Fatigue crack growth threshold: implications, determination and data evaluation

Int. J. Fatigue Vol. 19, Supp. No. 1, pp. S145–S149, 1997  1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0142–1123/97/$17.00+.00

PII: S0142-1123(97)00058-3

Fatigue crack growth threshold: implications, determination and data evaluation H. Do¨ker DLR, German Aerospace Research Establishment, Institute of Materials Research, D-51170 Ko¨ln, Germany

In standard test procedures the determination of fatigue crack growth threshold values is part of da/dN– ⌬K-curve determinations, and with one specimen only one threshold value can be determined. Thus, the complete characterization of the threshold behaviour of a certain material, for example, by ⌬Kth– R-curves, is extremely time consuming, specimen consuming and expensive. Based on experimental experience, a procedure is described which allows a complete ⌬Kth–R-curve determination with a single specimen. The threshold behaviour of a material can be described by four constants, which can easily be determined. A very simple analytical description of the threshold behaviour of materials is given, by which even quite complicated appearing ⌬Kth-R-curves can be represented.  1998 Elsevier Science Ltd. (Keywords: fatigue crack propagation; crack growth threshold; threshold determination; test methods; data evaluation; load ratio; standardization)

INTRODUCTION According to ASTM standard E 6471 the fatigue crack growth threshold ⌬Kth is defined as the asymptotic value of the stress-intensity factor range ⌬K at which the fatigue crack growth rate da/dN of long cracks approaches zero. Though fatigue crack growth threshold values have been determined for at least 30 years, no self-sufficient standard for the determination of threshold values independent from da/dN-curve determinations, could be established up to now. According to standard test procedures like ASTM E 647, the determination of fatigue crack growth threshold values is part of da/dN-curve determinations, and with one specimen only one threshold value can be determined. Thus, the complete characterization of the threshold behaviour of a certain material in accordance with existing standards, for example, by a ⌬Kth–R-curve, is extremely time consuming, specimen consuming and expensive. Therefore, in many cases where threshold data seems to be indispensable, only isolated values are determined, and sometimes, the threshold behaviour of a material is judged on the basis of a ⌬Kth-value determination at a load ratio, R, of 0.1. This proceedure is very questionable and it seems desirable to develop an independent and more economical procedure for the characterization of the threshold behaviour of materials. Based on the now 30 years of experience with threshold determinations, it is possible to propose such a procedure. First of all, threshold determination has to be decoupled from the da/dNcurve determination. This makes 10 or more threshold S145

determinations possible with one C(T)-specimen, if the specimen width is large enough. Then, by analysing the threshold data it is possible to reduce the number of threshold determinations necessary to completely characterize a material. GENERAL THRESHOLD BEHAVIOUR The threshold behaviour of a material is usually described by a ⌬Kth–R-curve, which may be influenced by environment and temperature. In many cases, ⌬Kth decreases with increasing positive R-ratio as shown in Figure 1(a). In Figure 1(b) the curve of Figure 1(a) is presented as a Kmax, th–R-diagram according to Schmidt and Paris2. Kmax, th is the Kmax value at threshold. In Figure 1(c) the same curve is shown as a ⌬Kth–Kmax-diagram according to Do¨ker et al.3. The threshold behaviour of Figure 1(a) and (b) was interpreted by Schmidt and Paris2 as a crack closure phenomenon according to Figure 1(d). To propagate a crack, it is thought that a stress–intensity factor range ⌬K ⬎ ⌬K0 is necessary. As long as Kmin of the loading cycle remains lower than the crack opening load Kop, ⌬Kth decreases linearly with increasing positive R according to: ⌬Kth = Kmax, 0·(1⫺R) for 0 ⱕ R ⱕ Rc

(1)

with Kmax, 0 being Kmax at threshold for R = 0. The effective threshold, ⌬K0, is: ⌬K0 = Kmax⫺Kop = Kmax, 0⫺Kop

(2)

For R ⬎ Rc, Kmin is higher than Kop, and ⌬Kth = ⌬K0.

S146

H. Do¨ker

Figure 1 (a–c) Fundamental threshold behaviour of metallic materials in different graphical presentations and (d) presentation of the crack closure threshold model of Schmidt and Paris2

For negative R-ratios ⌬Kth remains constant according to the definition of ⌬K in ASTM E 647 and equals Kmax, 0. The two diagrams in Figure 1(b) and 1(c) are much better suited to represent and understand the threshold behaviour of a corresponding material than the traditionally used ⌬Kth–R-diagram. Figure 1(b) and 1(c) show clearly, that under constant amplitude conditions a crack will grow, if at the same time Kmax ⬎ Kmax, 0 and ⌬K ⬎ ⌬K0. Thus, the threshold behaviour can be described by two independent constants only, namely Kmax, 0 and ⌬K0 (see also Vasudevan et al.4). Experience shows that ⌬K0 is mainly determined by the base material of an alloy, and that the variation within one alloy group is very small. ⌬K0 values for different base materials are proportional to their elastic moduli5 while Kmax, 0 can be strongly influenced by microstructure and environment. ⌬Kth–R-curves of the type shown in Figure 1(a) are often observed for steels and Al-alloys. In many cases, however, the observed ⌬Kth-R-curves deviate considerably from the behaviour shown in Figure 1(a). Neither is the decrease for low R-ratios linear nor is ⌬Kth constant for high R-ratios. Figure 2(a) shows the Kmax, th–R- and Figure 2(b) the ⌬Kth–Kmax-diagram of such a material, a X 2 Cr-Ni-Mo-17-12 steel at 550°C6 (tests performed at DLR). It can be seen, that the Kmax, th–R-curve at low R-ratios and the ⌬Kth–Kmaxcurve for higher Kmax-values can be represented by straight lines with the mathematical equations Kmax, th = Kmax, 0 + ␤R with ␤ ⱖ 0

(3)

⌬Kth = ⌬K0 + ␣Kmax with ␣ ⱕ 0

(4)

with Kmax, 0 and ⌬K0 being the intersection points of the two straight lines with the vertical axes, and ␣ and ␤ the respective slopes. Based on the simple relations

R = Kmin/Kmax and ⌬K = Kmax–Kmin applied to the threshold situation the ⌬Kth–R-curve can be calculated by ⌬Kth = (Kmax, 0 + ␤R) (1⫺R) from Equation (3)

(5)

and ⌬Kth =

1⫺R ⌬K0 from Equation (4) 1⫺R⫺␣

with R ⬍ 1

(6)

Equation (5) represents a downward oriented parabola, which intersects with the positive R-axis at R = 1 and with the ⌬Kth-axis at ⌬Kth = Kmax, 0 (see Figure 3). The curve of Equation (6) crosses the R-axis also at R = 1 and increases asymptotically to ⌬K0. Now only four independent constants (Kmax, 0, ⌬K0, ␣ and ␤) are necessary to describe the threshold behaviour of a material. Compared to the original understanding of Schmidt and Paris, Equations (5) and (6) give a first order approximation of the behaviour of a real material. Figure 3(a) shows the construction of the ⌬Kth–Rcurve. The real ⌬Kth-value is the maximum value of Equations (5) and (6). In region I of Figure 3(a) the threshold behaviour is controlled by Equation (6). For a real material, ⌬Kth can not decrease to 0, as the curve breaks off when Kmax reaches KIc, that is, R reaches (KIc–⌬Kth)/KIc. In region II ⌬Kth is controlled by Equation (5). This means that Kmax, th or, in the interpretation of Schmidt and Paris, that the crack opening load is influenced by Kmin. The transition point between region I and region II can be calculated by comparison of Equations (5) and (6). For negative R-ratios ⌬Kth decreases with decreasing R according to Equation (3), as for negative R-ratios ⌬K is defined as Kmax according to ASTM E 647 [region III in Figure 3(a)]. In region IV Equation (6) becomes relevant again and ⌬Kth remains nearly con-

Fatigue crack growth threshold

S147

Figure 2 (a) Kmax, th–R-diagram and (b) ⌬Kth–Kmax-diagram for a high temperature steel at 550°C; 䊊, measured at constant R; 왖, measured at constant Kmax (Huthmann and Gossmann6)

Figure 3

Construction of (a) a ⌬Kth–R-curve and (b) the influenc of a variation of the parameters ␣, ␤ and ⌬K0 on the ⌬Kth–R-curve

stant. An additional decrease of Kmin has no influence on ⌬Kth. In Figure 3(b) the influence of the magnitude of the four parameters Kmax, 0, ⌬K0, ␣ and ␤ is demonstrated. Its obvious that very different appearing ⌬Kth–R-curves can be described. R-independent threshold behaviour is observed, if ⌬K0 has nearly the same magnitude as Kmax, 0. If Kmax, 0 is much higher than ⌬K0 , the threshold behaviour is mainly Kmax controlled and the deviation at high R-ratios cannot be observed during threshold determinations with constant R-ratio, unless extremely high R-ratios are used. There are very few sources of threshold data in the negative and high positive R-ratio ranges. The data of Kemper et al.7 correlate very well with the curve shown in Figure 3(a). For positive R-ratios numerous data sets are available, and all checked data could be described according to Equations (3) and (4). This was for instance shown3 for data of Schmidt and Paris2. Figure 4 shows the ⌬Kth–R-curve for the data from Figure 26. Data of Davenport and Brook exhibit the same behaviour8. EXPERIMENTAL THRESHOLD DETERMINATION As shown in the preceding paragraph, four independent constant values (Kmax, 0, ⌬K0, ␣ and ␤) can describe the threshold behaviour of a material. These constants are well hidden in the ⌬Kth–R-curve, but can be easily accessed via the Kmax, th–R- and the ⌬Kth–Kmax-diagrams. According to Figure 2, about five threshold values have to be determined at low R-ratio (e.g. 0.1

Figure 4 ⌬Kth–R-diagram for the data from Figure 26; 䊊, measured at constant R; 왖, measured at constant Kmax

ⱕ R ⱕ 0.5…0.7) and about the same number at very high R-ratios, or better, at constant Kmax9. The R-ratios and the Kmax-values should be evenly distributed in the relevant R- and Kmax-regions. The lowest Kmax-value should be chosen ca 20% higher than Kmax at threshold for the highest R-ratio of the constant R tests. The highest Kmax-value has to be lower than the limit for crack growth under constant load for the investigated material in the applied environment. In Figure 2 the values corresponding to the open circles were determined at constant R, whereas the values corresponding to the closed symbols were determined at constant Kmax. It can be seen that the result of a threshold determination is independent of the applied method. By calculating the regression lines through the data

S148

Figure 5

H. Do¨ker

Fracture surface of a 50 mm–C(T) specimen after ⌬Kth-determinations

points, the four relevant constants to describe the threshold behaviour can be determined. For the low R-ratio tests the load shedding procedure of ASTM E 647 may be applied. In our laboratory we use the d.c.-potential drop technique to measure the crack length, and use the crack length signal to control Kmax and Kmin applied to the specimen. After precracking the specimen at low crack growth rates ( 苲 10⫺5 mm cycle⫺1) for ca 1 mm, the applied ⌬K is decreased by 5% every 0.1 mm until it takes more than 1 or 2 million cycles to pass the next 0.1 mm. This means that da/dN ⬍ 10⫺7 mm cycle⫺1 or 5 × 10⫺8 mm cycle⫺1 has been reached. After this, the procedure is repeated (pre-cracking and load shedding) for the next R- or Kmax-value. Figure 5 shows the fracture surface of such a specimen. The nice crack growth markings allow a post-mortem re-evaluation of the specimen. In order to avoid load interaction effects, it is recommended that for the single threshold determinations with one specimen R and Kmax increase with increasing crack length. According to ASTM E 647 the single threshold value is determined via an extrapolation procedure. The crack growth rates (equally spaced between 10⫺6 and 10⫺7 mm cycle⫺1) are plotted in a log da/dN– log ⌬K-diagram, and are extrapolated to 10⫺7 mm cycle⫺1. The corresponding ⌬K-value is taken as ⌬Kth. This evaluation procedure is simply based on the tradition of plotting da/dN-curves in log–logdiagrams. Figure 6 shows that it could be better to use a linear plot of the da/dN-data to determine ⌬Kthvalues. The straight line in Figure 6(a) looks, transferred to the log–log-diagram, startling similar to the low crack growth region of a log da/dN–log ⌬K-diagram. Vice versa the crack growth data points of Figure 6(b) taken from a real threshold measurement arrange on a straight line in Figure 6(a). The regression line through the data points can be easily extrapolated to da/dN = 0. The second advantage of the linear presentation of low crack growth rate data is, that it

Figure 6

is possible to determine an initial gradient of the da/dN–⌬K-curve, which is of interest for crack growth calculations. From the experience of numerous threshold determinations at the DLR laboratory one can state, that the crack growth rates for a valid extrapolation should lie between 2 × 10⫺7 and 5 × 10⫺8 mm cycle⫺1. It is clear from this procedure, that the crack growth rates measured during the single load shedding steps cannot be used as constant amplitude crack growth rates. Nevertheless, especially for the very low crack growth rates, the crack growth values corresponded well with correctly determined da/dN-curves. If an unknown alloy is to be characterized with respect to threshold behaviour and the region of low R-ratio behaviour is unknown, the following procedure can be useful for making a rough estimate of the threshold behaviour of this material (Figure 7): one threshold value at low R-ratio ( 苲 0.1) is determined or is taken from a da/dN-curve, and is entered into a ⌬Kth–R-diagram. Then, a straight line (line a in Figure

Figure 7 Diagram for the estimation of the threshold behaviour of an uncharacterized alloy from one data point and the known behaviour of an alloy with the same base material

4(a) Linear plot and (b) log–log plot of da/dn-data of a single threshold determination

Fatigue crack growth threshold

S149

7) is drawn through this data point and the point {R = 1; ⌬Kth = 0}. A second line (line b) is drawn parallel to the R-axis in a distance corresponding to ⌬K0 or to a ⌬Kth-value at high R-ratio of an alloy with the same base metal as the unknown alloy. Then, the region of low R-ratio behaviour is between R = 0 and R corresponding to the intersection point of these two lines. For metals, at positive R-ratios, line a can be expected to be the lower limit of threshold values of the unknown material.

between the single threshold determinations these should be performed in a sequence with increasing R and increasing Kmax. Between the single threshold determinations the crack should be propagated at the R-ratio or the Kmax-value of the next threshold determination at moderate crack growth rates for a distance that guaranties crack propagation along the whole crack front. A new effort should be undertaken to establish an autonomous crack growth threshold standard.

SUMMARY

REFERENCES

The fatigue crack growth threshold behaviour of metallic materials can be characterized to a first order approximation by two linear equations with four independent constants. The constants Kmax, 0 and ␣ can be easily determined from a Kmax, th–R-curve derived from threshold determinations at low R-ratios. The constants ⌬K0 and ␤, which describe the intrinsic cyclic threshold behaviour, can be derived from a ⌬Kth–Kmax-curve from ⌬Kth-determinations at high R-ratios or at constant Kmax in the same manner. The single threshold determination should render crack growth data between 2 × 10⫺7 and 5 × 10⫺8 mm cycle⫺1. These da/dN-data should be plotted in a da/dN–⌬K-diagram with linear axes. Then ⌬Kth should be determined as the intersection point between the regression line through the data points and the abscissa, which means an extrapolation to da/dN = 0. Several threshold determinations can be performed with one specimen. In order to avoid interaction effects

1 2 3 4 5 6 7 8 9

ASTM E 647, Annual Book of ASTM Standards, Section 3, Vol. 03.01. American Society of Testing and Materials, Philadelphia, PA, 1995, pp. 578–614. Schmidt, R.A. and Paris, P.C., Progress in Flaw Growth and Fracture Toughness Testing. ASTM STP 536. American Society of Testing and Materials, Philadelphia, PA, 1973, pp. 79–94. Do¨ker, H., Bachmann, V., Castro, D. E. and Marci, G., Zeitschrift fu¨r Werkstofftechnik, 1987, 18, 323. Vasudevan, A. K., Sandananda, K. and Louat, N., Materials Science and Engineering, 1994, A188, 1. Liaw, P. K., Leax, T. R. and Logsdon, W. A., Acta Metallurgica, 1983, 31, 1581. Huthmann, H. and Gossmann, O., In 23. Vortragsveranstaltung des DVM-Arbeitskreises Bruchvorga¨nge. Deutscher Verband fu¨r Materialforschung und -pru¨fung, Berlin, 1981, pp. 271–284. Kemper, H., Weiss, B. and Stickler, R., In Fatigue ’87, Vol. II, ed. R.O. Ritchie and E.A. Starke. EMAS, Cradley Heath, 1987, pp. 789–799. Davenport, R. T. and Brook, R., Fatigue of Engineering Materials and Structures, 1979, 1, 151. Do¨ker, H., Bachmann, V. and Marci, G., In Fatigue Thresholds, Vol. I, ed. J. Ba¨cklund, A. Blom and J. C. Beevers. EMAS, Cradley Heath, 1982, pp. 45–57.