Fatigue crack growth, free vibrations, and breathing crack detection of aluminium alloy and steel beams

Fatigue crack growth, free vibrations, and breathingcrack detection of aluminium alloy and steel beamsU Andreaus* and P Baragatti

‘Sapienza’ Universita di Roma, Dipartimento di Ingegneria strutturale e geotecnica, Via Eudossiana 18–00184, Roma

The manuscript was received on 9 February 2009 and was accepted after revision for publication on 8 June 2009.

DOI: 10.1243/03093247JSA527

Abstract: This paper deals with online controlled propagation and vibration-based detectionof fatigue cracks in metal beams constituted of two different materials: 6082-T651 aluminiumalloy and Fe430 steel. The study addresses the initiation and propagation of cracks in thestructures and their influence on the free-vibration dynamic response. One of the originalaspects is the introduction of an actual fatigue crack instead of – as is usual – a narrow slot.

First, the crack growth is predicted analytically by numerically integrating the Paris–Walkerequation. Then, three-point bending tests are performed to obtain edge transverse cracks; twooriginal control procedures enable the tests to be traced, the results of which are comparedwith the numerical predictions.

Second, free vibrations of undamaged and cracked cantilever beams are excited by hammerimpact. The experimental results are compared with the numerical solutions of a finite elementmodel including local flexibility increase at crack opening. The differences between thedynamic behaviours of the intact and cracked beams in terms of frequency and damping allowthe damage to be detected. Even if this is a ‘linear’ method, it seems to enable the crackpresence to be detected and to account for the so-called ‘breathing’ crack. These features openthe door to future developments towards nonlinear detection methods.

Keywords: Euler beam, fatigue crack growth, breathing crack, damage detection, freevibration

1 INTRODUCTION

Experimental tests and analytical methods for crack

detection in beams were proposed in the literature

by assuming damage as a narrow slot without

closing effects [1–3].

Among a considerable number of papers reported

in the technical literature, the detection of actual

fatigue cracks [4–6] has not been a research topic.

One possible reason is the difficulty to initiate and

propagate an appropriate fatigue crack [7] and the

technical simplicity to create a slot. Fatigue crack

and slot, even narrow, exhibit different mechanical

behaviour [8]. In fact, a slot has a measurable width

which prevents any interaction between the two

surfaces of the slot itself, and hence it remains open

during vibration. On the contrary, this interaction

strongly influences the dynamic response of a

damaged beam because of the closure effects

inherently related to the so-called ‘breathing crack’

[9]. Nevertheless, several researchers assumed in

their work that the crack in a structural element was

open and remained open during vibration. Such an

assumption was made to avoid the complexities that

resulted from the non-linear characteristics pre-

sented by introducing a breathing crack.

Thus, the first goal of this work was the generation

of one single-edge crack in simply supported beams

with rectangular uniform cross-section by means of

high-cycle fatigue loading; the transverse surface

crack extended uniformly along the width of the

beam and laterally had uniform depth. In more

detail, the crack was initiated with a tiny saw cut and

propagated to the desired depth by three-point-

bending tests; a servo-hydraulic machine (MTS 810)

*Corresponding author: Dipartimento di Ingegneria strutturale e

geotecnica, ‘Sapienza’ Universita di Roma, Via Eudossiana 18,

00184 Roma, Italy.

email: [email protected]

595

JSA527 J. Strain Analysis Vol. 44

under load control was used to this end. Further-

more, the experimental tests were numerically

predicted by applying the Paris–Walker growth law.

The depth of the crack was measured directly by a

travelling microscope and verified with two control

techniques which were originally designed and

implemented; the first one used strain gauges with

stepwise variability of electrical resistance as long as

the crack grew up; the second one was based on the

reduction of structural stiffness which was measured

online and compared with the results of an analyt-

ical model of the beam. The specimens of metallic

beams were made of two different materials:

aluminium alloy [10–12] and mild steel [13, 14].

The extension of the proposed forecasting pro-

cedures to real and more complex structures could

be considered, such as sandwich components [15, 16],

plates [17, 18], I-beams [19], and welded structures

[20, 21], as well as the application to other scenarios

of structural damage [22–24].

Once the crack had spread, one end of the beam

was clamped to a base which was bolted to a test

table; so far, the second aim of this paper was the

detection of the damage in the cantilever beams by

means of free vibration tests, performed by striking

the free end of the beams with an impact hammer

and recording the acceleration of the beam tips by a

PCB 50 g full-scale piezoelectric accelerometer. In-

deed, as observed at the beginning of this section,

the application of a detection method to an actual

fatigue crack represented an experimental challenge,

owing to the strong nonlinear behaviour of the

cracked beam. The spectral and decaying analysis of

acceleration time-histories allowed for damage

detection by evaluating the natural frequencies and

damping ratios respectively in the damaged beams

and by comparing their variations with respect to the

intact beams [25–28]. Measuring and considering

the changes of the above-mentioned modal para-

meters in the presence of nonlinear effects, due to

the breathing behaviour of an actual fatigue crack,

allowed the effectiveness of this linear detection

method to be improved, and simultaneously the

development of a fully nonlinear technique for

damage detection to be enhanced [29–32].

2 FATIGUE CRACK GROWTH

2.1 Material data

Two test specimens of 6082-T651 aluminium alloy

and Fe430 steel having a 20620 mm square cross-

section were considered. According to the UNI10002

European Standard for Testing Materials Tensile,

specimens with test circular sections of 9 mm

diameter were carved from a bar. The specimens

were then slowly pulled in tension at a constant

speed by a mechanical-screw-driven machine,

where they were gripped until failure occurred. The

tensile tests provided the experimental force–strain

diagrams; using cross-section nominal stress and

carrying out linear regression of the proportionality

range of the data enabled the values 69.5 and

208.7 GPa respectively of the Young’s moduli to be

estimated, whereas the values 210 and 305 MPa of

the yield stress corresponded to the plateau level.

The geometrical and mechanical characteristics of

the specimens provided from the factory and

experimental tests are given in Table 1. The canti-

lever beams had total lengths of 710 mm for the

aluminium alloy material and 530 mm for the steel

one.

2.2 Three-point bending test

2.2.1 Test description

With reference to the structures and materials

defined in Table 1, notches were first produced

along the upper edge of the beams in order to

facilitate crack initiation. Specifically, rectangular

grooves with U-shaped bottoms of approximately

0.5 mm width and 1 mm depth were machined by a

milling machine; moreover, micro-mechanical slides

were used to achieve optimal alignment and uniform

length along the width of the beam, without

applying any load except the pressure due to the

milling machine.

Both sides of the specimens were mirror polished,

in order to monitor the crack propagation by means

Table 1 Material data

Material6082-T651aluminium alloy Fe430 steel

Square cross-section (mm6mm) 20620 20620Length (mm) 710 530Mass density (Kg/m3) 2710 7850Young’s modulus (GPa) 69.5 208.7Poisson’s ratio 0.33 0.33Yield strength (MPa) 210 305Ultimate strength (MPa) 210 440Rupture strain (%) 11 40Fatigue crack growth thresholdDKTH (MPa m1/2)

3 8

Fracture toughness KIC

(MPa m1/2)29 70

Walker equation constant C1

(mm/cycle/MPa?m1/2)2.71610208 3.29610208

Walker equation constant m1 3.7 2.44Walker equation constant c 0.641 0.79

596 U Andreaus and P Baragatti

J. Strain Analysis Vol. 44 JSA527

of a travelling microscope. Second, the notched

beams were forced by the three-point bending tests,

in which the span between the supports was 300 mm

long and the load P was applied at mid-span on the

upper edge of the beam (Fig. 1). The fatigue cracks

were produced by applying to the specimens a

constant amplitude cyclic loading within a range, the

extreme values of which are denoted by Pmin and

Pmax. A servohydraulic testing machine (MTS 810)

was used, characterized by 500 kN capacity and

¡100 mm maximum displacement; the testing ma-

chine was driven by the MTS Test Star IIs software.

The loads and the loading point displacements were

measured by means of a displacement transducer

and a force transducer respectively that are integral

to the actuator for position measurement and

control and are coaxially mounted. In addition, an

external PCB piezoelectric load cell was applied in-

between the specimen and the actuator.

The precisions of measurements were equal to:

¡0.05 per cent, for the load;

¡0.1 per cent, for the displacement;

¡0.01 per cent, for the time.

Cracking was achieved under load control with a

stress intensity factor higher than the threshold but

well below the fracture toughness of the materials.

During cyclic loading, the crack growth was mon-

itored on both sides of the specimen by using both

the physical and analytical techniques of measure-

ment control described below.

In order to perform the numerical simulation of

the experimental fatigue tests at hand, the applied

loading was assumed to be cyclic with constant

values of the maximum and minimum values Pmax

and Pmin. For fatigue crack growth work, it was

convenient to use the load range DP (5 Pmax 2 Pmin)

and the stress ratio R (5 Pmin/Pmax).

It was considered to be a growing crack that

increases its length by an amount Da owing to the

application of a number of cycles DN. The rate of

growth with cycles can be characterized by the ratio

Da/DN, or, for small intervals, by the derivative

da/dN. Various empirical relationships were em-

ployed for characterizing da/dN, and one of the

most widely adopted equations is [33, 34]

da

dN~

C1

1{Rð Þm1 1{cð Þ DKð Þm~C DKð Þm ð1Þ

where the material properties C1 5 2.7161028 (mm/

cycle/MPa m1/2), m1 5 3.70, and c 5 0.641 for the

aluminium alloy, and C1 5 3.2961028 (mm/cycle/

MPa m1/2), m1 5 2.44, and c 5 0.79 for the steel, were

applied from reference [35]. Equation (1) shows the

dependency of the crack growth rate da/dN on the

stress ratio R and on the stress intensity range DK

(i.e. the maximum/minimum range for stress in-

tensity factor K during a loading cycle) [35]. The

expression assumed for DK was the following [36]

DK ~Kmax{Kmin ð2Þ

where

K ~sffiffiffiffiffiffiffiffiffiffi(p a)

pFF(s), s~6M=(bh2), M~PL=4

FF(s)~1:106{1:552 sz7:71 s2{13:53 s3

z14:23 s4

where DP is the cyclic load range; L, h, and b are

respectively the length, the height, and the width of

the beam; a and s 5 a/h are respectively the depth

and the severity of the crack.

Unfortunately, equation (1) cannot be integrated

in closed form, hence the necessity of numerical

integration, such as Simpson’s rule. In order to

accomplish this task, it is convenient to invert and

suitably discretize equation (1) in ‘n’ intervals

Daj 5 aj+1 2 aj (j 5 1, 2, …, n), within the range of

initial and final values ai and af of the crack size a.

The initial integral is substituted by a summation

N~

ðaf

ai

dN

da

� �da~

Xf

i

DNj~1

C

Xf

i

Daj

(DK )m ð3Þ

At low growth rates, the curve da/dN versus DK [37]

generally becomes steep and appears to approach a

vertical asymptote denoted DKTH (5 3 MPa m1/2 for

Fig. 1 Test set-up

Fatigue crack growth, free vibrations, and breathing crack detection 597


the aluminium alloy and 8 MPa m1/2 for the steel

[35]), which is called the ‘fatigue crack growth

threshold’ and is interpreted as a lower limiting

value of DK below which crack growth does not

ordinarily occur. At high growth rates, the curve

again approaches an asymptote corresponding to

K 5 KIC (5 29 MPa m1/2 for the aluminium alloy and

80 MPa m1/2 for the steel [35]); i.e. the ‘fracture

toughness’ due to a rapid unstable crack growth just

prior to final failure of the test specimen. Rapid

unstable growth at high DK sometimes involves fully

plastic yielding. In such cases, the use of DK for the

portion of the curve is improper as the theoretical

limitations of the K concept are exceeded.

2.2.2 Numerical analysis and experimental tests

The above-outlined numerical approach was applied

to the fatigue tests of aluminium alloy and steel

beams, in order to forecast with sufficient accuracy

the number of cycles and time durations required to

perform the experiments at hand. For both materials,

loading was applied at a frequency of 20 Hz and the

integration of equation (3) was accomplished between

ai 5 1.0 mm and af 5 6.5 mm. In more detail, as far as

aluminium alloy was concerned, by assuming DK 5

4.35 MPa m1/2 (.DKTH), Kmax 5 11.62 MPa m1/2 (, KIC),

and Pmin 5 250 N and Pmax 5 1600 N, about <1316103

cycles were predicted, corresponding to approxi-

mately 1 h 50 min (Fig. 2(a)). In the case of steel, DK 5

12.26 MPa m1/2 (.DKTH), Kmax 5 32.71 MPa m1/2 (, KIC),

and Pmin 5 700 N and Pmax 5 4500 N were assumed

and led to a number of cycles of about 1016103, and

to a total duration of approximately 1 h 25 min;

Fig. 2(b).

The three-point bending tests for fatigue crack

growth are described below up to the attainment of

the final lengths of 6.5 mm, both for the aluminium

alloy (a) and steel (b) cases. In the simplest form of a

fatigue crack growth rate test, a cyclic load was applied

between fixed maximum and minimum levels, as in

the above-outlined numerical simulation, to the

specimens already described. A closed-loop servo-

hydraulic testing machine (MTS 810) was utilized to

apply the cyclic load for these tests as already

illustrated in Fig. 1. As the test proceeded with the

crack growing, data of deflection at midspan versus

number of load cycles were recorded (Fig. 3). The

beam stiffness was calculated as load/deflection ratio

at mid-span versus number of load cycles, and plotted

in terms of number of load cycles (Fig. 4).

(a) The test took about 1386103 cycles (about 1 h

55 min). A slight decrease of the curve was

visible almost at the beginning of the test, and

the length of 6.5 mm was attained after 137 500

cycles, after a steep descent. Therefore, the

number of cycles required physically to propa-

gate the crack up to the final length was quite

similar to the values of the numerical prediction

(130 578) with approximately a 5 per cent error.

The deflection at mid-span increased by 30 per

cent (Table 2, Fig. 3(a)), and the stiffness de-

creased by 25 per cent (Table 2, Fig. 4(a)).

(b) The test required 1106103 cycles and lasted 1 h

30 min. Also, in this case the initial part of the

experimental curve showed a soft descent and

the size of 6.5 mm was reached after 5110 000

cycles with an abrupt fall. Once again, the

analytical computation provided a reasonable

approximation of the number of cycles (100 583)

actually needed for the crack to attain its final

length with approximately a 9 per cent error.

The deflection at mid-span augmented by 8 per

cent (Table 2, Fig. 3(b)), and the stiffness re-

duced by 16 per cent (Table 2, Fig. 4(b)).

Along the upper edge of the beam, the bottom of

the rectangular groove with the U-shaped bottom and

Fig. 2 Numerical integration of fatigue crack growthfor (a) 6082-T651 aluminium alloy and (b)Fe430 steel



the propagated crack are shown in Figs 5(a) and (b)

for the aluminium alloy and steel beams respectively;

it can be observed in both cases that the initiation

point of the crack is located at approximately the

centre of the groove bottom. The picture was captured

by 50 times zoom microscope with digital acquisition

system. The position of the details shown in Fig. 5 was

evidenced in the whole beam overview of Fig. 1.

2.3 Test controls

As is known from the theory, when the crack starts to

grow, the quantity DK increases and so the growth

rate da/dN also increases. Consequently, it was

necessary to trace carefully the propagation of the

crack during the test.

The numerical prediction and the online plotting

of the above-cited deflection and stiffness at mid-

span helped crack development to be understood.

Furthermore, a visual control was performed by a 50

times zoom travelling microscope, as shown in Fig. 6

(solid line) for the aluminium alloy beam. However,

the need for more effective procedures that could

control crack propagation with sufficient accuracy

suggested the implementation of new control tools.

In order to achieve this, two types of control

procedure were implemented, namely a physical

and an analytical one.

The first control experiment was actuated by an

electrical measurement, in particular via a ‘crack

propagation gauge’ (Vishay Micro-Measurements,

Fig. 3 Beam deflection at mid-span of the (a) alumi-nium alloy and (b) steel specimens

Fig. 4 Beam stiffness as load/deflection ratio at mid-span of the (a) aluminium and (b) steel speci-mens

Table 2 Numerical and experimental results concerning the crack growth tests

Material

Experimental (E) Numerical (N) Number of cycles offset (%)

Deflection(%)

Stiffness(%)

Number ofcycles

Actualtime

Numberof cycles

Predictedtime N versus E

Aluminium alloy 30.0 225.0 137 500 1 h 54 min 130 578 1 h 48 min 25.0Steel 8.0 216.0 110 000 1 h 31 min 100 583 1 h 23 min 28.6



TK-09-CPA02-005). This is a particular strain gauge

whose pattern consists of 20 resistor strands of

different lengths, installed with a solvent-thinned

adhesive over the crack propagation area of the

specimen. Each strand is insensitive to the elastic

deformation, unlike the standard strain gauge; on the

contrary, when bonded to the beam, the propagation

of the fatigue crack through the gauge pattern causes

successive open-circuiting of the strands, resulting in

an increase of total resistance and consequently of the

measured voltage. These electrical jumps allow the

time history of the crack propagation to be recorded

and a specific point in the fracturing process, i.e. the

progressive length of the crack, to be determined, as

shown in Fig. 6 (dashed line). Therefore, the last jump,

corresponding to the fractured strand positioned at

6.5 mm from the edge, occurred at 137 500 cycles, i.e.

1 h 54 min, the actual time reported in Table 2 for

aluminium alloy.

The second type of control was an analytical

procedure based on the elaboration of experimental

data; in particular, it was able analytically to

calculate the crack length (numerical output result)

on the basis of displacements v and forces P

(experimental input data) measured at the load

point. In order to accomplish it, two models were

formulated and then compared. On one hand, the

cracked beam was modelled as a simply supported

beam having a rotational spring of elastic constant

KR at mid-span. The expression of the deflection v as

a function of the geometry (half-span L and bending

stiffness B), of the load P, and of the value KR of the

spring stiffness is

v~{1

2

L

3Bz

1

2KR

� �PL2 ð4Þ

Rearranging equation (4) as a function of KR yields

KR~{3BPL2

2PL3z12Bvð5Þ

where KR depends on deflection v and load P (i.e.

experimental input data).

On the other hand, an expression which relates KR

to the crack length a was developed in the case of

plane stress under bending [38]

(a)

(b)

Fig. 5 Crack propagation in the (a) aluminium alloyand (b) steel specimens

Fig. 6 On-site microscope (solid line), electrical(dashed line), and analytical/experimental(dotted line) control for crack propagation inaluminium alloy fatigue test

KR~bh2E

0:6384{1:035 sz3:7201 s2{5:1773 s3z7:553 s4{7:332 s5z2:4909 s6ð Þ 72 p s2ð6Þ



where b and h are respectively the width and the

height of the beam cross-section, E is Young’s

modulus, and the non-dimensional severity of the

crack s is the ratio between the length a of the crack

(i.e. numerical output result) and the height h of the

cross-section.

The crack severity s and consequently the length a

of the crack were determined by equalizing the right-

hand members of equations (5) and (6) via least-

squares optimization based on the Gauss–Newton

method. In this way, during the fatigue growth test,

the numerical value of the crack length was

predicted on the basis of experimentally measured

displacements and loads. In other words, this

procedure allowed the crack growth to be evaluated

online by combining experimental measurements of

loads and displacements with the analytical model

of the cracked section based on linear fracture

mechanics concepts, equation (6), and with the

analytical model of the beam based on simple beam

theory with local flexibility.

As plotted in Fig. 6 (dotted line) for the aluminium

alloy specimen, the elaboration of the experimental

input data led to the calculation of the output crack

length. In particular, the length of 6.5 mm occurred

at 137 500 cycles, as reported in Table 2.

2.4 Validation of results

Combining on-site experimental measurements of

the effective length of the cracks by travelling

microscope, and the above-described procedures

for test control, enabled: the crack propagation to be

traced with sufficient accuracy during the tests; the

fairly good agreement between experimental and

numerical results to be observed; and the analytical

models used to simulate the growth of the propagat-

ing crack and the behaviour of the damaged beam to

be validated. Figure 6 shows the superposition of the

curves obtained respectively by:

(a) on-site microscope (solid line);

(b) electrical control (dashed line);

(c) analytical/experimental control (dotted line).

In other words, as far as the aluminium alloy

specimen was concerned, the experimental mea-

surements, reported in columns 4 and 5 of Table 2

and shown in Figs 3(a), 4(a), and 6, indicated that

the desired crack length of 6.5 mm was attained after

the repetition of 137 500 cycles at 20 Hz frequency

and hence after a time of 1 h 54 min, whereas the

numerical prediction performed by equation (3) had

forecast the corresponding values of 137 578 cycles

and 1 h 48 min, which were reported in columns 6

and 7 of Table 2 and shown in Fig. 2(a). Further-

more, the validity of the analytical models adopted

to simulate the behaviour of the damaged beam,

equation (4), and the flexibility of the cracked

section, equation (6), was confirmed by the agree-

ment of the above-mentioned results with those

obtained through the mixed analytical/experimental

method governed by equations (4)–(6).

Analogous considerations could be presented for

the steel specimen by comparing experimental

measurements and numerical predictions. The for-

mer were reported in columns 4 and 5 of Table 2 and

shown in Figs 3(b) and 4(b), indicating that the

desired crack length of 6.5 mm was attained after the

repetition of 110 000 cycles at 20 Hz frequency and

hence after a time of 1 h 31 min; the latter had

forecast by equation (3) the corresponding values of

100 583 cycles at a duration of 1 h 23 min (see

columns 6 and 7 of Table 2 and Fig. 2(b)).

3 FREE RESPONSE TO IMPACT LOADING

3.1 Experimental tests

The free-vibration dynamics of the cracked beams of

section 2 was then both experimentally and ana-

lytically investigated in order to detect the presence

of damage; the frequency reduction due to local

flexibility increase revealed the defective character-

istics of the structural elements under examination

[39–41]; moreover, numerical and physical results

were compared by means of a suitable finite element

model of the cracked beams, accounting for local

flexibility increase by means of a bilinear rotational

spring. The experimental set-up was described and

the impact hammer test results of cracked and intact

beams were studied and compared.

As far as the experimental tests were concerned,

an accurate preparation work was accomplished in

order to guarantee a simple but efficient overall set-

up and to ensure that the whole assembly gave

repeatable results when dismantled and reas-

sembled again [42]. The beam structures were tested

grounded, in other words fixed to a base sufficiently

rigid to provide the necessary grounding: a rigid steel

table fixed to the ground (i.e. the floor) by bolting

was used as a base. The flexibility of the base itself

and its dynamic behaviour over the frequency range

of the detection tests were studied by a specific FEM

model, constituted by 130 frame elements (Fig. 7),

and by a dedicated experimental free vibration test,

performed by a PCB instrumented impact hammer.



The experimental study revealed that the first

natural frequency was equal to 9.09 Hz and that the

second natural frequency was equal to 14.29 Hz,

while the FEM model exhibited slightly different

values, the offsets of which were less than 2 per cent

(Table 3). The frequency values were nearly equal to

one-fifth and one-third respectively of the first

natural frequencies of the aluminium alloy beam

and to one-ninth and one-sixth of the first natural

frequencies of the steel beam (the evaluation of

which will be detailed below). Free vibration tests

were performed in order to evaluate the natural

frequencies of the aluminium alloy and steel beams;

thus, the natural frequencies of the base were

checked to be far enough from the resonances of

the beams in order not to affect the dynamic

responses of the beams themselves.

The beams were clamped to the base by means of

two adjacent fixed hinges in order to obtain an

overall cantilever constraint in the longitudinal

plane as shown in the scheme of Fig. 8(a) and in

the picture of Fig. 8(b). The hinges were obtained by

machining a steel plate in four V-shape supports

with chamfered edges in order not to incise the

specimens under testing; every bolt above and

below the supports (Fig. 8) was tightened by torque

wrench. The cracked beams were constrained so that

the previously 6.5 mm propagated cracks were

positioned at one-fourth of the free length of the

beams (i.e. one-fourth of the beam length away the

second hinge). The specimens were struck by an

impact hammer (PCB) with a Teflon tip at their free

ends and the responses of check points were

recorded. The quality of the strike was checked via

visual monitoring of the hammer input signal during

the tests to avoid multiple impact instead of a single

impulse. The beams were instrumented by (Fig. 8):

(a) three PCB 50 g full-scale piezoelectric acceler-

ometers, positioned respectively next to the

crack (A3), at the mid-span (A2), and on the

tip (A1); the precision of the accelerometers was

equal to ¡0.6 per cent;

(b) one electrical resistance strain gauge (Tokyo

Sokki Kenkyujo Co., precision: ¡0.25 per cent)

next to the second hinge (E1).

Furthermore, the hammer was supplied by an

internal force sensor; therefore, the recorded signals

were the time-histories of the accelerometers, of the

strain gauge, and of the hammer itself.

The signal processing equipment provided the

records of all these signal data and was constituted by:

(a) PC;

(b) compact 4-slot AC-powered chassis NI SCXI-1000;

(c) 8-channel low-pass elliptical filter module NI

SCXI-1141;

(d) universal strain gauge input module NI SCXI-

1520;

(e) 16-bit, 200 kS/s E series multifunction DAQ for

PCMCIA, NI DAQCard-6036E;

Fig. 7 Solid model for the evaluation of the naturalfrequencies of the test table

Table 3 Evaluation of the lower natural frequencies ofthe test table

Experimentaltest FEM model FEM versus experiment

f (Hz) f (Hz) Offset (%)

I mode 9.09 9.16 0.8II mode 14.29 14.5 1.5

Fig. 8 Scheme (a) and picture (b) of the experimentalset-up



(f) PCB modular signal conditioner.

The sampling of all the experimental signals was

carried out with the following settings:

(a) a sampling frequency (called scan rate) equal to

5000 Hz; consequently, the precision of the time

measurements was equal to ¡2?1024 s (¡0.02

per cent);

(b) a sampling time lasting 20 s.

These set-ups guaranteed an optimal signal proces-

sing, in order to obtain precise experimental re-

sponses. Indeed, the experimental results were

shown in terms of frequency (via spectral analyses),

time histories, and phase planes, which needed a

clean signal to be evaluated correctly.

As for the description of the experimental results,

the free responses of the beams to impact loading

were analysed and showed as

(a) natural frequencies;

(b) structural damping ratios.

Any one of the recorded signals exhibited the same

characteristics, which follow; hence the presentation

of the results was limited to the tip acceleration, for

simplicity’s sake.

1. The fast Fourier transform (FFT) of the signals

was performed and enabled the system frequen-

cies to be calculated: the first natural frequency

was equal to 43.2 Hz for the healthy beam and

42.3 Hz for the damaged one in the case of the

aluminium alloy material, while the frequency

values were 84.7 Hz for the uncracked beam and

82.5 for the cracked one in the case of the steel.

These results were summarized in Table 4 (ex-

perimental uncracked, experimental cracked) and

plotted in Fig. 9, where the tip acceleration

spectra for the intact and damaged beams can

be easily compared in a normalized scale. A

noticeable decrease of the natural frequencies

occurred due to the crack presence: the percen-

tage decreases of the first frequency of the

cracked beams in comparison with the intact

ones were 2.1 per cent for the aluminium alloy

beam (Fig. 9(a)), and 2.6 per cent (slightly larger)

for the steel one (Fig. 9(b)), as shown in the sixth

column of Table 4.

2. The damping ratios j of the structural systems

were evaluated by the method of the logarithmic

decrements d between two peaks of the experi-

mental response curves, according to the follow-

ing expression [43]

j~dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4p2zd2p ~

(1=n)ln(x0=xn)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4p2z (1=n)ln(x0=xn)½ �2

q ð7Þ

where x0 is the amplitude of a chosen peak and xn

is the amplitude of a peak n periods away. As

Table 4 Experimental frequencies and damping ratios for the first in-plane bending mode

Material

Frequency (Hz) Damping (%) Offset EC versus ES (%)

Uncracked (ES) Cracked (EC) Uncracked (ES) Cracked (EC) Frequency Damping

Aluminium alloy 43.2 42.3 1.22 1.29 22.1 5.7Steel 84.7 82.5 0.81 0.86 22.6 6.2

Fig. 9 Experimental uncracked (dotted line) and cra-cked (solid line) responses for the first frequencyof the (a) aluminium alloy and (b) steel beams



reported in Table 4, the damping ratios increased

in the damaged beams: from 1.22 per cent of the

intact beam to 1.29 per cent of the cracked one for

the aluminium alloy material, showing a +5.7 per

cent increase rate; and from 0.81 per cent of the

uncracked beam to 0.86 per cent of the damaged

one for the steel material, performing +6.2 per

cent increase rate, as shown in column 7 of

Table 4.

3.2 Numerical modelling

In order to simulate and predict the experimental

results, the structural assemblage sketched in Fig. 8

was modelled as a mono-dimensional system and

discretized by means of Euler beam finite elements

with two degrees-of-freedom in the x–y plane nodes:

in particular, 19 finite elements and 21 nodes were

implemented for the aluminium alloy beam and 15

finite elements and 17 nodes for the steel ones.

Geometrical and material data were taken according

to Table 1 for both aluminium alloy and steel beams,

while the supports were represented by two fixed in-

plane hinges. The dynamic analyses were worked

out using the Adina 8.1 package.

The crack was ‘breathing’, i.e. the crack opened and

closed in the time-domain while the beam was

vibrating. Thus, the crack was idealized as a bilinear

spring, characterized by two different stiffnesses: the

stiffness of the intact beam, kc, when the crack is

closed, and a suitably lower stiffness, ko, when the

crack is open. The stiffness of the bilinear spring in the

opening phase, which was introduced into the FEM

model, was given by equation (6) (i.e. KR 5 ko) for

rectangular cross-sections [38], relating the reduced

value ko of the spring with the non-dimensional

severity s 5 a/h of the crack, the width b and the height

h of the cross-section, and the elasticity modulus E of

the material. Hence, the open spring stiffness ko was

equal to 40 KN m and the closed spring stiffness kc to

467 KN m for the aluminium alloy beam; the open

spring stiffness ko was equal to 115 KN m and the

closed bending stiffness kc to 1379 KN m for the steel

one. Therefore, the moment–rotation relationship of

the spring was piecewise linear and introduced

nonlinearity into the system.

The first natural frequency fb of this particular

system could be evaluated introducing the so-called

‘bilinear frequency’ [44]

fb~2fc fo

fczfoð8Þ

where fc and fo could be considered the first natural

frequencies of the two subsystems with a constantly

closed or open crack respectively, i.e. with linear

springs of kc and ko elastic constants.

An impulsive force was applied at the beam tips

and the problem of analysing the system free

responses was solved by implicit time integration,

via the Newmark method and the full Newton

iteration scheme.

Also, for this numerical case, the spectral analyses

of the response signals were performed via FFT and

the presentation of the normalized results is limited

to the acceleration of the beam tip (Fig. 10).

The first natural frequency of the intact beams was

just equal to the closed crack frequency fc, while the

first natural frequency of the cracked beams was equal

to the bilinear frequency fb. The values of these

frequencies are reported in Table 5 (numerical un-

cracked, numerical cracked): the first natural fre-

quency fc of the intact aluminium alloy beam was

equal to 43.1 Hz, while the value of the first bilinear

Fig. 10 Numerical (dashed line) and experimental(solid line) free vibration responses for thefirst frequency of the cracked (a) aluminiumalloy and (b) steel beams



frequency fb of the cracked beam was 42.1 Hz; the

corresponding results for the steel beams were 85.1 Hz

and 82.8 Hz respectively.

Furthermore, the percentage decreases of these

first frequencies owing to the crack occurrence are

reported in column 4 of Table 5: in other words, the

decrease rate was equal to 2.2 per cent for the

aluminium alloy beam and 2.7 per cent for the steel

one.

3.3 Comparison between experimental andnumerical results

The implementation and development of the nu-

merical models were conducted with a model

updating methodology. The tuning of the model

parameters were performed in order to obtain a

numerical model that could actually simulate the

experimental dynamics. This achievement allowed a

model to be generated and tuned that was effectively

predictive of the real behaviour of the system and

was able also to simulate the system responses in a

different and wider dynamic range.

The model updating was validated by the compar-

ison between the first numerical and experimental

frequencies of the cracked beams, as shown in

Fig. 10(a) for the aluminium alloy beam and in

Fig. 10(b) for the steel beam, and was highlighted

in Table 5 for the uncracked (column 5) and cracked

(column 6) beams of both materials. In fact, for both

aluminium alloy and steel beams, the differences

between the numerical and experimental values of

the first frequencies were no larger than 0.4 per cent

for both cases of intact and damaged beams.

The achievement of a dynamically reliable model

enabled the system dynamics to be predicted as in

very good agreement with experimental results. As far

as the aluminium alloy beam was concerned, the

numerical decrease of the first frequency of the

cracked beam in comparison with the intact one

(22.2 per cent), in column 4 of Table 5, was

approximately equal to the decrease (22.1 per cent)

experimentally measured (column 6 of Table 4). The

steel beam exhibited a quite similar behaviour; in fact,

numerical computing provided a decrease of 2.7 per

cent, in column 4 of Table 5, for the first natural

frequency of the damaged beam with respect to the

uncracked one, where the corresponding experimen-

tal result was 22.6 per cent (column 6 of Table 4).

4 CONCLUSION

The first main goal of this paper was the experi-

mental initiation and propagation of an actual

fatigue crack, in order subsequently to perform

damage detection. The tests were extended to beams

made up of two different materials, namely, alumi-

nium alloy and steel. Furthermore, the experimental

tests were reliably predicted by means of a numer-

ical model based on the integration of the Paris–

Walker equation; in addition to the on-site measure-

ments by travelling microscope, these tests were

constantly controlled by means of two original

parameters. More explicitly, during the fatigue

growth tests, the length of the crack was predicted

by matching experimental online displacements and

loads with the deflections calculated through an

analytical model; in fact, a simply supported beam

having a rotational spring at mid-span simulated the

local flexibility reduction due to the propagating

crack. Simultaneously, the discrete jumps of the

electrical signal denounced the damage progression

in a crack propagation gauge.

In the second part of the present work, the

structural system was subjected to impact loading,

and the subsequent free motion was experimen-

tally investigated and numerically simulated, ac-

counting for the presence of a breathing crack. A

damage detection method was used which was

based on the decrease of the natural frequencies

and on the increase of the damping in the

structural system. This method had been applied

already, and had shown some limitations from the

quantitative point of view. In fact, the variations of

certain parameters might be affected by distur-

bances contained in the response and due to

material imperfections, constraint conditions,

electrical signal noises, etc. Furthermore, the

described method could include the study of the

higher vibration modes and not only the first ones,

even if the detection efficiency might decrease as

frequency increases [39].

Table 5 Numerical frequencies for the first in-plane bending mode

Material

Frequency (Hz) Frequency offset (%)

Uncracked (NS) Cracked (NC) NC versus NS (%) NS versus ES (%) NC versus EC (%)

Aluminium alloy 43.1 42.1 22.2 20.3 20.4Steel 85.1 82.8 22.7 0.4 0.3



However, the present paper addressed not a narrow

slot, which is characterized by a permanent decrease

of local stiffness, but rather a ‘true’ fatigue crack,

which ‘breathes’ during vibration. A structural system

containing this type of crack exhibits strong nonlinear

behaviour because, unlike the narrow slot, the two

surfaces of the crack can interact at the closure time.

Thus, even if the adopted method allowed the

differences of two linear parameters, namely, natural

frequencies and damping, to be estimated quantita-

tively, it seemed to work well also in the nonlinear

scenario related to the breathing crack.

In summary, the authors’ intent was theoretically

to design and actually implement a unified and

integrated procedure which was able (a) to produce

an actual fatigue crack (not a saw cut, as usual) by

integrating in an original procedure both numerical

predictions and experimental controls, and (b) to

detect online a true breathing crack (not a slot, as

usual) by applying linear methods (modal para-

meters) to nonlinear phenomena (breathing cracks).

Generally speaking, the immediate development

of the methods for detection of ‘breathing’ cracks

seemed directly to investigate the nonlinear aspects

which characterize the forced response of the

structural elements to harmonic excitation. In this

way, one could shift from the merely quantitative

linear analysis to the qualitative (and also quantita-

tive) analysis of the nonlinear response, and hence

develop a fully ‘nonlinear’ detection method.

ACKNOWLEDGEMENTS

The tests were performed in the Laboratory forTesting Materials and Structures of the Departmentof Structural Engineering, ‘Sapienza’ University ofRome, Faculty of Engineering. The authors wish tothank the Laboratory staff for their help in designingand constructing specimens, supports, and testingset-up, and in performing tests and recording results.This research has been partially funded by ‘Progettodi Ateneo 2006’ C26A059503 and ‘Progetto di Uni-versita’ C26A07TELB of the ‘Sapienza’ University ofRome.

F Authors 2009

REFERENCES

1 Rizos, P. F., Aspragathos, N., and Dimarogonas,A. D. Identification of crack location and magni-tude in a cantilever beam from the vibrationmodes. J. Sound Vibr., 1990, 138(3), 381–388.

2 Kam, T. Y. and Lee, T. Y. Detection of cracks instructures using modal test data. Engng Fract.Mech., 1992, 42(2), 381–387.

3 Chen, X. F., He, Z. J., and Xiang, J. W. Experimentson crack identification in cantilever beams. Exp.Mechs., 2005, 45(3), 295–300.

4 Brandon, J. A. and Sudraud, C. An experimentalinvestigation into the topological stability of acracked cantilever beam. J. Sound Vibr., 1998,211(4), 555–569.

5 Leonard, F., Lanteigne, J., Lalonde, S., andTurcotte, Y. Free-vibration behaviour of a crackedcantilever beam and crack detection. Mech. Syst.Signal Proc., 2001, 15(3), 529–548.

6 Benfratello, S., Cacciola, P., Impollonia, N.,Masnata, A., and Muscolino, G. Numerical andexperimental verification of a technique for locat-ing a fatigue crack on beams vibrating underGaussian excitation. Engng Fract. Mech., 2007, 74,2992–3001.

7 Vallellano, C., Navarro, A., Garcıa-Lomas, F. J.,and Domınguez, J. On the estimation of micro-structural effects in the near-threshold fatigue ofsmall cracks. J. Strain Analysis, 2008, 43(5), 337–347. DOI: 10.1243/03093247JSA351.

8 Qiu, H., Hills, D. A., and Dini, D. Furtherconsideration of closure at the root of a sharpnotch. J. Strain Analysis, 2008, 43(5), 405–409. DOI:10.1243/03093247JSA404.

9 Chondros, T. G., Dimarogonas, A. D., and Yao, J.Vibration of a beam with a breathing crack. J.Sound Vibr., 2001, 239(1), 57–67.

10 Medina-Velarde, J. L., Petrinic, N., and Ruiz, C.Experimental study of fast ductile crack growth inaluminium alloy 2014 panels. J. Strain Analysis,2001, 36(6), 561–577. DOI: 10.1243/0309324011514719.

11 Jogi, B. F., Brahmankar, P. K., Nanda, V. S., andPrasad, R. C. Some studies on fatigue crack growthrate of aluminum alloy 6061. J. Mater. Proc.Technol., 2008, 201(1–3), 380–384.

12 Zhao, T., Zhang, J., and Jiang, Y. A study of fatiguecrack growth of 7075-T651 aluminium alloy. Int. J.Fatigue, 2008, 30(7), 1169–1180.

13 Kalnaus, S., Fan, F., Vasudevan, A. K., and Jiang,Y. An experimental investigation on fatigue crackgrowth of AL6XN stainless steel. Eng. Fract. Mech.,2008, 75(8), 2002–2019.

14 Nikitin, I. and Besel, M. Effect of low-frequency onfatigue behaviour of austenitic steel AISI 304 atroom temperature and 25 uC. Int. J. Fatigue, 2008,30(10–11), 2044–2049.

15 Foundoukos, N., Xie, M., and Chapman, J. C.Fatigue tests on steel–concrete–steel sandwichcomponents and beams. J. Constr. Steel Res.,2007, 63(7), 922–940.

16 Sha, J. B., Yip, T. H., and Wong, S. K. M. In situsurface displacement analysis of fracture andfatigue behaviour under bending conditions ofsandwich beam consisting of aluminium foamcore and metallic face sheets. Mater. Sci. Technol.,2006, 22(1), 51–60.



17 James, M. N. and Paterson, A. E. Fatigue perfor-mance of 6261-T6 aluminium alloy – Constant andvariable amplitude loading of parent plate andwelded specimens. Int. J. Fatigue, 1997, 19(1),S109–S118.

18 Deng, J. and Lee, M. M. K. Fatigue performance ofmetallic beam strengthened with a bonded CFRPplate. Composite Structs, 2007, 78(2), 222–231.

19 James, M. N., Lambrecht, H. O., and Paterson, A.E. Fatigue strength of welded cover plates on 6261aluminium alloy I-beams. Int. J. Fatigue, 1993,15(6), 519–524.

20 Menzemer, C. C. and Fisher, J. W. Fatigue designof welded aluminium structures. TransportationRes. Rec., 1993, 1393, 79–88.

21 Dexter, R. J. and Pilarski, P. J. Crack propagationin welded stiffened panels. J. Construct. Steel Res.,2002, 58(5–8), 1081–1102.

22 Lopez, J., Cuerno, C., Atienza, R., and Guemes, A.Damage detection in boundary conditions. Com-parison benchmark. Key Engng Mater., 2003, 245–246, 59–68.

23 Sohn, H., Farrar, C. R., Hemez, F. M., Shunk,D. D., Stinemates, D. W., and Nadler, B. R. Areview of structural health monitoring literature:1996–2001. Los Alamos Natn Lab. Rep., LA-13976-MS, 2003, 1–301.

24 Mengelkamp, G., Fritzen, C. P., Dietrich, G.,Richter, W., Cuerno, C., Lopez-Diez, J., andGuernes, A. Structural health monitoring of theARTEMIS satellite antenna using a smart structureconcept. Second European Workshop on Structuralhealth monitoring, 2004, 41, 1075–1082.

25 Chati, M., Rand, R., and Mukherjee, S. Modalanalysis of a cracked beam. J. Sound Vibr., 1997,207(2), 249–270.

26 Kisa, M. and Brandon, J. The effects of closure ofcracks on the dynamics of a cracker cantileverbeam. J. Sound Vibr., 2000, 238(1), 1–18.

27 Luzzato, E. Approximate computation of non-linear effects in a vibrating cracked beam. J. SoundVibr., 2003, 265(4), 745–763.

28 Luo, T.-L., Wu, J. S.-S., and Hung, J.-P. A general-ized algorithm for the study of bilinear vibrations ofcracked structures. Struct. Engng Mech., 2006,23(1), 1–13.

29 Weaver, R. and Sundermeyer, J. N. Nonlinearvibrations of cracked beams. Final Report – ArmyResearch Office’s Short-term Innovative ResearchProgram, 1995.

30 Tsyfansky, S. L. and Beresnevich, V. I. Non-linearvibration method for detection of fatigue cracks inaircraft wings. J. Sound Vibr., 2000, 236(1), 49–60.

31 Nandi, A. and Neogy, S. Modelling of a beam with abreathing edge crack and some observations forcrack detection. J. Vibr. Control, 2002, 8(5),673–693.

32 Bovsunovsky, A. P. and Bovsunovsky, O. Crackdetection in beams by means of the driving forceparameters variation at non-linear resonance vi-brations. Key Engng Mater., 2007, 347, 413–420.

33 Paris, P. C. The fracture mechanics approach tofatigue – An interdisciplinary approach. In TenthSagamore Army Conference on Materials Research(Eds J. J. Burks et al.), Syracuse, NY, 1964, pp.107–127 (Syracuse University Press, NY).

34 Walker, K. The effects of stress ratio during crackpropagation and fatigue for 2024-T3 and 7075-Y6aluminum. ASTM: Effects of Environment andComplex Load History on Fatigue Life, STM STP462, West Conshohocken, PA, 1970, pp. 1–14.

35 Dowling, N. E. Mechanical behavior of materials,second edition, 1999 (Prentice Hall, Upper SaddleRiver, New Jersey).

36 Tada, H., Paris, P. C., and Irwin, G. R. The stressanalysis of cracks handbook, 1973 (Del ResearchCorp., Hellertown PA).

37 Paris, P. C., Bucci, R. J., Wessel, E. T., Clark, W. G.,and Mager, T. R. Extensive study of low fatiguecrack growth rates in A533 and A508 Steels. InProceedings of the 1971 National Symposium onFracture mechanics, Part I, ASTM STP 513 (Amer-ican Society for Testing and Materials, WestConshohocken, PA), pp. 141–176.

38 Ostachowicz, W. M. and Krawczuk, M. Analysis of theeffect of cracks on the natural frequencies of acantilever beam. J. Sound Vibr., 1991, 150(2), 191–201.

39 Andreaus, U., Casini, P., and Vestroni, F. Fre-quency reduction in elastic beams due to a stablecrack: numerical results compared with measuredtest data. Eng. Trans., 2003, 51(1), 1–16.

40 Andreaus, U., Batra, R., and Porfiri, M. Vibrationsof cracked Euler–Bernoulli beams using meshlesslocal Petrov–Galerkin (MLPG) method. Comput.Modeling in Engng Sci. (CMES), 2005, 9(2), 111–13.

41 Chen, X., Zi, Y. Y., Li, B., and He, Z. J. Identifica-tion of multiple cracks using a dynamic mesh-refinement method. J. Strain Analysis, 2006, 41(1),31–39. DOI: 10.1243/030932405X30911.

42 Ewins, D. J. Modal testing: theory, practice andapplication, second edition, 2000 (Research StudiesPress Ltd, Hertfordshire).

43 Chopra, A. K. Dynamics of structures – theoryand applications to earthquake engineering, 1995(Prentice Hall, Englewood Cliffs, NJ).

44 Andreaus, U., Casini, P., and Vestroni, F. Non-linear dynamics of a cracked cantilever beamunder harmonic excitation. Int. J. Non-LinearMech., 2007, 42(3), 566–575.

APPENDIX

Notation

a crack size

af final crack size

ai initial crack size

aj current crack size

b cross-section width

B bending stiffness



C1 proportionality coefficient of the

stress intensity range in equation (3)

E Young’s modulus

fb bilinear frequency

fc first natural frequencies of the

sub-model with a constantly closed

crack

fo first natural frequencies of the

submodel with a constantly open

crack

F(.) geometry function in equation (2)

h cross-section height

kc elastic constant of the spring

simulating the closed crack

ko elastic constant of the spring

simulating the open crack

K stress intensity factor

KIC fracture toughness

Kmax stress intensity factor at Pmax

Kmin stress intensity factor at Pmin

KR stiffness of the rotational spring

L beam length

m exponent coefficient of the stress

intensity range in equation (3)

m1 exponent coefficient of the stress

ratio in equation (3)

N number of cycles

P mid-span load

Pmax maximum load

Pmin minimum load

R stress ratio

s crack severity

v mid-span deflection

x0 amplitude of a chosen peak

xn amplitude of a peak n periods away

c exponent coefficient of the stress

ratio in equation (3)

d logarithmic decrement between two

peaks of the experimental response

curve

DK stress intensity range

DKTH fatigue crack growth threshold

DP load range

j damping ratio



Documents

Fatigue crack growth, free vibrations, and breathing crack detection of aluminium alloy and steel beams