A fracture mechanics analysis of ultrasonic fatigue

0013-7944/82/030365-wu)3.0010 Pergamon Press Ltd.

Engineering Fracmre Mechanics Vol. 16, No. 3, pp. 365-371. 1982 Printed in Great Britain

A FRACTURE MECHANICS ANALYSIS FATIGUE

OF ULTRASONIC

W. HOFFELNER and P. GUDMUNDSON Brown Boveri Research Centre, CH-5405Baden, Switzerland Abstract-The method of ultrasonic fatigue finds increasing interest in materials science. Especially, fatigue crack growth rates near the threshold stress intensity range, AK,, can be determined with this method in reasonable times providing no frequency and corrosion effects exist. But for an accurate application of this technique it is necessary to improve the testing systems and also the determination of the dynamic cyclic stress intensity range, AK. In this paper, fatigue crack growth experiments at ultrasonic frequencies with diferent meun stresses and also the calculation of the dynamic stress intensity range with finite elements are treated. On this basis fatigue crack growth curvesat room temperature of the alloys Hastelloy X and IN 800 were measured and compared with results obtained at low frequencies. No significant influence of frequency couldbe foundin these materials.

1. INTRODUCTION fatigue testing, a specimen performs resonant oscillations with high displacement amplitudes. The resulting cyclic deformation can be used to determine the fatigue properties of the sample material. Fatigue limits are usually measured with cylindrical specimens forced to perform longitudinal resonant oscillations of about 20 kHz, which leads to cyclic tension-compression with maximum deformation in the middle of the specimen. Many papers concerning this type of investigations are reviewed in [ I]. The introduction of fracture mechanics concepts into materials science offered a new application for the method of ultrasonic fatigue. As shown in [24] longitudinally oscillating fracture mechanics samples could be used for studies of fatigue crack propagation (Au/AN) as a function of the cyclic stress intensity range AK near the fatigue threshold stress intensity (AKJ. For the determination of these crack growth curves two basic assumptions had to be made: As the experiments were carried out with a mean load of zero, for the determination of the cyclic stress intensity range AK only the positive half of the strain cycle was used. Assuming only this part contributing to crack propagation and neglecting possible crack closure effects this was set as equivalent to a cyclic deformation with R = 0 (R = gmin/umax ). The other problem is the determination of the cyclic stress intensity range AK in these oscillating systems. This was done in all cases by combining the static stress intensity factor with the stress obtained from the resonance solution without a crack. In the present investigation improvements in these two points will be treated in more detail. Possibilities of superposition of static loads are discussed and dynamic stress intensity factors are calculated using the finite element method. IN

ULTRASONIC

2. TESTING SYSTEMS Basically, ultrasonic fatigue testing systems consist of a piezoelectric transducer driven by a high power sine generator. Normally, the displacement amplitude at the end of the transducer is too small to produce the necessary cyclic deformation in the specimen. Therefore, one or more mechanical amplitude transformers, so called horns, are fastened to the transducer. The longitudinal natural frequency of all parts is 20 kHz and therefore the whole system oscillates with this frequency. A specimen with a natural frequency of 20 kHz is coupled onto this driving part. In the simplest case this is a longitudinally oscillating fracture mechanics sample vibrating without static mean load as already mentioned in the introduction. The superposition of a static mean stress offers a system which has been described for smooth specimens[&9]. For this purpose the specimen is screwed between two cyclindrical coupling pieces having zero displacement in their middle. In these displacement nodes the static load can be applied without affecting the oscillations. Such a system with a center notched specimen is shown in Fig. 1. The geometry of the specimen was similar to the geometry used for fatigue experiments under high alternating strains[lO]. Alternatively specimens were also 365

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coupled with bolts to appropriate coupling pieces. Although fatigue crack growth takes place this configuration does not oscillate very stably and besides longitudinal oscillations also other modes of oscillations (plate modes, bending modes) can occur. Another possib~ity offers the use of transverse oscillating samples as described in [S]. The principle of this method is shown in Fig. 2. A 3-point bend specimen with a transverse resonant frequency of 20 kHz is supported in the displacement nodes and coupled to a longitudinally oscillating dynamic actuator. The displacement amplitude can be measured with an electrodynamic probe as described in [7]. Finally, it should be pointed out that also other types of specimens can be cyclically deformed at ultrasonic frequencies provided they have the necessary natural frequency and that the boundary conditions can be fulfilled experimentally. 3. DETE~INATION OF THE STRESS INTENS~Y FACTOR As already mentioned in the introduction, in the case of zero mean stress the cyclic stress intensity range is usually calculated from the relation:

where E is Youngs modulus and Y is the quasistatic geometry factor which is sometimes~41 multipIied

R

-transduce

Fig. 1. Schematic drawing of an ultrasonic fatigue system for crack growth measurements on longitudinally oscillating samples at different mean stresses.

Fig. 2. Schematic drawing of an ultrasonic fatigue system for crack growth measurements on transversally oscillating 3-point bend samples at different mean stresses[5].

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361

with a correction accounting for the change in cross section due to crack propagation. The strain range A6 is determined either directly by strain gages or it is calculated from the range of displacement Au0 at the end of the sample via the relationship

where A is the wavelength of sound in the specimen material. In our case the dynamic stress intensity factors were calculated with the general purpose finite element program ADINA [ll]. Two different geometries were treated, the three-point-bend specimen and the centrally cracked bar. Both cases were treated as plane problems. Except for the crack tip region, 8-noded isoparametric elements were used. The crack tip was modeled by 6-noded isoparametric triangular elements having their mid-side nodes adjacent to the crack tip in the quarter point of this side (as shown in Fig. 3). These elements show the expected r-l” singularity in stress. The stress intensity factors were calculated from the crack tip opening displacement[l2]. Only the two nodes closest to the crack tip were used. This method of calculating Kr is consistent with the interpolating displacement function of the triangular element. With the description given in Fig. 3 we can write

where p denotes Poisson’s ratio. All calculations were performed with a lumped mass matrix, only positive R-values were considered. 3.1 Three-point bend specimens The geometry of the specimens is shown in Fig. 6. The dimensions correspond to the 20 kHz-bending mode. For four different crack lengths, calculations were performed for the static and the dynamic situation, respectively. For the static calculation, a concentrated force, applied at the middle of the specimen, was considered. Because of the symmetry, only one half of the specimen had to be considered. In order to calculate the stress intensity factor for infinitesimal cracks finite element calculations for a specimen without crack were carried out. In this case the specimen was modeled by 16 elements with 65 nodal points and 120 degrees of freedom. The stress intensity factors for a specimen with an infinitesimal crack were obtained according to the static solution for an edge crack: K1 = 1,12.5~& where co (computed without crack) is the stress at the point of the crack.

Y

t x

crack Fig. 3. Finite element model of the crack tip.

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The finite element model of non zero crack lengths consisted of 54 elements with 189 nodal points and 360-366 degrees of freedom, Fig. 4. The stress intensity factor can be written in the following way:

(4) where u. denotes the displacement in the load line. In Fig. 6, f2 is shown as a function of crack length in -I

, B

~ Fig. 4. Finite element mesh for the three point bend specimen (a/w = 0.25).

Crackregto!i --_-j,

Fig. 5. Finite element mesh for the central cracked bar (a/w = 0.5).

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the static and the dynamic case. The differences between the static and the dynamic case are due to the differences in the stress fields for a static and a dynamic situation respectively. 3.2 Center cracked bar The geometry of this specimen is similar to the specimen shown in Fig. 7. The finite element model consisted of 62-74 elements with 205-235 nodal points and 359428 degrees of freedom (Fig. 5). Again because of symmetry, only one quarter of the specimen was modeled. The stress intensity factor can be calculated from the following relationship:

where u,, is the displacement at the end of the specimen. The function f3 is shown in Fig. 7. The static case was taken from Ref. [13]. For the same reasons as for the three-point-bend specimen, there are differences between the static and the dynamic stress intensity factors. For comparison KI was also calculated from eqns (1) and (2) with a geometry factor Y as given in [ 131.It can be seen that this approach is a very good approximation of the dynamic case up to an a/w-ratio of 0.3. Similar results were obtained using the strain at the center of the specimen instead of the displacements uo. Therefore, the conclusion can be drawn that both quantities can be used satisfactorily for the calculation of the stress intensity factor in the situation of ultrasonic fatigue as far as center cracked bars and positive R-values are concerned. But it must be pointed out that these calculations can be applied to other situations only on a very limited basis. Especially in case of zero mean stress which is normally used for ultrasonic fatigue crack growth experiments the difference in compliance between tension and compression can strongly influence the

cl1

2

3

4

5

a/w

Fig. 6. Dynamic and static solutions of the geometry function h

for a three point bend specimen

(L = 65 mm, w = 20 mm, p = 0.3, E = 2.O.lo’MPa).

31

2

3

4

5

6

VW Fig. 7. Dynamic and static solutions of the geometry function h(a/w, w/L,p)for a center cracked bar. (2L=92mm,2w=20mm,~=0.35,E=1.3~1~MPa).

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displacement field which is not accounted for in the above approach. Troubles can also arise in the non-symmetric case as, for example, with single edge notched samples where higher order bending modes also occur with increasing crack length. 4. DET~A~ON OF FATIGUECRACKGROW CURVES With the 3-point bend testing machine shown in Fig. 2 Au/AN-AK-curvesof the nickelbase alloys IN 800 and Hastelloy X were determined at 20 kHz. The stress intensity factors were calculated from eqn (4). The results are shown in Figs. 8 and 9. For comparison, experiments with DCB-type specimens at low frequencies have also been performed. To make an accurate comparison at low frequencies, AK also was calculated from the displacement in the load line. Although there is a difference of several orders of mag~tude in frequency there is a nice agreement in the fatigue crack propagation behaviour for these alloys. In case of IN 800 the t~eshold seems to be slightly lower at 20 kHz than at 60 Hz. But at both frequencies no inthrence of mean stress on the fatigue crack growth behaviour between R = 0,l and R = 0,3 could be found. Also in the case of Hastelloy X no significant difference between crack

1

d 1

2 cycltc

4 S,,I?SS

6

810

~nterwly

40

20 range.

f!,K

60

80

100

[MN ti”“]

Fig. 8. Fatigue crack propagationrates of HastelloyX at differentfrequencies(Literaturedata from[14]).

Fig. 9. Fatigue crack propagationrates of IN-800at differentfrequencies.

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rates at low and high frequencies were detected (as shown in Fig. 8). Although for a]] these materials fatigue crack propagation rates seem to be relatively insensitive to frequency this should not be generalized to all materials. Crack propagation rates could be affected by strain rate sensitivity of plastic deformation and also by environmental interactions. Both effects are expected to be small in the case of IN 800 and Hastelloy X because these are austenitic materials where normally plastic deformation shows only low strain rate sensitivity at RT and they do not show any corrosive attack in laboratory air.

propagation

5. CONCLUSION The method of ultrasonic fatigue can be used for the determination of fatigue crack growth rates as a function of the cyclic stress intensity range-especially in the near-threshold region. For this purpose, testing equipment allowing superposition of static loads is required as well as an accurate knowledge of the dynamic stress intensity range. As shown for longitudinally and transversally oscillating samples, static loads can be conveniently applied in displacement nodes. The finite element method could be used successfully, for the determination of the dynamic cyclic stress intensity range. It was shown, that in case of longitudinally oscillating center cracked bars the commonly used approximation for AK works well for short cracks. But it must be pointed out that these considerations are valid only in case of R 2 0 and cannot be applied without modifications to R ~0. However, with the finite element calculations fatigue crack growth rates of the nickel-based alloys IN 800 and Hastelloy X were determined as a function of the cyclic stress intensity range at different mean stresses. A good agreement with values at low frequencies could be found. But it should not be concluded that this is a general materials behaviour. Finally it should be pointed out that our considerations are not limited to room temperature only. Similarly to investigations on smooth samples [9,15] crack propagation measurements can also be performed at elevated temperatures. But with increasing temperature the frequency effects are expected to become more and more pronounced. Acknowledgemen&-The authors would like to thank Prof. J. Carlsson for valuable discussions. This work was financed in part by the Swiss Federal Government and performed within the frame work of COST-action 50.

REFERENCES [I] L. E. Willertz, Ultrasonic fatigue. Znf.Met. Rev. 2, 65-77 (1980). [2] S. Purushothaman and J. K. Tien, Slow crystallographic fatigue crack growth in a nickel-base alloy. Met. Trans. A, 9A, 351-355 (1978). [3] St. Stanzl and R. Mitsche, High frequency fatigue of metals, crack initiation and propagation. Fracture 1977Conf. Proc. ICF 4, Waterloo, Canada, June 19-24, 1977,749-751. [4] B. Weiss, R. Stickler, J. Fembdck and K. Pfaffinger, High cycle fat&e and threshold behaviour of powder metallurgical MOand Mo-alloys. Fatigue Engng Material Structures 2(l), 73-84 (1979). [5] W. Hoffelner, Fatigue crack growth at 2OkHz-a new technique, /. Phys. E.: Sci. Instr. 13,617619 (1980). [6] E. A. Nappiras, Techniques and equipment for fatigue testing at very high frequencies. Proc. ASTM 59,691-710 (1959). [7] K. Kromp, W. Kromp, B. Weiss and K. L. Maurer, Fatigue testing of metals at ultrasonic frequency. Materialpriif. U(9), 297-301 (1973). [8] A. Horsewell and I. Hansson, Ultrasonic fatigue of an austenitic stainless steel. Fatigue Engng in Material Structures 2, 97-106 (1979). [9] W. Hoffelner and M. 0. Speidel, COST-SO,2nd round Final Rep., BBC-KLR (1981). [IO] W. Hoffelner and C. Wtithrich, Fatigue crack growth rates in center notched specimens at high strain amplitudes, ht. 1. Fracture 17, R87-R90 (1981). (1I] K. J. Bathe, ALMNA:A Finite Element Program for Automatic Dynamic Incremental Nonlinear Analysis. MIT (1978). [I21 C. F. Shih, H. G. deLorenzi and M. D. German, Crack extension modelling with singular quadratic isoparametric elements. Inr. J, Fracture 12,647651 (1976). U3] H. Tada, P. C. Paris and G. R. Irwin, The Stress Analysis of Cracks Handbook. Del Research Corporation Hellertown, Pennsylvania, 2.16-2.17(1973). [l4] S. R. Holdsworth, COST-SO,1st round Final Report, GECB (Whetstone) Rep. No. W/CML/l977/49. WI S. Purushotaman, J. P. Wallace and J. K. Tien, High-Power Ultrasonic Fatigue. Ultrasonicsht. Conf. Proc., IPC Press Surrey, England, pp. 244-249(1973). (Received 18 May 1981;received for publication 21 July 1981)