THE THRESHOLD STRESS INTENSITY RANGE IN FATIGUE

Transcription

1 Fatigue ofengineering Materials and Structures Vol. 1, pp Perpmon Press. Printed in Great Britain. Fatigue of Engineering Materials Ltd THE THRESHOLD STRESS INTENSITY RANGE IN FATIGUE R. T. DAVENPORT? C.E.G.B. Scientifrc Services, Wythenshawe, Manchester and R. BROOK Department of Metallurgy, University of Sheffield Abstract-The parameters are considered that control the threshold stress intensity in fatigue and relationships existing between these parameters are examined. A new relation is derived which is shown to be consistent with experimental data for a O.l5%C, 1,5%Mn steel. a N dajdn R Kmm Kmin AK AKT AKo KC K,C KICd u, y, m, n, p Exponents 4, x, A, B, C Constants Nomenclature Crack length Number of cycl7s Crack growth per cycle Stress intensity ratio in fatigue (= Kmi,/Kmax) Maximum stress intensity factor Minimum stress intensity factor Range of stress intensity factor Range of stress intensity factor at threshold Threshold stress intensity factor at R = 0 Fracture toughness Plane strain fracture toughness Fracture toughness ; equivalent energy criterion INTRODUCTION PARIS first proposed [ 13 that the relationship between fatigue crack growth rates (da/dn) and stress intensity range (AK) could be described by: da - = C(AK). dn t Formerly Research Fellow, Department of Metallurgy, University of Sheffield

2 152 R. T. DAVENPORT and R. BROOK However, it is now established that the true shape of the log-log relation between da/dn vs AK is not linear, but sigmoidal. The deviations from linearity of these plots can be attributed to the presence of upper and lower bounds to the stress intensity range, i.e. and da -+O asak+ak, (Threshold condition) (2a) dn da - + co as AK + (1 -R)K, (Instability condition). dn In order to account for these restrictions on the applicability of eqn. (1) certain empirical modifications to the simple power law have been proposed. These are adequately reviewed elsewhere [2,3] but can be generalised in the form : (2b) da C(AKm - AKY) -- (3) dn - ((1-R)K,-AK}P Often the values of m and p in eqn. (3) are unity; where this is the case and where AKT = 0 the relation reduces to the well-known Forman eqn. (4), but where AKT is non-zero, m is 1 and n and p are equal, then eqn. (3) becomes that proposed by Nicholson [5]. To make use of general equations of this type, the values of C, n, m, p, AKT, K, and R need to be known. The stress ratio, and the values of the constant and exponents in eqn. (3) can all be determined from numerical analysis of experimental fatigue crack propagation data. However, this analysis cannot be carried out unless values of AKT and K, are known beforehand. The value of K, can be regarded as being the plane strain fracture toughness K,,, a material property, the calculation of which is well defined and which is independent of load, specimen configuration, etc. However, the same is not true of AKT, which has been shown to vary according to loading conditions [6,7]. Thus the value of AKT is not simply defined and needs to be determined for each type of fatigue loading considered. Also nothing is immediately apparent about the parameters which control the threshold stress intensity factor and hence those that determine the variation of its value. In what follows it is intended to identify the main parameters which control the value of the threshold stress intensity factor and to suggest a mathematical correlation between these parameters ; this will enable predictions of fatigue threshold level to be made for a variety of fatigue-loading situations. AN ANALYTICAL APPROACH TO THRESHOLD For fatigue situations having constant amplitude and constant positive stress ratio a basic relation exists in terms of the stress intensity factors : This is true for all K,,, condition : AK = Kmax(l -R). (4) and 0 < R < 1, and therefore it must also hold at the threshold AKT = {Kmax(1-R)}T. (5 1 Equation (5) shows that the threshold condition can be brought about by varying the values of K,,, and R such that the L.H.S. of eqn. (4) becomes equal to the L.H.S. of eqn. (5).

3 The threshold stress intensity range in fatigue 153 Now, eqn. (5) reduces to: AKT = AKo where AK, is the threshold value of the stress intensity factor (and also the stress intensity range) at R = 0. By making certain assumptions about the properties of AKT it has been proposed that [S] : AKT = AKo( 1 - R), (7 1 which predicts that AKT is a linear function of R and R only. In fact, Klesnil and LukaS proposed [6] a more general form AKT = AKo(1 -R), (8 1 suggesting that AKT and R are related by a simple power law and which therefore implicitly requires that AKT has some K,,, dependence. Equation (5) shows that to calculate the threshold level both K,,, and R need to be known. However, it provides no information about the way in which the threshold level varies with K,,, and R ; to predict this we need a further relationship. If we assume that AKT is geometry independent then we can propose that the threshold level is a function only of the instantaneous value of K,,, and the stress ratio (R): AKT =f(k,ax, R). (9 ) Thus for any threshold condition, eqns. (5) and (9) will form a set of simultaneous equations, which if eqn. (9) was known could be solved for any constant amplitude, constant R, fatigue situation. The linear approximation provided by eqn. (7), in fact, grossly underestimates the value of AKT and thus forms a very conservative estimate of threshold level. If this approximation is rejected then the relation of Klesnil and LukaS can be used to estimate the value of AKT; however, the value of y must first be determined from a series of lengthy and expensive threshold studies. Also eqn. (8) does not precisely indicate the relative effects of K,,, and R on AKT. At the moment, therefore, it is necessary to choose between a severe underestimate of AKT or carry out lengthy tests to determine the empirical constant in the Klesnil and L)ukaS relation. McEvily [9], has suggested that the dependence of AKT on AK, and R can be described by : AKT = AK,J(l-R)(l+R)-. (10) Now by expanding (1 + R)- by means of the binomial expansion and neglecting terms higher than R, this equation becomes: AKT = AK,(1-2R)0 5. (11) It can be seen that this equation resembles the general form of the Klesnil and LukaS equation; of course, further expansion of eqn. (1 1 ) predicts a linear dependence according to (1 -R). This effect should be small at small R values although it represents a significant modification when R is large. The applicability of eqn. (10) has been tested and is considered later.

4 KC Kz( 154 R. T. DAVENPORT and R. BROOK Alternatively, Wei and McEvily [lo] proposed that: which can be reduced to: AKoKc(1-R) AKT = (1-R)Kc+RAKo Now if AKo/K,-(l -R) z 1 then eqn. (13) can be approximated by the Klesnil and LukaS equation with y = 1. It is clear that any theory of the fatigue threshold must satisfy the following requirements : (1) AKT-+AKo, R-+O (ii) AKT-+(b, R-+1 (iii) AKT -+ x, Kmax -+ (iv) AKT -+ AK,, K,,, -+ AKo where (b and x are constants which may in certain cases be zero. AKT can only depend on the mechanical variables AKo, Kc, K,,, and (1-R). On dimensional grounds, then, the relation between these must be of the form: AKT= AKof(AKo, Kc,Km,x, (I-R)]. (15) A simple power relationship of this kind that satisfies the requirement of eqn. (14), where 4 and x are zero, is: As will be described later, our experimental results strongly suggest that this is the appropriate fundamental relationship, with ct having a value close to one third. In practice, this equation is not particularly convenient because, in addition to material parameters Kc and AK,, it involves two variables K,,, and R. However, the relationship (5) between these variables shows that K,,, can be eliminated. A simple manipulation produces the following cubic equation for AKT in terms of R alone: AK; + AKi:AKT AK:.K,-(1-R) - (Kc- AKO) (Kc- AKO) This equation has only one real root [ll], whose value is: Now if -- AKT (1- R)Kc K:( 1 - R)2-3 AK, /2(Kc-AKo) [4(Kc-AKO) (Kc-AK0)3 KcU- R) Kz( 1 - R)2 + 3 /2(Kc-AKo) - [4(Kc-AKO) w ~ (Kc-AK,) R)2 4(Kc-AK,3)2 = 0. + (18)

5 The threshold stress intensity range in fatigue 155 (which implies AK, 4 Kc), then It can be seen that the proposed relation between the primary variables will reduce to the general form of the Klesnil and LukaS relation (eqn. 8) under certain limiting conditions. However, the value of the exponent (7) according to our relation would be 0.33, whereas the value suggested by Klesnil and LukaS was 071. To summarise, it seems that most of the equations proposed so far to describe the variation of AKT with stress ratio and maximum stress can be loosely approximated by a general form of the Klesnil and Lukas equation. Now the inadequacies of eqn. (S), which are discussed more fully in a following section, demonstrate that an equation of this type is only an approximation to a more complex form of relationship between 1, K,,,, K, and AK, (which are postulated as material properties for the purpose of this analysis), and the stress ratio. EXPERIMENTAL DETAILS The threshold level for fatigue crack propagation and the effect and extent of the threshold range were measured on a O.lS%C, 15%Mn steel in the 890 C water quenched 630 C tempered condition, giving a yield strength of 350 N/mm2. Three-point bend specimens of 12mm thickness (BS.5447) were tested at approx. 100Hz in an Amsler Vibrophore. Positive values of stress ratio were used throughout. The reversed plastic zone size for the maximum AK used, assuming plane stress conditions, was estimated to be less than 100pm. Crack growth was detected by a potential difference method, and was monitored continuously. Calibration showed the resolution of the equipment to be at least mm. The technique adopted was to reduce systematically the stress intensity at the crack tip until propagation ceased. The threshold was adjudged to have been reached when no crack growth was detected in lo7 cycles. Consideration of eqn. (4) shows that the threshold condition can be brought about, under conditions of fixed amplitude loading with constant positive R values, by either fixing the maximum K value and changing Kmin and thus R, or by fixing R and changing K,,,, or by changing both R and K,,, at the same time. The first two of these techniques were used for the threshold tests described here. Initially five specimens were tested at different but individually constant mean stress intensity levels. The values of stress ratio were then adjusted to bring about the threshold condition. Using this technique a full range of maximum stress intensities from K,,, x K, to K,,, = AKo and R values 0 < R < 1, was covered adequately. This provided a sufficiently wide range of values to reveal whether a simple correlation between stress ratio and threshold level existed. RESULTS A simple relationship between R and AKT was not found to exist. Instead a more complex relationship involving AKT, K,, K,,, and R derived earlier (eqn. 18) was found

6 ~ 156 R. T. DAVENPORT and R. BROOK 03 I/// / I i C EQU 23 0 EQU E EQU to describe accurately the trend of the experimental data obtained. Equation (18) simplifies to : Table 1. Threshold test results AK, Experimental MPa Jm R Kmax Experimental MPaJm Eqn. (22) Theory MPa Jm

7 The threshold stress intensity range in fatigue 157 where note that the effect of changes in R and K,,, on AKT are not obvious from this equation. However, the relationship can be written more concisely in terms of K,,, and R, which although not independent, does help to give a clearer indication of the effects of variation in their respective values, i.e. The data-fit provided by eqn. (22) is given in Table 1 and Fig. 1. Also shown in Fig. 1 are relationships suggested by others (eqns. (7), (lo)), a simple cube-root relationship mentioned earlier (eqn. (20)) and, for comparison, the simple square root relation Equation (22) predicts that for constant K,,,, as R increases then AKT decreases. It also follows that, for constant R, as K,,, increases, then AK, will again decrease. This is demonstrated clearly by the data obtained. The value of AK, was determined by carrying out fatigue test with Kmin < K,,, and establishing threshold conditions. It was not possible, with the technique used, to achieve exactly the condition Kmin = 0. The approximate value of AK, obtained was 105 MPa Jm. The value of Kc used to predict AKT corresponded to the equivalent energy Klc, [12-141, i.e. 151 MPa Jm. DISCUSSION From an examination of eqn. (21) it can be seen that a simple power law relating AK, and R does not exist. However, it has been shown that under certain conditions a simple power law can be used to give an approximate relationship between these two parameters. It has also been shown that the linear relation between R and AKT does not hold for this material. However, a severe conservative estimate can be obtained by assuming the existence of a linear dependency. The fact that AKT has some K,,, dependence indicates that as K,,, increases during a fatigue test than the value of AKT decreases. Consequently the reduction in crack growth rate, due to the threshold effect, will be less as AK increases. This effect is then enhanced at high AK x AKc by the catastrophic influence which increases the slope of the crack growth vs AK plot. Threshold therefore becomes very much less important as AK increases. The large number of correlations proposed for AKT relationships illustrates the lack of a complete understanding of the processes which cause the fatigue threshold to exist. However, empirical results often support the simple power law, first suggested by Klesnil

8 158 R. T. DAVENPORT and R. BROOK and LukaS, as a reasonable approximation to threshold variation. This fact is further supported here, with the proviso that certain conditions must apply to the values of AK, and K,. It is felt that the equation proposed here to correlate AKT adequately describes the effect of R and K,,, on the value of AKT. Furthermore it provides a means of assessing easily the effect of variations in R and K,,,, individually or together, on the threshold stress intensity factor. CONCLUSION The threshold stress intensity in fatigue (AKT) has been shown to depend in a complex manner on the fracture toughness (K,) of the material, the maximum stress intensity (KmaX) and the stress ratio (R). A cube-root relation is proposed which satisfies requirements that are basic to the concept of the fatigue threshold and also phenomenological data ; the latter is exemplified, in this instance, by the behaviour of a 0.15%C 1.5%Mn steel. Acknowledgements-The authors thank Professor G. W. Greenwood for providing facilities for this research in the Department of Metallurgy, University of Sheffield. Dr. R. T. Davenport is also grateful to the Health and Safety Executive of the Department of Health and Social Security for their financial support. REFERENCES [l] Paris, P. C. and Erdogan, F. A Critical Analysis of Crack Propagation Laws, J. Basic Engng 1973, 85D, [2] Ritchie, R. 0. Near Threshold Fatigue in Ultra-high Strength Steel; Influence of Load Ratio and Cyclic Strength, J. Engng Muter. & Design July 1977, [3] Hoeppener, D. W. and Krupp, W. E. Prediction of Component Life by Application of Fatigue Crack Growth Knowledge, Engng Fract. Mech. 1974, 6, [4] Forman, R. G. Study of Fatigue Crack Initiation from Flaws Using Fracture Mechanics Theory, Engng Fract. Mech. 1972, 4, [5] Nicholson, C. E. Influence of Mean Stress and Environment on Crack Growth, Conf. Proc. Mechanics and Mechanisms of Crack Growth, Cambridge, U.K. 1973, pp British Steel Corp. [6] Klesnil, M. and LukaS, P. Effect of Stress Cycle Asymmetry on Fatigue Crack Growth, Muter. Sci. Engng 1972,9, [7] Obianyor, D. F. and Miller, K. J. The Effect of Stress History on Fatigue Crack Retardation Behaviour, J. Strain Anal. 1978, 13 (I), [8] Masounave, J. and Bailon, J. P. The Dependence of the Threshold Stress Intensity on the Cyclic Stress Ratio in Fatigue of Ferritic-Pearlite Steels, Scripta metall. 1975, 9, [9] McEvily, A. J. Current Aspects of Fracture, Conf. Proc. Fatigue 77, Cambridge, U.K. 1977, pp. 1-9: Metals Society. [lo] Wei, R. P. and McEvily, A. J. Fracture Mechanics and Corrosion Fatigue, Conf. Proc. Corrosion Fatigue, Storrs, Conn., U.S.A. 1971, pp : N.A.C.E. [I 11 Bamard, S. and Child, J. M. Higher Algebra. MacMillan, London, [ 121 Witt, F. J. Equivalent Energy Procedures for Predicting Gross Plastic Fracture, 4th Natn Symp. on Fracture Mechanics, Carnegie-Mellon University, [13] Witt, F. J. and Mager, T. R. A Procedure for Determining Bounding Values on Fracture Toughness K,, at any Temperature, 5th Natn Symp. on Fracture Mechanics, University of Illinois [14] Harrison, R. P., Loosemore, K. and Milne, I. Assessment of Integrity of Structures Containing Defects, C.E.G.B. Research Report R/H/6 Revision 1, 1977.