cr, a general formulation of estimating fatigue crack propagation lives (N P can be expressed as follows: (.5n (.5n (. acr (. ai NP =.5n.64 n c x3.4 x ( x (.5n.5... (4 Once the initial circumference crack has been formed, the crack will propagate when an applied stress intensity range ( K near by the crack tip has approached to a threshold level ( K th so that the initial depth of the circumference crack (a i can be calculated from: K t K th a i = (... (5 The dimension of the crack depth (a in the above equation is in mm and the value of threshold level ( K th used in the analysis is equal to.4 MPa m.5. [5] The number of load cycles to form the initial circumference crack of a i deep (N i can be predicted using the equation ( and hence, the total number of load cycles to fatigue failure (N is as follows: verify predicted fatigue life behaviour obtained from the analytical approach. It can be seen in Figure 6 that the test arrangement will cause the notch tip of the specimen to be subjected to completely reversed stresses with a mean stress that is equal to zero. Due to these reversed stresses, a fatigue crack is formed and propagates from this notch tip. The test of fatigue crack propagation was also carried out according to the loading arrangement such as shown in Figure 6. The test load applied to the specimen in the fatigue crack propagation test caused the notch tip of the tested specimen to be affected by a cyclic stress amplitude of 456 MPa with the same stress ratio R. At each interval of test load cycles applied, the test was stopped and the depth of the circumference crack from the notch tip was measured by means of potential drop equipment. This measurement was done using a special probe, which was located at four points around the notched and smooth parts of the specimen, where each measuring point was separated by an angular distance of 9 o. The depth of the crack was determined on the basis of the difference between potential drops at both different parts of the specimen measured. Bending moment diagram Maximum bending moment N = N i + N P... (6 Test load Tension Specimen Test load The above equation is then used as a fundamental in analysing the number of load cycles to fatigue failure of the sample at each cyclic stress applied so that its fatigue life behaviour can be analytically predicted. In case of fatigue crack growth behaviour, the number of load cycles at each crack extension is estimated using the equation (4 with the value of a cr is replaced by the depth of each extended crack, where the correlation of the crack depth and the corresponding number of load cycles indicates the fatigue crack growth behaviour of the sample. XPRIMNTAL APPROACH In the experimental approach, fatigue and fatigue crack propagation tests were performed on a number of specimens, which were exactly the same in design as that of the sample shown in Figure. The specimens were tested on a rotating bending fatigue-testing machine and the test load applied can be seen in Figure 6. In the fatigue test, the specimens were tested under various levels of test load at a room temperature. In order to ascertain that the specimens fail during the test and also to minimize testing times, each level of test load was applied in such a way that the value of notch stresses on each specimen was still higher than that of the endurance limit for the notch specimen. Because the notch factor K f of the specimen is equal to., the endurance limit of the notch specimen is therefore 35/. 59 MPa so that notch stress amplitudes corresponding to the test load applied varied in-between 9 to 54 MPa. ach stress level was applied with a stress ratio R = - and at each stress level, at least 3 identical specimens were tested to failure. Scattered data of test results is statistically analysed and they are limited within a tolerable scatter band of % to 9%. On the basis of a mean value of the scattered data at each stress level, actual fatigue life behaviour (S-N diagram of the tested specimens is drawn and the diagram is then used to Bending stress nd support Compression distribution nd support Clamping device Clamping device Machine rotation Figure 6. Fatigue test load arrangement. RSULTS AND DISCUSSION φ.5 φ. 4 φ. 3 In order to assess the accuracy of the analytical model presented in this paper, its analytical predictions are verified by comparing them to actual values measured in the fatigue tests. Prior to the verification of analytical data, the accuracy of experimental results is statistically assessed and individual data of each tested specimen should be in a tolerable scatter band of % to 9%. Figure 7 clearly indicates that test data of all tested specimens at each applied stress level are still scattered within the tolerable scatter band of % to 9%. This means that all test data obtained is valid to be used to verify the analytical-predicted data. To enable in carrying out the verification, the mean value of the scattered test data at each stress level applied is selected to be compared to the analytical-predicted data such as shown in Figure 8. It is clearly seen in the figure that each analytical-predicted data is quite close to the corresponding experimental one. The analytical data, on average, deviates % only from the actual one and this is of course the analytical data as expected. Thus, it can be interpreted here that the fatigue life of the investigated material can actually be estimated by utilizing a combined approach of local strain and stress intensity analyses. The results obtained from both analyses represent the analytical fatigue life at each stress level 4

5 applied, which agrees well with the actual one of the investigated material. CYCLIC STRSS AMPLITUD, Sa ( MPa CYCLIC STRSS AMPLITUD, Sa ( MPa SPCIMN : V-NOTCHD ROUND BAR MATRIAL : CARBON STL TST LOAD : ROTATING BNDING 5% 9% % Figure 7. Tolerable scatter band of test data. Figure 8. Verification of analytical-predicted data. During fatigue damage process of the investigated material, its fatigue crack growth behaviour is also analytically predicted using equation (4. In the Figure 9, the accuracy of the equation (4 is assessed by comparing its results to experimental ones. In the tests, the actual crack growths were monitored from the notch tip up to the stage of final fatigue fracture. The verification of the analytical crack growth in Figure 9 also shows that both analytical and experimental behaviour of fatigue crack growth has the same tendency, where the analytical behaviour is relatively close to the experimental one. It is. x NUMBR OF LOAD CYCLS, N ( CYCLS STRSS RATIO, R = S min / S max = - XPRIMNTAL DATA SPCIMN : V-NOTCHD ROUND BAR MATRIAL : CARBON STL TST LOAD : ROTATING BNDING. x 5 STRSS RATIO, R = S min / S max = - XPRIMNTAL DATA ANALYTICAL-PRDICTD DATA NUMBR OF LOAD CYCLS, N ( CYCLS believed that the relatively small deviation between both behaviour is likely affected by a variation e.g. in actual dimensions of each tested specimen. The variation is discounted in the analytical approach and the analysis is always based on nominal dimensions. CRACK LNGTH, a (mm 4 3 Figure 9. Fatigue crack growth behaviour. CONCLUSION MATRIAL : CARBON STL TST LOAD : ROTATING BNDING R : - This paper has presented an analytical model to evaluate fatigue life and fatigue crack propagation behaviour of a carbon steel material subjected to rotating bending loads. The fatigue life is analytically estimated by adding the crack formation and propagation lives. The investigated material is designed in the form of a V-notched round bar and the crack formation lives are basically obtained from analysing a local strain range at the notch tip. Meanwhile, the lives consumed by the initial crack to propagate to the final stage of fatigue failure are estimated by analysing a well-known relationship of stress intensity and fatigue crack propagation rate as formulated by Paris. This Paris formulation is also used to calculate the crack extension at each increase of load cycles, where the data obtained is then plotted to describe the fatigue crack propagation behaviour. The accuracy of analytical-predicted results is verified by the ones measured in the fatigue tests on a number of V-notched round bar specimens of a carbon steel material. The test results are statistically evaluated and their scattered data is limited within an acceptable scatter limit of % to 9%. The mean value of scattered test data is used to verify the analytical behaviour of fatigue life and fatigue crack propagation. It has been found that the analytical model presented can estimate the actual fatigue behaviour of the investigated material very well with a mean deviation of analytical predictions from actual values is % only x 4 NUMBR OF LOAD CYCLS, N (Cycles XPRIMNTAL DATA ANALYTICAL-PRDICTD DATA 5

6 NOMNCLATUR a a cr a i c K f K IC K t n N N i N P R α f K D UTS y : fatigue crack length : critical fatigue crack length : initial fatigue crack length : material constant : modulus of elasticity : notch factor : fracture toughness : stress concentration factor : material constant : fatigue life : fatigue crack formation lives : fatigue crack propagation lives : stress ratio : fatigue ductility exponent : fatigue ductility coefficient : stress intensity factor range : strain range : stress range : endurance limit : ultimate tensile strength : yield strength Behaviour of Cracked Al 4-T3 Components Reinforced By Graphite/poxy Composite Materials, Proceedings of Seminar On Fatigue Fracture Mechanics, 7-8 ( Translated from Indonesian Language. RFRNCS [] R.. Peterson, 974, Stress Concentration Factors, John Wiley Sons, Inc., 9-. [] Dieter Dengel, August 975, Die Arc Sin P - Transformation in infaches Verfahren Zur Grafischen Und Rechnerischen Auswertung Geplanter Wöhlerversuche, Zeitschrift für Werkstofftechnik, 6. Jahrgang, Heft 8, [3] T.G.F. Gray, February 977, Convenient Closed Form stress Intensity Factor Common Crack Configurations, International Journal of Fracture, Vol. 3, No., [4] Terance V. Duggan and James Byrne, 979, Fatigue As A Design Criterion, The Macmillan Press Ltd., 46-. [5] SDU, April 984, Fatigue Crack Propagation Rates and Threshold Stress Intensity Factors In High Alloy and Corrosion Resistant (Stainless Steel, SDU-Item Number 843, 47. [6] K. Hellan, 985, Introduction To Fracture Mechanics, McGraw Hill Book Co., [7] H.L. wald and R.J.H. Wanhill, 986, Fracture Mechanics, dward Arnold Ltd. [8] Harkali Setiyono, February 989, Fatigue Life of A Carbon Steel Material Affected by A Shot Peening Process, Proceedings of National Symposium on Fatigue of Materials and Structures, ( Translated from Indonesian Language. [9] Harkali Setiyono, 9- February 998, Fatigue 6