An alternative method of decreasing K FCGR testing

International Journal of Fatigue 22 (2000) 593–600 www.elsevier.com/locate/ijfatigue

An alternative method of decreasing ⌬K FCGR testing S. Sivaprasad a, S. Tarafder b

a,*

, M. Tarafder a, K.K. Ray

b

a Fatigue and Fracture Group, National Metallurgical Laboratory, Jamshedpur-831 007, India Department of Metallurgical and Materials Engineering, Indian Institute of Technology, Kharagpur-721 302, India

Received 28 July 1999; received in revised form 1 March 2000; accepted 1 March 2000

Abstract The conventional technique of decreasing ⌬K FCGR testing often presents experimental difficulties in terms of the length of crack that must be accommodated and the time required to reach low values of ⌬K. Addressing these issues, an alternative method of conducting such tests has been proposed. A relation according to which ⌬K must be decreased has been derived assuming that the monotonic plastic zone size can be decreased at a constant rate with increase in crack length without adversely affecting the integrity of the FCGR data. Through experimental assessment of the alternative method and examination of crack closure effects, such an asumption has been vindicated. It has been shown that this method is particularly suitable for high strength materials and that it provides great advantages when conducting FCGR tests at low frequencies. The employment of the method for obtaining threshold regime corrosion fatigue crack growth data has been demonstrated for HSLA steels.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Fatigue test methods; Corrosion fatigue; HSLA steels

1. Introduction Decreasing ⌬K fatigue crack growth rate (FCGR) tests are conventionally conducted by a load-shedding procedure as documented in the ASTM standard E 647 [1, pp. 569–596]. In this procedure, loads are progressively decreased as the crack length increases during a FCGR test such that the ⌬K envelope of the test is forced to follow the relation ⌬K⫽⌬KoeC(a−ao)

1 dr ⫽C⬘ r da

(2)

where C⬘ is a constant. The monotonic plastic zone size r is given by [3]

(1)

In the above equation, ⌬Ko and ao are the stress intensity factor range and crack length respectively with which the test is started. C is a negative constant, the value of which has been standardised at ⫺0.08 mm⫺1 so as to ensure that there are no delay effects originating from the progressive unloading. The procedure for conducting decreasing ⌬K FCGR tests, as described above, was proposed by Saxena et al. [2]. It is based on the requirement that the monotonic plastic zone associated with Kmax of the fatigue cycle * Corresponding author. Fax: +91-0657-426527. E-mail address: [email protected] (S. Tarafder).

should be decreased ‘at a rate such that the fractional change in the (plastic) zone size remains constant with increase in a’. Mathematically, this requirement can be expressed as

r⫽

冉 冊

1 Kmax 2p sy

2

(3)

where sy is the yield stress of the material under test. Substitution of Eq. (3) into Eq. (2), and integration after rearrangement yields Eq. (1) for FCGR tests under constant load ratio, R. It may be noted that the constants in Eqs. (1) and (2) are related as C=C⬘/2. A schematic of the reduction of ⌬K with increase in crack length as per Eq. (1) during a decreasing ⌬K FCGR test is shown in Fig. 1. Tests are normally started with a sufficiently high ⌬Ko that will ensure easy initiation of fatigue cracks. It can be visualised from the figure that in order to grow down to low values of ⌬K, fatigue cracks of substantial lengths have to be accom-

0142-1123/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 0 ) 0 0 0 2 9 - 3

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monotonic plastic zone attending the crack tip. The situation during decreasing ⌬K FCGR testing is similar, and in order to minimise retardation effects, loads must be decreased such that the fractional change in the monotonic plastic zone size is small. This condition can be written as −⌬r ⫽x r

Fig. 1. Schematic reduction of ⌬K with a as per conventional procedure represented by Eq. (1) and the proposed method represented by Eq. (10).

modated in the specimen. As an example, initiating a FCGR test at a ⌬Ko of 25 MPa√m, and using the recommended value of C, the fatigue crack would have to be grown through 苲20 mm in order to attain a ⌬K of 苲5 MPa√m. Moreover, as lower ⌬K values are achieved cracks have to be grown through larger and larger increments in order to produce a given reduction in ⌬K. Due to this, and because crack growth rates decrease as the ⌬K is lowered, longer time intervals are required to produce progressive reductions in ⌬K. Conventional decreasing ⌬K FCGR tests therefore demand that (i) larger specimens are used so that long fatigue cracks can be accommodated, and (ii) tests be conducted at high frequencies to cut down on the testing time. To the experimentalist, these are often not viable options, as for example when the product or component size restricts dimensions of specimens, or when FCGR tests are to be conducted in corrosive media at low frequencies to study corrosion–fatigue behaviour. In order to alleviate the above experimental problems associated with conventional decreasing ⌬K FCGR testing, an alternative methodology for conducting such tests has been proposed in this paper. Further, the application of the method has been experimentally implemented in order to ensure that the FCGR data obtained is free from unwanted retardation effects, notwithstanding the faster decrement rate of ⌬K. 2. Alternative method 2.1. Analytical derivation of ⌬K envelope Studies on effect of overloads on fatigue crack growth have clearly shown that reduction in the cyclic amplitudes of concatenated blocks of fatigue cycles results in retardation of crack growth rates [4–7]. This retardation is proportional to the relative decrease in the size of the

(4)

where x is a constant which is smaller than unity, and ⌬r denotes the change in the plastic zone size between reduction of loads, its sign indicating a decremental change. Retardation after the imposition of overloads is known to be operative through a distance that is proportional to the extent of the overload plastic zone [6,7]. Crack growth rates recover to their original levels only after the fatigue crack has been grown out through this distance. For the case of decreasing ⌬K FCGR testing, a similar situation may arise if crack growth retardation effects are brought about by reduction of cyclic amplitudes as the crack grows. In order to prevent accumulation of such retardation effects significantly affecting crack growth rates, it is necessary that cracks be grown out through multiples of prior plastic zone dimensions before subsequent reductions of cyclic amplitudes. This requirement can be stated as ⌬a⫽yr

(5)

in which ⌬a is the crack growth increment between reductions of load, and y is a constant greater than unity. Eqs. (4) and (5) are applicable when a stepped loadshedding scheme is being implemented manually to conduct decreasing ⌬K FCGR tests. In the limiting case of continuously decreasing fatigue cycling amplitudes in an automated test procedure, the rate of change of plastic zone size with crack length can be obtained by combining Eqs. (4) and (5) to give dr x ⫽⫺ da y

(6)

Differentiation of the plastic zone size relation in Eq. (2) and substitution into Eq. (6) yields dKmax x Kmax ⫽⫺ ps2y da y

(7)

which on integration with initial limits of ao and Kmaxo (the Kmax of the fatigue cycle at start of test) produces

冑

Kmax⫽ K 2maxo−Q(a−ao)

(8)

with x Q⫽2ps2y y

(9)

S. Sivaprasad et al. / International Journal of Fatigue 22 (2000) 593–600

to represent the path of change of Kmax of the fatigue cycles with crack growth. From Eq. (8), for the case of tests with constant R-ratio, the variation of ⌬K with crack growth can be written as

冑

⌬K⫽ ⌬K 2o−Q(1−R)2(a−ao)

(10)

with Q given by Eq. (9). Eq. (10) thus represents the ⌬K envelope to be used as per the proposed alternative method. 2.2. Mathematical form of derived ⌬K envelope The basic difference between the exponentially decreasing conventional procedure [Eq. (1)] and the proposed methodology [Eq. (10)] lies in the rate at which the monotonic plastic zone size is allowed to decrease with growth of the fatigue crack. For the conventional technique this is given by Eq. (2); dr/da is proportional to r; i.e. as r decreases with progressive reduction of ⌬K, the magnitude of dr/da is continuously decreased. In contrast, for the proposed method, dr/da remains constant throughout the test, as shown by Eq. (6). It is contended that the philosophy based on which the rate of decrease of the monotonic plastic zone size is controlled in the conventional procedure is overly conservative. As the derivation of the proposed method reveals, it should be logically acceptable to decrease the plastic zone size at a constant rate during FCGR testing. This is, of course, subject to experimental verification, which is presented later. The conventional procedure for conducting decreasing ⌬K FCGR tests employs an exponentially decaying ⌬K envelope, which is apparent from inspection of Eq. (1). In comparison, the nature of the proposed ⌬K envelope in Eq. (10) is unclear. Eq. (10) can be re-written as

冋

Q(1−R)2 ⌬K 2o ⌬K 2⫽⫺4 (a⫺ao)⫺ 4 Q(1−R)2

册

(11)

This is in the image of the general equation of a parabola and hence the proposed alternative ⌬K envelope is parabolic in nature. In Fig. 1, the proposed parabolic ⌬K envelope is shown. It is immediately apparent that, for appropriate choice of controlling parameters, it should be possible to achieve lower ⌬K levels at smaller amounts of crack extension by this methodology, in comparison to the conventional procedure. 2.3. Factors affecting the rate of ⌬K reduction The rate of reduction of ⌬K as per the proposed methodology is essentially controlled by the parameter Q in Eq. (10). It is apparent that as Q, given by Eq. (9), increases, ⌬K is reduced at a faster rate. The value of Q, in turn, is dependent on the various parameters on the right hand side of Eq. (9).

595

The quantities x and y in Eq. (9), originating from the conditions given by Eqs. (4) and (5) respectively, are basically arbitrary. As an informed estimate, x can be taken as 0.1 and y as 10, giving x/y=0.01. It may be noted that decrement of the plastic zone size by 10% (x=0.1) corresponds to reduction of ⌬K or load by 3.16%. It may also be noted that after reduction of ⌬K, the crack has to be grown out through sufficient distance (y=10 implies 10 plastic zone sizes) in order to preclude the accumulation of retardation effects. In the proposed formulation, the ratio x/y is important and not the individual values of x and y. Thus x can be taken as 0.02 and y as 2 to give the same value of x/y of 0.1. When conducting tests under computer control, it can be expected that load reductions are implemented using very fine steps maintaining a given value of x/y, so that the rate of reduction of plastic zone size is ⫺x/y according to Eq. (6). The constant 2p in Eq. (9) comes from the coefficient 1/2p in the plastic zone size relation in Eq. (3) that is for a state of plane stress. For the case of FCGR testing, a high degree of stress triaxiality can be assumed to exist. A plane strain plastic zone size is obtained by replacing 1/2p with 1/6p in Eq. (3)[3]. The constant 2p in the relation for Q [Eq. (9)] may therefore be replaced by 6p. This would increase the value of Q and consequently permit an enhanced rate of ⌬K reduction. Conversely, if 2p is retained in Eq. (9), the actual rate of decrease of plastic zone size with crack length will be three times slower than what is sought to be controlled through the ratio x/y. It is clear from Eq. (9) that the yield stress, sy, of the material under test affects the value of Q. It can be readily appreciated that for higher strength materials, ⌬K can be reduced at a faster rate since the value of Q increases. From Eq. (10) it is obvious that the R-ratio at which a test is conducted also affects the rate of reduction of ⌬K. For the same value of Q, ⌬K will be reduced at a lower rate for higher values of R. This is due to the larger size of the monotonic plastic zones at higher R-ratios, and the consequent need for growing out the crack through a larger distance prior to reduction of loads. Based on the discussions made above, the relation for Q [Eq. (9)] can be stated more explicitly for experimental use. Taking x/y as 0.01, retaining the coefficient 2p and introducing a user preference parameter t for controlling the rate of ⌬K reduction, Eq. (9) can be re-written as Q⫽0.0628 s2yt

(12)

The experimentalist may use a suitable value of t to exercise control on the rate of ⌬K reduction. Considerations of plane strain plastic zone size and variations in the values of x and y can thus be implemented through suitable choice of the value of t.

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Table 1 Chemical composition (wt%) of HSLA steels used Steel

C

Mn

P

S

N

Si

Cr

Mo

Al

Nb

Ni

Cu

HSLA-80 HSLA-100

0.05 0.06

1.00 0.84

0.009 0.011

0.001 0.003

0.01 0.008

0.34 0.25

0.61 0.74

0.51 0.58

0.025 0.023

0.037 0.03

1.77 3.47

1.23 1.54

3. Experimental verification As stated earlier, it is imperative that the proposed method of conducting a decreasing ⌬K FCGR test at an accelerated rate be experimentally implemented in order to establish that unwanted retardation effects are not introduced. Tests were therefore carried out using the proposed method and the conventional technique, and the FCGR data compared. Comparisons were also made with respect to crack closure behaviour in the two cases. In addition to tests conducted in air, comparative assessments were also carried out for corrosion fatigue crack growth—a situation for which the proposed methodology would be especially suitable. Details of the tests and their results are presented below.

et al. [8] was used by the software for calculation of crack lengths. The software performed on-line crack closure measurements following the recommendations of ASTM task group E 24.04.04 [9]. The FCGR tests were conducted in air and in 3.5% NaCl solution. The containment and loading arrangement used for the corrosion fatigue (CFCGR) tests is shown in Fig. 2. All tests were carried out with R=0.1, and at frequency of 10 Hz for tests in air and at 1 Hz for tests in 3.5% NaCl solution. For conventional decreasing ⌬K tests, Eq. (1) was used to control the ⌬K envelope, using C=⫺0.08 mm⫺1. Tests based on the proposed method employed Eq. (10), with Q calculated from Eq. (12) using t=1, 2 and 3, and the appropriate value of sy. The calculated values of Q are listed in

3.1. Experimental procedure Two varieties of Cu-strengthened HSLA steels, designated here as HSLA-80 and HSLA-100, were employed for carrying out FCGR tests. The chemical composition and mechanical properties of the steels are given in Tables 1 and 2y respectively. The steels were available in the form of strips in the quenched and tempered condition. Standard SENB specimens in L–T orientation, of width 20 mm and thickness 10 mm were fabricated from the strips. The FCGR tests were conducted on a 100kN servohydraulic closed loop testing machine. The machine was equipped with a digital controller, interfaced to a computer. Tests were controlled using software in which the desired ⌬K reduction scheme could be implemented. Crack lengths were monitored by the software, employing the compliance technique. Crack mouth displacements for compliance computation were measured using a 5 mm gauge length COD gauge at knife edges affixed across the crack mouth. A location independent compliance crack length relation proposed by Tarafder

Fig. 2.

Containment and loading arrangement for CFCGR tests.

Table 2 Mechanical properties of HSLA steels used Steel

sYS (MPa)

sUTS (MPa)

%El

%RA

na

Hardness (VHN)

Charpy energy (J)

HSLA-80 HSLA-100

650 840

715 884

24.2 21.6

75.8 73.5

0.12 0.08

250 300

218 192

a

Obtained from s=ken, s=true stress, e=true strain, in plastic range

S. Sivaprasad et al. / International Journal of Fatigue 22 (2000) 593–600

597

Table 3 Values of Q used for FCGR tests as per the proposed method Steel

sy (MPa)

t

Qa (MPa2)

HSLA-80

682.51

HSLA-100

862

1 2 3 1 2 3

29 252.6 58 505.3 87 757.9 46 663.2 93 326.3 139 989.5

a

Q=0.0628s2yt

Table 3 for both the steels. While tests were conducted for all the three values of t for obtaining FCGR data in air, tests in NaCl medium were conducted only for t=2 in the HSLA-100 steel. 3.2. Results and discussion The expected variation of ⌬K with crack extension during FCGR tests based on the method proposed in this paper is shown in Fig. 3 for t=1, 2 and 3, for both HSLA80 and HSLA-100 steels. The ⌬K envelope for the conventional technique is also shown. The ⌬K values are normalised with respect to a ⌬Ko of 25 MPa√m. It is clear from the figure that for a given crack extension, it is possible to achieve lower ⌬K levels in the material of higher strength. For the lower strength material at lower values of t (and hence Q), the proposed method may not lead to any advantage in terms of acceleration of the decreasing ⌬K FCGR test. This is apparent in the case of HSLA-80 with t=1 in Fig. 3; it can be seen that the

Fig. 3. Normalised ⌬K envelope obtained for various values of t in HSLA-80 and HSLA-100 steels (⌬Ko=25 MPa√m).

Fig. 4. FCGR data of HSLA-80 steel, obtained by using the proposed method with various values of t, and the conventional technique.

use of conventional technique leads to the achievement of lower ⌬K levels throughout the allowable range of crack extension. The proposed method can therefore be said to be particularly suitable for materials of relatively higher strength. Figs. 4 and 5 show Paris plots of the FCGR data obtained using the proposed method with t=1, 2 and 3 for HSLA-80 and HSLA-100 steels respectively. The data obtained by employing the conventional technique are

Fig. 5. FCGR data of HSLA-100 steel, obtained by using the proposed method with various values of t, and the conventional technique.

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also included in the plots. As a first impression, it appears from the plots that the data obtained by the proposed method is compatible with the FCGR determined by the conventional technique. Data from the various tests lie within a small scatter band, which is thought to be acceptable. On closer inspection, it is found that data from the fastest test by the proposed method (t=3) in the HSLA-80 steel (Fig. 4) is pronouncedly on the lower side of the scatter band. This is perhaps a manifestation of retardation due to excessively rapid reduction of loads. In order to clarify if retardation processes are active in tests employing the proposed method of decreasing ⌬K, the crack closure behaviour during the tests was examined. Figs. 6 and 7 show plots of Kcl/Kmax against the applied ⌬K for the various tests conducted, for HSLA-80 and HSLA-100 steels respectively. Kcl is the crack closure stress intensity factor that is determined on-line by the testing software, corresponding to the load at 2% deviation from the open crack (i.e. linear) load– COD slope. It can be seen from Fig. 7 that for the HSLA-100 steel, Kcl/Kmax for all the tests with various t values follow the same path with reduction of ⌬K. Hence for the higher strength steel, variation of the rate of plastic zone size reduction, controlled by changing t, does not affect crack closure behaviour, and the FCGR data obtained by the proposed method can be said to be free from retardation effects. For the lower strength HSLA-80 steel, however, it is evident from Fig. 6 that at the fastest rate of plastic zone size reduction (t=3), crack closure levels are higher, indicating that retardation effects have been manifested. It may be therefore be prudent to restrict the value of t to ⱕ2 for this steel.

Fig. 6. Crack closure behaviour in HSLA-80 steel during decreasing ⌬K FCGR tests using proposed and conventional methods.

Fig. 7. Crack closure behaviour in HSLA-100 steel during decreasing ⌬K FCGR tests using proposed and conventional methods.

It may be pointed out that the experimental verification of the proposed method has been carried out within mainly the Paris regime of fatigue crack growth. Limitations of specimen size prohibiting the accommodation of fatigue cracks of sufficient length and the inability to conduct tests at higher frequencies and/or prolonged time periods prevented the verification to be continued to lower levels of ⌬K. The results of the CFCGR tests conducted in NaCl medium on the HSLA-100 steel are presented in Fig. 8. It can be seen that for the common regime of ⌬K in tests conducted by the conventional technique and the proposed method, the crack growth rates are comparable. Additionally it is evident that with the proposed method it is possible to achieve lower levels of ⌬K for essentially the same extent of crack growth. While for the conventional test, crack growth rates were restricted to the Paris regime, the CFCGR data from the test by the proposed method cover a large part of the threshold regime as well. At 1 Hz test frequency, the conventional test took 68 h to complete. It is estimated that in order to grow down to ⌬K of 苲6.5 MPa√m using the conventional testing procedure, the test would have to continue for 苲120 h (not considering that it may not be possible to accommodate the crack length in the specimen). In comparison, the test conducted according to the proposed method required 21 h. It is thus demonstrated that the proposed method of decreasing ⌬K FCGR testing provides tremendous experimental advantage, especially when testing at low frequencies, as during corrosion fatigue crack growth testing. It is well known that FCGR in the threshold regime

S. Sivaprasad et al. / International Journal of Fatigue 22 (2000) 593–600

Q⫽0.0628 s2yt

Fig. 8. Corrosion fatigue crack growth behaviour of the HSLA-100 steel obtained by decreasing ⌬K FCGR tests conducted by conventional and proposed methods.

can be substantially affected by the occurrence of crack closure. Closure can originate from various sources; wedging of reaction products, including oxides, between the crack faces is known to be a potent mechanism for it [10]. The manifestation of such mechanism is governed by the time available for reaction. A shorter test duration obtained using the proposed method may therefore result in lower amount of closure to be exhibited. FCGR tests conducted in aqueous environments are known to be influenced by hydrogen. The interaction of hydrogen with the fracture process zone is a complex phenomenon, with the rate determining step often being a diffusion controlled process like hydrogen transport to the crack tip or hydrogen entry into the material [11]. The duration of test will therefore determine the intensity of the influence of hydrogen. The effect of hydrogen may manifest itself as a change in the resistance of the material to crack growth, and also through alteration in the fracture surface morphology giving rise to altered closure processes.

4. Conclusions An alternative method of decreasing ⌬K FCGR testing has been proposed in this paper. In this method, the decrease of ⌬K with crack growth during a test is controlled by the relation

冑

⌬K⫽ ⌬K 2o−Q(1−R)2(a−ao) where

(13)

599

(14)

t being chosen by the experimentalist to control the speed of the test. The method has essentially been derived from considerations of a constant rate of decrease of the monotonic plastic zone size with increase in crack length. The proposed method has been implemented on two varieties of HSLA steel, and it is shown that by judicious selection of the parameter t in Eq. (14), the integrity of FCGR data obtained can be assured. The method is particularly suitable for use with materials of high strength and in situations where limitations of specimen size are imposed or when the time required for experimentation can be expected to be long. The use of the method to obtain threshold regime corrosion fatigue crack growth data has been demonstrated. It is reiterated that the method proposed in this paper has not been developed to replace the conventional technique documented by ASTM standard E647. It merely provides an alternative in situations where it is difficult or impossible to implement the procedures laid down in the standard.

Acknowledgements This investigation has been carried out as part of the work for the Office of Naval Research, USA, under grant No. N00014-95-1-0015. The authors gratefully acknowledge the research support provided by Dr V.R. Ranganath in this work.

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