An Experimental Rig to Investigate Fatigue Crack ... - Springer Link

Meccanica 38: 19–31, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

An Experimental Rig to Investigate Fatigue Crack Growth Under Dynamic Loading CHEE-HOE FOONG, MARIAN WIERCIGROCH and WILLIAM F. DEANS Centre for Applied Dynamics Research, Department of Engineering, University of Aberdeen, Fraser Noble Building, King’s College; AB24 3UE, Aberdeen, Scotland, U.K. (Accepted: 10 July 2002) Abstract. A novel experimental rig capable of generating a versatile dynamic loading has been designed and tested to overcome the shortcomings of conventional fatigue testing machines such as the difficulty in providing zero crossing aperiodic loading. The main principle of this new design is based on two, single degree of freedom based excited oscillators, where inertial forces act on a specially designed specimen. By changing the natural frequency of the oscillator, the extent of the preloads and pattern of the excitation signal on the shaker, the rig provides a new and robust means of fatigue testing, particularly for aperiodic loading. Key words: Fatigue, Experimental rig, Dynamic loading.

1. Introduction Despite the advancement of technology many uncontrolled failures such as collapsing of bridges, aircraft crashes, sinking of ships and rails cracking still occur. A surface flaw present in a structure will form a crack under the influence of dynamic loading. This crack will then propagate by a further increase of a number of stress cycles and eventually reach a stage where fracture and failure occur. Current theories, assuming constant amplitude cyclic loading and steady crack growth, are used. Throughout previous decades, much research has been published on the phenomenon of fatigue crack growth with respect to a number of stress cycles, but until now no single law has given a satisfactory solution to predict fatigue life. Even though the well-known Paris law [25] is used by many designers, this law does not predict fatigue life when random loading is concerned. Furthermore, the law only represents the intermediate stage of the crack growth curve. The whole fatigue process consists of the initial cracking, intermediate crack propagation and final fracture. It must be considered that progressive loading interacting with crack growth is not well understood. In practice, structures do not experience constant amplitude loading. In order to establish a crack growth law that matches the real life situation, it is necessary to analyse the problem using a different approach. An alternative way to deal with fatigue problem is to analyse the problem from a dynamic point of view. Careful mathematical modelling reveals that fatigue must be studied in the context of dynamic crack propagation, which in turn, can be only fully understood if treated as a nonlinear dynamic system [34]. Introducing comprehensive dynamic loading, it is possible to study the crack growth phenomenon in the time domain rather than with respect to the number of stress cycles. It is essential to monitor crack growth in the time domain when aperiodic, chaotic and stochastic loading occurs. Under aperiodic loading, the stress amplitude

20

Chee-Hoe Foong et al.

applied to the cracked structure varies with respect to time. Therefore analysing crack growth in term of a number of loading cycles does not provide satisfactory information to model the crack growth law. To verify a new mathematical model it is necessary to carry out extensive experimental studies. Most fatigue tests are carried out on conventional fatigue testing machines but several disadvantages are present. Hence to overcome these deficiencies, a novel experimental rig was designed, which uses inertial forces generated by oscillating masses in order to provide the required loading pattern. In this arrangement, a cracked specimen is held in between two masses, on the top and bottom. Preloading forces are applied to these masses to ensure that good contact is accomplished between the masses and the specimen. Furthermore, the loading is able to oscillate across the zero mean load level. The rig is capable of providing dynamic periodic and aperiodic responses to the cracked specimen giving the ability to study the crack growth phenomenon in a more realistic way. This paper presents details of the proposed experimental rig and provides some experimental results, obtained from the preliminary tests. 2. Literature Review Many structures and components are subjected to vibration and, if surface flaws are present, unexpected failure may occur. Throughout the last three decades, many investigators have examined the dynamic behaviour of cracked structures and components such as beams, rotating shafts and turbine blades, etc. The presence of cracks in structures and components introduces a local flexibility that affects the dynamics or stability characteristics. Without any further action, the crack will start to propagate and may eventually lead to widespread structural damage and possible danger to human life. Hence, it is crucial to understand the vibration characteristics of cracked structures and components and the obtained vibration signature is useful for modelling a versatile crack growth law regardless of whether the loading is periodic or aperiodic. The discontinuity in the stiffness of structures and components caused by a local defect has been known since the 1940s by Kirmsher [23] and Thomson [32]. Analysis of the stress and strain in the vicinity of the crack tip was carried out by Irwin [22] who established the stress intensity factor as a function of the remote stress, geometry and crack length. The displacement of a cracked elastic structure as a function of strain energy release rate [31], related to the stress intensity factor, was computed to give the local flexibility (inverse of stiffness) of the cracked region [24]. Knowing the existence of a crack would affect the vibrating characteristic of structures, a substantial amount of literature has been published. Petroski [26] modelled the effects of a crack on an elastic beam and deduced that the vibration amplitude of the cracked beam was three times higher than the uncracked one. Dimarogonas and Papadopoulos [14] used the Paris energy equation [31] to obtain the local flexibility of a cracked shaft subjected to bending. Changes in the eigenvalues and eigenvectors of a cracked cantilever beam have been studied by Yuen [35]. A nondestructive testing method to identify the crack location and size was developed by Rizos et al. [27]. They considered that a cracked cantilever beam consists of two members, left and right of the crack, and expressions for the vibration modes of these parts were derived. By measuring the vibrating amplitudes at two points on the structure, the crack location and size could be obtained from the expressions of the vibration modes. Rizos et al. also claimed that their method of detecting damage in structures was more reliable than using

An Experimental Rig to Investigate Fatigue Crack Growth

21

the natural frequency approach since relatively high errors would be expected for measuring small changes in natural frequencies. The theory of a Euler–Bernoulli beam containing one or more pairs of symmetric cracks was developed by Christides and Barr [7]. Using the Hu-Washizu variation principle, independent assumptions were required for the displacement, strain, stress and momentum fields. The change of the stress and strain field induced by the crack were shown to decay exponentially away from the crack tip. The decay parameter had to be determined experimentally. Warburton [33] pointed out, however, that the decay parameter could be determined from the local flexibility approach. Chondros et al. [4–6] employed the Hu-Washizu–Barr variation principle to derive the differential equation and the boundary conditions of the continuous cracked bar. Recently, Fernández et al. [16] proposed a simplified method to evaluate the fundamental frequency of a cracked Euler–Bernoulli beam. The crack was represented by a hinge-elastic spring. The analytical results were in good agreement with the numerical results obtained from finite element method. Naturally, cracks in a vibrating structure will start to propagate when resonance occurs. Fatigue crack propagation in resonating structural members was studied by Dentsoras and Dimarogonas [10–13]. One of the shortcomings of Paris law is that it only applies to a single mode of loading and, in a situation when combined loading is present [30], only the strain energy density factor could only be utilised. Dentsoras and Dimarogonas then applied Sih’s theory to a cylindrical shaft and, since the strain energy density factor is related to the stress intensity factor, the remote stress of the stress intensity factor is expressed as a function of the natural frequency and the material’s damping. Integrating the crack growth rate numerically, and plotting the number of stress cycles against the crack depth ratio, Dentsoras and Dimarogonas realised that the crack propagation rate was drastically reduced with lower values of material damping factor. If the damping factor was low enough, no sign of crack growth was observed. They postulated that the material damping factor was the main influence on crack arrest. The work discussed so far assumed that the cracks remained open during vibration. The crack will remain open only if the mean load, which causes a static deflection on the beam, was higher than the amplitude of vibration. In this circumstance, a cracked beam could be modelled as a linear spring since, for a given crack depth, the equivalent spring constant remains the same for both directions of loading. However, if no static load or even a small mean load is applied, the crack will open and close depending on the sign of the vibrating cycles. This situation will cause the beam to behave in a nonlinear manner. This arises because when the crack is closed, the beam is treated as an uncracked beam and when it is open the natural frequencies of the beam drops thus causing the spring constant to be different for both directions of loading. In this phenomenon, the nonlinear characteristic of the crack is considered as a bilinear spring. Gudmundson [20] studied the effects of a closing crack experimentally and concluded that the natural frequencies of a beam containing closing crack were almost identical to the uncracked one. Ibrahim et al. [21] modelled a fatigue crack using the bondgraph technique to obtain the frequency–response functions of a cantilever beam. Their numerical results agreed with Gudmundson’s experimental results. Shen and Chu [28] introduced a contact parameter into the assumed stress, strain and displacement expressions of the cracked beam theory proposed by Shen and Pierre [29]. Applying the Galerkin method, a bilinear equation of motion for each vibration mode of a simply supported beam was obtained. The existence of fatigue crack was then analysed, based on the spectrum pattern in the time and frequency domain. Chu and Shen [8] extended their

22


studies to identify the existence of a fatigue crack by proposing a closed-form solution based on two square wave functions to model the stiffness change of a cracked beam under low frequency of excitation. From their previous study, Shen and Chu [28] realised that instead of using a constant amplitude harmonic force to excite the bilinear oscillator, a bilinear forcing function should be incorporated in the solution of a bilinear oscillator to represent the cracked beam. Investigation of a breathing (opening and closing) crack was also carried out by Collins et al. [9], Abrahim and Brandon [1], Friswell and Penny [18], and recently, by Chati and Mukherjee [3]. Despite numerous studies having been carried out to obtain the vibration characteristic of crack structures, nothing has been published on the prediction of the fatigue life of the crack structures under random loading using a dynamics approach. 3. Rig Design Under the influence of vibrations, when surface imperfections such as cracks are present in structures and components, in most cases, the cracks will open and close causing the structure and component to vibrate across its neutral axis. As most structures and components will experience bending rather than tension loading when subjected to vibrations during service, it was decided that fatigue testing should be carried out on specimens which are subjected to bending. To establish a robust mathematical model, extensive experimental studies are needed. Fatigue tests on bending specimens on conventional fatigue testing machine pose several limitations, namely, (a) the bending specimen cannot vibrate across the zero mean load level, (b) the rollers of the bending fixture have a tendency to lose contact with the specimen when aperiodic loading is applied and (c) when modelling is concerned, the load interacting with the cracked specimen cannot be observed and studied comprehensively during the experiment. Taking the above limitations into consideration, a novel experimental rig was designed and is depicted in Figure 1. The first thought to overcome the above limitations is that the rig must be comprised of masses and springs in order to induce the required dynamic loading on the bending specimen. To allow vibrations across the zero mean load level, both ends of the bending specimen must be fixed. To prevent stress concentrations from being induced at the ends of the specimen, metal pins are used to hold the specimen on the specimen support. Furthermore, one side of the bending specimen is slotted to allow the pin to slide in order to prevent buckling. Two masses attached with leaf springs, one at the top and the other at the bottom of the specimen provide inertial forces from base excitation to allow the bending specimen to be loaded across the zero mean load level. A pair of leaf springs are sandwiched in between each mass, which are bolted on tower B, so that the masses will oscillate in a vertical manner. In this case, the force induced by the mass will act vertically on the specimen. To prevent the masses from losing contact with the specimen, preloading forces are needed. This was done by preloading the leaf springs on the top and the bottom of the masses. These preloading forces are equal and opposite in direction so that no load is present on the bend specimen before vibration. Besides considering the limitations mentioned above, the rig must be designed to carried out fatigue tests on different materials. To achieve this, several parameters must be varied in accordance to the type of materials being tested. Hence, tower A and B are allowed to


23

Figure 1. Experimental rig: (a) isometric view; (b) close-up photograph.

slide along the base plate. With this flexibility, the natural frequency of mass A and B, and the preloading forces can be varied. Furthermore, the rig is capable of conducting test on specimen regardless of its geometry by sliding the specimen support along the base plate. From the fatigue point of view, the stress intensity factor, which is initially obtained experimentally, is one of the main parameters to controlling the rate of fatigue crack growth. Fatigue

24


cracking will not exist if the value of the stress intensity factor is less than the threshold value for the material and if it is too high the crack will propagate too fast, causing difficulties in observing fatigue crack growth during the experiment. Therefore the amount of mass required to produce sufficient force on the bending specimen, regardless of the types of materials being tested must be calculated reasonably accurately. In order to calculate the required force which is based on the critical stress intensity factor of the material, equation (1) which is obtained from the British Standard (BS7448 [2]) was applied. KBW 1.5 S × f (a/W )

Fmax =

(1)

where

a f W

=

3(a/W )0.5 [1.99 − (a/W )(1 − a/W )(2.15 − 3.93a/W + 2.7a 2 /W 2 )] 2(1 + 2a/W )(1 − a/W )1.5

(2)

In equation (1), a is the crack length, Fmax is the maximum force, K is the stress intensity factor; a function of remote stress and crack length, and B, W and S is the thickness, the width and the loading span of the specimen, respectively. To ensure that the rig is able to induce fatigue cracking on steels, the value of the critical stress intensity factor for mild steel was used to calculate the quantity of the mass required. The value of the critical stress intensity factor of mild steel is taken from Table 4.31 of Frost et al. [19]. The specimen width, thickness and span was chosen to be W = 40 mm, B = 20 mm and S = 184 mm. And the crack length to width ratio is assumed to be 0.3. Substituting the relevant values to equation (1), the maximum force needed to apply to the mild steel specimen is calculated and is approximately equivalent to 2 kN. The mass required to induce an inertial force of 2 kN with amplitude of 1 mm and excitation frequency of 30 Hz is approximately 56 kg. The amplitude of excitation is limited to 1 mm to prevent extensive plasticity occurring in the vicinity of the crack tip. It is impractical to use such a large mass in the design. Therefore, an alternate approach to design the rig was needed. Instead of calculating the appropriate mass, the value of the mass was set at 4 kg. Reducing the geometry of the bending specimen to 20 mm in width and 10 mm in thickness, the span of the specimen was calculated to be 400 mm. The length of the leaf springs was calculated by applying an approximation developed by Emans [15] for the type of boundary conditions utilised in the design. Hence, the spring stiffness equation is, R = 12

432 y 3 EI yEI + L3 35 L5

(3)

4. Experimental Setup and Preliminary Results The schematic of the experimental layout is depicted in Figure 2. The experimental rig was mounted on a dynamic shaker which is controlled by the function generator. The operational frequency of the dynamic shaker range is from 5–3000 Hz. The base displacement (y) and acceleration (y) ¨ were measured by an LVDT and accelerometer, respectively. The signal from the LVDT was passed through the LVDT signal conditioner and then to the data acquisition system. The data acquisition system digitised data at a rate of 5 kHz. The LVDT was


25

Figure 2. Schematic view of experimental layout.

calibrated to give the actual displacement and is shown in Figure 3(a). The accelerations of mass A (y¨a ) and B (y¨b ) were measured by accelerometers. The forces induced by mass A (fa ) and B (fb ) were measured by force transducers. The signals from the accelerometers and force transducers mentioned above were passed through charge amplifiers, and monitored on oscilloscopes before being captured on the data acquisition system. The relative displacement between the cracked specimen and the base was measured by the eddy current probe on the assumption of small angular displacement. The probe was positioned some distance horizontally away from the center of the cracked specimen and approximately 1 mm above the cracked specimen. To obtained the actual displacement, the probe was calibrated and a best fit curve was obtained as shown in Figure 3(b). The signal was then passed through the proximitor and then to the data acquisition system. The propagation of crack growth was measured by the ACPD (alternating current potential difference) crack growth monitor. Current of 0.5 A from the monitor was passed across the overall length of the specimen, and the signal from the voltage lead (position in between the crack) was then fed back to the monitor. When the crack starts to propagate, the voltage changes (due to the change in resistance) and the signal was then fed into the data acquisition system. The actual crack growth in millimeters was then calibrated, and the curve is depicted in Figure 3(c).

26


Figure 3. Calibration curves: (a) LVDT; (b) eddy current probe; (c) crack growth monitor.

Figure 4. Time history under free vibration of spring length 30 mm of mass B.


27

Figure 5. Damped natural frequency v.s. length of spring of mass A.

Figure 6. Time histories: (a) relative displacement between the probe and specimen surface; (b) base displacement.

The natural frequencies of mass A and mass B with different spring lengths were obtained from experiment by simply tapping the masses. It was calibrated with spring lengths from 30 to 100 mm in steps of 5 mm. The time history of mass B under free vibration for spring length of 30 mm is shown in Figure 4. Applying the well-known formula of Francis et al. [17], δ=

2π ξ 1 y0 ln = n yn 1 − ξ2

(4)

to obtain the damping factor ξ ; where δ is the logarithmic decrement, y0 and yn represent the initial amplitude and the amplitude after n cycles, respectively, the damped natural frequencies

28


Figure 7. (a) Crack growth curve of aluminium type 2024 (b) a close-up photograph of crack development on aluminium specimen type 2024 before total collapse.

of the masses were calculated. Figure 5 shows the characteristics curve of reduction of the damped natural frequency (ωd ) with respect to the increase of spring length for mass A. Fatigue tests were carried out on aluminium (type 2024) bend specimens under the influence of harmonic excitation. The frequency and amplitude of the base excitation were set according to the amplitude of the bend specimen. Several parameters were varied during the fatigue tests in order to determine the optimum rate of crack propagation. It is impossible to observe the crack growth phenomenon if the crack is propagating too fast, and if it is too slow a longer time is needed for the specimen to fail completely. The parameters that were varied during the tests were the frequency and amplitude of the dynamic shaker, the spring stiffness of the leaf springs and the preloading forces (to make sure the masses are always in contact with the specimen). As the eddy current probe is positioned some distance away from the center of the specimen, to maintain the deflection of the cracked specimen at its mid-span to 1 mm, the relative displacement between the face of the probe and the specimen under harmonic excitation was kept constant at 0.48 mm, at that particular position. This could be done by observing the amplitude from the oscilloscope and at the same time controlling the frequency and amplitude of the dynamic shaker. However, due to time constraint, the distance between the face of the probe and the specimen was controlled to 0.27 mm. As a result the specimen was oscillating with an amplitude of more than 1 mm. A typical time history sense by the eddy


29

current probe is shown in Figure 6(a), where Figure 6(b) depicts the time history of the base displacement. By comparing Figures 6(a) and (b), it is clearly seen that the response follows the excitation, however, due to the opening and closing of a crack the response is not entirely symmetrical around zero displacement. Referring to Figure 7(a), the crack in the aluminum specimen started to propagate at approximately 8000 s. The duration of the fatigue test for the aluminium specimen to fail completely is approximately 10 000 s. It clearly indicates that the initial stage of fatigue crack growth consumed about 80% of the whole process, which is expected during fatigue testing. A close-up photograph of the cracked specimen is shown in Figure 7(b). From Figure 7(b), the direction of crack propagation is only approximately 1◦ with reference to the crack plane, furthermore, no substantial amount of plasticity effect could be observed even though the cracked specimen had deflected more than 1 mm. 5. Concluding Remarks To overcome the shortcomings of conventional fatigue testing machines such as the difficulty in providing zero crossing aperiodic loading, a novel experimental rig to investigate fatigue crack growth under the influence of dynamic loading was designed and tested. Beside overcoming the deficiencies of conventional fatigue testing machines, the rig plays an important part in studying fatigue when fatigue crack growth is treated as a nonlinear dynamic system. Inertial forces provided by two, linear oscillators via based excitation are utilised to induce fatigue cracking on bend specimen. The flexibility of the rig allows the natural frequency of oscillator, the extent of the preloads, the pattern of the excitation signal on the shaker and the geometry of the specimen to be changed. Therefore, the rig have provided a new means of fatigue testing regardless of periodic or aperiodic loading. It also enable to carry out fatigue testing on a range of materials and specimen with different geometry. Successful fatigue crack growth tests were obtained for the aluminium type 2024 specimen investigated in the first instance. The direction of crack was propagating almost along the crack plane, furthermore, no substantial plasticity could be observed along the cracked edge. Further work is now underway to model the rig and further fatigue testing on a range of materials will be carried out. Acknowledgements The authors would like to kindly acknowledge support from University of Aberdeen Research Committee Grant S97104. The authors would also like to express their thanks to the workshop chief mechanical technician Mr John Manley for his helpful advice on the rig design. References 1. 2. 3.

Abrahim, O.N.L. and Brandon, J.A., ‘A piecewise linear approach for the modelling of a breathing crack’, in: 12th International Seminar On Modal Analysis, Leuven, Belgium, 1992, pp. 417–431. British Standard, ‘Method for determination of KIC , critical CTOD and critical J values of metallic materials’, in: Fracture Mechanics Toughness Tests, British Standard BS7448, Part 1, 1991. Chati, M.R. and Mukherjee, S., ‘Model analysis of a cracked beam’, J. Sound Vib. 207(2) (1997) 249–270.

30 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

Chee-Hoe Foong et al. Chondros, T.G., Dimarogonas, A.D. and Yao, J., ‘Longitudinal vibration of a bar with a breathing crack’, Eng. Fract. Mech. 61 (1998a) 503–518. Chondros, T.G., Dimarogonas, A.D. and Yao, J., ‘Longitudinal vibration of a continuous cracked bar’, Eng. Fract. Mech. 61 (1998b) 593–606. Chondros, T.G., Dimarogonas, A.D. and Yao, J., ‘A continuous cracked beam vibration theory’, J. Sound Vib. 215(1) (1998c) 17–34. Christides, S. and Barr, A.D.S., ‘One-dimensional theory of cracked Bernoulli–Euler beams. Int. J. Mech. Sci. 26 (1984) 639–648. Chu, Y.C. and Shen, M.-H.H., ‘Analysis of forced bilinear oscillators and the application to cracked beam dynamics’, AIAA J. 30(10) (1992) 2512–2519. Collins, K.R., Plaut, R.H. and Wauer, J., ‘Free and forced longitudinal vibrations of a cantilever bar with a crack’, J. Vib. Acoust. 114 (1992) 171–177. Dentsoras, A.J. and Dimarogonas, A.D., ‘Resonance controlled fatigue crack propagation’, Eng. Fract. Mech. 17(4) (1983a) 381–386. Dentsoras, A.J. and Dimarogonas, A.D., ‘Resonance controlled fatigue crack propagation in a beam under longitudinal vibrations’, Int. J. Fracture 23 (1983b) 15–22. Dentsoras, A.J. and Dimarogonas, A.D., ‘Resonance controlled fatigue crack propagation in cylindrical shafts under combined loading’, in: ASME Winter Annual Meeting, Boston, 1983c. Dentsoras, A.J. and Dimarogonas, A.D., ‘Fatigue crack propagation in resonating structures’, Eng. Fract. Mech. 34(3) (1989) 721–728. Dimarogonas, A.D. and Papadopoulos, C.A., ‘Vibration of cracked shafts in bending’, J. Sound Vib. 91(4) (1983) 583–593. Emans, J., Dynamics of Controllable Piecewise Linear Oscillator, Meng Thesis, University of Aberdeen, Scotland, 1999, pp. 245–260. Fernández-Sáez, J., Rubio, L. and Navarro, C., ‘Approximate calculation of the fundamental frequency for bending vibrations of cracked beams’, J. Sound Vib. 225(2) (1999) 345–352. Francis, S.T., Ivan, E.M. and Rolland, T.H., Mechanical Vibrations: Theory and Applications, Prentice Hall, Englewood Cliffs, New Jersey, 1978, pp. 78–80. Friswell, M.L. and Penny, J., ‘A simple nonlinear model of a cracked beam’, in: Proc. 10th International Modal Analysis Conference, 1, San Diego, USA, 1992, pp. 516–521. Frost, N.E., Marsh, K.J. and Pook, L.P., Metal Fatigue, Oxford University Press, London, 1974, pp. 245–260. Gudmundson, P., ‘The dynamic behavior of slender structures with cross-sectional cracks’, J. Mech. Phys. Solids 31(4) (1983) 329–345. Ibrahim, A., Ismail, F. and Martin, H.R., ‘Modelling of the dynamics of a continuous beam including nonlinear fatigue crack’, J. Modal Anal. 2 (1987) 76–82. Irwin, G.R., ‘Analysis of stresses and strains near the end of a crack traversing a plate’, J. Appl. Mech.-T. ASME 24 (1957) 361–364. Kirmsher, P.G., ‘The effect of discontinuities on the natural frequency of beams’, P. Am. Soc. Test. Mater. 44 (1944) 897–904. Liebowitz, H., Vanderveldt, H. and Harris, D.W., ‘Carrying capacity of notched column’, Int. J. Solids Struct. 3 (1967) 489–500. Paris, P.C., Gomez, M.P. and Anderson, W.E., ‘A rational theory of fatigue’, Trend Eng. 13(1) (1961) 9–14. Petroski, H.J., ‘Simple static and dynamic models for the cracked elastic beam’, Int. J. Fracture 17(4) (1981) R71–R76. Rizos, P.F., Aspragathos, N. and Dimarogonas, A.D., ‘Identification of crack location and magnitude in a cantilever beam from the vibration modes’, J. Sound Vib. 138(3) (1990) 381–388. Shen, M.-H.H. and Chu, Y.C., ‘Vibrations of beams with a fatigue crack’ Computers and Structures 45(1) (1992) 379–393. Shen, M.-H.H. and Pierre, C., ‘Free vibrations of beams with a single-edge crack’, J. Sound Vib. 170(2) (1994) 237–259. Sih, G.C., ‘Strain energy density factor applied to mixed mode problems’, Eng. Fract. Mech. 10 (1974) 305–321. Tada, H., Paris, P.C. and Irwin, G.R., The Stress Analysis of Cracks Handbook, Del Research Corporation, Hellertown, Pennsylvania, 1973.

An Experimental Rig to Investigate Fatigue Crack Growth 32. 33. 34. 35.

31

Thomson, W.J., ‘Vibration of slender bars with discontinuities in stiffness’, J. Appl. Mech.-T. ASME 71 (1949) 203–207. Warburton, G.B., ‘Letter to the Editor’, Int. J. Mech. Sci. 29 (1987) 443–446. Wiercigroch, M. and de Kraker, B. (eds), Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities, World Scientific, London, New York, Singapore, 2000. Yuen, M.F., ‘A numerical study of the eigenparameters of a damaged cantilever’, J. Sound Vib. 103 (1985) 301–310.