4.3 veriﬁcation of the mode shapes - eth z · 4.3 veriﬁcation of the mode shapes 67 4.3...

4.3 Verification of the mode shapes 67

4.3 Verification of the mode shapes

Since the modelled mode shapes are being used for the evaluation of the stressstate within the cross-section containing the crack, it is desirable to measure theseshape functions during the fatigue test and to compare the results with the shapefunctions obtained through the modelling process. It would also be interesting toknow how the cracked region affects the mode shapes of, not only the first, butalso the higher modes in order to estimate the applicability of the quite simplecrack model with its two discontinuity parameters for other vibration problemscontaining cracks or other defects. One important question is whether there isindeed a kink in the symmetric mode shapes (compare also to Fig. 2.5) orwhether the weakening effect of the crack influences the mode shape more in a“smeared sense” over a larger region near the crack rather than just in a verylocalized manner.

To answer these questions, a scanning laser interferometer system (SLIF consist-ing of: Polytec OFV 050 Optical Scanning Head, OFV 3001S Vibrometer Con-troller, Polytec software including the controlling PC) was used to obtain the firstseven modes of vibration of the beam containing a fatigue crack.

Fig. 4.11 Top view of first mode of transverse vibration measured with a scanninginterferometer system. The system was excited with a linear sweep signal; theresponse was analysed with the complex FFT.

68 4 Fatigue testing results with beams

For this purpose the fatigue test was interrupted in intervals as to have represen-tations of the mode shapes for various crack length values throughout the fatiguetest. The beam (aluminum alloy 2024-T3, notch depth of 2 mm) was then excitedwith a periodic recurring sweep signal that included at least the frequency rangeof the desired modes, e.g. from 100 Hz up to 10 kHz. For this alternate method ofexcitation, neither the beam nor any kind of hardware involved in fatigue testingneeded to be repositioned, thereby ensuring the same conditions for continuingfatigue testing after the interval of using the scanning interferometer. Only thearbitrary waveform generator (HP 33120A) connected to the SLIF system had tobe connected to the excitation coil rather than the voltage controlled oscillator(KH 5920). The response signal of the velocity of a good number of grid pointson the beam’s top surface was then recorded with this system and transformedfrom the time into the frequency domain by means of the complex Fast FourierTransformation (FFT). The individual first few modes of the system were finallyextracted and displayed in diagrams for a side view illustration which are listedin Fig. 4.12 on the right hand column.

As an indication of how the setup appeared from the view of the digital camera,the first mode is represented in Fig. 4.11 by a grey-scale distribution in additionto the beam. The camera was built into in the SLIF sensor head for easy scanningpoint selections and other evaluation purposes. Although the modes looked evenbetter in a colour-scale representation, it was a better approach to quantify themode in a side-view illustration in order to compare them directly with the mod-elled mode shapes. For the crack length of 6 mm, evaluated from the frequencyof mode 1, the first seven mode shapes with arbitrary amplitudes were modelledby keeping in mind that they should represent the realistic “non-linear” mode.Therefore, the two mode shape values - one for the half-cycle with a completelyclosed crack (where the crack length is essentially the notch depth), and the otherone for the open-crack state - were combined into an “average” mode shape. Alsothe corrected higher amplitude for the open-crack state according to Section 3.4.3is considered in this analysis. These mode shape diagrams are then listed on theleft hand side of Fig. 4.12 for a direct qualitative comparison with the measuredshape functions. Already the first measured mode of vibration shows the “kink”at the cracked section as predicted. This is more pronounced for the higher modesof symmetric vibration. As expected, mode 4, where the crack is not loaded inter-nally by any bending moment as it contains a node exactly at the crack locationin the middle, remains unchanged, regardless of how deep the crack is.

4.3 Verification of the mode shapes 69

Fig. 4.12 Side view of mode shapes, comparison of theoretical model (left hand side) tomeasured shapes (right hand side), relative crack length

a/b

=0.6.

0123

- 0.6

- 0.4

- 0.2

0

0.2

0.4

0.6

- 1

0

1

2

Mode 1

Mode 3

Mode 4

Mode 5

Mode 7

0-0.1-0.2

0.2

0.1 0.2

0.1-0.1-0.2 0

4

2

0

-2

-4

0-0.1-0.2 0.1 0.2

0-0.1-0.2 0.1 0.2

0-0.1-0.2 0.1 0.2

-2-1

-1

0

1

-0.5

0.5

1.5


4.4 Influence of temperature on the resonant frequency

This section is dedicated to the question of how ambient temperature changesduring fatigue testing have an influence on the resonant frequency and whetherthere is an increase in the temperature of the specimen during the vibration, bothglobally and locally, at the crack.

4.4.1 Influence of ambient temperature changes on the resonant frequency

To answer the first part of the question, two beam specimens without a notch, oneout of aluminum alloy (AlMgSi 1) and the other one made of steel (St 37), wereexcited to resonate in their first mode of bending vibration in a climate chamber(CTS -40/200) with temperature and humidity control. Every 30 minutes the tem-perature was increased by 5 K (Kelvin). Just prior to performing the next temper-ature increase, the stabilized frequency of vibration corresponding to a specifictemperature value was recorded. It turned out that a waiting period of at least 15minutes was necessary for the new temperature to spread evenly throughout thebeam’s core. After this period, the frequency remained very stable. In order tocompare the frequency versus time characteristics of both materials, the respec-tive frequency values were normalized by the measurement at the reference tem-perature of 20°C and combined in the diagram of Fig. 4.13.

The results indicate that between 15°C and 65°C the resonant frequency drops ina linear manner with increasing temperature. Therefore, it seemed appropriate toperform a linear least-square fit analysis. The slope of each line gives the relativechange of resonant frequency per 1 K increase:

(4.1)

For example, this means that an increase in ambient temperature of 1 K in an alu-minum beam vibrating at 300 Hz would reduce the frequency by about 0.074 Hz.

aluminum: ∆ff 20°C( )--------------------- 1⁄ K 2.46 10

4–1K⁄

steel: ∆ff 20°C( )--------------------- 1K⁄

⋅–

1.35 104–

1K⁄⋅–

=

=

4.4 Influence of temperature on the resonant frequency 71

Fig. 4.13 Resonant frequency of vibrating beams at different temperatures.

This seems insignificant at first, but if one considers the high stability in resonantfrequency of about 0.001 Hz, the accuracy in crack length determination wouldstill be affected considerably. Fig. 4.9 indicates that a change in frequency of 40Hz (from the value 320 Hz) corresponds to an increase in crack length of 5 mm.An uncertainty of 1 K in the specimen would result in an uncertainty in fre-quency measurements that translates into an inaccuracy of a crack length estima-tion of almost 10 micrometers. When looking at small crack lengths, thisinaccuracy would be even higher, because the slope of the curve in Fig. 4.9 is lesssteep for smaller cracks. But as ambient temperature changes only occur veryslowly, the resolution in crack length measurements and the determination ofcrack growth rates would hardly be affected.

Brem

[10] investigated the factors that have an influence on the resonant fre-quency of slender beams (aluminum alloys and steel) with changing temperature.Most of the results in Section 4.4.2 and Section 4.4.3 are based on his work. Herealized that the temperature dependent elastic modulus is the major contributingfactor and influences the frequency about 20 times more than one would expectfrom the changing geometry due to thermal expansion effects.

20 30 40 50 60

0.99

0.992

0.994

0.996

0.998

1

Temperature T [°C]

Steel St 37

Aluminum Al-Mg-Si 1

Freq

uenc

y ra

tio f

[T]

/ f [

20°C

]


4.4.2 Analysis of heat generation in vibrating beams

Since temperature has a strong influence and the frequency of vibration is quitehigh in comparison to conventional fatigue testing, it needs to be analysedwhether there is any considerable heat generation within the beam, either glo-bally, through the internal material damping characteristics, or locally, throughplastic deformation due to the advancing crack.

In order to estimate temperature distributions, the material is assumed to be linearviscoelastic. The elastic modulus in this case can be described by the complexmodulus :

, (4.2)

where the real part is often called “storage modulus” and the imaginary part,“loss modulus”, is responsible for the energy dissipation in the form of heat gen-eration. For metals, the ratio is usually very small.

Assuming the above, the internal heat generation rate per unit volume wascalculated as a function of position along the longitudinal axis of the beam for thefirst mode of bending vibration and for a given typical amplitude of 2 mm at theends of the beam. Eq. (4.3) gives a value for which is the average of thisheat generation rate for one cross-section of the beam:

, (4.3)

where is the period of the vibration, and the experimentally deter-

mined amplitude of transverse vibration. For the case of a steel beam and an

assumed value of this function is plotted in Fig. 4.14.

E* ω( )

E* ω( ) E' ω( ) iE'' ω( )+ E* eiϕ

= =

ϕ E= '' E'⁄ 1«

q ξ( )

q ξ( )

q ξ( )E''π

12T p-------------

w ,ξξ ξ( ) b⋅

l2

-----------------------------2

=

)

T p w ξ( ))

ϕ 1 104–

⋅=


Fig. 4.14 Internal heat generation rate per unit volume for a steel beam (length 500 mm,cross-section mm, , amplitude of 2 mm at the ends).

Fig. 4.15 Temperature distribution for a steel beam (length 500 mm, cross-section mm, , amplitude of 2 mm at the ends).

longitudinal axis ξ=x/l [-]

inte

rnal

hea

t gen

erat

ion

rate

qd

ot(ξ

) [

W/m

^3]

0 0.2 0.4 0.6 0.8 10

500

1000

1500

2000

10 10× ϕ 1 104–⋅=

0 0.2 0.4 0.6 0.8 1ξ=x/l [-]

20.001

20.002

20.003

20.004

20.005

20.006

T [° C]

10 10×ϕ 1 10

4–⋅=


In order to obtain a steady state temperature distribution, this heat generation rateis included in the heat conduction equation, in which the convection effect withthe ambient air temperature ( ) is considered by assuming a con-

stant convection coefficient . For the aluminum beam, with an amplitude of

vibration of mm at the ends, the value of is estimated 25.3 W/(K/m), as anexample. The resulting differential equation Eq. (4.4):

(4.4)

where is the thermal conductivity of the beam material, can be solved ana-

lytically. The boundary conditions are listed in Eq. (4.5).

(4.5)

The temperature gradient is zero at the center of the beam due to symme-try. It is assumed that there is no heat conduction on both ends of the beam andthat the heat convection on the square sides of the beam is negligible compared tothe area along the longitudinal axis, therefore the temperature gradient is alsozero.

As expected, according to Fig. 4.15 the distribution in temperature is, for anypart of the beam, slightly higher than the ambient temperature and is highesttoward the center of the beam, where the largest values of stress and strain arelocated. The calculated increase in temperature is very small and an experimentalvalidation of these results was performed, in view of the uncertainty in and

possibly .

Tair 20°C=

α

2± α

ξ2

2

∂

∂T ξ( ) C1 T ξ( ) Tair–( )⋅– C2 w ,ξξ ξ( )[ ]

2+ 0 ,

C14 α l

2

λMat b----------------- , C2

πE'' b2

12λMat T p l2

---------------------------------- , ==

=)

λMat

T ξ( )∂ξ∂

--------------ξ 0=

0 T ξ( )∂ξ∂

--------------ξ 0.5=

0 T ξ( )∂ξ∂

--------------ξ 1=

0.=,=,=

T∂ ξ∂⁄

α

ϕ


4.4.3 Temperature measurements during the fatigue test

During the vibration and fatigue tests, the temperature is measured near the endof the beam, at the node of the first bending mode of vibration, and close to thecrack via temperature-sensitive resistance elements (PT100), in order to moni-tor the heat generation in close vicinity of the crack. These elements were eitherglued or fastened with thin rubber bands onto the beam surface. Careful handlingof the thin electric wires that connect these temperature elements to the controlunit was necessary, since they are very prone to breaking in the course of vibra-tion. For this reason, they are also partly glued to the beam surface. The wiresleave the beam preferably at the position of a node where they are least likely toconstantly bend back and forth.

The steady-state temperature differences between the center and the end positionof the vibrating beam (aluminum and steel) without a notch were measured for avariety of amplitudes. The factor in Eq. (4.2) was obtained in order to matchthe measured temperature difference with the calculated temperature difference.It was concluded [10] that the factor for the analysis of heat generation would

have to be higher ( , ) than originally

assumed especially for steel, in order to match the experimental results.

Adams [2] studied the damping characteristics of various metals on specimensvibrating in its free-free longitudinal mode. He determined the energy dissipationfrom the rate of temperature increase at different sections of the specimen. For awide range of plain carbon steels, Adams estimates the amplitude-independentdamping between 0.05% and 0.15%. At high stress amplitudes of around

MPa, however, he noticed a considerably higher damping up to 4% forcarbon steels. This observation might explain, why, in this present thesis, a highervalue for the factor has to be used in order to obtain a realistic internal heat

generation rate per unit volume (Eq. (4.3)), that results in a temperature dis-tribution of the beam similar to the experiment.

During the fatigue test, it was also observed that the temperature differencebetween the crack location and the position of the nodes was considerably higherfor steel (maximum of 1°C) than for aluminum (maximum of 0.3°C). The tem-perature increase at the nodes, in comparison with the air temperature, turned outto be by far smaller than the temperature difference within the beam.

ϕ

ϕ

ϕSteel 2.8 102–

⋅= ϕAlu 4.4 103–

⋅=

30 200–

ϕ

q ξ( )


Summarizing the results in this section, it is important to occasionally check theheat generation within the beam during fatigue testing and to interpret the meas-ured frequency values accordingly, i.e. if necessary, to correct them for tempera-ture changes, especially when the absolute crack length needs to be determinedvery accurately. With the described measurement technique, it was not possibleto distinguish between the part of heat generation due to crack advance and thepart due to material damping effects. However, the analysis of the cracked regionwith an infrared camera (with a temperature resolution of about 0.5 °C and a spa-tial resolution of less than 1 mm) did not reveal any substantial temperatureincrease at the crack front in relation to its close vicinity.

Regarding temperature measurements in combination with fatigue testing, onereference to a very recent publication is worth mentioning. La Rosa and Risitano[42] report a methodology for the determination of the fatigue limit which isbased on the increase in temperature when the stress reaches values above thefatigue limit due to plastic deformations. The fatigue limit in this context is to beunderstood as the maximum stress amplitude for which a cyclically loaded struc-ture reaches an infinite service life. La Rosa and Risitano observed the surface ofa single specimen or more complex components with two thermal infrared scan-ning cameras and realized that while the stresses are below the fatigue limit, thereis only a very small temperature increase, but as soon as the phase of global plas-tic macro-deformation was reached, they noticed a sudden increase. This effectwas more pronounced the more the stress exceeded the fatigue limit. They fur-thermore presented a way to extrapolate the fatigue limit with a minimum ofthree temperature measurements at different stress levels. Although this methodis probably best applicable to specimens, where a relatively large region under-goes plastic deformation before a crack eventually forms, it is certainly an inter-esting approach. Their research also suggests that considerable temperaturechanges are only to be expected if a larger region undergoes plastic deformations.However, in our case, where fatigue crack growth is very local, the plastic regionis rather small and global stresses should remain well below the fatigue limit.

4.5 Damping characteristics during fatigue testing 77

4.5 Damping characteristics during fatigue testing

During the fatigue test, not only does the frequency of vibration change, but alsothe damping characteristics. This occurs through crack face interaction, where byrepeated contact and friction the damping increases considerably in vibratingbeams with increasing crack length. The radiation of energy takes place in somepart through acoustic emission, especially in specimens made of the aluminumalloy 2024-T3, and can make fatigue testing with high crack growth ratesunpleasant to the human ear at times.

One positive side effect of the increased damping characteristics for fatigue test-ing is that the voltage amplitude at the driving coil does not need constant adjust-ments for constant crack growth rates. If there was no change in the damping,one would need continuously lower electromagnetic forces to have the crackadvance at constant growth rates with increased crack length. But because of thedamping changes, these effects even out to some extent. As a result, there were anumber of fatigue tests performed without ever changing the voltage amplitudeof the frequency generator signal connected to the actuating coil.

4.5.1 Damping Measurement Techniques

In Section 3.3 the theory of extracting the damping characteristics (Q-factor)from the phase versus frequency diagram were described. This technique hasbeen applied at the Institute of Mechanics with success for determining the vis-cosity of fluids [22], the influence of the material’s microstructure on thedynamic response of materials [55], and evaluating vibration characteristics ofmicrosamples [46], [63].

However, for the characterization of damping during the fatigue test, a few disad-vantages were noticed with this technique. It is desirable to obtain the Q-factorfor a beam that is vibrating with a typical amplitude during a fatigue experiment.But since the exact determination of the resonant frequency is of crucial impor-tance to this technique, and since the frequency might slightly shift because of anadvancing crack, one would have to reduce the amplitude significantly, which isnot an option if the damping is dependent on the amplitude of vibration. In addi-tion, one would have to wait for the averaging procedure that is needed to meas-ure the stabilized frequency for at least three different phase arguments. Of


course, this is time-consuming and these intervals of lower amplitude of vibra-tion could have an influence on the crack growth properties in general, such ascrack closure effects, healing effects, or changes in the plastic zone distributionahead of the crack tip.

As an alternative, it turned out to be sufficient and appropriate to evaluate thedamped free vibration and to determine the damping coefficient whichdescribes the exponential decay of the peak values of displacement according toEq. (4.6). In the literature (see [80] and Section 1.8) one often comes across theterm logarithmic decrement , the natural logarithm of the ratio of two succes-sive peaks of displacement (or velocity), which provides a simple definition forthe damping of a free oscillator and can also be applied experimentally in a sim-ple way. The relation between the logarithmic decrement and the damping

coefficient is given in Eq. (4.7) for completeness.

(4.6)

(4.7)

Applying this damping measurement technique during the fatigue test means thatthe excitation source must be turned off only for a short interval (10-20 seconds).During this interval, the decay of the amplitude versus time is measured and anexponential fit algorithm is applied to estimate the damping coefficient . The Q-factor is then obtained through

, (4.8)

in which the natural frequency of the undamped vibration should be consid-

ered, although in practice the resonant frequency corresponding to the maxi-

mum amplitude of vibration was taken. However, Q-factors in vibrating beams

including cracks turned out to be between and , which means that there

δ

Λ

Λ

δ

q t( ) q eδt–

ωt ϕ+( )sin=

Λqi

qi 1+------------ln 2π

δω---- 2π

δ

ω02

δ2

–

----------------------= = =

δ

Qω02δ------

π f 0δ

---------= =

f 0

f R

103

104

4.5 Damping characteristics during fatigue testing 79

is a negligible influence on the measurement of Q, whether or is chosen

(Eq. (4.9)).

(4.9)

Furthermore, with this decay technique it was observed that the damping charac-teristics at high amplitudes (i.e. as high as needed for crack advances) practicallydid not differ from the characteristics for an amplitude that measures only 10% ofthe previous value. Q-factor measurements with this simple technique were veryconsistent, reliable, and quick to perform, while disturbing the fatigue experi-ment in the least amount [50]. To demonstrate the quality of this procedure,eleven successive measurements of the Q-factor in an aluminum alloy beam wereperformed. The mean value was 4282, and the standard deviation turned out to beonly 19.8. With the phase-locked loop approach (Eq. (3.4)) using the phase ver-sus frequency information, results of repeated Q-factor measurements were notas consistent and contained a higher degree of variation.

4.5.2 Results of damping measurements

Q-factor measurements were performed in intervals during the fatigue tests ofslender beams ( mm) with two different aluminum alloys. Theresults in Fig. 4.16 and Fig. 4.17 indicate that, for both specimens, the Q-factorstrongly depends on the crack length and decreases, as damping increases, con-siderably for increasing crack length values. Since the Q-factor drop is very pro-nounced in the beginning stage of crack growth, measuring the dampingcharacteristics could provide an additional way of identifying small cracks orimperfections in vibrating structures throughout their service life.

f 0 f R

f R f 0 1 1

2Q2

----------–=

10 10 400××


Fig. 4.16 Q-factor as a function of relative crack length (material: aluminum alloy AlMgSi1,beam mm, initial notch depth ratio 0.2).

Fig. 4.17 Q-factor as a function of relative crack length (material: aluminum alloy 2024-T3,beam mm, initial notch depth ratio 0.3).

0.4 0.5 0.6 0.7 0.8500

1000

1500

2000

2500

3000Q

-fac

tor

mea

sure

men

t [-]

crack length ratio ζ=a/b

10 10 400××

0.35 0.4 0.45 0.5 0.55

4000

6000

8000

10000

crack length ratio ζ=a/b

Q-f

acto

r m

easu

rem

ent [

-]

10 10 400××

4.6 Control of fatigue crack growth rate 81

4.6 Control of fatigue crack growth rate

In order to have a fatigue test running for hours or even days without supervision,a high degree of automation is required. One goal was to actually control thecrack growth rate during testing. As a first approach, the amplitude of the voltagecontrolled oscillator was continuously corrected in order to achieve a more orless constant decrease of the resonant frequency versus time. The control algo-rithm is implemented in the LabVIEW file that performs the control of instru-ments and the data acquisition. The user can specify the desired slope of thefrequency decrease, control parameters, and an upper limit of the voltage ampli-tude as a safety precaution to ensure for example that the coil is not overheating.Once the control of the decreasing frequency of vibration is achieved in a stableand reliable manner, the next step would be to define a crack growth rate profileover the entire fatigue experiment and to control the amplitude accordingly byevaluating the modelled relationship between frequency and crack length duringtesting.

Fig. 4.18 Example for a linear decrease of resonance frequency versus time achieved byamplitude control. In the diagram to the left the control parameters were not yetoptimized, indicated by oscillating effects in crack growth rates.

In Fig. 4.18 two diagrams representing the controlled resonant frequency versustime illustrate the results of this amplitude control. In both cases, the overallslope of the decreasing frequency is achieved in accordance with the previouslyspecified target slope. The control parameters for the test represented in the dia-gram to the left were not optimized yet and the proportional control constant wasset too high, which led to an oscillation in the slope of the otherwise fairlystraight curve. The diagram to the right documents that the goal of a linear

0 500 1000 1500 2000 2500 3000 3500

304

306

308

310

312

314

316

time [s]

reso

nant

fre

quen

cy [

Hz]

0 500 1000 1500 2000 2500 3000260

265

270

275

280

285

time [s]

reso

nant

fre

quen

cy [

Hz]


decrease in frequency with respect to time was achieved.

4.7 Representation in fatigue diagrams and comparison with results from the literature

For the estimation of the remaining lifetime of a component with an initial cracklength, it is important to know how many cycles this component might be able towithstand under certain loading conditions to reach a final or critical cracklength. This is one of the reasons that fatigue testing data is plotted as crackadvance per cycle versus the stress intensity factor range. In a logarithmic “loglog”-plot of this established fatigue diagram the region of stable crack growth ischaracterized by a linear curve (compare to Section 3.4.4). This region is oftenreferred to as the power-law behaviour and is described by the Paris-law [53](Eq. (3.12)). Many studies on fatigue in the literature are dedicated to this power-law behaviour and other empirical equations describing the crack growth behav-iour have been developed, including R-ratio effects and the region of rapid crackgrowth. However, about 80-90% of the service life of dynamically loaded com-ponents is spent in the near-threshold region, where the existing crack is shortand the crack growth rate is small.

The results of 10 fatigue tests with aluminum beams (2024-T3, initial crack depthof 2 mm) under the same testing conditions (ambient air, stress ratio )are summarized in Fig. 4.19. The diagrams were obtained using the evaluationprocedure described in Chapter 3. The near-threshold region down to a very low

crack growth rate of m/cycle is represented as well as the linear region

of stable crack growth up to m/cycle. The threshold value accord-

ing to the experimental data is about , which agrees

well with the threshold values typically found in near-threshold fatigue crack

growth tests. Taylor and Jianchun [77] list values between 2 and 4 for

aluminum alloys in general and the value 3.4 specifically for the alloy

2024-T3 at the stress ratio of . Also the fatigue data displayed in

Fig. 4.20 (ASM Specialty Handbook [17]) indicate -values between 3 and

R 1–=

1 1011–

⋅

1 107–

⋅ ∆Kth

∆Kth 3.2 106 Pa m⋅=

MPa m

MPa m

R 0.1=

∆Kth

4.7 Representation in fatigue diagrams and comparison with results from the literature 83

4 for the same type of alloy at . Stanzl et al. [73] investigatedthe environmental influence of humid air, dried air and vacuum on the near-threshold fatigue crack growth behaviour of the alloy 2024-T3 using an ultra-sonic resonance fatigue testing technique and obtained the threshold values

2.1 for humid air, 2.3 for dried air, and 3.3 for vac-uum.

Fig. 4.19 Fatigue diagram for the aluminum alloy 2024-T3 as obtained from the PLLresonance technique.

In order to compare the experimental results for the region of stable crack growthwith reference values in the literature, the well-established relation from Speidel[71] (Eq. (3.13), indicated by the upper straight line) is included in the fatiguediagram of Fig. 4.19. This line, which is a good approximation for fatigue data ofvarious metals at , when the stress intensity factor is normalized by the

MPa m R 0.33=

MPa m MPa m MPa m

10 -11

10 -10

10- 9

10- 8

10- 7

Stress Intensity Factor Range ∆K [MPa m ]1/2

Cra

ck G

row

th R

ate

da/

dN [

m/c

ycle

]

da/dN=5.1 10 (∆K/E)6 3.5

da/dN=3.5 10 (∆K/E)8 4

1.5 2 3 5 7 10

R 0=


elastic modulus, is found to form an upper bound for the crack growth rates pre-sented here for a given -value. It should be pointed out that the fatigue tests in

this work are performed with completely reversed loading (i.e. ),whereas fatigue testing in the literature is often done with a positive stress ratioR, usually between 0.1 and 0.9. For higher stress ratios it is generally observed([11], [28]) that the line in the fatigue diagram is shifted towards higher crackgrowth rates.

This effect is largely attributed to the crack closure phenomenon, which meansthat for lower R-ratios the crack closes partially due to plasticity, surface rough-ness or the formation of an oxide-layer. Since Elber [27] started to concentratehis research on crack closure effects about 30 years ago, many theories andempirical model have been developed, some of them by introducing an effectivestress intensity factor range , which is dependent on the stress ratio. Paris

et al. [54] summarize major research contributions of the past three decades onthe modelling of near-threshold behaviour in a historical perspective and praisethe very simple new model of Hertzberg whose findings have not yet been pub-lished but are currently in press. According to reference [54] he predicts a thresh-old corner at a location in the fatigue diagram that only depends on the Burgersvector and the elastic modulus and approximates crack growth rates versus theeffective stress intensity factor range in a power law that only contains

those two parameters. Paris et al. compare this model with fatigue data of twoaluminum alloys (but only for ) and notice an excellent agreement.

The stress ratio can influence the crack growth rates by as much as one order ofmagnitude. Another linear curve with the equation

(4.10)

is included in the diagram (Fig. 4.19) to reflect a best data fit for the region gov-erned by the power-law behaviour. Interestingly, Paris and Erdogan [53] list anexponent of 4 as well, that best suited their experimental data. However, for vari-ous materials this exponent might vary as much as from 2 to 7.

∆K

R 1–=

∆Keff

∆Keff

R 0.1=

dadN------- 3.5 10

8 ∆KE

--------

4,

dadN------- in [m/cycle],

∆KE

-------- in m[ ]⋅=

4.7 Representation in fatigue diagrams and comparison with results from the literature 85

Fig. 4.20 Fatigue crack growth data from the reference volume “ASM Specialty Handbook –Aluminum and Aluminum Alloys” [17], page 39.

The results represented in Fig. 4.19 are in excellent agreement throughout thewhole range compared with results for the same aluminum alloy 2024-T3 with arelatively low stress ratio of , which are displayed in Fig. 4.20. Fur-thermore, comparing the results with the tests performed by Stanzl et al. [73](with completely reversed loading, ) indicate an excellent agreement for

the power-law region, whereas the results for crack growth rates below m/cycles in the present work would be in between the curves for the humid air andvacuum conditions that are published in their paper. Hahn and Simon [38] com-

R 0.33=

R 1–=

109–


piled the results of 22 different investigations on cyclic crack growth, mainly forthe 2024-T3 and 7075-T6 high strength aluminum alloys and presented the scat-ter bands for each material in which the majority of the testing data was located.Although most of this fatigue data contains relatively fast crack growth rates

from up to m/cycles, the present data fits well in their correspond-ing scatter band portion of the material 2024-T3.

In summary, it is demonstrated that fatigue testing data obtained with the PLLbased resonance technique for beams are highly comparable to various fatiguedata sources in the literature. This confirms that the proposed technique offers areliable and simple alternative to fatigue testing performed with conventionalequipment and established evaluation techniques.

4.8 Non-linearity effects due to closing of the crack

As mentioned previously in Section 2.4 Actis and Dimarogonas [1] reportedpeaks at multiples of the resonant frequency in the frequency spectrum of vibrat-ing beams containing cracks. Man et al. [45] monitored the slot depth in a beamby exciting the structure with an impact hammer and analysing the frequencyshift of the first four modes. However, since the beams contained saw cuts with aflat root and a width of 1.25 mm, the vibration is essentially linear and no extrapeaks are present in their experimental analysis of the power frequency spectrum.

In this section, frequency spectrum measurements will be presented that areobtained through a linear sweep excitation, as well as by analysing the frequencyresponse when only the first mode of vibration is excited.

4.8.1 FFT-spectrum analysis of the vibrating cracked structure

The FFT power spectrum1 was analysed in an aluminum beam (2024-T3) after afatigue crack with the relative crack length of a/b=0.6 was created. The beamwas excited with a periodic linear sweep signal ranging from 100 Hz up to 10

1. The power spectrum is the signal power represented on a logarithmic vertical scale versusthe frequency on the horizontal axis.

109–

3 10⋅5–

4.8 Non-linearity effects due to closing of the crack 87

kHz and the FFT power spectrum was obtained by evaluating the velocity timesignals of the scanned data points by means of the scanning laser interferometersystem (SLIF). The measured mode shape functions corresponding to the peaksin Fig. 4.21 are already displayed on the right hand side of Fig. 4.12 (the datarepresented in these two figures are based on the same experiment and speci-men).

Fig. 4.21 FFT power spectrum (Decibel scale of velocity) analysed with the scanning laserinterferometer (SLIF), relative crack length a/b=0.6, initial notch depth

.

Apparently, there are no peaks in this power spectrum diagram that could beattributed to multiples of the frequency of mode 1 (290 Hz), even though thecrack length of 6 mm in a beam with a square cross-section of mm wasvery high. In contrary to the peaks corresponding to mode 1 through 7, there is noexplanation for the other peaks with a magnitude of less than -50 dB.

Another approach of measuring the frequency spectrum without interrupting thefatigue test was undertaken by recording the time signal of the velocity at onelocation of the beam followed by an FFT-evaluation (Fig. 4.22), while the beamwas excited to vibrate in the first mode of bending vibration with a sufficientlyhigh amplitude for fatigue crack advances. Now this FFT power spectrum con-

mod

e 1

mod

e 2

mod

e 3

mod

e 4

mod

e 5

mod

e 6

mod

e 7

a0 b⁄ 0.2=

10 10×


tains a large number of regularly spaced peaks at exactly the multiples of the res-onant frequency.

Fig. 4.22 FFT-power spectrum with peaks at multiples of the resonant frequency (314.2 Hz)of the first mode.

Interestingly, these peaks were already present at an early stage of fatigue crackgrowth and the character of these peaks did not change significantly for anincreasing crack depth [61]. In this respect, the behaviour is different from whatActis and Dimarogonas [1] observed, where peaks were found to be more pro-nounced as crack length increased. However, when the amplitude of vibration isreduced significantly (from 0.95 mm to 0.48 mm at the free end), most of thepeaks at multiple frequency values, except for the ones at the double and triplefrequency, disappear (see Fig. 4.23).

frequency [Hz]

velo

city

, dB

sca

le


Fig. 4.23 FFT-power spectrum illustrating the dependency of the “higher harmonics” on theamplitude of vibration. Amplitude is 0.95 mm at the end of the beam.

Fig. 4.24 FFT-power spectrum as Fig. 4.23. Amplitude at the end of the beam is 0.48 mm.

frequency [Hz]

velo

city

, dB

sca

le

frequency [Hz]

dB s

cale

of

velo

city


No attempt will be made at this point regarding the explanation of this behaviour.Prior to any detailed conclusions, the measurement method which includes theuse of a laser interferometer system, needs to be carefully checked. Usually thissystem is used for very small oscillations in the nanometer and micrometer rangeand a considerable degree of knowledge and experience is available for thismeasurement range, but some difficulties were noticed to obtain a clean signal(without noise and other disturbances that could not be completely reduced) ofdisplacement or velocity versus time for large amplitudes higher than 1 mm. Asan alternative, the response signal was measured electrodynamically with a sen-sor coil and in this case, no peaks at multiple frequencies were noticed. However,the quality of the signal to noise ratio was perhaps insufficient to resolve smallamplitudes at higher frequencies.

4.8.2 Outlook and suggestions for further measurements of non-linearity effects

One suggestion to check the reliability of the measurements given in Fig. 4.22and Fig. 4.23 would be to apply “backup” techniques other than the laser interfer-ometer and the coil sensors. The use of an accelerometer comes to mind, but thisshould be designed in a way as to be most sensitive for frequencies higher thanthe exciting frequency of the beam.

Ruotolo et al. [58] analysed the vibrational response of a cracked cantilever beamto harmonic forcing using finite-element modelling with a closing crack.Theysuggested the use of higher order frequency response functions (FRFs) andshowed that these functions are highly dependent upon the size and the positionof the crack.

In fact, very recently, an increasing number of papers were published concerningthe use of higher-order spectra such as the bispectrum or the trispectrum [48],[15] and its relevance to study non-linear effects of vibration. The bispectrum isthe decomposition of the third moment of a signal [57]. Fackrell et al. [29], [30]describe how to interpret the bispectral analysis in theory and with applicationsto a cantilever beam with different support conditions. When the beam was onlyloosely fixed to the support, the bicoherence function, which is a normalizedform of the bispectrum, between input and output of a beam system dropped con-siderably due to the non-linearity influence of the support. This is because the


stiffness of the support varies during the one cycle of vibration. They furtherdemonstrated that this effect was rather independent of the location of the accel-erometer. Therefore, this method provides potential for non-destructive conditionmonitoring.

Rivola and White [57] investigated the performance of a bilinear oscillator andanalysed the oscillator’s response numerically. They concluded that the magni-tude of the super harmonics are very small when the stiffness ratio does not differvery much from the value 1. However, the bicoherence plot provides a muchgreater sensitivity to weak non-linearities than a power spectrum could possiblyprovide, even for the stiffness ratio of 0.95 and 0.9. The authors conclude thatbicoherence analysis provides a sensitive tool to detect whether there are non-lin-earities present in the system, indicating structural damage or other faults, butalso that there are further difficulties in interpreting the information contained inthe bicoherence. Boltezar and Hammond [8] present an extensive experimentalstudy on the response of multi-degree-of-freedom mechanical systems consistingof an array of pendulums that are connected by springs with rather strong non-linear characteristics.

The reason why higher order statistics (HOS) such as the bispectral analysis pro-vides a powerful tool to detect non-linearities is that a HOS-analysis yields infor-mation about a signal’s non-Gaussianity [15]. A Gaussian input into a linearsystem will lead to Gaussian output, while the same input into a non-linear sys-tem will lead to non-Gaussian output. Since the HOS filters out white noise,which is also Gaussian, and other information related to a linear system, it is verysensitive to non-linearity effects. The use of higher order statistics as a nonde-structive monitoring technique will certainly gain more importance in the future,and the author suggests to apply this technique to fatigue testing of beams as anadditional way to correlate the degree of non-linearity with the size of the crack.

5 Fatigue testing of microstructures

Micromechanical structures often reveal defects or stress concentrations whichcan cause cracks to initiate and propagate. Very small crack growth rates willquickly lead to fatigue failure of dynamically loaded small-scale structures. Thusa high sensitivity in crack length measurements is necessary to monitor the crackpropagation.

One cannot presume that the fatigue failure mechanisms investigated in speci-mens of macroscopic scale and often with continuum-mechanical assumptions,can be extended to significantly smaller structures with dimensions of a fewmicrometers. Cracks that would be considered small in regular-sized specimenswould already be large compared to the specimen thickness of a micrometer-sized structure. Also, the influence of surface conditions on the overall mechani-cal properties is expected to be more pronounced.

The fatigue testing technique outlined in the two previous chapters was success-fully applied to testing of microstructures ([65], [25]). The method of using thefrequency of vibration for crack length estimations is very generic in principleand can be applied to any size of a structure. The only restriction is that suitablemeans of the excitation must be provided which are sufficient to load the inter-ested region of the structure in such a way that fatigue failure will entail. Thisstructure could, for example, be the wing of a large aircraft or, as in the followingcase, a microbeam that is less than a millimeter in length and 100 micrometers orless in width. It is obvious that a microstructure with these dimensions cannot betested in a conventional fatigue testing machine, which is usually designed toaccommodate a limited range of specimen sizes and loading conditions. It alsolacks suitable sensors to determine the crack length with a useful resolution andaccuracy.

Mazza [46] and Schiltges [63] noticed in their quasi-static tension and torsionexperiments of microstructures that the proper alignment of the loaded specimenis of crucial importance and that new actuators and sensors had to be developedsince commercially available ones did not fulfil the requirements. It is thus notalways feasible to just “down-scale” conventional testing facilities into themicrometer range. In addition, various problems of specimen handling andprocessing need to be solved beforehand.

94 5 Fatigue testing of microstructures

In the case of fatigue testing of microstructures, measuring the crack length witha conventional method, i.e. a travelling microscope would result in a resolutionof 10 µm in the best case, which is certainly not an option for a 30 µm wide spec-imen. Also, any method for determining the crack opening displacement (COD)cannot be realized. Force measurements according to the quasi static experimentsof Mazza and Schiltges could not be applied in a similar way for dynamicallyloaded structures since the weight scale can not be used for measuring fast tran-sient processes.

However, the resonance technique with a phase-locked loop control offers greatpotential for fatigue tests with small-scale structures. Since the resolution isgiven in relation to the specimen width, a change in crack depth in the order of

the spacing of the atomic lattice (2 ... m) could be resolved for micro-beams with a cross-sectional width of about 10 µm, if the relative stability of

of the controlled resonant frequency will also be achieved in micro-structures. This extremely high resolution in crack length would be very benefi-cial for studying the mechanisms of fatigue crack growth on a microscopic level.

5.1 LIGA Samples

5.1.1 Manufacturing process and sample design

The pure nickel and nickel-iron alloy samples for the fatigue tests were manufac-tured with the LIGA-technique that uses a deep X-ray lithography process whichis explained in detail by Ehrfeld and Lehr [26]. The samples were produced at theInstitut für Mikrotechnik Mainz, Germany in cooperation with Abel (see Mazza etal. [47]).

The microsample and its fabrication must meet several criteria. The dimensionsof the microbeam, and especially of the cross-section, should be accurate and notvary more than a few micrometers from sample to sample. The design of thesample should be suitable for static as well as dynamic (vibration) testing and thehandling and alignment be kept as simple as possible. For these reasons largerplates of about mm are integrated into the structure on either end of themicrobeam (see Fig. 5.1). This addition to the “main” microstructures has the

3 1010–

×

2.5 105–

×

5 7×

5.1 LIGA Samples 95

advantage that for vibration tests, the variety of loading of the microbeam portionis increased considerably depending on the excited mode, although in this workonly the first bending mode was considered for fatigue testing. It is beneficialwhen the resonant frequency is reduced to a level for which the amplitudes ofvibration turn out to be sufficiently large to load the structure in a way suitablefor fatigue testing. The two end plates of the sample are connected by a protec-tive frame which protects the microbeam from loading or even fracture duringthe fabrication and handling process. Only before mounting or clamping one orboth end plates to the testing apparatus, is the frame removed in a “micro workbench” outlined by Mazza [46]. This device allows both end plates to be clampedsimultaneously thereby mechanically isolating the microbeam while removingthe protecting frame. After this removal, extreme caution is necessary not todestroy or bend the microbeam before starting the test. The protective frame alsocontains test structures that are indicators for the quality of the manufacturingprocess and for internal stress detection.

The thickness for both the microbeam and the plates is the same and variesbetween 120-200 µm, depending on the fabrication parameters for each wafercontaining several microsamples with different dimensions of the microbeam.Aside from the pure nickel samples, nickel-iron alloys with a varying degree ofiron-content from 6% to 53% were realized. The elastic parameters for these dif-ferent materials such as Young’s modulus, the elastic limit, and shear moduli werepreviously determined with both vibration tests and tensile tests by Mazza et al.[47]. Schiltges [63] developed a mixed numerical experimental technique andused an iterative least square procedure to determine sets of four elastic constantsout of the total number of six constants for the orthotropic materials silicon aswell as for the various Ni and NiFe-alloys of the LIGA structures. He utilized aPLL resonance technique and included torsional modes of vibration to obtain twoshear moduli in addition to the two elastic moduli in both plane directions of theslim microstructures.

Mazza [47] obtained a difference between the experimentally determined Young’smodulus and the techniques of vibration testing and tensile testing of less than4% in general, or even down to 1.5% for the case of pure nickel. Also the scatter-ing of the individual measurements of the elastic moduli was only about 2%. Thisis a good confirmation for the validity of the developed and applied methods anda viable basis for further modelling of the fatigue-cracked structure by usingthese proven parameters.


5.1.2 Geometry and notch shapes

Fig. 5.1 Dimensions of the notched LIGA-specimens used for fatigue testing.

The LIGA-technique allows the design of complex types of geometries includingsmooth transitions from the microbeam into the attached plates and the design ofvarious types of notches, such as rounded or V-notched (see Fig. 5.1). This isvery beneficial to either reducing stress concentrations near transitions or, in thecase of fatigue testing, to enforce crack initiation exactly at the root radius of thenotch. The slots in the plates are necessary for the distribution of the etchingsolution in order to separate the sample from the substrate. The lateral dimen-sions of LIGA samples are accurate within +/- 0.5 µm, whereas the thicknessmight vary considerably from wafer to wafer, resulting in batches of sampleswith about the same thickness. A constant thickness within a maximum error ofabout 1 µm is given for the microbeam section of the sample, but near the plateedges and around the slots the thickness might be reduced by as much as 10 µm,which should be taken into account for any vibration modelling. Particularly the

notc

hed

mic

robe

am

14 m

m

5 mm

LIGA specimen microbeam, rounded notch

thickness: 120-180 µm

width: 100 µm

depth: 50 µm radius: 10 µm

notch:

microbeam, V-notch

thickness: 120-180 µm

width: 100 µm

depth: 20 µm V-notch:

opening angle: 90

protective frame

test

str

uctu

res

o

550

µm

550

µm

5.1 LIGA Samples 97

thickness must be measured individually for every sample being tested. For thefatigue tests, this was done after the test in order not to risk any damage to thefragile microbeam prior to testing.

For this purpose, a scanning profilometer (UBM Measurement System) based onthe laser focusing principle was used with a resolution in height profile measure-ment of 10 nm within a 100 µm range. A simple but very effective procedure tomeasure the thickness of the specimen at various locations has been proposedand explained in detail by Mazza and Schiltges ([46], [63]), and was applied tospecimens investigated in this work as well. To summarize their approach: themicrobeam is placed on a sharp edge (i.e. a silicon test structure) such that it isonly supported by this edge and the edge of the specimen’s plate, and thereby thesample is inclined at a very small angle. Next, the two-dimensional profile of theregion around the support is scanned. The difference in height between the sup-port and the specimen divided by the cosine of the inclination angle then definesthe thickness. With this approach, systematic errors resulting from the surfacecontact or contaminations between specimen and support could be reduced to agreat extent. The most suitable method to measure the width and other lateraldimensions proved to be the analysis of the level of reflection rather than theanalysis of the profile scan, because the change in reflection is very abrupt at theedge of a sample. For the profile scan, the UBM system is forced to determine theheight based on signals with poor reflection values near the edges, which is there-fore not the most reliable option. The evaluation of one scan with the profilom-eter delivers both results - the profile distribution as well as the level of reflection.

Examples of the microbeam sections of three different LIGA-samples being usedfor fatigue testing are displayed in Fig. 5.2. The specimen in (a) has a V-typenotch with a depth of 20 µm. The photograph in (b) was taken after fatigue test-ing with a visible fatigue crack that was initiated from the rounded notch with anotch radius of 20 µm. A relatively long (1 mm) and very slender microbeamwith a width of 40 µm is represented in (c).


Fig. 5.2 Photographs of microbeams (length 0.5–1 mm) with different geometry: V-notch(a), rounded notch with a visible crack (b), without notch (c).

5.2 Experimental setup for fatigue testing of microstructures

In general, the same principle used in the fatigue testing of beams is applied fortesting microbeams - the sample is excited to vibrate at resonance with an ampli-tude that is suitable to induce a crack and to ensure crack growth in a controlledmanner.

(a)(c)

(b)

5.2 Experimental setup for fatigue testing of microstructures 99

After detaching the specimen from its protective frame (Fig. 5.1), one plate of theLIGA-sample is fastened onto one side of a cubic adapter element with a two-component glue and then carefully screwed onto the round adapter, ensuring thatthe specimen is positioned along a vertical plane. The stacked piezoelement (PIPhysik Instrumente, Type P 843.60, low voltage piezoelement, elongation range0-90 µm, see Fig. 5.3) is mounted on a support unit, allowing for coarse and finevertical positioning. This piezoelement is connected to the output of the piezo-controller (LVPZT-amplifier, PI Physik Instrumente, Type E-505.00 and Servo-Controller E-509.S1) which amplifies the signal produced by the frequency gen-erator. The frequency range needed for fatigue testing was 40-300 Hz, which iswell below the resonant frequency 6 kHz of the stacked piezoelement. Therefore,this device is suitable for providing the dynamic excitation of the sample.Although the piezoelement offers an elongation range of up to 90 µm, in mostcases of fatigue testing, an amplitude of only 1 µm or less was needed to obtainlarge deflections of up to a few millimeters of the lower plate, due to the high Q-factor.

The vibration of the lower free moving end plate, which is an in-plane bendingmotion as a result of the sample and notch geometry, is monitored by a laseropticdisplacement sensor based on the triangulation principle (for an illustration of themotion see Fig. 5.4 and Fig. 5.10). This sensor offers a resolution in positionmeasurements of 0.5 µm within a range of 10 mm. The analog output of the sen-sor amplitude of vibration is measured by a multimeter and read from the PC.The support of the piezoelement can be positioned vertically, whereas the lasersensor is positioned in both directions of the horizontal plane by means ofmicrometer translation stages, thus allowing for precise alignment of the point ofmeasurement.


1.Stacked piezoelement (Physik Instrumente, Type: P 843.60).2.Adapter element with suitable threads.3.Cubic element screwed into the adapter element.4.LIGA-specimen.5.Laseroptical displacement sensor mounted onto a translation stage.

Fig. 5.3 Stacked piezoelement with LIGA sample mounted to it.

5.2.1 Various approaches for phase-locked loop control

The instrumentation setup for the realization of the analog phase-locked loopcontrol is essentially the same compared to fatigue testing of beams, except forthe actuator design. However, when testing at frequencies lower than about 100Hz, the stability of this analog control was not found to be stable enough. In addi-tion, a digital control with much greater stability was therefore developed withinthe laboratory software LabVIEW which will be outlined in the next section. Theprinciple (see Fig. 5.4) of the PLL remains the same, however. The phase differ-ence between the input and output signal is compared to a target phase difference

1 23

4

5


and the excitation frequency is corrected accordingly to reach this target. Themain difference is that the lock-in amplifier is now only used as a phase meterand the computer utilizing a digital algorithm replaces the analog PI-controller.

Fig. 5.4 Diagram of the setup for fatigue testing of microstructures. Depending on thetesting frequency, the PLL might be realized with an analog or digital control.

5.2.2 Phase locked loop with digital control implemented in LabVIEW

Since the devices that were essential for the application of an analog PLL-control(in detail the voltage controlled oscillator KH 5920, the PI-controller and thelock-in-amplifier EG&G 5210) could not provide sufficient stability in resonantvibrations below 100 Hz, a substitution or an alternative solution had to be found.In order to realize a digital PLL control and also to have a high degree of interac-tive control during the fatigue experiment, the PI-controller was replaced by adigital control algorithm implemented into the LabVIEW application file in

laser displacement sensorLIGA specimen

low voltage piezoelement

excitation

AAAAAAAAAAAA

amplifier

vibration

personalcomputer

multimeter 1 multimeter 2

frequency generator

lock-in amplifier

∆ϕ analog

∆ϕ digital

set new frequency

counter

f f


charge of the data acquisition and control of the experiment. The lock-in-ampli-fier was used only to obtain the phase difference value between the excitation andresponse signal, while the voltage controlled oscillator was substituted by the fre-quency generator (Stanford Research DS 345). This provides sinusoidal signalsthat are very accurate and stable in frequency.

Fig. 5.5 Demonstration of the digital phase control: Frequency and phase differencebetween excitation and displacement versus time of a microstructure (no crackgrowth). In a sequence, the target phase differences of 220°, 230°, 250° and 200°were selected and also achieved with this setup.

0 20 60 80

104.6

104.8

105

105.2

105.4

time [s]

freq

uenc

y [H

z]

0 20 60 80200

210

220

230

240

250

time [s]

phas

e di

ffer

ence

[°

(deg

ree)

]


The input (or error) e(t) into the digital controller is the deviation of the currentphase difference from the desired phase difference. The output u(t) is, in thiscase, the frequency. The analog transfer function D(s) for the PI control and thecorresponding time domain relation between the input e(t) and output u(t) are:

(5.1)

If one performs a finite-difference approximation using backward differencesbased on the differential form of Eq. (5.1), the output for the next interval is

given by:

(5.2)

where T is the time between two measurements and the index k the number ofelapsed intervals. This very simple algorithm allows to calculate the new valuefor the frequency that the frequency generator needs to be adjusted to, based onthe previous value as well as on the current and previous phase difference value.An algorithm for a proportional-integral-differential (PID) control was imple-mented in a very similar way, but for stabilizing the frequency at resonance, therewas no noticeable advantage in using the additional derivative of this type ofcontroller. In practise, it was also easier to optimize only the two parameters

and instead of the three that would be necessary for a PID control.

Although during one control step there are three commands to be performedinvolving the communication between the laboratory PC and three instrumentsover a GPIB interface bus, this communication takes place within only 0.02 sec-onds. This means that the frequency could be adjusted 50 times within one sec-ond to achieve the desired phase difference. Although this rate would still not besufficient for any fast transient processes, it proved to be very satisfactory for the

D s( ) U s( )E s( )------------ K p

Ki

s-----

u t( )

+

K pe t( ) Ki e t .d

0

t

∫+

= =

=

uk

u Kie t( ) K pe t( )uk

+uk 1– K p KiT+( )ek K pek 1––+

==

K p

Ki


task of adjusting the resonant frequency during fatigue experiments in micro-structures with a high stability and a frequency of vibration below 100 Hz. Theuser can also adjust the parameters and during the test, should the need

for it arise.

Fig. 5.5 shows the control characteristics of this digital phase control, when amicrostructure is forced to vibrate in a way as to achieve certain values of phasedifference. The frequency and phase difference between excitation and displace-ment versus time is displayed, but there is no crack growth. In a sequence, thetarget phase differences 220°, 230°, 250° and 200° were selected and achievedwith this setup within about 5 seconds.

1. Support of stacked piezoelement and vertical positioning (coarse and fine).2. Stacked piezoelement (PI P 843.60).3. Laseroptical displacement sensor: Micro-Optronic NCDT series 2000.4. Lock-in amplifier: EG&G Princeton Applied Research 5210.5. 2 Multimeters: Fluke 8842.6. Frequency counter: Keithley 776.7. Digital function generator: Stanford Research DS 345.

Fig. 5.6 Setup for fatigue testing of microstructures.

K p Ki

1

23

5

6

7

4

5.3 Modelling the vibrating microstructure 105

A photograph of the setup for fatigue testing of microstructures is presented inFig. 5.6 with the instruments listed below. The support (1) for the stacked piezoe-lement (2) allows coarse and fine vertical positioning. The laseroptic displace-ment sensor is horizontally fine-positioned for a maximum degree of signalreflection from the side surface of the lower plate. The instruments (4) through(7) have been described previously and are used for the digital implementation ofthe phase-locked loop.

5.3 Modelling the vibrating microstructure

In comparison to the slender beams that were simply supported on strings (seeSection 2.2 for the modelling), the major differences, apart from the overalldimensions, are the support conditions and the large mass and inertia provided bythe plates.

Fig. 5.7 Model for the microstructure containing a crack with the rigid support on one side.

In the model displayed in Fig. 5.7, the microstructure itself is considered to bethe only flexible portion of the structure; the plates can be assumed as rigid bod-ies. That this is a valid assumption, especially for the first mode of bending vibra-tion, was pointed out by Mazza [46] and Schiltges [63] through vibration testsand finite-element modelling of the whole structure. For this reason one can con-sider the part of the microbeam whose connecting plate is glued to the adapter of

AAAAAA

ϑ

ϕ

R

B

L

m , I

l l

t (thickness)

dS

1 2

Pl Pl


the piezoelement as being clamped like a cantilever beam. The resulting geomet-ric boundary conditions for the case of transverse vibration are that the displace-ment and the slope at the fixed end remain zero.

In analogy to the modelling of the simple beam outlined in Section 2.2, the crackmodel containing the discontinuity parameters and , together with the corre-sponding compliance matrix S (according to Eq. (2.1)), acts as the linking ele-ment connecting the two portions of the microbeam. These are modelled by thepartial differential equations describing longitudinal and transverse vibration(Euler-Bernoulli theory, compare to Eq. (2.17)). The solution of these differentialequations yields the modes of vibration and the eigenfrequencies as a function ofcrack length in the same way as for larger beams.

5.3.1 Equivalent mass and inertia properties of the lower plate

In order to obtain the dynamic boundary conditions resulting from the mass andthe moment of inertia of the lower plate, the change of momentum according toNewton’s law and the change of angular momentum needs to be formulated forthe lower plate, with the dimensions and the geometry as depicted in Fig. 5.7. Forthis task, it is essential to calculate the mass and moment of inertia with respect tothe center of gravity of this plate. The calculation of these values will not be pre-sented in detail in this thesis since it is a straightforward analytical procedure.Basically, a rectangular plate with the dimension is considered as the ele-mentary shape, which is further reduced by the two quarter-circles with theradius R. When the geometry of the plate and the density of the NiFe-alloy mate-rial are known, the desired values such as mass and inertia of the plate are welldefined.

As pointed out earlier in Section 5.1.2, apart from measuring the thickness of themicrobeam, it is necessary to obtain also the average thickness of the plate whichmight have a different value compared to the microbeam due to the manufactur-ing process. But since the determination of an average thickness of the plate isdifficult to realize and provides a considerable error source, measuring the weightof the microstructure, which is in the order of 50 to 100 mg, might be an alterna-tive approach. This weight measurement could either be achieved before mount-ing the sample on the adapter, or after the fatigue test for the lower plate only.

ϕ ϑ

B L×

5.3 Modelling the vibrating microstructure 107

5.3.2 Correcting the measured amplitude of vibration for the open-crack-state

Also for the case of vibrating microstructures, the crack opens and closes, andtherefore, the measured amplitude of vibration must be corrected for an accuratecalculation of the stress intensity factor. This is outlined in this section with thereferences to the corresponding Section 3.4.3 in mind, because the approach isvery similar.

Fig. 5.8 Shape function for the first bending mode, relative crack depth 0.4, microbeamlength 550 µm.

Fig. 5.8 shows an example of the calculated mode shape (transverse displace-ment) of a microbeam (length 550 µm) with a relative crack depth of 0.4. Asexpected, the slope is zero at the position mm and at mm themode shape curve contains the typical “kink” or discontinuity in its slope. Whenthe crack closes for the duration of one half-cycle, the displacement amplitude atthe position , where the “lower” free plate is connected to the micro-beam, will reach a smaller value. The slope at that position will also be smaller.

- 0.0002 - 0.0001 0 0.0001 0.00020

0.001

0.002

0.003

0.004

0.005

longitudinal axis [m]

shap

e fu

nctio

n [-

]

x 0.275–= x 0=

x 0.275=


Because the amplitude of vibration is measured at the edge of the vibrating plate(the laser measurement point is visible in Fig. 5.10), the displacement and theslope for both the open and the closed-crack state, need to be considered to calcu-late the ratio of maximum transverse deflection in order to correct the measuredamplitude for the open-crack state.

In comparison to Eq. (3.8) the formulation for the potential energy fromthe bending deformation and the deflection of the “equivalent” rotational springis slightly modified, because the two microbeam portions are deformed differ-ently and one cannot assume symmetry as with the simply supported beams anymore:

(5.3)

The correcting factor in order for the potential energy to be equal for bothextreme states of transverse deflection is then obtained in the same way asexplained in Section 3.4.3 and by evaluating Eq. (3.9). When this correction fac-tor is known for any crack length value a, the ratio at the position of the vibratingplate (laser displacement sensor position), is given by:

, (5.4)

where is the distance from the position at to the measurement point

of the laser.

The result of this analysis for the laser measurement position mm is

applied to a V-notched microbeam (length 550 µm, cross-section 100 x 145 µm,relative notch depth 0.2) and is plotted in the diagram of Fig. 5.9.

U a( )

U a( )12--- EI W1,xx x a,( )[ ]

2x

12--- EI W2,xx x a,( )[ ]

2x

12Sbb a( )--------------------+ W1,x x 0= a,( ) W2,x x 0= a,( )–[ ]

2 .

d

0

l2

∫+d

l1–

0

∫=

ϒ a( )

ϒ a( ) ϒ a( )W x l2 a,=( ) W ,x x l2 a,=( ) dLa+

W x l2 a0,=( ) W ,x x l2 a0,=( ) dLa+------------------------------------------------------------------------------------------=

dLa x l2=

dLa 4=

5.4 Fatigue testing results 109

Fig. 5.9 Ratio of maximum transverse deflection at the position of the laser measurementfor the open-crack state compared to the closed-crack state, microbeam dimensions

µm, .

5.4 Fatigue testing results

In this section, fatigue testing results with LIGA-microstructures with differentdimensions and different types of notches are summarized. It was also possible tointroduce fatigue cracks in smooth microbeams without any notch and to controlthe fatigue crack growth in a stable manner.

The photograph in Fig. 5.10 clearly demonstrates the first mode of vibration of anunnotched microstructure (the microbeam is 1 mm long but only 30 µm wide)and the large amplitudes (about 1 mm) that are achieved with the actuating pie-zoelement and that are necessary for a crack to be initiated in this type of speci-men. The laser measurement position and its trace during the vibration is visibleas a bright line on the right hand side of the lower vibrating plate of the LIGA-sample. Fortunately, the vibration of the whole microsample remains purely inone plane, which makes it possible for the laser beam to be always positioned atthe side surface of the plate, although its thickness measures only some value

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91

1.5

2

2.5

3

3.5

crack length ratio [-]

ampl

itude

rat

io a

t las

er p

ositi

on [

-]

100 145 550×× a0 b⁄ 0.2=


between 120 and 180 µm. This is quite remarkable and underlines the high qual-ity of the LIGA manufacturing process. In fact, the author was prepared to pro-vide a different way of measuring the amplitude of this in-plane motion, but wassurprised to find out that this was not necessary, since for almost all of the fatiguetested specimens, except for one which already showed signs of irregularitiesfrom the etching process, the use of the laseroptic displacement sensor proved tobe a reliable measurement technique.

Fig. 5.10 Vibrating unnotched LIGA sample, length of microbeam is 1 mm. The sample isglued to the cubic-shaped adapter element, which is screwed onto thepiezoelement.

5.4.1 Fatigue testing of unnotched microbeams

An example for the measured resonant frequency with respect to time during afatigue experiment is given in Fig. 5.11. The dimensions of the unnotched LIGAspecimen (NiFe 13%) are µm. The crack was initiated and prop-agated in a cross-section about 20 µm below the transition from the upper plateinto the unnotched microbeam, that is, the crack started in a region where the

28 145 300××


bending moment was expected to be largest within the microbeam. The voltageamplitude remained constant on the actuating piezoelement. Using the modelleddecreasing function of the resonant frequency versus the crack length, the cracklength for each measured frequency was determined and added to the diagram inFig. 5.11.

Fig. 5.11 Resonant frequency and crack length of a LIGA specimen (NiFe, 13% Fe) duringthe fatigue test, dimension of unnotched microbeam µm, constantvoltage amplitude on actuating piezoelement.

The crack growth rate (see Fig. 5.12) during the first 200 seconds varied between

m/cycle and increased to a maximum rate of about m/cycle. This rate of crack growth rate is very low and would be considered tobelong to the threshold region in the characteristic fatigue diagram. Even thoughthis fatigue test took only a few minutes, these low rates of crack growth weredetectable. Therefore fatigue testing with microstructures offers a great potentialto study threshold fatigue characteristics within a reasonable amount of time. Thechance of environmental effects, such as temperature or humidity changes duringnight and day cycles, on the testing results are significantly reduced when the testcan be performed within minutes, or at least within a few hours.

0 100 200 300 400

45

50

55

60

65

6

8

10

12

14

16

reso

nant

fre

quen

cy [

Hz]

time [s]cr

ack

leng

th [

µm

]

frequency

crack length

28 145 300××

2 … 4 1011–

⋅ 2 109–

⋅


Fig. 5.12 Crack growth rate of the unnotched LIGA specimen corresponding to theexperimental data of Fig. 5.11.

Considering the high stability in frequency, the resolution of crack length meas-urements is in the range of 10 nm. This value in crack length resolution was esti-mated from the fact that during the shift of 20 Hz in resonant frequency with thestability and resolution in frequency measurements of 0.005 Hz the crack lengthincreased by about 14 µm. If one assumes the same resolution in crack lengthmeasurements in relation to the total crack advance as the resolution in frequencymeasurement in relation to the frequency shift, this resolution in crack length isbelow 5 nm on average. As with beam specimens, the slope of the frequency ver-sus crack length curve is smaller in the beginning of the crack growth and bigger,i.e. steeper, for more advanced crack length values. Therefore, the resolution incrack length measurements might be somewhat lower than this average value inthe beginning of the fatigue experiment and, in general, higher the larger thecrack grows.

The scanning electron microscope (SEM) photographs in Fig. 5.13 illustrate thefracture surface of the unnotched microbeam after the fatigue test. For an inter-pretation of these SEM pictures, the author is inclined to state the following: the

0 100 200 300 400

0

5 · 10- 10

1 · 10- 9

1.5 · 10- 9

2 · 10- 9

time [s]

da/d

N [

m/c

ycle

]


crack apparently started at the upper long side, initially grew faster on its righthand side portion as indicated by the higher magnification in Fig. 5.13, thenmoved to the lower side, where a narrow rough region indicates the final ruptureof the specimen. Interestingly, this small narrow region is marked off by linesthat are very much parallel to the longer sides of the cross-section of themicrobeam. If the crack would indeed grow in such a parallel manner in general,or if this was just coincidental cannot be answered with certainty, since not all ofthe fracture surfaces of other specimens contain different regions of roughnessthat are so easily distinguishable as in this case and the effort to produce SEM-photographs was not undertaken for more than a few samples.

Fig. 5.13 Fracture surface of unnotched specimen ( µm), SEM photographs, magnification (left), magnification (right).

The following photographs in Fig. 5.14 through Fig. 5.16 document variousstages before, during and after the fatigue test of a microsample. Fig. 5.14 showsthe microsample (NiFe 25%) containing a very slender 1 mm long microbeamwith the cross-section of µm prior and after the detachment from theprotective frame. The two pictures in Fig. 5.15, taken during the final stage offatigue testing, indicate that an edge crack has been formed that grew from theposition as pointed out by the arrow until the lower plate eventually fell off. Theleft picture in Fig. 5.15 clearly illustrates the asymmetric oscillation and indicatesthat the amplitude for the open-crack state is significantly higher than for theclosed-crack state. This also confirms the importance of the procedure to correct

28 145×500 × 1500 ×

40 120×


the amplitude accordingly (outlined in Section 5.3.2). Fig. 5.16 shows that thelower portion of the specimen finally broke off during the fatigue test.

For this unnotched specimen, before there was any noticeable decrease in reso-nant frequency, the amplitude of vibration was increased in small steps (inputvoltage into the piezo-amplifier up to 1.7 V) during two hours of testing until themeasured amplitude at the position of the laser (3 mm above the lower edge ofthe lower plate) reached a value of 1.5 mm. The resonant frequency suddenlydropped significantly from its starting value (60.75 Hz down to 15.8 Hz) andafter 15 more minutes, the fatigue test was terminated by the final rupture of thelower plate.

After a crack has been initiated on the smooth surface, this newly formed crackprovides a significant stress concentration. If further fatigue crack growth isdesired in a more or less controlled manner, one has to reduce the amplitude ofvibration accordingly. For these reasons, fatigue tests with smooth microbeamsare more difficult to perform than with notched microbeams, which seem more“predictable” in terms of the amplitude of excitation needed for a crack to propa-gate from the notch and also because there is no need to wait an undefined dura-tion of time until a crack would finally form.

Fig. 5.14 LIGA sample with unnotched microbeam before (left) and after (right) removingthe protective frame (dimensions µm).40 120 1000××


Fig. 5.15 LIGA sample vibrating during final stage of fatigue. The crack is faintly visible asa hair-line.

Fig. 5.16 Rupture during final stage of fatigue.

5.4.2 Fatigue testing of notched microbeams

A number of fatigue tests were performed with notched LIGA specimens with aV-notch and a rounded notch (compare Fig. 5.2 for sample and notch dimen-sions). Since most of these microbeams have a width of 100 µm and are 550 µmlong, the resonant frequency is considerably higher compared to the very slenderunnotched microbeams, and also easier to control.

Fig. 5.17 shows the SEM photograph of the fracture surface of a notchedmicrobeam (length 550 µm, cross-section µm, notch depth 50 µm,notch radius 10 µm). The dark rectangle represents the smooth side wall of thenotch, whereas the remaining portion of the crack surface can be divided into aregion of fatigue crack growth and final rupture. The four horizontal lines within

crack

100 123×


the fatigued area growth mark off regions of different plane heights. This effectof lines perpendicular to the notch were observed on fracture surfaces of the sim-ple beam specimens also, especially when the notch was too blunt for the crack tostart growing within one plane.

Fig. 5.17 Fracture surface of notched specimen (cross-section µm), SEMphotographs, magnification (left), magnification (right).

Fig. 5.18 Microbeam with a V-notch before (left) and after the fatigue experiment (right),crack from V-notch is visible due to a strong light source from the very left.

The results of another fatigue experiment with a V-notched microbeam (NiFe12%) are presented in Fig. 5.18 and Fig. 5.19. As indicated by the photograph tothe right in Fig. 5.18 the crack, which is visible through the positioning of thelight source from the very left, and almost parallel to the specimen surface, grewfrom the V-notch in an almost straight line perpendicular to the longitudinal axisof the microbeam. In order to examine the fracture surface of this specimen, the

100 123×500 × 1500 ×


remaining fracture surface of the microbeam was broken apart manually (seeFig. 5.19).

Fig. 5.19 Microbeam with a V-notch after the fatigue experiment and breaking the specimenapart (left), cross-section (right).

This fatigue test took about 6000 seconds and is characterized by slow crack

growth rates in the range of m/cycle. The decrease in frequencyversus time is plotted in the diagram of Fig. 5.20.

Fig. 5.20 Resonant frequency of V-notched specimen displayed in Fig. 5.19.

2 … 6 1011–

⋅

0 1000 2000 3000 5000 6000

140

160

180

200

220

240

260

280

time [s]

freq

uenc

y [H

z]


5.4.3 Representation in the fatigue diagram

The fatigue testing results were processed and analysed in the way described inSection 5.3, with the intended objective of representing the data in a fatigue dia-gram. The microbeams with the V-notch and the length of 550 µm were chosenfor this analysis for several reasons. As pointed out earlier, this type of specimenturned out to be the one with the most consistent behaviour in terms of stabilityof vibration control, manufacturing quality and specimen handling (the V-notched microbeam with the width of 100 µm is obviously sturdier than a 30 or40 µm wide microbeam). Also, there were simply more specimens of this typeavailable without any visible defects caused by the etching or handling process.

Fig. 5.21 Fatigue diagram combining the results of 5 V-notched specimens.

1 1.5 2 3

5. 10 -12

1. 10 -11

5. 10 -11

1. 10 -10

5. 10 -10

1. 10 - 9

stress intensity factor range ∆K [MPa m ]1/2

crac

k gr

owth

rat

e da

/dN

[m

/cyc

le]


Fig. 5.21 summarizes the processing analysis of fatigue testing results of 5 V-notched specimens. The range of crack growth is quite large, from a few picome-ters per cycle up to 1 nanometer/cycle. The displayed region of this fatigue dia-gram would be considered to belong to the near-threshold regime of fatigue on a

macroscopic scale. Also, the values between 1 and 3 are associatedwith a stress intensity factor range near or at the threshold. This may be com-pared to Taylor’s and Jianchun’s [77] compilation of near-threshold fatigue dataof metallic alloys.

Looking at the data in Fig. 5.21, it is difficult to determine whether one could orshould interpret a trend line or curve from such a limited amount of tests. Usuallythis region of the fatigue diagram is characterized by a large amount of scatterthrough microstructural and/or environmental effects and difficulties associatedwith the testing technique at very low crack growth rates. Nevertheless, it isencouraging to discover, that the results of 5 different specimens can be localizedin a relatively confined region in this diagram. The crack growth rate, as well asthe stress intensity factor range, show values in the order of magnitude where onewould expect them to be from the viewpoint of conventional macroscopic fatiguetesting and its associated large amount of data.

The lower left corner in Fig. 5.21 contains testing results of very slow crackgrowth rates of only a few picometers per cycle. Interestingly, these data pointsalmost form vertical lines that are parallel to each other. The resonant frequencyand the amplitude of vibration during the beginning stage of this experiment areshown in Fig. 5.22 and Fig. 5.23.

Considering the stepwise increase in amplitude, one might attribute this behav-iour to a limited resolution in amplitude measurements. However, the resolutionof the laseroptic displacement sensor is m, and therefore lower than the dis-tance between two successive steps in Fig. 5.23.

The repeated drops in resonant frequency, according to Fig. 5.22, coincide withthe sudden increases in amplitude of vibration. The model for the cracked micro-structure with respect to this particular specimen indicates an increase in cracklength of around 0.1 m for an average drop of 0.2 Hz, as seen in Fig. 5.22. Ifthe discontinuous reduction of the resonant frequency is indeed a result of crackgrowth, it can be concluded that a rapid crack advance is followed by an intervalof slow crack growth below 20 picometers per cycle.

MPa m

0.5 µ

µ


Fig. 5.22 Resonant frequency of a V-notched microbeam during a fatigue test. Crack growthrates are in the range of 1 to 20 picometers per cycle.

Fig. 5.23 Amplitude of vibration of a V-notched microbeam corresponding to Fig. 5.22.

400 600 800 1000 1200 1400 1600251

252

253

254

255

256

257

time [s]

reso

nant

fre

quen

cy [

Hz]

400 600 800 1000 1200 1400 1600

0.0375

0.04

0.0425

0.045

0.0475

0.05

0.0525

0.055

time [s]

ampl

itude

of

vibr

atio

n [m

m]


This phenomenon may be explained by the need for the crack to subsequentlybreak through a barrier in order to propagate through the material. In polycrys-tals, for example, the grain boundaries are identified to act as a barrier for micro-crack growth [28]. The question then arises, which microstructural mechanism orbarriers would be responsible for this present behaviour? There are many theoriesavailable regarding the nucleation and propagation of cracks (Suresh [75], Ellyin[28]). Some of them presume a specific material (mostly metals) or at least a typ-ical polycrystalline structure, others are developed for single crystals. Also, thereare categories for the type of crack growth, for example short, near threshold, orlong crack growth. It is then a matter of definition whether a crack is consideredshort, be it in relation to the plastic zone size, grain size or specimen dimensions.Therefore, one cannot directly transfer existing theories and observations withregular sized specimens to micrometer-sized specimens.

In summary, the PLL resonance technique, in combination with fatigue testing ofmicrostructures, provides a powerful tool to study the growth of fatigue crackswith nanometer resolution and crack growth rates down to a few picometers percycle.

6 Conclusions and Outlook

An experimental technique for the fatigue testing of resonating structures wasdeveloped in which the crack length is continuously determined with very highresolution. The stability of the vibrations, as a result of the phase-locked loopcontrol (e.g. stability in frequency of Hz with respect to 300 Hz) and theaccuracy of the non-linear model for the cracked structure, makes this techniquevery reliable. There are many advantages compared to conventional fatigue test-ing machines. The hardware required for fatigue testing of slender beams couldbe reduced to a minimum, mainly consisting of the specimen itself, the support atthe nodes with strings, and the coils and magnets for the excitation. Operatingexactly at resonance and low damping entail low energy inputs into the system.Since the system is supported at its nodes of vibration, there is no extra frictiongenerated in bearings. Also, force measurements are not necessary as the internalloading is calculated from the deformation state. The automated control ofdevices and data acquisition combined with the high testing frequency, assist infatigue testing within a reasonable duration of time. For macroscopic slenderbeams of 0.01 m width, the crack length resolution of 0.1 mm on average, is atleast one order of magnitude better than any of the conventional methodsdescribed in ASTM Standard E 647-93.

The crack model is suitable and very accurate for crack length and crack growthrate conversions from the frequency measurements in both a qualitative and aquantitative sense. However, it is essential to include the non-linearity effect inthe model for the cracked vibrating structure. If the cracks were treated asnotches with no crack face interaction, the linear model would deliver a consider-ably lower frequency than the non-linear model for a given crack length. In otherwords, the linear model would severely underestimate the crack size for a givenmeasured frequency. Fig. 4.9 points out that this discrepancy could amount to asmuch as 1 mm difference for a 10 mm wide beam specimen. Another indicationfor the quality of the crack model is the excellent agreement of the modelledmode shapes with measurements made by a scanning laser interferometer sys-tems. This underlines, to some extent, the validity of the approach to determinestresses near the crack through the mode shape that has been scaled to the currentamplitude of vibration.

It is important to observe possible heat generation during fatigue testing, both

0.001±

124 6 Conclusions and Outlook

near the crack as well as along the vibrating structure, and to interpret the meas-ured frequency values accordingly, that is if necessary, to correct them for tem-perature changes, especially when the absolute crack length needs to bedetermined very accurately.

The results in Section 4.7 demonstrate that fatigue testing data obtained with thePLL resonance technique are highly comparable to various fatigue data sourcesin the literature and that this method is very suitable for the investigation of near-threshold behaviour, due to its sensitivity at very low crack growth rates and theconvenience of testing at high frequencies. For studying very low crack growthrates, the author suggests to test at even higher frequencies by choosing a beamgeometry with a smaller aspect ratio (length divided by the width), although theacoustic emission might certainly become an annoying issue in the kHz-region.Future endeavours could be undertaken in order to achieve a more complete dataset from automated testing under various environmental conditions, such ashumid versus dry air or even in the vacuum. Walker [79] has already demon-strated the feasibility of vibration testing in a vacuum chamber with accuratetemperature control. Since the operating frequency is a function of specimengeometry alone, and not restricted by the specifications of a testing machine, thismethod offers a great potential for studying the influence of the testing frequencyon the crack growth mechanism, such as the formation of an oxide layer underdifferent environmental conditions.

All fatigue experiments so far were performed under completely reversed load-ing conditions, which means that the stress ratio remained at . It shouldtherefore be investigated whether the design of specimens or the support could bemodified to accommodate a static pre-load. An interesting task would then be toextend the non-linear model to predict a realistic frequency when the crack closesonly for a small fraction within one cycle.

The presented method can be implemented in the testing of various vibratingstructures with different geometries as it was demonstrated with fatigue testing ofLIGA-microsamples. Also, the crack model could easily be included into exist-ing models for continuous structures as a linking element, for example as a spe-cial element type in a finite-element representation.

Fatigue tests were successfully performed with piezoelectrically excited micro-structures and cracks were initiated in both types, notched and unnotched LIGA-specimens. Very small crack growth rates were measured from 1 nanometer per

R 1–=


cycle down to only a few picometers/cycle. With microbeams and cross-sectionaldimensions of m, an exceptional high crack length resolution of lessthan 5 nanometers was achieved.

However, research with fatigue testing of microstructures is still in the beginningstages and therefore, many advances are to be expected in the near future. A care-ful design of microsamples manufactured with high precision, specifically withfatigue tests in mind, seems very promising. Instead of having microstructuresvibrate in a bending mode, one could probably think of specimens designed for alongitudinal or other modes of vibration. Since the absolute resolution in cracklength measurement with the resonance technique is generally given in relationto the specimen size, which means it will become higher with decreasing speci-men size, more insight into the micro-mechanisms of fatigue initiation and near-threshold crack growth behaviour will be gained.

28 145× µ

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Curriculum Vitae

Dirk H. Schlums

Born on 27 April 1969 in Braunschweig, Germany

Citizen of Germany

Aug. 1975–June 1978 Primary school in Dorsten, Westfalen, Germany.

Sept. 1978–June 1981 Primary school in Braunschweig, Germany.

Aug. 1981–May 1988 Ricarda-Huch High school, Braunschweig.

June 1988–Sept. 1988 Practical training, Selwig & Lange, Braunschweig.

Oct. 1988–July 1994 Degree “Diplom-Ingenieur” in Mechanical Engineering from the Technical University of Braunschweig, graduated with distinction.

Sept. 1991–Aug. 1992 Exchange student at the University of Waterloo, Ontario, Canada, Department of Mechanical Engineering.

Sept. 1993–Dec. 1993 Engineer intern, Audi AG, Ingolstadt, Germany.

Sept. 1994–Jan. 2000 Doctoral candidate and teaching assistant at the Institute of Mechanical Systems (formerly Institute of Mechanics) at the Swiss Federal Institute of Technology Zurich.

Feb. 2000–Present Product Manager, Rieter Machine Works Ltd., Winterthur, Switzerland.

4.3 veriﬁcation of the mode shapes - eth z · 4.3 veriﬁcation of the mode shapes 67 4.3...

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