Research ArticleThe Effects of Breathing Behaviour on Crack Growth ofa Vibrating Beam

Wenguang Liu 1 andMark E Barkey2

1School of Aeronautic Manufacturing Engineering Nanchang Hangkong University Nanchang Jiangxi 330063 China2Department of Aerospace Engineering and Mechanics The University of Alabama Tuscaloosa AL 35487 USA

Correspondence should be addressed to Wenguang Liu liuwg14nchueducn

Received 10 October 2017 Revised 28 January 2018 Accepted 7 February 2018 Published 11 March 2018

Academic Editor Mahmoud Bayat

Copyright copy 2018 Wenguang Liu and Mark E Barkey This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The effects of breathing behaviour on the dynamic response and crack growth are studied through a cracked cantilever beamThe main goal is to reveal the coupling mechanism of dynamic response and crack growth by employing a plain single-degree-of-freedom (SDOF) lumped system with the breathing crack stiffness and friction damping The friction damping loss factor isderived by using Coulomb friction model and energy principle Natural frequency dynamic stress dynamic stress intensity factor(DSIF) and crack growth are analyzed by case studies in the end Results indicate that not only does the stiffness oscillates duringcrack growth corresponding to the physically open and closed states of the crack but also stiffness and friction damping oscillatenonlinearly with crack growth This behaviour induces not only nonlinear dynamic response but also nonlinear crack growth Itprovides an approximate description of the nonlinearities introduced by the presence of a breathing crack Therefore it can beemployed to improve the prediction precision of the crack identification and crack growth life of a cracked cantilever beam

1 Introduction

Cantilever beam-like structures such as aircraftwings engineblades and rocket bodies are widely used in the aerospaceengineering field Cracks often generate on the surfaceof a beam when the beam-like structure is subjected tovibratory loading conditions In order to ensure the safetyand reliability of these beam-like structures study on thedynamic response and crack growth is important to preventcatastrophic failures now and in the future

According to whether the crack is in the opening orclosing state during working time the crack model is clas-sified into two types open crack models or breathing crackmodels As the name implies open crackmodels are regardedas keeping the crack in the open state during motionFor example Biswal et al [1] Krawczuk and Ostachowicz[2] Papadopoulos and Dimarogonas [3] Ostachowicz andKrawczuk [4] Chondros et al [5 6] Heydari et al [7] andBehera et al [8] developed different kinds of open crackmodels so as to simplify the dynamic analysis for crackedbeams However the open crack model is not accurate for

a beam having huge excitation due to the crack closureeffect Especially when a beam is at its resonance state evena very tiny force may stimulate a large dynamic responseThus Cheng et al [9] Chondros et al [10] Douka andHadjileontiadis [11] Kisa and Brandon [12] Surace et al [13]Chatterjee [14] Ruotolo et al [15] and Nandi and Neogy[16] proposed breathing crack models so as to consider thetransition state between the fully open and fully closed stateof a crack Compared with the open crack models breathingcrack modes are in better agreement with cracks in practicalengineering systems That is why most of the recent studiesabout the dynamic response and crack growth mainly focuson the breathing cracked beams In this article two breathingcrack models are employed to derive the stiffness model ofthe breathing cracked beam One is using the square wavefunction to express the stiffness model of a cracked beamand another is using the harmonic function The squarewave function model considers that breathing process haseither fully open or fully closed two states But the harmonicfunction model takes the transition state between the fullyopen crack state and fully closed state into account

HindawiShock and VibrationVolume 2018 Article ID 2579419 12 pageshttpsdoiorg10115520182579419

2 Shock and Vibration

L

h

b

a

F

x

z

o

l

y

z

o①

③

②

Figure 1 Model of a cracked cantilever beam where 119865 is an exciting force and 119886 is the crack depth

Through these crack models the vibrational responseof the cracked beam had been studied by many authorsOstachowicz and Krawczuk [4] presented a means to analyzethe effects of open cracks upon the natural frequencies ofa cracked cantilever beam Cheng et al [9] investigated anonlinear crack model to analyze the vibrational responseof a cracked beam Douka and Hadjileontiadis [11] and Liuand Barkey [17] studied the dynamic response of a breathingcracked beam so as to reveal the nonlinear behaviour Kisaand Brandon [12] Zeng et al [18] Zhang et al [19] andMa et al [20] created a finite element model to analyze thevibrational response of a cracked beam Ruotolo et al [15]analyzed the dynamic response of a cracked beam underharmonic forcing by using a so-called closing crack modelto represent the cracked element Liu et al [21] proposed athree-segment beam model having local flexibilities due tocracks to analyze the dynamic response of a beam having aclosed fully embedded crack Shih and Chen [22] examinedthe coupling effect of the fatigue crack growth and dynamicresponse of a cracked shaft with straight-edge or ellipticalarc cracks considered Dentsoras and Dimarogonas [23ndash25]studied the crack propagation of a vibrating beam underdifferent excitation Results indicate that the stiffness value ofa cracked beamwhen the closure effects are considered variesnonlinearly It is changed periodically when a periodical forceacts on the beam The impacts of breathing behaviour onthe stiffness of the cracked beam are reflected by a time-varied stiffness model However most of dynamic responseanalysis assumes that the crack remains static The dynamicresponse varyingwith crack severity is ignoredwhen carryingout crack growth analysis and the coupling effects betweenthe dynamic response and the crack growth are generally notconsidered

The coupling of the crack growth and the dynamicresponse is a complicated but important mechanism to thedynamic design and the damage tolerant design of a crackedbeamworking in vibration environmentHowever at presentfew works have been done concerning the coupling effectsof the fatigue crack growth and the dynamic response Mostof the present works that have been done ignore the factthat the crack severity changes during vibration analysisIn fact not only will the crack depth increase but alsowill the microslip of the crack surfaces that are producedwhen the closure effect is considered for a vibrating beam

In this case the friction force between the crack surfacesis introduced by the normal pressure forces This frictionforce will do work over the microslide distance so thatthe energy dissipation is produced When friction energydissipation happens damping is introducedWhen a crackedbeam works in a resonance condition the friction dampingwill play a very important role in the dynamic responseand crack propagation This scenario happens frequently inequipment and structures Therefore resonance is one of thecommon reasons of equipment failures Because the crackalters dynamic properties of the vibrating beam the dynamicresponse analysis is often used to calculate the dynamic forceapplied to the cracked section of beam These observationsimply that the breathing behaviour is a complicated but areal very important factor to the damage tolerant designfor a beam Therefore it is vital to understand the dynamicresponse changes caused by the crack growth and its effectupon the crack growth But the previous research pays littleattention to this problem In order to discover themechanismof friction damping and its effect on the dynamic responseand crack growth the critical detail is to set up a properfriction damping model for the breathing crack

The present work is to analyze the first mode frequencyof the cracked beam by the mentioned two stiffness modelsand Galerkin method to derive a friction damping model fora breathing crack by energy principle and Coulomb frictionmodel and to discuss the dynamic stress response stressintensity factor and crack growth with friction damping lossfactor included by case studies in the end

2 Vibration Equation of a Cracked Beam

21 Single-Degree-of-Freedom (SDOF) System In this articlean Euler-Bernoulli cantilever beam of 304 stainless steel(304SS) is considered The material of the beam is assumedto be isotropic and homogenous And a straight surface crackis assumed to be on the top edge The coordinate geometryand dimensions of the beam are shown in Figure 1

Ignoring the structural damping and the crack dampingthe transverse dynamic equation of an elastic beam in the 119909-119911plane under a concentrated sine force 119865 can be expressed as

11986411986812059741199081205971199094 + 12058811986012059721199081205971199052 = 1198650 sin (120596119905) 120575 (119909 minus 119871) (1)

Shock and Vibration 3

Here119908 is the transverse displacement119909 is the coordinate 119905 istime 119864 is elastic modulus of material 120588 is density of material119868 is the area moment of inertia of the beamrsquos cross sectionabout 119910-axis 119860 is area of the cross section 1198650 is amplitudeof the external harmonic force 120596 is excitation frequency and120575 is the Dirac function

Based on the composition of a forced dynamic responsethe first mode does the most contribution Therefore onlythe first vibration mode of the cracked cantilever beam isconsidered in this article Suppose that 119882(119909) is the vibrationamplitude at 119909 position of the neutral axis of the crosssection of beam and 119879(119905) is amplitude-time function atany position The equation of motion of a single-degree-of-freedom (SDOF) system for the intact beam throughGalerkin method can be expressed as

119898lowast 12059721198791205971199052 + 119896lowast119879 = 119865lowast (2)

Here 119898lowast is the generalized mass 119896lowast is the generalizedstiffness and 119865lowast is the generalized force They are given by

119898lowast = 120588119860int1198710

1198822 (119909) 119889119909119896lowast = 119864119868int119871

0

1205974119882(119909)1205971199094 119882(119909) 119889119909119865lowast = 1198650 sin (120596119905) int119871

0119882(119909) 120575 (119909 minus 119871) 119889119909

(3)

Based on the boundary condition of a cantilever beamthe vibration shape function of the first mode is given by [26]

119882(119909) = (cos 18751119871 119909 minus cosh 18751119871 119909)minus 07341 (sin 18751119871 119909 minus sinh 18751119871 119909)

(4)

Supposing that the damping of the beam composed ofthe material damping 120574119898 and the crack damping 120574119891 then theSDOF vibration equation of the cracked beam with dampingcan be modified as

119898lowast 12059721198791205971199052 + 119888lowast 120597119879120597119905 + 119896lowast119879 = 119865lowast (5)

Here 119888lowast is the damping coefficient it is given by

119888lowast = (120574119898 + 120574119891)radic119898lowast119896lowast (6)

Therefore the cracked cantilever beam is simplified intoa SDOF system as shown in Figure 2

22 Stiffness Model of a Cracked Beam When there is aperiodic force 119865 acting at the free end of the cantilever beamas shown in Figure 1 the cracked beam will be stimulated tovibrate and the crack will alternate between open and closedstatesThis phenomenon is defined as the so-called breathingbehaviour It can be regarded as that the stiffness value of the

Flowast

mlowast

clowastklowast

Figure 2 Equivalent SDOF model of the cracked cantilever beam

breathing cracked beamvarieswith time between the stiffnessof the beam as the crack is fully closed and the stiffness as thecrack is fully open Thus the stiffness value of the breathingcracked beam can be expressed by

119896br (119905) = 119896119900 + 120573 (119905) (119896119888 minus 119896119900) (7)

Here 119905 is the time 120573(119905) is the breathing function 119896119900 is thestiffness of the beam as crack is fully open 119896119888 is the stiffnesswhen the crack is fully closed 119896br is the stiffness for thebreathing cracked beam and 119896119888 is regarded as equal to thestiffness of the intact beam 119896lowast in this article

In order to understand and simulate the open and closedstate of the breathing crack a square wave function is usedduring analysis The square wave function is given by

120573 (119905) = 1 0 lt 120596119905 lt 1205870 120587 lt 120596119905 lt 2120587 (8)

Here the function means that in one vibration cycle when0 lt 120596119905 lt 120587 the crack is in fully close state and when 120587 lt120596119905 lt 2120587 the crack is in fully open state That implied thatthe breathing process of the crack is divided into two statesabsolutely by the square wave function either fully open orfully close with no transition states between themThis is notthe real situation in the beam but it is helpful to understandthe frequency of the breathing cracked beam

If there is a sine excitation force 119865 applied to the beamit is more reasonable to use a harmonic breathing functionthan to use a square wave breathing function When thebeam vibrates predominantly at its fundamental mode theharmonic function is given by

120573 (119905) = 12 + 12 sin (120596119905) (9)

4 Shock and Vibration

Through (9) and (7) the stiffness model of the breathingcracked beam can be written as

119896br (119905) = (119896119888 + 119896119900)2 + (119896119888 minus 119896119900)2 sin (120596119905) (10)

From (10) it indicates that the stiffness of the breathingcracked beam is composed of the stiffness of open crackedbeam and the stiffness of closed cracked beam It can be seenfrom Figure 1 that in case of 120596119905 = 119899120587 (119899 = 1 2 3 ) theaxis 119909 of the beam stays at A position and the crack is inhalf open state In case of 120596119905 = 2119899120587 minus 1205872 the axis 119909 of thebeam reachesB position and the crack is fully open In caseof 120596119905 = 2119899120587 + 1205872 the axis 119909 of the beam reachesC positionand the crack is fully closed

Ignoring the shear force a cracked cantilever beam can beregarded as two intact beams that are connected by a torsionalspring as shown in Figure 3 Therefore the stiffness of thetorsional spring 119896119879 due to crack can be obtained from theequation developed by Dimarogonas et al [27] as

119896119879 = 119864119887ℎ272120587 (119871 minus 119897)2 (1 minus ]2)Φ (11)

Here 119897 is the distance of the crack from the fixed end 119871 119887 andℎ are the respective length width and height of the beam ]is the Poisson ratio of the material The function 120601 is given by

120601 = 0629 (119886ℎ)2 minus 1047 (119886ℎ)3 + 4602 (119886ℎ)4

minus 9975 (119886ℎ)5 + 20295 (119886ℎ)6 minus 32993 (119886ℎ)7

+ 47041 (119886ℎ)8 minus 40693 (119886ℎ)9 + 196 (119886ℎ)10

(12)

Therefore the stiffness of the open cracked beam can becalculated by

119896119900 = 119896119879119896119888119896119879 + 119896119888 (13)

3 Friction Damping Model

31 Sliding Friction Model During crack propagation thecrack tip material will yield and then separate When thematerial of the crack tip fractures the crack will advanceAccording to the knowledge of materials mechanics theyield failure of ductile materials often happens related tothe maximum shear stress And the maximum shear stressusually appears on the incline plane which forms a 45∘ angleto the cross section of the cracked beam under tension stressstate formost of steelsTherefore test results of specimenwithsmooth surface often yield to some 45∘ cross inclination stripsas shown in Figure 4

O

l

L

b

h

kT

Figure 3 Stiffness model of the cracked cantilever beam

Slip line

Figure 4 Yield slip line

Take the Euler-Bernoulli beam having a breathing crackas the research object as shown in Figure 1 Suppose that thecrack growth is just straight ahead driven by the dynamicresponse it could be conceived that the cross-yield slip linewith 45∘ near the crack tip probably will appear on the surfaceas shown in Figure 5

Before creating the sliding friction model for the breath-ing crack some assumptions should be given at first asfollows

(1) The dynamic response of the cracked beam is stable(2) The energy dissipation of the cracked beam is pro-

duced just by the friction of the crack surface(3) The friction force at the semicrack depth as 119876 point

as shown in Figure 5 is employed to calculate the dissipatedenergy

(4) The energy dissipation happens only when the crackis in the compressed zone

(5) The friction force of the crack surfaces satisfies withthe Coulomb model law

(6) The total storage strain energy of the intact beam isequal to that of the breathing cracked beam

Based on these given assumptions the surface slidefriction model of the breathing crack is set up as shown inFigure 6 In the frictional model 119865119899 is the normal pressingforce which is perpendicular to the cross section of beamand 119865119891 is the friction force between the crack surfaces Itis estimated by the normal stress of 119876 point a positionof semicrack depth The relative microsliding distance 119889depends on the crack open distance (COD) at the semicrackdepth position

32 Damping Loss Factor According to the geometry of thecracked beam as shown in Figure 5 the nominal stress of 119876point as the crack is fully close can be estimated by Hookersquoslaw It is given by

120590119899 = 05119864 (ℎ minus 119886) 12059721199081205971199092100381610038161003816100381610038161003816100381610038161003816119909=119897 (14)

Shock and Vibration 5

lz

xo

Crack surface

Slip line

Enlarged

Q

F

Q a

a

b

ho

z

y

a2

Figure 5 Crack growth road

Therefore the normal pressing force is obtained

119865119899 = 119886119887120590119899 (15)

Because the breathing crack will be opened graduallywhen it is in intensive zone and will be closed when it is in acompressed zone the value of pressing forcemay exist duringthe half cycle relative to the effect of external harmonic forceand the average pressing force 119865av in one cycle is given by

119865av = 112058710038161003816100381610038161003816100381610038161003816int120587

0119865119899 sin (120596119905) 119889 (120596119905)10038161003816100381610038161003816100381610038161003816 =

2119865119899120587 (16)

If the frictional coefficient of crack surfaces for 304SS is 120583the friction force between the crack surfaces can be derivedby the sliding friction model It is given by

119865119891 = 120583119865av cos 45∘ (17)

In each cycle the breathing process of the crack from theopen state to the close state includes 4 phases If position Ais the initial position as shown in Figure 1 a total breathingprocess includes A rarr B rarr A rarr C rarr A Only whenthe crack is in the compressed zone the friction force willbe produced Based on the work-energy theorem for systemsthe energy dissipation introduced by the sliding friction forcebetween the crack surfaces in one vibration cycle is given by

119882119862 = 2119865119891119889 (18)

It can be seen from Figure 6 that the relative slidingdistance between the crack surfaces depends on COD of 119876

d

COD

QFn

45∘

Fn

Ff

Figure 6 Friction model of crack surface

point An empirical formula for COD with 1 accuracy forany 119886ℎ is given by Tada et al [28]

COD = 4120590119899119886119864 119881(119886ℎ) (19)

Here the function 119881(119886ℎ) is given by

119881(119886ℎ) = 08 minus 17 (119886ℎ) + 24 (119886ℎ)2 + 066(1 minus 119886ℎ)2 (20)

Figure 6 shows that the relation between COD and 119889 is

119889 = COD sdot cos 1205874 (21)

Therefore the energy dissipation caused by the frictionforce within one vibration cycle is obtained

119882119862 = 4120587 12058311986411988621198871205902119899119881(119886ℎ) (22)

According to the energy principle and the assumptionsthe strain energy storage in the cracked beam having rectan-gular cross section is given by

119881max = int1198710

1198641198682 (12059721199081205971199092 )2 119889119909

+ int1198710

12 (1 + ]) 1198641198682119860 (12059731199081205971199093 )2 119889119909

(23)

For a cracked beam the friction damping loss factor 120574119891can be defined as

120574119891 = 1198821198622120587119881max (24)

6 Shock and Vibration

Therefore the damping loss factor caused by the frictionof the crack surfaces is obtained

120574119891 = 21205831198862119887 (ℎ minus 119886)2 119881 (119886ℎ) ((12059721199081205971199092)10038161003816100381610038161003816119909=119897)21205872 (119868 int119871

0(12059721199081205971199092)2 119889119909 + (24 (1 + ]) 1198682119860) int119871

0(12059731199081205971199093)2 119889119909) (25)

4 Vibration Response Analysis

41Mode Frequency If the generalized stiffness 119896lowast is replacedby the stiffness of the open cracked beam 119896119900 then the firstmode frequency of the open cracked beam is also obtainedTherefore the first mode frequency of the cracked beamcorresponding to fully open and fully closed states can bepredicted approximately by the following equation

120596119900119888 = radic119896119900119888119898lowast (26)

Here 120596119900 is the frequency of the open cracked beam and 120596119888 isthe frequency of the closed cracked beam

If the breathing behaviour is considered the stiffness ofthe cracked beam varies with time within one vibration cycleIt is impossible to plug the stiffness model of a breathingcracked beam into the frequency equation (26) to analyzethe mode frequency of the breathing cracked beam directlyBut within one cycle no matter which kind of the breathingfunction is used the cycle period time will not change Ifthe square wave function is used the vibration cycle timeincludes two parts The prehalf cycle corresponds to the fullyopen crack and the posthalf cycle corresponds to the fullyclosed crack Therefore the relation of the sine breathingbehaviour and the square wave breathing behaviour mustobey

2120587120596br= 120587120596119900 +

120587120596119888 (27)

Here 120596br is the frequency of the breathing cracked beam Itis given by

120596br = 2120596119900120596119888120596119900 + 120596119888 (28)

42 Dynamic Stress Intensity Factor (DSIF) For a vibratingcracked beam the crack propagation speed relies on notonly the amplitude of dynamic stress response but also thecrack depth For an edge cracked beam as shown in Figure 1ignoring the shear force the beam endures bending loadmainlyThe crackwill open as type I crack In real engineeringpractice mode I crack maybe is the most common crackmodel and it is very sensitive to dynamic responseThereforethe mode I crack is considered in this article Based on thestrength theory and Hookersquos law the nominal dynamic bulk

stress at the top edge of the cracked beamunder pure bendingstates can be estimated approximately by

120590119889 = 119864ℎ2 1205972119908 (119909 119905)1205971199092100381610038161003816100381610038161003816100381610038161003816119909=119897 (29)

Plugging the dynamic displacement response into thestress equation we obtain

120590119889 = 119864120578119885stℎ2 1205972119882(119909)1205971199092

100381610038161003816100381610038161003816100381610038161003816119909=119897 (30)

Here 119885st is the static displacement it is the ratio of thegeneralized force and the stiffness of the cracked beam 120578 isthe amplify ratio of displacement response it is given by [26]and expressed by

120578 = 1radic(1 minus 1205822)2 + ((120574119898 + 120574119891) 120582)2

(31)

Here 120582 is the frequency ratio When 120582 is near to 1 the beamis near or at resonance the amplify ratio is very large and thedamping ratio plays important role in the resonance responseand crack growth

Based on the definition of stress intensity factor (SIF) infracture mechanics the SIF of the beam under pure bendingcan be defined by empirical formula

119870119868119889 = 120590119889radic120587119886119865119868 (119886ℎ) (32)

Here 119870119868119889 is SIF 120590119889 is the nominal dynamic bulk stress119865119868(119886ℎ) is crack shape function and it is given by manyauthors in approximate expression as [28]

119865119868 (119886ℎ) = 1122 minus 140 (119886ℎ) + 733 (119886ℎ)2

minus 1308 (119886ℎ)3 + 140 (119886ℎ)4 (33)

The shape function is claimed to be of engineeringaccuracy for any 119886ℎ lt 065 Crack Growth Analysis

51 The Modified Forman Equation The parameters such asthe crack closure effect stress ratio SIF range critical SIF andthe threshold of SIF are considered for the model of crack

Shock and Vibration 7

growth which is called themodified FormanModel [21]Thismodel is employed for crack growth analysis in this study andshown as

119889119886119889119873 = 119861(1 minus 1198911 minus 119903 Δ119870)119899 [1 minus Δ119870thΔ119870]119901[1 minus Δ119870 (1 minus 119903)119870119888]119902 (34)

Here 119903 is the stress ratio 119870119888 is the critical SIF Δ119870th isthe threshold SIF and Δ119870 is the stress intensity range 119873 isthe numbers of vibration cycle 119861 119899 119901 and 119902 are fatigueproperties of materials 119891 is closure function and given byNewman Jr [29]119891

=

max (1199031198980 + 1198981119903 + 11989821199032 + 11989831199033) 119903 ge 01198980 + 1198981119903 0 gt 119903 ge minus21198980 minus 21198981 119903 lt minus2

(35)

where

1198980 = (0825 minus 034120572 + 0051205722) [cos(120587119878max(21205900) )]1120572

1198981 = (0415 minus 0071120572) 119878max1205900 1198982 = 1 minus 1198980 minus 1198981 minus 11989831198983 = 21198980 + 1198981 minus 1

(36)

The threshold SIF is given by

Δ119870th = Δ119870th0 ( 119886119886 + 1198860)05 ( 4120587 tan (1 minus 119903)) (37)

Here Δ119870th0 is the threshold SIF when 119903 is 0 1198860 is the initialcrack depth

In these formulas 120572 is the constraint coefficient for planestress or plane strain problem 119878max1205900 is the ratio of themaximum stress and stress flow For the constant amplitudeloading the increment of crack growth can be calculated byintegrating (34) Because the stress relies on the crack depththe exciting frequency and the damping the amplitude ofdynamic stress response is assumed to be a constant valuewithin one cycle The incremental crack growth in each cyclecan be estimated by (34) Then the crack depth after 119894 timescycle can be calculated by the superposition method andexpressed as

119886119894 = 1198860 + 119894sum119895=1

Δ119886119895 (38)

Here 119886119894 is the total crack depth after 119894 times cycle and Δ119886119895 isthe crack incremental of the 119895th step vibration

The incremental of crack growth within one vibrationcycle is calculated by employing the modified Forman crackgrowth equation It is expressed by

Δ119886 = 119861(1 minus 1198911 minus 119903 Δ119870)119899 [1 minus Δ119870thΔ119870]119901[1 minus Δ119870 (1 minus 119903)119870119888]119902 (39)

Table 1 Material properties of 304SS

Youngrsquos modulus 119864 (MPa) 204 times 105

Poissonrsquos ratio ] 03Mass density 120588 (kgm3) 7860Yield stress 120590119904 (MPa) 2758Ultimate stress 120590119887 (MPa) 620Fracture toughness 119870119862 (MPasdotmm05) 7645Threshold value Δ119870th0 (MPasdotmm05) 1216

Table 2 Crack growth properties of 304SS

119861 6 times 10minus10119901 025119902 025119899 30119878max1205900 03120572 25

The coupling analysis can be done by the above math-ematical equations The stiffness is changed gradually withcrack growth first The damping is also introduced by thefriction energy dissipation from crack surfaces Both of themaffect the dynamic response greatly On the contrary thecrack propagation ratio is also affected by the change ofdynamic response

52 Failure Criteria If the crack depth extends to a certainvalue or the dynamic stress is too large alternative failuremodes may occur to the beam Therefore it is especiallyimportant to work out reasonable criteria to indicate thefailure Three failure criteria are considered and shown asfollows

(1) Geometrical Limit The failure occurs automatically if thecrack depth is larger than the half height of the beam becauseSIF is not suitable when used to estimate the fatigue life incrack growth equation

(2) Critical SIF The fracture occurs if the maximum SIF islarger than critical SIF (119870119868119889max ge 119870119862)(3) Fracture Failure The static fracture failure occurs if themaximum value of the bulk dynamic stress is larger than theultimate strength stress

6 Results and Discussion

A cantilever beam of 304SS is considered as 1m in length012m in width 015m in height and a straight-edge crackwith distance 119897 = 05m from the fixed end The force thatacts on the free end of the cracked beam is a sine excitationThe amplitude of the force is 1198650 = 500N and the stress ratio is119903 = minus1 The materials of the beam are shown in Table 1 Thecrack growth properties of the beam are shown in Table 2

Because the damping properties in the stainless steelsubjected to deformation and temperature the maximum

8 Shock and Vibration

07

075

08

085

09

095

1

Freq

uenc

y ra

tio

01 02 03 04 050Crack depth ratio ah

brc by ABAQUSbrc by Equation (28)oc by Equation (26)

Figure 7 Frequencies change with crack severity

damping loss factor is not bigger than 001 [30] and thedamping loss factor 120574119898 of 304SS is assumed to be 0005 Inorder to investigate the effects of breathing behaviour on thedynamic response and the crack growth themode frequencyfriction damping loss factor dynamic stress dynamic stressintensity and crack growth are analyzed

Figure 7 displays the relation of the mode frequency ratioand the crack depth ratio Results indicate that the predictedfrequency of breathing cracked beam by (28) is in goodagreement with the frequency calculated by ABAQUS Thedecrease ratio of the frequency of the breathing cracked beam120596br120596119888 is smaller than that of the open cracked beam120596119900120596119888 Itis implied that if the frequency of the force acted on the beamis near to the resonance frequency the breathing crackedbeam is subjected to a bigger response because the resonanceregion of the breathing cracked beam is wider than that ofthe open cracked beam Furthermore the crack growth speedof the breathing crack is faster than that of the open cracknear the resonance condition Therefore using a breathingcrack mode to carry out dynamic analysis or crack growthcalculation is more accurate than using an open crack model

Figure 8 investigates the friction damping loss factor byusing different friction coefficient Results indicate that thefriction damping loss factor varies with not only the crackseverity but also the friction coefficient With the increaseof the crack depth or the friction coefficient the frictiondamping loss factor increases That implied that the energydissipated by the friction force is increasing during crackgrowth It is very helpful to reduce the dynamic responseof the resonant beam Thus the crack growth ratio of thebreathing crack is reduced too

Figure 9 shows the maximum dynamic stress of thebreathing cracked beam It is compared with that of the opencracked beam near the resonance region without considering

0

0005

001

0015

002

0025

003

Fric

tion

dam

ping

loss

fact

or

01 02 03 04 050Crack depth ratio ah

= 01

= 03

= 05

f

Figure 8 Friction damping loss factor with crack severity changeunder different friction coefficients

Breathing crack c = 098

Open crack c = 098Breathing crack c = 099

Open crack c = 099Breathing crack c = 1

Open crack c = 1

01 02 03 04 050Crack depth ratio ah

0

20

40

60

80

100

120

140

160

180

Max

imum

dyn

amic

stre

ssd

max

(MPa

)

Figure 9 The maximum dynamic stress with crack severity change(120583 = 0)

the friction damping Results indicate that the stress responseamplitude varies with crack depth Resonance vibrationhappens when the crack reaches a certain depth with onefixed exciting frequencyThe maximum stress happens at theresonant point And themaximumdynamic stress of the opencracked beam is smaller than that of the breathing crackedbeam with the same exciting frequency This implies that the

Shock and Vibration 9

0

20

40

60

80

100

120

140

160

180

Max

imum

dyn

amic

stre

ssd

max

(MPa

)

01 02 03 04 050Crack depth ratio ah

c = 098 = 01

c = 098 = 03

c = 099 = 01

c = 099 = 03

c = 1 = 01

c = 1 = 03

c = 098 = 05

c = 099 = 05

c = 1 = 05

Figure 10Themaximumdynamic stress with crack severity changeof breathing cracked beam

closure effect changes the stiffness of the breathing crackedbeam so that the maximum stress response also changes

Figure 10 presents the effect of the friction damping on themaximum dynamic stress of a breathing cracked beam as thebeam working under resonance condition It can be shownthat the maximum stress of the beam is greatly reduced bythe friction damping This implied that the friction dampingis an important factor to reduce the crack growth ratio

Figure 11 discovers the relation of the maximum dynamicSIFs of the open cracked beam and that of the breathingcracked beam without friction damping It is shown that themaximum dynamic SIF of the breathing cracked beam islarger than that of the open cracked beam without frictiondamping considered under the same exciting frequency Buteach of them happens at different crack depths

Figure 12 studies the effects of friction damping loss factoron themaximum dynamic SIF of the breathing cracked beamat resonance It can be seen that dynamic SIFs are verysensitive to the friction damping Comparing Figure 10 withFigure 12 we can find that the change law of the dynamicstress is different from that of the dynamic SIF The reasonis because the crack depth also plays an important role in SIFduring crack growth

Figures 13 and 14 examine the relation of the crack depthwith the vibration cycles It can be seen that the crackgrowth ratio of the open cracked beam is slower than thatof the breathing cracked beam if the friction damping is notconsidered The reason is because the decrease ratio of the

Breathing crack c = 098

Open crack c = 098

Breathing crack c = 099

Open crack c = 099

Breathing crack c = 1

Open crack c = 1

0

200

400

600

800

1000

1200

1400

1600

1800

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

Figure 11 The maximum dynamic SIF with crack severity changefor different frequency (120583 = 0)

0

500

1000

1500

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 12 The maximum dynamic SIF with crack severity changeof breathing cracked beam

mode frequency of the breathing cracked beam is slower thanthat of the open cracked beam This means the breathingcracked beam has a wider resonance region Considering

10 Shock and Vibration

01 015 02 025 03 035 04 045 05

Breathing crack c = 098

Breathing crack c = 099

Breathing crack c = 1

Open crack c = 098

Open crack c = 099

Open crack c = 1

Crack depth ratio ah

times104

0

1

2

3

4

5

Num

bers

of v

ibra

tion

cycl

e N

Figure 13 Crack depth with vibration cycles for different frequency(120583 = 0)

the friction damping the crack growth ratio is reduced bythe friction damping for a breathing cracked beam obviouslywhen the beam comes into resonance region as shown inFigure 14

Figure 15 shows the relation of the decrease ratio offrequency ratio and the numbers of vibration cycles It can beseen that the decrease ratio of frequency of the open crackedbeam is faster than that of the breathing cracked beam at thefirst phase of the crack growth when the exciting frequency is098 times of themode frequency of the intact beam But aftercertain numbers of vibration cycle the decrease ratio of thefrequency ratio of the breathing cracked beam is faster thanthat of the open cracked beam

7 Conclusions

In this article the stiffness model and the friction dampingmodel of a breathing cracked beam are derived The firstmode frequency is analyzed by using a SDOF model whichis simplified by the Galerkin method The dynamic stressis derived and the crack growth is studied The effects ofthe breathing behaviour on the dynamic response and crackgrowth are revealed so as to analyze the coupling mechanismof the crack growth anddynamic response approximatelyThemain conclusions are obtained

(1) During vibration analysis of a cracked beam the crackclosure effect between opening and closing states must beconsidered A breathing crack model is better than the opencrackmodel for dynamic response and crack growth analysisA sinusoidal breathing function is useful to simulate the

0

2

4

6

8

10

Num

bers

of v

ibra

tion

cycl

e N

times104

015 02 025 03 035 04 045 0501Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 14 Breathing crack depth with vibration cycles for differentfriction coefficient and frequency

times104

088

09

092

094

096

098

1

Freq

uenc

y ra

tio

br

cor

o

c

1 2 3 4 50Numbers of vibration cycle N

oc

brc = 0brc = 02

brc = 04

Figure 15 Frequency ratio with vibration cycles (120596120596119888 = 098)

transition state between the fully open state and the fullyclosed state It meets the object facts and the physical naturalof crack It may be closer to the real motion of the breathingcrack than that of the squarewave function Using a breathingcrack model is more realistic than using an opening crackmodel

Shock and Vibration 11

(2) Based on energy principle and Coulomb frictionmodel the friction model is set up and friction damping lossfactor is derived Friction damping not only plays importantrole in the maximum dynamic response but also playsimportant role in the crack growth Especially to resonanceresponse and crack growth friction damping effect is veryobvious

(3)TheGalerkinmethod is used to simplify the breathingcracked beam into a SDOF systemThe first mode frequencyequation is derived approximately The first mode frequencychanges with the crack severity The frequency decrease ratioof the breathing cracked beam is slower than that of theopen cracked beam If the closure behaviour of the breathingcrack is not considered the degree of crack growth may beunderestimated

(4) The modified Forman equation is employed to simu-late the crack growth A crack growth life analytical methodis proposed with the coupling effect of vibration and fatiguecrack growth included Breathing behaviour of the crack isimportant to crack growthThe dynamical properties play animportant role in crack growthWhen the exciting frequencyis near to the first mode frequency the crack growth ratio isvery fastThe crack growth life of the breathing cracked beamis smaller than that of the open cracked beam

Nomenclature

119886 1198860 Crack depth initial crack depth119886119894 Δ119886119895 The 119894th step crack depth the 119895th step crackincremental119860 Area of cross section119887 Width of beam cross section119861 Fatigue properties of materials119888lowast Coefficient of modal damping119862 Flexibility of the intact beam119862119879 Flexibility due to the presence of the crack119862119900 Total flexibility of the open cracked beam119889 Sliding distance between crack surfaces119864 Youngrsquos modulus119891 Closure function of breathing crack119865 Harmonic load119865lowast 1198650 The generalized force loading amplitude119865119899 119865119891 Normal pressing force friction force119865av Average pressing force119865119868 Crack shape functionℎ Height of the beam cross section119868 Area moment of inertial of the beam crosssection119896lowast The generalized stiffness119896br Stiffness of the breathing cracked beam119896119888 Stiffness of the close cracked beam119896119900 Stiffness of the open cracked beam119896119879 Stiffness of the torsional spring119870119868119889 Dynamic stress intensity factor119870119868119889max The maximum dynamic stress intensity factor119870119862 Fracture toughnessΔ119870th The threshold of stress intensity factorΔ119870 The range of stress intensity factor119897 Distance of crack from the fixed end

119871 Length of the beam119898lowast The generalized mass119872 Moment119899 119901 119902 Fatigue properties of materials119873 Numbers of vibration cycle119876 Shear force119903 Stress ratio119905 Time119879 Amplitude-time function119881max The strain energy119882119862 Energy dissipation119908 Lateral deflection in 119911 direction of beam119882(119909) Modal function119909 119910 119911 The coordinate119885st Static displacement120588 ] Density of material Poissonrsquos ratio120583 Friction coefficient120596 Excitation frequency120596119888 Frequency of the close cracked beam120596119900 Frequency of the open cracked beam120574119898 Material damping120574119891 Frictional damping120578 Amplify ratio120582 Frequency ratio120573 The breathing function120575 The Dirac function120590119899 120590119889 Nominal stress normal bulk dynamic stress120590119887 120590119904 Ultimate stress yield stress

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research work is supported by the National NaturalScience Foundation of China (51565039)

References

[1] A R Biswal T Roy R K Behera S K Pradhan and P KParida ldquoFinite element based vibration analysis of a nonpris-matic Timoshenko beam with transverse open crackrdquo ProcediaEngineering vol 144 pp 226ndash233 2016

[2] MKrawczuk andWMOstachowicz ldquoModelling and vibrationanalysis of a cantilever composite beam with a transverse opencrackrdquo Journal of Sound and Vibration vol 183 no 1 pp 69ndash891995

[3] C A Papadopoulos and A D Dimarogonas ldquoCoupled longi-tudinal and bending vibrations of a rotating shaft with an opencrackrdquo Journal of Sound and Vibration vol 117 no 1 pp 81ndash931987

[4] W M Ostachowicz and M Krawczuk ldquoAnalysis of the effect ofcracks on the natural frequencies of a cantilever beamrdquo Journalof Sound and Vibration vol 150 no 2 pp 191ndash201 1991

[5] T G Chondros andA D Dimarogonas ldquoVibration of a crackedcantilever beamrdquo Journal of Vibration andAcoustics vol 120 no3 pp 742ndash746 1998

12 Shock and Vibration

[6] T G Chondros A D Dimarogonas and J Yao ldquoA continuouscracked beam vibration theoryrdquo Journal of Sound and Vibrationvol 215 no 1 pp 17ndash34 1998

[7] M Heydari A Ebrahimi and M Behzad ldquoContinuous modelfor flexural vibration analysis of Timoshenko beams with avertical edge crackrdquo Archive of Applied Mechanics vol 85 no5 pp 601ndash615 2015

[8] R K Behera A Pandey and D R Parhi ldquoNumerical andexperimental verification of a method for prognosis of inclinededge crack in cantilever beam based on synthesis of modeshapesrdquo Procedia Technology vol 14 pp 67ndash74 2014

[9] S M Cheng A S J Swamidas X J Wu and W WallaceldquoVibrational response of a beamwith a breathing crackrdquo Journalof Sound and Vibration vol 225 no 1 pp 201ndash208 1999

[10] T G Chondros A D Dimarogonas and J Yao ldquoVibration of abeam with a breathing crackrdquo Journal of Sound and Vibrationvol 239 no 1 pp 57ndash67 2001

[11] E Douka and L J Hadjileontiadis ldquoTime-frequency analysis ofthe free vibration response of a beam with a breathing crackrdquoNDT amp E International vol 38 no 1 pp 3ndash10 2005

[12] M Kisa and J Brandon ldquoEffects of closure of cracks on thedynamics of a cracked cantilever beamrdquo Journal of Sound andVibration vol 238 no 1 pp 1ndash18 2000

[13] C Surace R Ruotolo and D Storer ldquoDetecting nonlinearbehaviour using the volterra series to assess damage in beam-like structuresrdquo Journal of Theoretical and Applied Mechanicsvol 49 no 3 pp 905ndash926 2011

[14] AChatterjee ldquoNonlinear dynamics anddamage assessment of acantilever beamwith breathing edge crackrdquo Journal of Vibrationand Acoustics vol 133 no 5 Article ID 051004 2011

[15] R Ruotolo C Surace P Crespo and D Storer ldquoHarmonicanalysis of the vibrations of a cantilevered beam with a closingcrackrdquo Computers amp Structures vol 61 no 6 pp 1057ndash10741996

[16] A Nandi and S Neogy ldquoModelling of a beam with a breathingedge crack and some observations for crack detectionrdquo Journalof Vibration and Control vol 8 no 5 pp 673ndash693 2002

[17] W Liu and M E Barkey ldquoNonlinear vibrational response ofa single edge cracked beamrdquo Journal of Mechanical Science ampTechnology vol 31 no 11 pp 5231ndash5243 2017

[18] J Zeng H MaW Zhang and BWen ldquoDynamic characteristicanalysis of cracked cantilever beams under different cracktypesrdquo Engineering Failure Analysis vol 74 pp 80ndash94 2017

[19] W Zhang H Ma J Zeng S Wu and B Wen ldquoVibrationresponses analysis of an elastic-support cantilever beam withcrack and offset boundaryrdquo Mechanical Systems and SignalProcessing vol 95 pp 205ndash218 2017

[20] HMa J Zeng Z Lang L Zhang Y Guo and BWen ldquoAnalysisof the dynamic characteristics of a slant-cracked cantileverbeamrdquo Mechanical Systems and Signal Processing vol 75 pp261ndash279 2016

[21] J Liu W D Zhu P G Charalambides Y M Shao Y F Xuand X M Fang ldquoA dynamic model of a cantilever beam with aclosed embedded horizontal crack including local flexibilitiesat crack tipsrdquo Journal of Sound and Vibration vol 382 pp 274ndash290 2016

[22] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[23] A Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagationrdquo Engineering FractureMechanics vol17 no 4 pp 381ndash386 1983

[24] A J Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagation in a beam under longitudinal vibra-tionsrdquo International Journal of Fracture vol 23 no 1 pp 15ndash221983

[25] A J Dentsoras andADDimarogonas ldquoFatigue crack propaga-tion in resonating structuresrdquo Engineering Fracture Mechanicsvol 34 no 3 pp 721ndash728 1989

[26] M Lalanne P Berthier and J D HagopianMechanical Vibra-tions for Engineers John Wiley amp Sons New York NY USA1983

[27] A D Dimarogonas S A Paipetis and T G ChondrosAnalyticalMethods inRotorDynamics SpringerDordrechtTheNetherlands 2013

[28] H Tada P C Paris andG R IrwinTheStress Analysis of CracksHandbook The American Society of Mechanical Engineers2000

[29] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984

[30] R Mulyukov S Mikhailov R Zaripova and D SalimonenkoldquoDamping properties of 18Cr-10Ni stainless steel with submi-crocrystalline structurerdquoMaterials Research Bulletin vol 31 no6 pp 639ndash645 1996

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Shock and Vibration 3

Here119908 is the transverse displacement119909 is the coordinate 119905 istime 119864 is elastic modulus of material 120588 is density of material119868 is the area moment of inertia of the beamrsquos cross sectionabout 119910-axis 119860 is area of the cross section 1198650 is amplitudeof the external harmonic force 120596 is excitation frequency and120575 is the Dirac function

Based on the composition of a forced dynamic responsethe first mode does the most contribution Therefore onlythe first vibration mode of the cracked cantilever beam isconsidered in this article Suppose that 119882(119909) is the vibrationamplitude at 119909 position of the neutral axis of the crosssection of beam and 119879(119905) is amplitude-time function atany position The equation of motion of a single-degree-of-freedom (SDOF) system for the intact beam throughGalerkin method can be expressed as

119898lowast 12059721198791205971199052 + 119896lowast119879 = 119865lowast (2)

Here 119898lowast is the generalized mass 119896lowast is the generalizedstiffness and 119865lowast is the generalized force They are given by

119898lowast = 120588119860int1198710

1198822 (119909) 119889119909119896lowast = 119864119868int119871

0

1205974119882(119909)1205971199094 119882(119909) 119889119909119865lowast = 1198650 sin (120596119905) int119871

0119882(119909) 120575 (119909 minus 119871) 119889119909

(3)

Based on the boundary condition of a cantilever beamthe vibration shape function of the first mode is given by [26]

119882(119909) = (cos 18751119871 119909 minus cosh 18751119871 119909)minus 07341 (sin 18751119871 119909 minus sinh 18751119871 119909)

(4)

Supposing that the damping of the beam composed ofthe material damping 120574119898 and the crack damping 120574119891 then theSDOF vibration equation of the cracked beam with dampingcan be modified as

119898lowast 12059721198791205971199052 + 119888lowast 120597119879120597119905 + 119896lowast119879 = 119865lowast (5)

Here 119888lowast is the damping coefficient it is given by

119888lowast = (120574119898 + 120574119891)radic119898lowast119896lowast (6)

Therefore the cracked cantilever beam is simplified intoa SDOF system as shown in Figure 2

22 Stiffness Model of a Cracked Beam When there is aperiodic force 119865 acting at the free end of the cantilever beamas shown in Figure 1 the cracked beam will be stimulated tovibrate and the crack will alternate between open and closedstatesThis phenomenon is defined as the so-called breathingbehaviour It can be regarded as that the stiffness value of the

Flowast

mlowast

clowastklowast

Figure 2 Equivalent SDOF model of the cracked cantilever beam

breathing cracked beamvarieswith time between the stiffnessof the beam as the crack is fully closed and the stiffness as thecrack is fully open Thus the stiffness value of the breathingcracked beam can be expressed by

119896br (119905) = 119896119900 + 120573 (119905) (119896119888 minus 119896119900) (7)

Here 119905 is the time 120573(119905) is the breathing function 119896119900 is thestiffness of the beam as crack is fully open 119896119888 is the stiffnesswhen the crack is fully closed 119896br is the stiffness for thebreathing cracked beam and 119896119888 is regarded as equal to thestiffness of the intact beam 119896lowast in this article

In order to understand and simulate the open and closedstate of the breathing crack a square wave function is usedduring analysis The square wave function is given by

120573 (119905) = 1 0 lt 120596119905 lt 1205870 120587 lt 120596119905 lt 2120587 (8)

Here the function means that in one vibration cycle when0 lt 120596119905 lt 120587 the crack is in fully close state and when 120587 lt120596119905 lt 2120587 the crack is in fully open state That implied thatthe breathing process of the crack is divided into two statesabsolutely by the square wave function either fully open orfully close with no transition states between themThis is notthe real situation in the beam but it is helpful to understandthe frequency of the breathing cracked beam

If there is a sine excitation force 119865 applied to the beamit is more reasonable to use a harmonic breathing functionthan to use a square wave breathing function When thebeam vibrates predominantly at its fundamental mode theharmonic function is given by

120573 (119905) = 12 + 12 sin (120596119905) (9)

4 Shock and Vibration

Through (9) and (7) the stiffness model of the breathingcracked beam can be written as

119896br (119905) = (119896119888 + 119896119900)2 + (119896119888 minus 119896119900)2 sin (120596119905) (10)

From (10) it indicates that the stiffness of the breathingcracked beam is composed of the stiffness of open crackedbeam and the stiffness of closed cracked beam It can be seenfrom Figure 1 that in case of 120596119905 = 119899120587 (119899 = 1 2 3 ) theaxis 119909 of the beam stays at A position and the crack is inhalf open state In case of 120596119905 = 2119899120587 minus 1205872 the axis 119909 of thebeam reachesB position and the crack is fully open In caseof 120596119905 = 2119899120587 + 1205872 the axis 119909 of the beam reachesC positionand the crack is fully closed

Ignoring the shear force a cracked cantilever beam can beregarded as two intact beams that are connected by a torsionalspring as shown in Figure 3 Therefore the stiffness of thetorsional spring 119896119879 due to crack can be obtained from theequation developed by Dimarogonas et al [27] as

119896119879 = 119864119887ℎ272120587 (119871 minus 119897)2 (1 minus ]2)Φ (11)

Here 119897 is the distance of the crack from the fixed end 119871 119887 andℎ are the respective length width and height of the beam ]is the Poisson ratio of the material The function 120601 is given by

120601 = 0629 (119886ℎ)2 minus 1047 (119886ℎ)3 + 4602 (119886ℎ)4

minus 9975 (119886ℎ)5 + 20295 (119886ℎ)6 minus 32993 (119886ℎ)7

+ 47041 (119886ℎ)8 minus 40693 (119886ℎ)9 + 196 (119886ℎ)10

(12)

Therefore the stiffness of the open cracked beam can becalculated by

119896119900 = 119896119879119896119888119896119879 + 119896119888 (13)

3 Friction Damping Model

31 Sliding Friction Model During crack propagation thecrack tip material will yield and then separate When thematerial of the crack tip fractures the crack will advanceAccording to the knowledge of materials mechanics theyield failure of ductile materials often happens related tothe maximum shear stress And the maximum shear stressusually appears on the incline plane which forms a 45∘ angleto the cross section of the cracked beam under tension stressstate formost of steelsTherefore test results of specimenwithsmooth surface often yield to some 45∘ cross inclination stripsas shown in Figure 4

O

l

L

b

h

kT

Figure 3 Stiffness model of the cracked cantilever beam

Slip line

Figure 4 Yield slip line

Take the Euler-Bernoulli beam having a breathing crackas the research object as shown in Figure 1 Suppose that thecrack growth is just straight ahead driven by the dynamicresponse it could be conceived that the cross-yield slip linewith 45∘ near the crack tip probably will appear on the surfaceas shown in Figure 5

Before creating the sliding friction model for the breath-ing crack some assumptions should be given at first asfollows

(1) The dynamic response of the cracked beam is stable(2) The energy dissipation of the cracked beam is pro-

duced just by the friction of the crack surface(3) The friction force at the semicrack depth as 119876 point

as shown in Figure 5 is employed to calculate the dissipatedenergy

(4) The energy dissipation happens only when the crackis in the compressed zone

(5) The friction force of the crack surfaces satisfies withthe Coulomb model law

(6) The total storage strain energy of the intact beam isequal to that of the breathing cracked beam

Based on these given assumptions the surface slidefriction model of the breathing crack is set up as shown inFigure 6 In the frictional model 119865119899 is the normal pressingforce which is perpendicular to the cross section of beamand 119865119891 is the friction force between the crack surfaces Itis estimated by the normal stress of 119876 point a positionof semicrack depth The relative microsliding distance 119889depends on the crack open distance (COD) at the semicrackdepth position

32 Damping Loss Factor According to the geometry of thecracked beam as shown in Figure 5 the nominal stress of 119876point as the crack is fully close can be estimated by Hookersquoslaw It is given by

120590119899 = 05119864 (ℎ minus 119886) 12059721199081205971199092100381610038161003816100381610038161003816100381610038161003816119909=119897 (14)

Shock and Vibration 5

lz

xo

Crack surface

Slip line

Enlarged

Q

F

Q a

a

b

ho

z

y

a2

Figure 5 Crack growth road

Therefore the normal pressing force is obtained

119865119899 = 119886119887120590119899 (15)

Because the breathing crack will be opened graduallywhen it is in intensive zone and will be closed when it is in acompressed zone the value of pressing forcemay exist duringthe half cycle relative to the effect of external harmonic forceand the average pressing force 119865av in one cycle is given by

119865av = 112058710038161003816100381610038161003816100381610038161003816int120587

0119865119899 sin (120596119905) 119889 (120596119905)10038161003816100381610038161003816100381610038161003816 =

2119865119899120587 (16)

If the frictional coefficient of crack surfaces for 304SS is 120583the friction force between the crack surfaces can be derivedby the sliding friction model It is given by

119865119891 = 120583119865av cos 45∘ (17)

In each cycle the breathing process of the crack from theopen state to the close state includes 4 phases If position Ais the initial position as shown in Figure 1 a total breathingprocess includes A rarr B rarr A rarr C rarr A Only whenthe crack is in the compressed zone the friction force willbe produced Based on the work-energy theorem for systemsthe energy dissipation introduced by the sliding friction forcebetween the crack surfaces in one vibration cycle is given by

119882119862 = 2119865119891119889 (18)

It can be seen from Figure 6 that the relative slidingdistance between the crack surfaces depends on COD of 119876

d

COD

QFn

45∘

Fn

Ff

Figure 6 Friction model of crack surface

point An empirical formula for COD with 1 accuracy forany 119886ℎ is given by Tada et al [28]

COD = 4120590119899119886119864 119881(119886ℎ) (19)

Here the function 119881(119886ℎ) is given by

119881(119886ℎ) = 08 minus 17 (119886ℎ) + 24 (119886ℎ)2 + 066(1 minus 119886ℎ)2 (20)

Figure 6 shows that the relation between COD and 119889 is

119889 = COD sdot cos 1205874 (21)

Therefore the energy dissipation caused by the frictionforce within one vibration cycle is obtained

119882119862 = 4120587 12058311986411988621198871205902119899119881(119886ℎ) (22)

According to the energy principle and the assumptionsthe strain energy storage in the cracked beam having rectan-gular cross section is given by

119881max = int1198710

1198641198682 (12059721199081205971199092 )2 119889119909

+ int1198710

12 (1 + ]) 1198641198682119860 (12059731199081205971199093 )2 119889119909

(23)

For a cracked beam the friction damping loss factor 120574119891can be defined as

120574119891 = 1198821198622120587119881max (24)

6 Shock and Vibration

Therefore the damping loss factor caused by the frictionof the crack surfaces is obtained

120574119891 = 21205831198862119887 (ℎ minus 119886)2 119881 (119886ℎ) ((12059721199081205971199092)10038161003816100381610038161003816119909=119897)21205872 (119868 int119871

0(12059721199081205971199092)2 119889119909 + (24 (1 + ]) 1198682119860) int119871

0(12059731199081205971199093)2 119889119909) (25)

4 Vibration Response Analysis

41Mode Frequency If the generalized stiffness 119896lowast is replacedby the stiffness of the open cracked beam 119896119900 then the firstmode frequency of the open cracked beam is also obtainedTherefore the first mode frequency of the cracked beamcorresponding to fully open and fully closed states can bepredicted approximately by the following equation

120596119900119888 = radic119896119900119888119898lowast (26)

Here 120596119900 is the frequency of the open cracked beam and 120596119888 isthe frequency of the closed cracked beam

If the breathing behaviour is considered the stiffness ofthe cracked beam varies with time within one vibration cycleIt is impossible to plug the stiffness model of a breathingcracked beam into the frequency equation (26) to analyzethe mode frequency of the breathing cracked beam directlyBut within one cycle no matter which kind of the breathingfunction is used the cycle period time will not change Ifthe square wave function is used the vibration cycle timeincludes two parts The prehalf cycle corresponds to the fullyopen crack and the posthalf cycle corresponds to the fullyclosed crack Therefore the relation of the sine breathingbehaviour and the square wave breathing behaviour mustobey

2120587120596br= 120587120596119900 +

120587120596119888 (27)

Here 120596br is the frequency of the breathing cracked beam Itis given by

120596br = 2120596119900120596119888120596119900 + 120596119888 (28)

42 Dynamic Stress Intensity Factor (DSIF) For a vibratingcracked beam the crack propagation speed relies on notonly the amplitude of dynamic stress response but also thecrack depth For an edge cracked beam as shown in Figure 1ignoring the shear force the beam endures bending loadmainlyThe crackwill open as type I crack In real engineeringpractice mode I crack maybe is the most common crackmodel and it is very sensitive to dynamic responseThereforethe mode I crack is considered in this article Based on thestrength theory and Hookersquos law the nominal dynamic bulk

stress at the top edge of the cracked beamunder pure bendingstates can be estimated approximately by

120590119889 = 119864ℎ2 1205972119908 (119909 119905)1205971199092100381610038161003816100381610038161003816100381610038161003816119909=119897 (29)

Plugging the dynamic displacement response into thestress equation we obtain

120590119889 = 119864120578119885stℎ2 1205972119882(119909)1205971199092

100381610038161003816100381610038161003816100381610038161003816119909=119897 (30)

Here 119885st is the static displacement it is the ratio of thegeneralized force and the stiffness of the cracked beam 120578 isthe amplify ratio of displacement response it is given by [26]and expressed by

120578 = 1radic(1 minus 1205822)2 + ((120574119898 + 120574119891) 120582)2

(31)

Here 120582 is the frequency ratio When 120582 is near to 1 the beamis near or at resonance the amplify ratio is very large and thedamping ratio plays important role in the resonance responseand crack growth

Based on the definition of stress intensity factor (SIF) infracture mechanics the SIF of the beam under pure bendingcan be defined by empirical formula

119870119868119889 = 120590119889radic120587119886119865119868 (119886ℎ) (32)

Here 119870119868119889 is SIF 120590119889 is the nominal dynamic bulk stress119865119868(119886ℎ) is crack shape function and it is given by manyauthors in approximate expression as [28]

119865119868 (119886ℎ) = 1122 minus 140 (119886ℎ) + 733 (119886ℎ)2

minus 1308 (119886ℎ)3 + 140 (119886ℎ)4 (33)

The shape function is claimed to be of engineeringaccuracy for any 119886ℎ lt 065 Crack Growth Analysis

51 The Modified Forman Equation The parameters such asthe crack closure effect stress ratio SIF range critical SIF andthe threshold of SIF are considered for the model of crack

Shock and Vibration 7

growth which is called themodified FormanModel [21]Thismodel is employed for crack growth analysis in this study andshown as

119889119886119889119873 = 119861(1 minus 1198911 minus 119903 Δ119870)119899 [1 minus Δ119870thΔ119870]119901[1 minus Δ119870 (1 minus 119903)119870119888]119902 (34)

Here 119903 is the stress ratio 119870119888 is the critical SIF Δ119870th isthe threshold SIF and Δ119870 is the stress intensity range 119873 isthe numbers of vibration cycle 119861 119899 119901 and 119902 are fatigueproperties of materials 119891 is closure function and given byNewman Jr [29]119891

=

max (1199031198980 + 1198981119903 + 11989821199032 + 11989831199033) 119903 ge 01198980 + 1198981119903 0 gt 119903 ge minus21198980 minus 21198981 119903 lt minus2

(35)

where

1198980 = (0825 minus 034120572 + 0051205722) [cos(120587119878max(21205900) )]1120572

1198981 = (0415 minus 0071120572) 119878max1205900 1198982 = 1 minus 1198980 minus 1198981 minus 11989831198983 = 21198980 + 1198981 minus 1

(36)

The threshold SIF is given by

Δ119870th = Δ119870th0 ( 119886119886 + 1198860)05 ( 4120587 tan (1 minus 119903)) (37)

Here Δ119870th0 is the threshold SIF when 119903 is 0 1198860 is the initialcrack depth

In these formulas 120572 is the constraint coefficient for planestress or plane strain problem 119878max1205900 is the ratio of themaximum stress and stress flow For the constant amplitudeloading the increment of crack growth can be calculated byintegrating (34) Because the stress relies on the crack depththe exciting frequency and the damping the amplitude ofdynamic stress response is assumed to be a constant valuewithin one cycle The incremental crack growth in each cyclecan be estimated by (34) Then the crack depth after 119894 timescycle can be calculated by the superposition method andexpressed as

119886119894 = 1198860 + 119894sum119895=1

Δ119886119895 (38)

Here 119886119894 is the total crack depth after 119894 times cycle and Δ119886119895 isthe crack incremental of the 119895th step vibration

The incremental of crack growth within one vibrationcycle is calculated by employing the modified Forman crackgrowth equation It is expressed by

Δ119886 = 119861(1 minus 1198911 minus 119903 Δ119870)119899 [1 minus Δ119870thΔ119870]119901[1 minus Δ119870 (1 minus 119903)119870119888]119902 (39)

Table 1 Material properties of 304SS

Youngrsquos modulus 119864 (MPa) 204 times 105

Poissonrsquos ratio ] 03Mass density 120588 (kgm3) 7860Yield stress 120590119904 (MPa) 2758Ultimate stress 120590119887 (MPa) 620Fracture toughness 119870119862 (MPasdotmm05) 7645Threshold value Δ119870th0 (MPasdotmm05) 1216

Table 2 Crack growth properties of 304SS

119861 6 times 10minus10119901 025119902 025119899 30119878max1205900 03120572 25

The coupling analysis can be done by the above math-ematical equations The stiffness is changed gradually withcrack growth first The damping is also introduced by thefriction energy dissipation from crack surfaces Both of themaffect the dynamic response greatly On the contrary thecrack propagation ratio is also affected by the change ofdynamic response

52 Failure Criteria If the crack depth extends to a certainvalue or the dynamic stress is too large alternative failuremodes may occur to the beam Therefore it is especiallyimportant to work out reasonable criteria to indicate thefailure Three failure criteria are considered and shown asfollows

(1) Geometrical Limit The failure occurs automatically if thecrack depth is larger than the half height of the beam becauseSIF is not suitable when used to estimate the fatigue life incrack growth equation

(2) Critical SIF The fracture occurs if the maximum SIF islarger than critical SIF (119870119868119889max ge 119870119862)(3) Fracture Failure The static fracture failure occurs if themaximum value of the bulk dynamic stress is larger than theultimate strength stress

6 Results and Discussion

A cantilever beam of 304SS is considered as 1m in length012m in width 015m in height and a straight-edge crackwith distance 119897 = 05m from the fixed end The force thatacts on the free end of the cracked beam is a sine excitationThe amplitude of the force is 1198650 = 500N and the stress ratio is119903 = minus1 The materials of the beam are shown in Table 1 Thecrack growth properties of the beam are shown in Table 2

Because the damping properties in the stainless steelsubjected to deformation and temperature the maximum

8 Shock and Vibration

07

075

08

085

09

095

1

Freq

uenc

y ra

tio

01 02 03 04 050Crack depth ratio ah

brc by ABAQUSbrc by Equation (28)oc by Equation (26)

Figure 7 Frequencies change with crack severity

damping loss factor is not bigger than 001 [30] and thedamping loss factor 120574119898 of 304SS is assumed to be 0005 Inorder to investigate the effects of breathing behaviour on thedynamic response and the crack growth themode frequencyfriction damping loss factor dynamic stress dynamic stressintensity and crack growth are analyzed

Figure 7 displays the relation of the mode frequency ratioand the crack depth ratio Results indicate that the predictedfrequency of breathing cracked beam by (28) is in goodagreement with the frequency calculated by ABAQUS Thedecrease ratio of the frequency of the breathing cracked beam120596br120596119888 is smaller than that of the open cracked beam120596119900120596119888 Itis implied that if the frequency of the force acted on the beamis near to the resonance frequency the breathing crackedbeam is subjected to a bigger response because the resonanceregion of the breathing cracked beam is wider than that ofthe open cracked beam Furthermore the crack growth speedof the breathing crack is faster than that of the open cracknear the resonance condition Therefore using a breathingcrack mode to carry out dynamic analysis or crack growthcalculation is more accurate than using an open crack model

Figure 8 investigates the friction damping loss factor byusing different friction coefficient Results indicate that thefriction damping loss factor varies with not only the crackseverity but also the friction coefficient With the increaseof the crack depth or the friction coefficient the frictiondamping loss factor increases That implied that the energydissipated by the friction force is increasing during crackgrowth It is very helpful to reduce the dynamic responseof the resonant beam Thus the crack growth ratio of thebreathing crack is reduced too

Figure 9 shows the maximum dynamic stress of thebreathing cracked beam It is compared with that of the opencracked beam near the resonance region without considering

0

0005

001

0015

002

0025

003

Fric

tion

dam

ping

loss

fact

or

01 02 03 04 050Crack depth ratio ah

= 01

= 03

= 05

f

Figure 8 Friction damping loss factor with crack severity changeunder different friction coefficients

Breathing crack c = 098

Open crack c = 098Breathing crack c = 099

Open crack c = 099Breathing crack c = 1

Open crack c = 1

01 02 03 04 050Crack depth ratio ah

0

20

40

60

80

100

120

140

160

180

Max

imum

dyn

amic

stre

ssd

max

(MPa

)

Figure 9 The maximum dynamic stress with crack severity change(120583 = 0)

the friction damping Results indicate that the stress responseamplitude varies with crack depth Resonance vibrationhappens when the crack reaches a certain depth with onefixed exciting frequencyThe maximum stress happens at theresonant point And themaximumdynamic stress of the opencracked beam is smaller than that of the breathing crackedbeam with the same exciting frequency This implies that the

Shock and Vibration 9

0

20

40

60

80

100

120

140

160

180

Max

imum

dyn

amic

stre

ssd

max

(MPa

)

01 02 03 04 050Crack depth ratio ah

c = 098 = 01

c = 098 = 03

c = 099 = 01

c = 099 = 03

c = 1 = 01

c = 1 = 03

c = 098 = 05

c = 099 = 05

c = 1 = 05

Figure 10Themaximumdynamic stress with crack severity changeof breathing cracked beam

closure effect changes the stiffness of the breathing crackedbeam so that the maximum stress response also changes

Figure 10 presents the effect of the friction damping on themaximum dynamic stress of a breathing cracked beam as thebeam working under resonance condition It can be shownthat the maximum stress of the beam is greatly reduced bythe friction damping This implied that the friction dampingis an important factor to reduce the crack growth ratio

Figure 11 discovers the relation of the maximum dynamicSIFs of the open cracked beam and that of the breathingcracked beam without friction damping It is shown that themaximum dynamic SIF of the breathing cracked beam islarger than that of the open cracked beam without frictiondamping considered under the same exciting frequency Buteach of them happens at different crack depths

Figure 12 studies the effects of friction damping loss factoron themaximum dynamic SIF of the breathing cracked beamat resonance It can be seen that dynamic SIFs are verysensitive to the friction damping Comparing Figure 10 withFigure 12 we can find that the change law of the dynamicstress is different from that of the dynamic SIF The reasonis because the crack depth also plays an important role in SIFduring crack growth

Figures 13 and 14 examine the relation of the crack depthwith the vibration cycles It can be seen that the crackgrowth ratio of the open cracked beam is slower than thatof the breathing cracked beam if the friction damping is notconsidered The reason is because the decrease ratio of the

Breathing crack c = 098

Open crack c = 098

Breathing crack c = 099

Open crack c = 099

Breathing crack c = 1

Open crack c = 1

0

200

400

600

800

1000

1200

1400

1600

1800

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

Figure 11 The maximum dynamic SIF with crack severity changefor different frequency (120583 = 0)

0

500

1000

1500

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 12 The maximum dynamic SIF with crack severity changeof breathing cracked beam

mode frequency of the breathing cracked beam is slower thanthat of the open cracked beam This means the breathingcracked beam has a wider resonance region Considering

10 Shock and Vibration

01 015 02 025 03 035 04 045 05

Breathing crack c = 098

Breathing crack c = 099

Breathing crack c = 1

Open crack c = 098

Open crack c = 099

Open crack c = 1

Crack depth ratio ah

times104

0

1

2

3

4

5

Num

bers

of v

ibra

tion

cycl

e N

Figure 13 Crack depth with vibration cycles for different frequency(120583 = 0)

the friction damping the crack growth ratio is reduced bythe friction damping for a breathing cracked beam obviouslywhen the beam comes into resonance region as shown inFigure 14

Figure 15 shows the relation of the decrease ratio offrequency ratio and the numbers of vibration cycles It can beseen that the decrease ratio of frequency of the open crackedbeam is faster than that of the breathing cracked beam at thefirst phase of the crack growth when the exciting frequency is098 times of themode frequency of the intact beam But aftercertain numbers of vibration cycle the decrease ratio of thefrequency ratio of the breathing cracked beam is faster thanthat of the open cracked beam

7 Conclusions

In this article the stiffness model and the friction dampingmodel of a breathing cracked beam are derived The firstmode frequency is analyzed by using a SDOF model whichis simplified by the Galerkin method The dynamic stressis derived and the crack growth is studied The effects ofthe breathing behaviour on the dynamic response and crackgrowth are revealed so as to analyze the coupling mechanismof the crack growth anddynamic response approximatelyThemain conclusions are obtained

(1) During vibration analysis of a cracked beam the crackclosure effect between opening and closing states must beconsidered A breathing crack model is better than the opencrackmodel for dynamic response and crack growth analysisA sinusoidal breathing function is useful to simulate the

0

2

4

6

8

10

Num

bers

of v

ibra

tion

cycl

e N

times104

015 02 025 03 035 04 045 0501Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 14 Breathing crack depth with vibration cycles for differentfriction coefficient and frequency

times104

088

09

092

094

096

098

1

Freq

uenc

y ra

tio

br

cor

o

c

1 2 3 4 50Numbers of vibration cycle N

oc

brc = 0brc = 02

brc = 04

Figure 15 Frequency ratio with vibration cycles (120596120596119888 = 098)

transition state between the fully open state and the fullyclosed state It meets the object facts and the physical naturalof crack It may be closer to the real motion of the breathingcrack than that of the squarewave function Using a breathingcrack model is more realistic than using an opening crackmodel

Shock and Vibration 11

(2) Based on energy principle and Coulomb frictionmodel the friction model is set up and friction damping lossfactor is derived Friction damping not only plays importantrole in the maximum dynamic response but also playsimportant role in the crack growth Especially to resonanceresponse and crack growth friction damping effect is veryobvious

(3)TheGalerkinmethod is used to simplify the breathingcracked beam into a SDOF systemThe first mode frequencyequation is derived approximately The first mode frequencychanges with the crack severity The frequency decrease ratioof the breathing cracked beam is slower than that of theopen cracked beam If the closure behaviour of the breathingcrack is not considered the degree of crack growth may beunderestimated

(4) The modified Forman equation is employed to simu-late the crack growth A crack growth life analytical methodis proposed with the coupling effect of vibration and fatiguecrack growth included Breathing behaviour of the crack isimportant to crack growthThe dynamical properties play animportant role in crack growthWhen the exciting frequencyis near to the first mode frequency the crack growth ratio isvery fastThe crack growth life of the breathing cracked beamis smaller than that of the open cracked beam

Nomenclature

119886 1198860 Crack depth initial crack depth119886119894 Δ119886119895 The 119894th step crack depth the 119895th step crackincremental119860 Area of cross section119887 Width of beam cross section119861 Fatigue properties of materials119888lowast Coefficient of modal damping119862 Flexibility of the intact beam119862119879 Flexibility due to the presence of the crack119862119900 Total flexibility of the open cracked beam119889 Sliding distance between crack surfaces119864 Youngrsquos modulus119891 Closure function of breathing crack119865 Harmonic load119865lowast 1198650 The generalized force loading amplitude119865119899 119865119891 Normal pressing force friction force119865av Average pressing force119865119868 Crack shape functionℎ Height of the beam cross section119868 Area moment of inertial of the beam crosssection119896lowast The generalized stiffness119896br Stiffness of the breathing cracked beam119896119888 Stiffness of the close cracked beam119896119900 Stiffness of the open cracked beam119896119879 Stiffness of the torsional spring119870119868119889 Dynamic stress intensity factor119870119868119889max The maximum dynamic stress intensity factor119870119862 Fracture toughnessΔ119870th The threshold of stress intensity factorΔ119870 The range of stress intensity factor119897 Distance of crack from the fixed end

119871 Length of the beam119898lowast The generalized mass119872 Moment119899 119901 119902 Fatigue properties of materials119873 Numbers of vibration cycle119876 Shear force119903 Stress ratio119905 Time119879 Amplitude-time function119881max The strain energy119882119862 Energy dissipation119908 Lateral deflection in 119911 direction of beam119882(119909) Modal function119909 119910 119911 The coordinate119885st Static displacement120588 ] Density of material Poissonrsquos ratio120583 Friction coefficient120596 Excitation frequency120596119888 Frequency of the close cracked beam120596119900 Frequency of the open cracked beam120574119898 Material damping120574119891 Frictional damping120578 Amplify ratio120582 Frequency ratio120573 The breathing function120575 The Dirac function120590119899 120590119889 Nominal stress normal bulk dynamic stress120590119887 120590119904 Ultimate stress yield stress

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research work is supported by the National NaturalScience Foundation of China (51565039)

References

[1] A R Biswal T Roy R K Behera S K Pradhan and P KParida ldquoFinite element based vibration analysis of a nonpris-matic Timoshenko beam with transverse open crackrdquo ProcediaEngineering vol 144 pp 226ndash233 2016

[2] MKrawczuk andWMOstachowicz ldquoModelling and vibrationanalysis of a cantilever composite beam with a transverse opencrackrdquo Journal of Sound and Vibration vol 183 no 1 pp 69ndash891995

[3] C A Papadopoulos and A D Dimarogonas ldquoCoupled longi-tudinal and bending vibrations of a rotating shaft with an opencrackrdquo Journal of Sound and Vibration vol 117 no 1 pp 81ndash931987

[4] W M Ostachowicz and M Krawczuk ldquoAnalysis of the effect ofcracks on the natural frequencies of a cantilever beamrdquo Journalof Sound and Vibration vol 150 no 2 pp 191ndash201 1991

[5] T G Chondros andA D Dimarogonas ldquoVibration of a crackedcantilever beamrdquo Journal of Vibration andAcoustics vol 120 no3 pp 742ndash746 1998

12 Shock and Vibration

[6] T G Chondros A D Dimarogonas and J Yao ldquoA continuouscracked beam vibration theoryrdquo Journal of Sound and Vibrationvol 215 no 1 pp 17ndash34 1998

[7] M Heydari A Ebrahimi and M Behzad ldquoContinuous modelfor flexural vibration analysis of Timoshenko beams with avertical edge crackrdquo Archive of Applied Mechanics vol 85 no5 pp 601ndash615 2015

[8] R K Behera A Pandey and D R Parhi ldquoNumerical andexperimental verification of a method for prognosis of inclinededge crack in cantilever beam based on synthesis of modeshapesrdquo Procedia Technology vol 14 pp 67ndash74 2014

[9] S M Cheng A S J Swamidas X J Wu and W WallaceldquoVibrational response of a beamwith a breathing crackrdquo Journalof Sound and Vibration vol 225 no 1 pp 201ndash208 1999

[10] T G Chondros A D Dimarogonas and J Yao ldquoVibration of abeam with a breathing crackrdquo Journal of Sound and Vibrationvol 239 no 1 pp 57ndash67 2001

[11] E Douka and L J Hadjileontiadis ldquoTime-frequency analysis ofthe free vibration response of a beam with a breathing crackrdquoNDT amp E International vol 38 no 1 pp 3ndash10 2005

[12] M Kisa and J Brandon ldquoEffects of closure of cracks on thedynamics of a cracked cantilever beamrdquo Journal of Sound andVibration vol 238 no 1 pp 1ndash18 2000

[13] C Surace R Ruotolo and D Storer ldquoDetecting nonlinearbehaviour using the volterra series to assess damage in beam-like structuresrdquo Journal of Theoretical and Applied Mechanicsvol 49 no 3 pp 905ndash926 2011

[14] AChatterjee ldquoNonlinear dynamics anddamage assessment of acantilever beamwith breathing edge crackrdquo Journal of Vibrationand Acoustics vol 133 no 5 Article ID 051004 2011

[15] R Ruotolo C Surace P Crespo and D Storer ldquoHarmonicanalysis of the vibrations of a cantilevered beam with a closingcrackrdquo Computers amp Structures vol 61 no 6 pp 1057ndash10741996

[16] A Nandi and S Neogy ldquoModelling of a beam with a breathingedge crack and some observations for crack detectionrdquo Journalof Vibration and Control vol 8 no 5 pp 673ndash693 2002

[17] W Liu and M E Barkey ldquoNonlinear vibrational response ofa single edge cracked beamrdquo Journal of Mechanical Science ampTechnology vol 31 no 11 pp 5231ndash5243 2017

[18] J Zeng H MaW Zhang and BWen ldquoDynamic characteristicanalysis of cracked cantilever beams under different cracktypesrdquo Engineering Failure Analysis vol 74 pp 80ndash94 2017

[19] W Zhang H Ma J Zeng S Wu and B Wen ldquoVibrationresponses analysis of an elastic-support cantilever beam withcrack and offset boundaryrdquo Mechanical Systems and SignalProcessing vol 95 pp 205ndash218 2017

[20] HMa J Zeng Z Lang L Zhang Y Guo and BWen ldquoAnalysisof the dynamic characteristics of a slant-cracked cantileverbeamrdquo Mechanical Systems and Signal Processing vol 75 pp261ndash279 2016

[21] J Liu W D Zhu P G Charalambides Y M Shao Y F Xuand X M Fang ldquoA dynamic model of a cantilever beam with aclosed embedded horizontal crack including local flexibilitiesat crack tipsrdquo Journal of Sound and Vibration vol 382 pp 274ndash290 2016

[22] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[23] A Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagationrdquo Engineering FractureMechanics vol17 no 4 pp 381ndash386 1983

[24] A J Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagation in a beam under longitudinal vibra-tionsrdquo International Journal of Fracture vol 23 no 1 pp 15ndash221983

[25] A J Dentsoras andADDimarogonas ldquoFatigue crack propaga-tion in resonating structuresrdquo Engineering Fracture Mechanicsvol 34 no 3 pp 721ndash728 1989

[26] M Lalanne P Berthier and J D HagopianMechanical Vibra-tions for Engineers John Wiley amp Sons New York NY USA1983

[27] A D Dimarogonas S A Paipetis and T G ChondrosAnalyticalMethods inRotorDynamics SpringerDordrechtTheNetherlands 2013

[28] H Tada P C Paris andG R IrwinTheStress Analysis of CracksHandbook The American Society of Mechanical Engineers2000

[29] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984

[30] R Mulyukov S Mikhailov R Zaripova and D SalimonenkoldquoDamping properties of 18Cr-10Ni stainless steel with submi-crocrystalline structurerdquoMaterials Research Bulletin vol 31 no6 pp 639ndash645 1996

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4 Shock and Vibration

Through (9) and (7) the stiffness model of the breathingcracked beam can be written as

119896br (119905) = (119896119888 + 119896119900)2 + (119896119888 minus 119896119900)2 sin (120596119905) (10)

From (10) it indicates that the stiffness of the breathingcracked beam is composed of the stiffness of open crackedbeam and the stiffness of closed cracked beam It can be seenfrom Figure 1 that in case of 120596119905 = 119899120587 (119899 = 1 2 3 ) theaxis 119909 of the beam stays at A position and the crack is inhalf open state In case of 120596119905 = 2119899120587 minus 1205872 the axis 119909 of thebeam reachesB position and the crack is fully open In caseof 120596119905 = 2119899120587 + 1205872 the axis 119909 of the beam reachesC positionand the crack is fully closed

Ignoring the shear force a cracked cantilever beam can beregarded as two intact beams that are connected by a torsionalspring as shown in Figure 3 Therefore the stiffness of thetorsional spring 119896119879 due to crack can be obtained from theequation developed by Dimarogonas et al [27] as

119896119879 = 119864119887ℎ272120587 (119871 minus 119897)2 (1 minus ]2)Φ (11)

Here 119897 is the distance of the crack from the fixed end 119871 119887 andℎ are the respective length width and height of the beam ]is the Poisson ratio of the material The function 120601 is given by

120601 = 0629 (119886ℎ)2 minus 1047 (119886ℎ)3 + 4602 (119886ℎ)4

minus 9975 (119886ℎ)5 + 20295 (119886ℎ)6 minus 32993 (119886ℎ)7

+ 47041 (119886ℎ)8 minus 40693 (119886ℎ)9 + 196 (119886ℎ)10

(12)

Therefore the stiffness of the open cracked beam can becalculated by

119896119900 = 119896119879119896119888119896119879 + 119896119888 (13)

3 Friction Damping Model

31 Sliding Friction Model During crack propagation thecrack tip material will yield and then separate When thematerial of the crack tip fractures the crack will advanceAccording to the knowledge of materials mechanics theyield failure of ductile materials often happens related tothe maximum shear stress And the maximum shear stressusually appears on the incline plane which forms a 45∘ angleto the cross section of the cracked beam under tension stressstate formost of steelsTherefore test results of specimenwithsmooth surface often yield to some 45∘ cross inclination stripsas shown in Figure 4

O

l

L

b

h

kT

Figure 3 Stiffness model of the cracked cantilever beam

Slip line

Figure 4 Yield slip line

Take the Euler-Bernoulli beam having a breathing crackas the research object as shown in Figure 1 Suppose that thecrack growth is just straight ahead driven by the dynamicresponse it could be conceived that the cross-yield slip linewith 45∘ near the crack tip probably will appear on the surfaceas shown in Figure 5

Before creating the sliding friction model for the breath-ing crack some assumptions should be given at first asfollows

(1) The dynamic response of the cracked beam is stable(2) The energy dissipation of the cracked beam is pro-

duced just by the friction of the crack surface(3) The friction force at the semicrack depth as 119876 point

as shown in Figure 5 is employed to calculate the dissipatedenergy

(4) The energy dissipation happens only when the crackis in the compressed zone

(5) The friction force of the crack surfaces satisfies withthe Coulomb model law

(6) The total storage strain energy of the intact beam isequal to that of the breathing cracked beam

Based on these given assumptions the surface slidefriction model of the breathing crack is set up as shown inFigure 6 In the frictional model 119865119899 is the normal pressingforce which is perpendicular to the cross section of beamand 119865119891 is the friction force between the crack surfaces Itis estimated by the normal stress of 119876 point a positionof semicrack depth The relative microsliding distance 119889depends on the crack open distance (COD) at the semicrackdepth position

32 Damping Loss Factor According to the geometry of thecracked beam as shown in Figure 5 the nominal stress of 119876point as the crack is fully close can be estimated by Hookersquoslaw It is given by

120590119899 = 05119864 (ℎ minus 119886) 12059721199081205971199092100381610038161003816100381610038161003816100381610038161003816119909=119897 (14)

Shock and Vibration 5

lz

xo

Crack surface

Slip line

Enlarged

Q

F

Q a

a

b

ho

z

y

a2

Figure 5 Crack growth road

Therefore the normal pressing force is obtained

119865119899 = 119886119887120590119899 (15)

Because the breathing crack will be opened graduallywhen it is in intensive zone and will be closed when it is in acompressed zone the value of pressing forcemay exist duringthe half cycle relative to the effect of external harmonic forceand the average pressing force 119865av in one cycle is given by

119865av = 112058710038161003816100381610038161003816100381610038161003816int120587

0119865119899 sin (120596119905) 119889 (120596119905)10038161003816100381610038161003816100381610038161003816 =

2119865119899120587 (16)

If the frictional coefficient of crack surfaces for 304SS is 120583the friction force between the crack surfaces can be derivedby the sliding friction model It is given by

119865119891 = 120583119865av cos 45∘ (17)

In each cycle the breathing process of the crack from theopen state to the close state includes 4 phases If position Ais the initial position as shown in Figure 1 a total breathingprocess includes A rarr B rarr A rarr C rarr A Only whenthe crack is in the compressed zone the friction force willbe produced Based on the work-energy theorem for systemsthe energy dissipation introduced by the sliding friction forcebetween the crack surfaces in one vibration cycle is given by

119882119862 = 2119865119891119889 (18)

It can be seen from Figure 6 that the relative slidingdistance between the crack surfaces depends on COD of 119876

d

COD

QFn

45∘

Fn

Ff

Figure 6 Friction model of crack surface

point An empirical formula for COD with 1 accuracy forany 119886ℎ is given by Tada et al [28]

COD = 4120590119899119886119864 119881(119886ℎ) (19)

Here the function 119881(119886ℎ) is given by

119881(119886ℎ) = 08 minus 17 (119886ℎ) + 24 (119886ℎ)2 + 066(1 minus 119886ℎ)2 (20)

Figure 6 shows that the relation between COD and 119889 is

119889 = COD sdot cos 1205874 (21)

Therefore the energy dissipation caused by the frictionforce within one vibration cycle is obtained

119882119862 = 4120587 12058311986411988621198871205902119899119881(119886ℎ) (22)

According to the energy principle and the assumptionsthe strain energy storage in the cracked beam having rectan-gular cross section is given by

119881max = int1198710

1198641198682 (12059721199081205971199092 )2 119889119909

+ int1198710

12 (1 + ]) 1198641198682119860 (12059731199081205971199093 )2 119889119909

(23)

For a cracked beam the friction damping loss factor 120574119891can be defined as

120574119891 = 1198821198622120587119881max (24)

6 Shock and Vibration

Therefore the damping loss factor caused by the frictionof the crack surfaces is obtained

120574119891 = 21205831198862119887 (ℎ minus 119886)2 119881 (119886ℎ) ((12059721199081205971199092)10038161003816100381610038161003816119909=119897)21205872 (119868 int119871

0(12059721199081205971199092)2 119889119909 + (24 (1 + ]) 1198682119860) int119871

0(12059731199081205971199093)2 119889119909) (25)

4 Vibration Response Analysis

41Mode Frequency If the generalized stiffness 119896lowast is replacedby the stiffness of the open cracked beam 119896119900 then the firstmode frequency of the open cracked beam is also obtainedTherefore the first mode frequency of the cracked beamcorresponding to fully open and fully closed states can bepredicted approximately by the following equation

120596119900119888 = radic119896119900119888119898lowast (26)

Here 120596119900 is the frequency of the open cracked beam and 120596119888 isthe frequency of the closed cracked beam

If the breathing behaviour is considered the stiffness ofthe cracked beam varies with time within one vibration cycleIt is impossible to plug the stiffness model of a breathingcracked beam into the frequency equation (26) to analyzethe mode frequency of the breathing cracked beam directlyBut within one cycle no matter which kind of the breathingfunction is used the cycle period time will not change Ifthe square wave function is used the vibration cycle timeincludes two parts The prehalf cycle corresponds to the fullyopen crack and the posthalf cycle corresponds to the fullyclosed crack Therefore the relation of the sine breathingbehaviour and the square wave breathing behaviour mustobey

2120587120596br= 120587120596119900 +

120587120596119888 (27)

Here 120596br is the frequency of the breathing cracked beam Itis given by

120596br = 2120596119900120596119888120596119900 + 120596119888 (28)

42 Dynamic Stress Intensity Factor (DSIF) For a vibratingcracked beam the crack propagation speed relies on notonly the amplitude of dynamic stress response but also thecrack depth For an edge cracked beam as shown in Figure 1ignoring the shear force the beam endures bending loadmainlyThe crackwill open as type I crack In real engineeringpractice mode I crack maybe is the most common crackmodel and it is very sensitive to dynamic responseThereforethe mode I crack is considered in this article Based on thestrength theory and Hookersquos law the nominal dynamic bulk

stress at the top edge of the cracked beamunder pure bendingstates can be estimated approximately by

120590119889 = 119864ℎ2 1205972119908 (119909 119905)1205971199092100381610038161003816100381610038161003816100381610038161003816119909=119897 (29)

Plugging the dynamic displacement response into thestress equation we obtain

120590119889 = 119864120578119885stℎ2 1205972119882(119909)1205971199092

100381610038161003816100381610038161003816100381610038161003816119909=119897 (30)

Here 119885st is the static displacement it is the ratio of thegeneralized force and the stiffness of the cracked beam 120578 isthe amplify ratio of displacement response it is given by [26]and expressed by

120578 = 1radic(1 minus 1205822)2 + ((120574119898 + 120574119891) 120582)2

(31)

Here 120582 is the frequency ratio When 120582 is near to 1 the beamis near or at resonance the amplify ratio is very large and thedamping ratio plays important role in the resonance responseand crack growth

Based on the definition of stress intensity factor (SIF) infracture mechanics the SIF of the beam under pure bendingcan be defined by empirical formula

119870119868119889 = 120590119889radic120587119886119865119868 (119886ℎ) (32)

Here 119870119868119889 is SIF 120590119889 is the nominal dynamic bulk stress119865119868(119886ℎ) is crack shape function and it is given by manyauthors in approximate expression as [28]

119865119868 (119886ℎ) = 1122 minus 140 (119886ℎ) + 733 (119886ℎ)2

minus 1308 (119886ℎ)3 + 140 (119886ℎ)4 (33)

The shape function is claimed to be of engineeringaccuracy for any 119886ℎ lt 065 Crack Growth Analysis

51 The Modified Forman Equation The parameters such asthe crack closure effect stress ratio SIF range critical SIF andthe threshold of SIF are considered for the model of crack

Shock and Vibration 7

growth which is called themodified FormanModel [21]Thismodel is employed for crack growth analysis in this study andshown as

119889119886119889119873 = 119861(1 minus 1198911 minus 119903 Δ119870)119899 [1 minus Δ119870thΔ119870]119901[1 minus Δ119870 (1 minus 119903)119870119888]119902 (34)

Here 119903 is the stress ratio 119870119888 is the critical SIF Δ119870th isthe threshold SIF and Δ119870 is the stress intensity range 119873 isthe numbers of vibration cycle 119861 119899 119901 and 119902 are fatigueproperties of materials 119891 is closure function and given byNewman Jr [29]119891

=

max (1199031198980 + 1198981119903 + 11989821199032 + 11989831199033) 119903 ge 01198980 + 1198981119903 0 gt 119903 ge minus21198980 minus 21198981 119903 lt minus2

(35)

where

1198980 = (0825 minus 034120572 + 0051205722) [cos(120587119878max(21205900) )]1120572

1198981 = (0415 minus 0071120572) 119878max1205900 1198982 = 1 minus 1198980 minus 1198981 minus 11989831198983 = 21198980 + 1198981 minus 1

(36)

The threshold SIF is given by

Δ119870th = Δ119870th0 ( 119886119886 + 1198860)05 ( 4120587 tan (1 minus 119903)) (37)

Here Δ119870th0 is the threshold SIF when 119903 is 0 1198860 is the initialcrack depth

In these formulas 120572 is the constraint coefficient for planestress or plane strain problem 119878max1205900 is the ratio of themaximum stress and stress flow For the constant amplitudeloading the increment of crack growth can be calculated byintegrating (34) Because the stress relies on the crack depththe exciting frequency and the damping the amplitude ofdynamic stress response is assumed to be a constant valuewithin one cycle The incremental crack growth in each cyclecan be estimated by (34) Then the crack depth after 119894 timescycle can be calculated by the superposition method andexpressed as

119886119894 = 1198860 + 119894sum119895=1

Δ119886119895 (38)

Here 119886119894 is the total crack depth after 119894 times cycle and Δ119886119895 isthe crack incremental of the 119895th step vibration

The incremental of crack growth within one vibrationcycle is calculated by employing the modified Forman crackgrowth equation It is expressed by

Δ119886 = 119861(1 minus 1198911 minus 119903 Δ119870)119899 [1 minus Δ119870thΔ119870]119901[1 minus Δ119870 (1 minus 119903)119870119888]119902 (39)

Table 1 Material properties of 304SS

Youngrsquos modulus 119864 (MPa) 204 times 105

Poissonrsquos ratio ] 03Mass density 120588 (kgm3) 7860Yield stress 120590119904 (MPa) 2758Ultimate stress 120590119887 (MPa) 620Fracture toughness 119870119862 (MPasdotmm05) 7645Threshold value Δ119870th0 (MPasdotmm05) 1216

Table 2 Crack growth properties of 304SS

119861 6 times 10minus10119901 025119902 025119899 30119878max1205900 03120572 25

The coupling analysis can be done by the above math-ematical equations The stiffness is changed gradually withcrack growth first The damping is also introduced by thefriction energy dissipation from crack surfaces Both of themaffect the dynamic response greatly On the contrary thecrack propagation ratio is also affected by the change ofdynamic response

52 Failure Criteria If the crack depth extends to a certainvalue or the dynamic stress is too large alternative failuremodes may occur to the beam Therefore it is especiallyimportant to work out reasonable criteria to indicate thefailure Three failure criteria are considered and shown asfollows

(1) Geometrical Limit The failure occurs automatically if thecrack depth is larger than the half height of the beam becauseSIF is not suitable when used to estimate the fatigue life incrack growth equation

(2) Critical SIF The fracture occurs if the maximum SIF islarger than critical SIF (119870119868119889max ge 119870119862)(3) Fracture Failure The static fracture failure occurs if themaximum value of the bulk dynamic stress is larger than theultimate strength stress

6 Results and Discussion

A cantilever beam of 304SS is considered as 1m in length012m in width 015m in height and a straight-edge crackwith distance 119897 = 05m from the fixed end The force thatacts on the free end of the cracked beam is a sine excitationThe amplitude of the force is 1198650 = 500N and the stress ratio is119903 = minus1 The materials of the beam are shown in Table 1 Thecrack growth properties of the beam are shown in Table 2

Because the damping properties in the stainless steelsubjected to deformation and temperature the maximum

8 Shock and Vibration

07

075

08

085

09

095

1

Freq

uenc

y ra

tio

01 02 03 04 050Crack depth ratio ah

brc by ABAQUSbrc by Equation (28)oc by Equation (26)

Figure 7 Frequencies change with crack severity

damping loss factor is not bigger than 001 [30] and thedamping loss factor 120574119898 of 304SS is assumed to be 0005 Inorder to investigate the effects of breathing behaviour on thedynamic response and the crack growth themode frequencyfriction damping loss factor dynamic stress dynamic stressintensity and crack growth are analyzed

Figure 7 displays the relation of the mode frequency ratioand the crack depth ratio Results indicate that the predictedfrequency of breathing cracked beam by (28) is in goodagreement with the frequency calculated by ABAQUS Thedecrease ratio of the frequency of the breathing cracked beam120596br120596119888 is smaller than that of the open cracked beam120596119900120596119888 Itis implied that if the frequency of the force acted on the beamis near to the resonance frequency the breathing crackedbeam is subjected to a bigger response because the resonanceregion of the breathing cracked beam is wider than that ofthe open cracked beam Furthermore the crack growth speedof the breathing crack is faster than that of the open cracknear the resonance condition Therefore using a breathingcrack mode to carry out dynamic analysis or crack growthcalculation is more accurate than using an open crack model

Figure 8 investigates the friction damping loss factor byusing different friction coefficient Results indicate that thefriction damping loss factor varies with not only the crackseverity but also the friction coefficient With the increaseof the crack depth or the friction coefficient the frictiondamping loss factor increases That implied that the energydissipated by the friction force is increasing during crackgrowth It is very helpful to reduce the dynamic responseof the resonant beam Thus the crack growth ratio of thebreathing crack is reduced too

Figure 9 shows the maximum dynamic stress of thebreathing cracked beam It is compared with that of the opencracked beam near the resonance region without considering

0

0005

001

0015

002

0025

003

Fric

tion

dam

ping

loss

fact

or

01 02 03 04 050Crack depth ratio ah

= 01

= 03

= 05

f

Figure 8 Friction damping loss factor with crack severity changeunder different friction coefficients

Breathing crack c = 098

Open crack c = 098Breathing crack c = 099

Open crack c = 099Breathing crack c = 1

Open crack c = 1

01 02 03 04 050Crack depth ratio ah

0

20

40

60

80

100

120

140

160

180

Max

imum

dyn

amic

stre

ssd

max

(MPa

)

Figure 9 The maximum dynamic stress with crack severity change(120583 = 0)

the friction damping Results indicate that the stress responseamplitude varies with crack depth Resonance vibrationhappens when the crack reaches a certain depth with onefixed exciting frequencyThe maximum stress happens at theresonant point And themaximumdynamic stress of the opencracked beam is smaller than that of the breathing crackedbeam with the same exciting frequency This implies that the

Shock and Vibration 9

0

20

40

60

80

100

120

140

160

180

Max

imum

dyn

amic

stre

ssd

max

(MPa

)

01 02 03 04 050Crack depth ratio ah

c = 098 = 01

c = 098 = 03

c = 099 = 01

c = 099 = 03

c = 1 = 01

c = 1 = 03

c = 098 = 05

c = 099 = 05

c = 1 = 05

Figure 10Themaximumdynamic stress with crack severity changeof breathing cracked beam

closure effect changes the stiffness of the breathing crackedbeam so that the maximum stress response also changes

Figure 10 presents the effect of the friction damping on themaximum dynamic stress of a breathing cracked beam as thebeam working under resonance condition It can be shownthat the maximum stress of the beam is greatly reduced bythe friction damping This implied that the friction dampingis an important factor to reduce the crack growth ratio

Figure 11 discovers the relation of the maximum dynamicSIFs of the open cracked beam and that of the breathingcracked beam without friction damping It is shown that themaximum dynamic SIF of the breathing cracked beam islarger than that of the open cracked beam without frictiondamping considered under the same exciting frequency Buteach of them happens at different crack depths

Figure 12 studies the effects of friction damping loss factoron themaximum dynamic SIF of the breathing cracked beamat resonance It can be seen that dynamic SIFs are verysensitive to the friction damping Comparing Figure 10 withFigure 12 we can find that the change law of the dynamicstress is different from that of the dynamic SIF The reasonis because the crack depth also plays an important role in SIFduring crack growth

Figures 13 and 14 examine the relation of the crack depthwith the vibration cycles It can be seen that the crackgrowth ratio of the open cracked beam is slower than thatof the breathing cracked beam if the friction damping is notconsidered The reason is because the decrease ratio of the

Breathing crack c = 098

Open crack c = 098

Breathing crack c = 099

Open crack c = 099

Breathing crack c = 1

Open crack c = 1

0

200

400

600

800

1000

1200

1400

1600

1800

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

Figure 11 The maximum dynamic SIF with crack severity changefor different frequency (120583 = 0)

0

500

1000

1500

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 12 The maximum dynamic SIF with crack severity changeof breathing cracked beam

mode frequency of the breathing cracked beam is slower thanthat of the open cracked beam This means the breathingcracked beam has a wider resonance region Considering

10 Shock and Vibration

01 015 02 025 03 035 04 045 05

Breathing crack c = 098

Breathing crack c = 099

Breathing crack c = 1

Open crack c = 098

Open crack c = 099

Open crack c = 1

Crack depth ratio ah

times104

0

1

2

3

4

5

Num

bers

of v

ibra

tion

cycl

e N

Figure 13 Crack depth with vibration cycles for different frequency(120583 = 0)

the friction damping the crack growth ratio is reduced bythe friction damping for a breathing cracked beam obviouslywhen the beam comes into resonance region as shown inFigure 14

Figure 15 shows the relation of the decrease ratio offrequency ratio and the numbers of vibration cycles It can beseen that the decrease ratio of frequency of the open crackedbeam is faster than that of the breathing cracked beam at thefirst phase of the crack growth when the exciting frequency is098 times of themode frequency of the intact beam But aftercertain numbers of vibration cycle the decrease ratio of thefrequency ratio of the breathing cracked beam is faster thanthat of the open cracked beam

7 Conclusions

In this article the stiffness model and the friction dampingmodel of a breathing cracked beam are derived The firstmode frequency is analyzed by using a SDOF model whichis simplified by the Galerkin method The dynamic stressis derived and the crack growth is studied The effects ofthe breathing behaviour on the dynamic response and crackgrowth are revealed so as to analyze the coupling mechanismof the crack growth anddynamic response approximatelyThemain conclusions are obtained

(1) During vibration analysis of a cracked beam the crackclosure effect between opening and closing states must beconsidered A breathing crack model is better than the opencrackmodel for dynamic response and crack growth analysisA sinusoidal breathing function is useful to simulate the

0

2

4

6

8

10

Num

bers

of v

ibra

tion

cycl

e N

times104

015 02 025 03 035 04 045 0501Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 14 Breathing crack depth with vibration cycles for differentfriction coefficient and frequency

times104

088

09

092

094

096

098

1

Freq

uenc

y ra

tio

br

cor

o

c

1 2 3 4 50Numbers of vibration cycle N

oc

brc = 0brc = 02

brc = 04

Figure 15 Frequency ratio with vibration cycles (120596120596119888 = 098)

transition state between the fully open state and the fullyclosed state It meets the object facts and the physical naturalof crack It may be closer to the real motion of the breathingcrack than that of the squarewave function Using a breathingcrack model is more realistic than using an opening crackmodel

Shock and Vibration 11

(2) Based on energy principle and Coulomb frictionmodel the friction model is set up and friction damping lossfactor is derived Friction damping not only plays importantrole in the maximum dynamic response but also playsimportant role in the crack growth Especially to resonanceresponse and crack growth friction damping effect is veryobvious

(3)TheGalerkinmethod is used to simplify the breathingcracked beam into a SDOF systemThe first mode frequencyequation is derived approximately The first mode frequencychanges with the crack severity The frequency decrease ratioof the breathing cracked beam is slower than that of theopen cracked beam If the closure behaviour of the breathingcrack is not considered the degree of crack growth may beunderestimated

(4) The modified Forman equation is employed to simu-late the crack growth A crack growth life analytical methodis proposed with the coupling effect of vibration and fatiguecrack growth included Breathing behaviour of the crack isimportant to crack growthThe dynamical properties play animportant role in crack growthWhen the exciting frequencyis near to the first mode frequency the crack growth ratio isvery fastThe crack growth life of the breathing cracked beamis smaller than that of the open cracked beam

Nomenclature

119886 1198860 Crack depth initial crack depth119886119894 Δ119886119895 The 119894th step crack depth the 119895th step crackincremental119860 Area of cross section119887 Width of beam cross section119861 Fatigue properties of materials119888lowast Coefficient of modal damping119862 Flexibility of the intact beam119862119879 Flexibility due to the presence of the crack119862119900 Total flexibility of the open cracked beam119889 Sliding distance between crack surfaces119864 Youngrsquos modulus119891 Closure function of breathing crack119865 Harmonic load119865lowast 1198650 The generalized force loading amplitude119865119899 119865119891 Normal pressing force friction force119865av Average pressing force119865119868 Crack shape functionℎ Height of the beam cross section119868 Area moment of inertial of the beam crosssection119896lowast The generalized stiffness119896br Stiffness of the breathing cracked beam119896119888 Stiffness of the close cracked beam119896119900 Stiffness of the open cracked beam119896119879 Stiffness of the torsional spring119870119868119889 Dynamic stress intensity factor119870119868119889max The maximum dynamic stress intensity factor119870119862 Fracture toughnessΔ119870th The threshold of stress intensity factorΔ119870 The range of stress intensity factor119897 Distance of crack from the fixed end

119871 Length of the beam119898lowast The generalized mass119872 Moment119899 119901 119902 Fatigue properties of materials119873 Numbers of vibration cycle119876 Shear force119903 Stress ratio119905 Time119879 Amplitude-time function119881max The strain energy119882119862 Energy dissipation119908 Lateral deflection in 119911 direction of beam119882(119909) Modal function119909 119910 119911 The coordinate119885st Static displacement120588 ] Density of material Poissonrsquos ratio120583 Friction coefficient120596 Excitation frequency120596119888 Frequency of the close cracked beam120596119900 Frequency of the open cracked beam120574119898 Material damping120574119891 Frictional damping120578 Amplify ratio120582 Frequency ratio120573 The breathing function120575 The Dirac function120590119899 120590119889 Nominal stress normal bulk dynamic stress120590119887 120590119904 Ultimate stress yield stress

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research work is supported by the National NaturalScience Foundation of China (51565039)

References

[1] A R Biswal T Roy R K Behera S K Pradhan and P KParida ldquoFinite element based vibration analysis of a nonpris-matic Timoshenko beam with transverse open crackrdquo ProcediaEngineering vol 144 pp 226ndash233 2016

[2] MKrawczuk andWMOstachowicz ldquoModelling and vibrationanalysis of a cantilever composite beam with a transverse opencrackrdquo Journal of Sound and Vibration vol 183 no 1 pp 69ndash891995

[3] C A Papadopoulos and A D Dimarogonas ldquoCoupled longi-tudinal and bending vibrations of a rotating shaft with an opencrackrdquo Journal of Sound and Vibration vol 117 no 1 pp 81ndash931987

[4] W M Ostachowicz and M Krawczuk ldquoAnalysis of the effect ofcracks on the natural frequencies of a cantilever beamrdquo Journalof Sound and Vibration vol 150 no 2 pp 191ndash201 1991

[5] T G Chondros andA D Dimarogonas ldquoVibration of a crackedcantilever beamrdquo Journal of Vibration andAcoustics vol 120 no3 pp 742ndash746 1998

12 Shock and Vibration

[6] T G Chondros A D Dimarogonas and J Yao ldquoA continuouscracked beam vibration theoryrdquo Journal of Sound and Vibrationvol 215 no 1 pp 17ndash34 1998

[7] M Heydari A Ebrahimi and M Behzad ldquoContinuous modelfor flexural vibration analysis of Timoshenko beams with avertical edge crackrdquo Archive of Applied Mechanics vol 85 no5 pp 601ndash615 2015

[8] R K Behera A Pandey and D R Parhi ldquoNumerical andexperimental verification of a method for prognosis of inclinededge crack in cantilever beam based on synthesis of modeshapesrdquo Procedia Technology vol 14 pp 67ndash74 2014

[9] S M Cheng A S J Swamidas X J Wu and W WallaceldquoVibrational response of a beamwith a breathing crackrdquo Journalof Sound and Vibration vol 225 no 1 pp 201ndash208 1999

[10] T G Chondros A D Dimarogonas and J Yao ldquoVibration of abeam with a breathing crackrdquo Journal of Sound and Vibrationvol 239 no 1 pp 57ndash67 2001

[11] E Douka and L J Hadjileontiadis ldquoTime-frequency analysis ofthe free vibration response of a beam with a breathing crackrdquoNDT amp E International vol 38 no 1 pp 3ndash10 2005

[12] M Kisa and J Brandon ldquoEffects of closure of cracks on thedynamics of a cracked cantilever beamrdquo Journal of Sound andVibration vol 238 no 1 pp 1ndash18 2000

[13] C Surace R Ruotolo and D Storer ldquoDetecting nonlinearbehaviour using the volterra series to assess damage in beam-like structuresrdquo Journal of Theoretical and Applied Mechanicsvol 49 no 3 pp 905ndash926 2011

[14] AChatterjee ldquoNonlinear dynamics anddamage assessment of acantilever beamwith breathing edge crackrdquo Journal of Vibrationand Acoustics vol 133 no 5 Article ID 051004 2011

[15] R Ruotolo C Surace P Crespo and D Storer ldquoHarmonicanalysis of the vibrations of a cantilevered beam with a closingcrackrdquo Computers amp Structures vol 61 no 6 pp 1057ndash10741996

[16] A Nandi and S Neogy ldquoModelling of a beam with a breathingedge crack and some observations for crack detectionrdquo Journalof Vibration and Control vol 8 no 5 pp 673ndash693 2002

[17] W Liu and M E Barkey ldquoNonlinear vibrational response ofa single edge cracked beamrdquo Journal of Mechanical Science ampTechnology vol 31 no 11 pp 5231ndash5243 2017

[18] J Zeng H MaW Zhang and BWen ldquoDynamic characteristicanalysis of cracked cantilever beams under different cracktypesrdquo Engineering Failure Analysis vol 74 pp 80ndash94 2017

[19] W Zhang H Ma J Zeng S Wu and B Wen ldquoVibrationresponses analysis of an elastic-support cantilever beam withcrack and offset boundaryrdquo Mechanical Systems and SignalProcessing vol 95 pp 205ndash218 2017

[20] HMa J Zeng Z Lang L Zhang Y Guo and BWen ldquoAnalysisof the dynamic characteristics of a slant-cracked cantileverbeamrdquo Mechanical Systems and Signal Processing vol 75 pp261ndash279 2016

[21] J Liu W D Zhu P G Charalambides Y M Shao Y F Xuand X M Fang ldquoA dynamic model of a cantilever beam with aclosed embedded horizontal crack including local flexibilitiesat crack tipsrdquo Journal of Sound and Vibration vol 382 pp 274ndash290 2016

[22] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[23] A Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagationrdquo Engineering FractureMechanics vol17 no 4 pp 381ndash386 1983

[24] A J Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagation in a beam under longitudinal vibra-tionsrdquo International Journal of Fracture vol 23 no 1 pp 15ndash221983

[25] A J Dentsoras andADDimarogonas ldquoFatigue crack propaga-tion in resonating structuresrdquo Engineering Fracture Mechanicsvol 34 no 3 pp 721ndash728 1989

[26] M Lalanne P Berthier and J D HagopianMechanical Vibra-tions for Engineers John Wiley amp Sons New York NY USA1983

[27] A D Dimarogonas S A Paipetis and T G ChondrosAnalyticalMethods inRotorDynamics SpringerDordrechtTheNetherlands 2013

[28] H Tada P C Paris andG R IrwinTheStress Analysis of CracksHandbook The American Society of Mechanical Engineers2000

[29] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984

[30] R Mulyukov S Mikhailov R Zaripova and D SalimonenkoldquoDamping properties of 18Cr-10Ni stainless steel with submi-crocrystalline structurerdquoMaterials Research Bulletin vol 31 no6 pp 639ndash645 1996

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Shock and Vibration 5

lz

xo

Crack surface

Slip line

Enlarged

Q

F

Q a

a

b

ho

z

y

a2

Figure 5 Crack growth road

Therefore the normal pressing force is obtained

119865119899 = 119886119887120590119899 (15)

Because the breathing crack will be opened graduallywhen it is in intensive zone and will be closed when it is in acompressed zone the value of pressing forcemay exist duringthe half cycle relative to the effect of external harmonic forceand the average pressing force 119865av in one cycle is given by

119865av = 112058710038161003816100381610038161003816100381610038161003816int120587

0119865119899 sin (120596119905) 119889 (120596119905)10038161003816100381610038161003816100381610038161003816 =

2119865119899120587 (16)

If the frictional coefficient of crack surfaces for 304SS is 120583the friction force between the crack surfaces can be derivedby the sliding friction model It is given by

119865119891 = 120583119865av cos 45∘ (17)

In each cycle the breathing process of the crack from theopen state to the close state includes 4 phases If position Ais the initial position as shown in Figure 1 a total breathingprocess includes A rarr B rarr A rarr C rarr A Only whenthe crack is in the compressed zone the friction force willbe produced Based on the work-energy theorem for systemsthe energy dissipation introduced by the sliding friction forcebetween the crack surfaces in one vibration cycle is given by

119882119862 = 2119865119891119889 (18)

It can be seen from Figure 6 that the relative slidingdistance between the crack surfaces depends on COD of 119876

d

COD

QFn

45∘

Fn

Ff

Figure 6 Friction model of crack surface

point An empirical formula for COD with 1 accuracy forany 119886ℎ is given by Tada et al [28]

COD = 4120590119899119886119864 119881(119886ℎ) (19)

Here the function 119881(119886ℎ) is given by

119881(119886ℎ) = 08 minus 17 (119886ℎ) + 24 (119886ℎ)2 + 066(1 minus 119886ℎ)2 (20)

Figure 6 shows that the relation between COD and 119889 is

119889 = COD sdot cos 1205874 (21)

Therefore the energy dissipation caused by the frictionforce within one vibration cycle is obtained

119882119862 = 4120587 12058311986411988621198871205902119899119881(119886ℎ) (22)

According to the energy principle and the assumptionsthe strain energy storage in the cracked beam having rectan-gular cross section is given by

119881max = int1198710

1198641198682 (12059721199081205971199092 )2 119889119909

+ int1198710

12 (1 + ]) 1198641198682119860 (12059731199081205971199093 )2 119889119909

(23)

For a cracked beam the friction damping loss factor 120574119891can be defined as

120574119891 = 1198821198622120587119881max (24)

6 Shock and Vibration

Therefore the damping loss factor caused by the frictionof the crack surfaces is obtained

120574119891 = 21205831198862119887 (ℎ minus 119886)2 119881 (119886ℎ) ((12059721199081205971199092)10038161003816100381610038161003816119909=119897)21205872 (119868 int119871

0(12059721199081205971199092)2 119889119909 + (24 (1 + ]) 1198682119860) int119871

0(12059731199081205971199093)2 119889119909) (25)

4 Vibration Response Analysis

41Mode Frequency If the generalized stiffness 119896lowast is replacedby the stiffness of the open cracked beam 119896119900 then the firstmode frequency of the open cracked beam is also obtainedTherefore the first mode frequency of the cracked beamcorresponding to fully open and fully closed states can bepredicted approximately by the following equation

120596119900119888 = radic119896119900119888119898lowast (26)

Here 120596119900 is the frequency of the open cracked beam and 120596119888 isthe frequency of the closed cracked beam

If the breathing behaviour is considered the stiffness ofthe cracked beam varies with time within one vibration cycleIt is impossible to plug the stiffness model of a breathingcracked beam into the frequency equation (26) to analyzethe mode frequency of the breathing cracked beam directlyBut within one cycle no matter which kind of the breathingfunction is used the cycle period time will not change Ifthe square wave function is used the vibration cycle timeincludes two parts The prehalf cycle corresponds to the fullyopen crack and the posthalf cycle corresponds to the fullyclosed crack Therefore the relation of the sine breathingbehaviour and the square wave breathing behaviour mustobey

2120587120596br= 120587120596119900 +

120587120596119888 (27)

Here 120596br is the frequency of the breathing cracked beam Itis given by

120596br = 2120596119900120596119888120596119900 + 120596119888 (28)

42 Dynamic Stress Intensity Factor (DSIF) For a vibratingcracked beam the crack propagation speed relies on notonly the amplitude of dynamic stress response but also thecrack depth For an edge cracked beam as shown in Figure 1ignoring the shear force the beam endures bending loadmainlyThe crackwill open as type I crack In real engineeringpractice mode I crack maybe is the most common crackmodel and it is very sensitive to dynamic responseThereforethe mode I crack is considered in this article Based on thestrength theory and Hookersquos law the nominal dynamic bulk

stress at the top edge of the cracked beamunder pure bendingstates can be estimated approximately by

120590119889 = 119864ℎ2 1205972119908 (119909 119905)1205971199092100381610038161003816100381610038161003816100381610038161003816119909=119897 (29)

Plugging the dynamic displacement response into thestress equation we obtain

120590119889 = 119864120578119885stℎ2 1205972119882(119909)1205971199092

100381610038161003816100381610038161003816100381610038161003816119909=119897 (30)

Here 119885st is the static displacement it is the ratio of thegeneralized force and the stiffness of the cracked beam 120578 isthe amplify ratio of displacement response it is given by [26]and expressed by

120578 = 1radic(1 minus 1205822)2 + ((120574119898 + 120574119891) 120582)2

(31)

Here 120582 is the frequency ratio When 120582 is near to 1 the beamis near or at resonance the amplify ratio is very large and thedamping ratio plays important role in the resonance responseand crack growth

Based on the definition of stress intensity factor (SIF) infracture mechanics the SIF of the beam under pure bendingcan be defined by empirical formula

119870119868119889 = 120590119889radic120587119886119865119868 (119886ℎ) (32)

Here 119870119868119889 is SIF 120590119889 is the nominal dynamic bulk stress119865119868(119886ℎ) is crack shape function and it is given by manyauthors in approximate expression as [28]

119865119868 (119886ℎ) = 1122 minus 140 (119886ℎ) + 733 (119886ℎ)2

minus 1308 (119886ℎ)3 + 140 (119886ℎ)4 (33)

The shape function is claimed to be of engineeringaccuracy for any 119886ℎ lt 065 Crack Growth Analysis

51 The Modified Forman Equation The parameters such asthe crack closure effect stress ratio SIF range critical SIF andthe threshold of SIF are considered for the model of crack

Shock and Vibration 7

growth which is called themodified FormanModel [21]Thismodel is employed for crack growth analysis in this study andshown as

119889119886119889119873 = 119861(1 minus 1198911 minus 119903 Δ119870)119899 [1 minus Δ119870thΔ119870]119901[1 minus Δ119870 (1 minus 119903)119870119888]119902 (34)

Here 119903 is the stress ratio 119870119888 is the critical SIF Δ119870th isthe threshold SIF and Δ119870 is the stress intensity range 119873 isthe numbers of vibration cycle 119861 119899 119901 and 119902 are fatigueproperties of materials 119891 is closure function and given byNewman Jr [29]119891

=

max (1199031198980 + 1198981119903 + 11989821199032 + 11989831199033) 119903 ge 01198980 + 1198981119903 0 gt 119903 ge minus21198980 minus 21198981 119903 lt minus2

(35)

where

1198980 = (0825 minus 034120572 + 0051205722) [cos(120587119878max(21205900) )]1120572

1198981 = (0415 minus 0071120572) 119878max1205900 1198982 = 1 minus 1198980 minus 1198981 minus 11989831198983 = 21198980 + 1198981 minus 1

(36)

The threshold SIF is given by

Δ119870th = Δ119870th0 ( 119886119886 + 1198860)05 ( 4120587 tan (1 minus 119903)) (37)

Here Δ119870th0 is the threshold SIF when 119903 is 0 1198860 is the initialcrack depth

In these formulas 120572 is the constraint coefficient for planestress or plane strain problem 119878max1205900 is the ratio of themaximum stress and stress flow For the constant amplitudeloading the increment of crack growth can be calculated byintegrating (34) Because the stress relies on the crack depththe exciting frequency and the damping the amplitude ofdynamic stress response is assumed to be a constant valuewithin one cycle The incremental crack growth in each cyclecan be estimated by (34) Then the crack depth after 119894 timescycle can be calculated by the superposition method andexpressed as

119886119894 = 1198860 + 119894sum119895=1

Δ119886119895 (38)

Here 119886119894 is the total crack depth after 119894 times cycle and Δ119886119895 isthe crack incremental of the 119895th step vibration

The incremental of crack growth within one vibrationcycle is calculated by employing the modified Forman crackgrowth equation It is expressed by

Δ119886 = 119861(1 minus 1198911 minus 119903 Δ119870)119899 [1 minus Δ119870thΔ119870]119901[1 minus Δ119870 (1 minus 119903)119870119888]119902 (39)

Table 1 Material properties of 304SS

Youngrsquos modulus 119864 (MPa) 204 times 105

Poissonrsquos ratio ] 03Mass density 120588 (kgm3) 7860Yield stress 120590119904 (MPa) 2758Ultimate stress 120590119887 (MPa) 620Fracture toughness 119870119862 (MPasdotmm05) 7645Threshold value Δ119870th0 (MPasdotmm05) 1216

Table 2 Crack growth properties of 304SS

119861 6 times 10minus10119901 025119902 025119899 30119878max1205900 03120572 25

The coupling analysis can be done by the above math-ematical equations The stiffness is changed gradually withcrack growth first The damping is also introduced by thefriction energy dissipation from crack surfaces Both of themaffect the dynamic response greatly On the contrary thecrack propagation ratio is also affected by the change ofdynamic response

52 Failure Criteria If the crack depth extends to a certainvalue or the dynamic stress is too large alternative failuremodes may occur to the beam Therefore it is especiallyimportant to work out reasonable criteria to indicate thefailure Three failure criteria are considered and shown asfollows

(1) Geometrical Limit The failure occurs automatically if thecrack depth is larger than the half height of the beam becauseSIF is not suitable when used to estimate the fatigue life incrack growth equation

(2) Critical SIF The fracture occurs if the maximum SIF islarger than critical SIF (119870119868119889max ge 119870119862)(3) Fracture Failure The static fracture failure occurs if themaximum value of the bulk dynamic stress is larger than theultimate strength stress

6 Results and Discussion

A cantilever beam of 304SS is considered as 1m in length012m in width 015m in height and a straight-edge crackwith distance 119897 = 05m from the fixed end The force thatacts on the free end of the cracked beam is a sine excitationThe amplitude of the force is 1198650 = 500N and the stress ratio is119903 = minus1 The materials of the beam are shown in Table 1 Thecrack growth properties of the beam are shown in Table 2

Because the damping properties in the stainless steelsubjected to deformation and temperature the maximum

8 Shock and Vibration

07

075

08

085

09

095

1

Freq

uenc

y ra

tio

01 02 03 04 050Crack depth ratio ah

brc by ABAQUSbrc by Equation (28)oc by Equation (26)

Figure 7 Frequencies change with crack severity

damping loss factor is not bigger than 001 [30] and thedamping loss factor 120574119898 of 304SS is assumed to be 0005 Inorder to investigate the effects of breathing behaviour on thedynamic response and the crack growth themode frequencyfriction damping loss factor dynamic stress dynamic stressintensity and crack growth are analyzed

Figure 7 displays the relation of the mode frequency ratioand the crack depth ratio Results indicate that the predictedfrequency of breathing cracked beam by (28) is in goodagreement with the frequency calculated by ABAQUS Thedecrease ratio of the frequency of the breathing cracked beam120596br120596119888 is smaller than that of the open cracked beam120596119900120596119888 Itis implied that if the frequency of the force acted on the beamis near to the resonance frequency the breathing crackedbeam is subjected to a bigger response because the resonanceregion of the breathing cracked beam is wider than that ofthe open cracked beam Furthermore the crack growth speedof the breathing crack is faster than that of the open cracknear the resonance condition Therefore using a breathingcrack mode to carry out dynamic analysis or crack growthcalculation is more accurate than using an open crack model

Figure 8 investigates the friction damping loss factor byusing different friction coefficient Results indicate that thefriction damping loss factor varies with not only the crackseverity but also the friction coefficient With the increaseof the crack depth or the friction coefficient the frictiondamping loss factor increases That implied that the energydissipated by the friction force is increasing during crackgrowth It is very helpful to reduce the dynamic responseof the resonant beam Thus the crack growth ratio of thebreathing crack is reduced too

Figure 9 shows the maximum dynamic stress of thebreathing cracked beam It is compared with that of the opencracked beam near the resonance region without considering

0

0005

001

0015

002

0025

003

Fric

tion

dam

ping

loss

fact

or

01 02 03 04 050Crack depth ratio ah

= 01

= 03

= 05

f

Figure 8 Friction damping loss factor with crack severity changeunder different friction coefficients

Breathing crack c = 098

Open crack c = 098Breathing crack c = 099

Open crack c = 099Breathing crack c = 1

Open crack c = 1

01 02 03 04 050Crack depth ratio ah

0

20

40

60

80

100

120

140

160

180

Max

imum

dyn

amic

stre

ssd

max

(MPa

)

Figure 9 The maximum dynamic stress with crack severity change(120583 = 0)

the friction damping Results indicate that the stress responseamplitude varies with crack depth Resonance vibrationhappens when the crack reaches a certain depth with onefixed exciting frequencyThe maximum stress happens at theresonant point And themaximumdynamic stress of the opencracked beam is smaller than that of the breathing crackedbeam with the same exciting frequency This implies that the

Shock and Vibration 9

0

20

40

60

80

100

120

140

160

180

Max

imum

dyn

amic

stre

ssd

max

(MPa

)

01 02 03 04 050Crack depth ratio ah

c = 098 = 01

c = 098 = 03

c = 099 = 01

c = 099 = 03

c = 1 = 01

c = 1 = 03

c = 098 = 05

c = 099 = 05

c = 1 = 05

Figure 10Themaximumdynamic stress with crack severity changeof breathing cracked beam

closure effect changes the stiffness of the breathing crackedbeam so that the maximum stress response also changes

Figure 10 presents the effect of the friction damping on themaximum dynamic stress of a breathing cracked beam as thebeam working under resonance condition It can be shownthat the maximum stress of the beam is greatly reduced bythe friction damping This implied that the friction dampingis an important factor to reduce the crack growth ratio

Figure 11 discovers the relation of the maximum dynamicSIFs of the open cracked beam and that of the breathingcracked beam without friction damping It is shown that themaximum dynamic SIF of the breathing cracked beam islarger than that of the open cracked beam without frictiondamping considered under the same exciting frequency Buteach of them happens at different crack depths

Figure 12 studies the effects of friction damping loss factoron themaximum dynamic SIF of the breathing cracked beamat resonance It can be seen that dynamic SIFs are verysensitive to the friction damping Comparing Figure 10 withFigure 12 we can find that the change law of the dynamicstress is different from that of the dynamic SIF The reasonis because the crack depth also plays an important role in SIFduring crack growth

Figures 13 and 14 examine the relation of the crack depthwith the vibration cycles It can be seen that the crackgrowth ratio of the open cracked beam is slower than thatof the breathing cracked beam if the friction damping is notconsidered The reason is because the decrease ratio of the

Breathing crack c = 098

Open crack c = 098

Breathing crack c = 099

Open crack c = 099

Breathing crack c = 1

Open crack c = 1

0

200

400

600

800

1000

1200

1400

1600

1800

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

Figure 11 The maximum dynamic SIF with crack severity changefor different frequency (120583 = 0)

0

500

1000

1500

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 12 The maximum dynamic SIF with crack severity changeof breathing cracked beam

mode frequency of the breathing cracked beam is slower thanthat of the open cracked beam This means the breathingcracked beam has a wider resonance region Considering

10 Shock and Vibration

01 015 02 025 03 035 04 045 05

Breathing crack c = 098

Breathing crack c = 099

Breathing crack c = 1

Open crack c = 098

Open crack c = 099

Open crack c = 1

Crack depth ratio ah

times104

0

1

2

3

4

5

Num

bers

of v

ibra

tion

cycl

e N

Figure 13 Crack depth with vibration cycles for different frequency(120583 = 0)

the friction damping the crack growth ratio is reduced bythe friction damping for a breathing cracked beam obviouslywhen the beam comes into resonance region as shown inFigure 14

Figure 15 shows the relation of the decrease ratio offrequency ratio and the numbers of vibration cycles It can beseen that the decrease ratio of frequency of the open crackedbeam is faster than that of the breathing cracked beam at thefirst phase of the crack growth when the exciting frequency is098 times of themode frequency of the intact beam But aftercertain numbers of vibration cycle the decrease ratio of thefrequency ratio of the breathing cracked beam is faster thanthat of the open cracked beam

7 Conclusions

In this article the stiffness model and the friction dampingmodel of a breathing cracked beam are derived The firstmode frequency is analyzed by using a SDOF model whichis simplified by the Galerkin method The dynamic stressis derived and the crack growth is studied The effects ofthe breathing behaviour on the dynamic response and crackgrowth are revealed so as to analyze the coupling mechanismof the crack growth anddynamic response approximatelyThemain conclusions are obtained

(1) During vibration analysis of a cracked beam the crackclosure effect between opening and closing states must beconsidered A breathing crack model is better than the opencrackmodel for dynamic response and crack growth analysisA sinusoidal breathing function is useful to simulate the

0

2

4

6

8

10

Num

bers

of v

ibra

tion

cycl

e N

times104

015 02 025 03 035 04 045 0501Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 14 Breathing crack depth with vibration cycles for differentfriction coefficient and frequency

times104

088

09

092

094

096

098

1

Freq

uenc

y ra

tio

br

cor

o

c

1 2 3 4 50Numbers of vibration cycle N

oc

brc = 0brc = 02

brc = 04

Figure 15 Frequency ratio with vibration cycles (120596120596119888 = 098)

transition state between the fully open state and the fullyclosed state It meets the object facts and the physical naturalof crack It may be closer to the real motion of the breathingcrack than that of the squarewave function Using a breathingcrack model is more realistic than using an opening crackmodel

Shock and Vibration 11

(2) Based on energy principle and Coulomb frictionmodel the friction model is set up and friction damping lossfactor is derived Friction damping not only plays importantrole in the maximum dynamic response but also playsimportant role in the crack growth Especially to resonanceresponse and crack growth friction damping effect is veryobvious

(3)TheGalerkinmethod is used to simplify the breathingcracked beam into a SDOF systemThe first mode frequencyequation is derived approximately The first mode frequencychanges with the crack severity The frequency decrease ratioof the breathing cracked beam is slower than that of theopen cracked beam If the closure behaviour of the breathingcrack is not considered the degree of crack growth may beunderestimated

(4) The modified Forman equation is employed to simu-late the crack growth A crack growth life analytical methodis proposed with the coupling effect of vibration and fatiguecrack growth included Breathing behaviour of the crack isimportant to crack growthThe dynamical properties play animportant role in crack growthWhen the exciting frequencyis near to the first mode frequency the crack growth ratio isvery fastThe crack growth life of the breathing cracked beamis smaller than that of the open cracked beam

Nomenclature

119886 1198860 Crack depth initial crack depth119886119894 Δ119886119895 The 119894th step crack depth the 119895th step crackincremental119860 Area of cross section119887 Width of beam cross section119861 Fatigue properties of materials119888lowast Coefficient of modal damping119862 Flexibility of the intact beam119862119879 Flexibility due to the presence of the crack119862119900 Total flexibility of the open cracked beam119889 Sliding distance between crack surfaces119864 Youngrsquos modulus119891 Closure function of breathing crack119865 Harmonic load119865lowast 1198650 The generalized force loading amplitude119865119899 119865119891 Normal pressing force friction force119865av Average pressing force119865119868 Crack shape functionℎ Height of the beam cross section119868 Area moment of inertial of the beam crosssection119896lowast The generalized stiffness119896br Stiffness of the breathing cracked beam119896119888 Stiffness of the close cracked beam119896119900 Stiffness of the open cracked beam119896119879 Stiffness of the torsional spring119870119868119889 Dynamic stress intensity factor119870119868119889max The maximum dynamic stress intensity factor119870119862 Fracture toughnessΔ119870th The threshold of stress intensity factorΔ119870 The range of stress intensity factor119897 Distance of crack from the fixed end

119871 Length of the beam119898lowast The generalized mass119872 Moment119899 119901 119902 Fatigue properties of materials119873 Numbers of vibration cycle119876 Shear force119903 Stress ratio119905 Time119879 Amplitude-time function119881max The strain energy119882119862 Energy dissipation119908 Lateral deflection in 119911 direction of beam119882(119909) Modal function119909 119910 119911 The coordinate119885st Static displacement120588 ] Density of material Poissonrsquos ratio120583 Friction coefficient120596 Excitation frequency120596119888 Frequency of the close cracked beam120596119900 Frequency of the open cracked beam120574119898 Material damping120574119891 Frictional damping120578 Amplify ratio120582 Frequency ratio120573 The breathing function120575 The Dirac function120590119899 120590119889 Nominal stress normal bulk dynamic stress120590119887 120590119904 Ultimate stress yield stress

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research work is supported by the National NaturalScience Foundation of China (51565039)

References

[1] A R Biswal T Roy R K Behera S K Pradhan and P KParida ldquoFinite element based vibration analysis of a nonpris-matic Timoshenko beam with transverse open crackrdquo ProcediaEngineering vol 144 pp 226ndash233 2016

[2] MKrawczuk andWMOstachowicz ldquoModelling and vibrationanalysis of a cantilever composite beam with a transverse opencrackrdquo Journal of Sound and Vibration vol 183 no 1 pp 69ndash891995

[3] C A Papadopoulos and A D Dimarogonas ldquoCoupled longi-tudinal and bending vibrations of a rotating shaft with an opencrackrdquo Journal of Sound and Vibration vol 117 no 1 pp 81ndash931987

[4] W M Ostachowicz and M Krawczuk ldquoAnalysis of the effect ofcracks on the natural frequencies of a cantilever beamrdquo Journalof Sound and Vibration vol 150 no 2 pp 191ndash201 1991

[5] T G Chondros andA D Dimarogonas ldquoVibration of a crackedcantilever beamrdquo Journal of Vibration andAcoustics vol 120 no3 pp 742ndash746 1998

12 Shock and Vibration

[6] T G Chondros A D Dimarogonas and J Yao ldquoA continuouscracked beam vibration theoryrdquo Journal of Sound and Vibrationvol 215 no 1 pp 17ndash34 1998

[7] M Heydari A Ebrahimi and M Behzad ldquoContinuous modelfor flexural vibration analysis of Timoshenko beams with avertical edge crackrdquo Archive of Applied Mechanics vol 85 no5 pp 601ndash615 2015

[8] R K Behera A Pandey and D R Parhi ldquoNumerical andexperimental verification of a method for prognosis of inclinededge crack in cantilever beam based on synthesis of modeshapesrdquo Procedia Technology vol 14 pp 67ndash74 2014

[9] S M Cheng A S J Swamidas X J Wu and W WallaceldquoVibrational response of a beamwith a breathing crackrdquo Journalof Sound and Vibration vol 225 no 1 pp 201ndash208 1999

[10] T G Chondros A D Dimarogonas and J Yao ldquoVibration of abeam with a breathing crackrdquo Journal of Sound and Vibrationvol 239 no 1 pp 57ndash67 2001

[11] E Douka and L J Hadjileontiadis ldquoTime-frequency analysis ofthe free vibration response of a beam with a breathing crackrdquoNDT amp E International vol 38 no 1 pp 3ndash10 2005

[12] M Kisa and J Brandon ldquoEffects of closure of cracks on thedynamics of a cracked cantilever beamrdquo Journal of Sound andVibration vol 238 no 1 pp 1ndash18 2000

[13] C Surace R Ruotolo and D Storer ldquoDetecting nonlinearbehaviour using the volterra series to assess damage in beam-like structuresrdquo Journal of Theoretical and Applied Mechanicsvol 49 no 3 pp 905ndash926 2011

[14] AChatterjee ldquoNonlinear dynamics anddamage assessment of acantilever beamwith breathing edge crackrdquo Journal of Vibrationand Acoustics vol 133 no 5 Article ID 051004 2011

[15] R Ruotolo C Surace P Crespo and D Storer ldquoHarmonicanalysis of the vibrations of a cantilevered beam with a closingcrackrdquo Computers amp Structures vol 61 no 6 pp 1057ndash10741996

[16] A Nandi and S Neogy ldquoModelling of a beam with a breathingedge crack and some observations for crack detectionrdquo Journalof Vibration and Control vol 8 no 5 pp 673ndash693 2002

[17] W Liu and M E Barkey ldquoNonlinear vibrational response ofa single edge cracked beamrdquo Journal of Mechanical Science ampTechnology vol 31 no 11 pp 5231ndash5243 2017

[18] J Zeng H MaW Zhang and BWen ldquoDynamic characteristicanalysis of cracked cantilever beams under different cracktypesrdquo Engineering Failure Analysis vol 74 pp 80ndash94 2017

[19] W Zhang H Ma J Zeng S Wu and B Wen ldquoVibrationresponses analysis of an elastic-support cantilever beam withcrack and offset boundaryrdquo Mechanical Systems and SignalProcessing vol 95 pp 205ndash218 2017

[20] HMa J Zeng Z Lang L Zhang Y Guo and BWen ldquoAnalysisof the dynamic characteristics of a slant-cracked cantileverbeamrdquo Mechanical Systems and Signal Processing vol 75 pp261ndash279 2016

[21] J Liu W D Zhu P G Charalambides Y M Shao Y F Xuand X M Fang ldquoA dynamic model of a cantilever beam with aclosed embedded horizontal crack including local flexibilitiesat crack tipsrdquo Journal of Sound and Vibration vol 382 pp 274ndash290 2016

[22] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[23] A Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagationrdquo Engineering FractureMechanics vol17 no 4 pp 381ndash386 1983

[24] A J Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagation in a beam under longitudinal vibra-tionsrdquo International Journal of Fracture vol 23 no 1 pp 15ndash221983

[25] A J Dentsoras andADDimarogonas ldquoFatigue crack propaga-tion in resonating structuresrdquo Engineering Fracture Mechanicsvol 34 no 3 pp 721ndash728 1989

[26] M Lalanne P Berthier and J D HagopianMechanical Vibra-tions for Engineers John Wiley amp Sons New York NY USA1983

[27] A D Dimarogonas S A Paipetis and T G ChondrosAnalyticalMethods inRotorDynamics SpringerDordrechtTheNetherlands 2013

[28] H Tada P C Paris andG R IrwinTheStress Analysis of CracksHandbook The American Society of Mechanical Engineers2000

[29] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984

[30] R Mulyukov S Mikhailov R Zaripova and D SalimonenkoldquoDamping properties of 18Cr-10Ni stainless steel with submi-crocrystalline structurerdquoMaterials Research Bulletin vol 31 no6 pp 639ndash645 1996

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

6 Shock and Vibration

Therefore the damping loss factor caused by the frictionof the crack surfaces is obtained

120574119891 = 21205831198862119887 (ℎ minus 119886)2 119881 (119886ℎ) ((12059721199081205971199092)10038161003816100381610038161003816119909=119897)21205872 (119868 int119871

0(12059721199081205971199092)2 119889119909 + (24 (1 + ]) 1198682119860) int119871

0(12059731199081205971199093)2 119889119909) (25)

4 Vibration Response Analysis

41Mode Frequency If the generalized stiffness 119896lowast is replacedby the stiffness of the open cracked beam 119896119900 then the firstmode frequency of the open cracked beam is also obtainedTherefore the first mode frequency of the cracked beamcorresponding to fully open and fully closed states can bepredicted approximately by the following equation

120596119900119888 = radic119896119900119888119898lowast (26)

Here 120596119900 is the frequency of the open cracked beam and 120596119888 isthe frequency of the closed cracked beam

If the breathing behaviour is considered the stiffness ofthe cracked beam varies with time within one vibration cycleIt is impossible to plug the stiffness model of a breathingcracked beam into the frequency equation (26) to analyzethe mode frequency of the breathing cracked beam directlyBut within one cycle no matter which kind of the breathingfunction is used the cycle period time will not change Ifthe square wave function is used the vibration cycle timeincludes two parts The prehalf cycle corresponds to the fullyopen crack and the posthalf cycle corresponds to the fullyclosed crack Therefore the relation of the sine breathingbehaviour and the square wave breathing behaviour mustobey

2120587120596br= 120587120596119900 +

120587120596119888 (27)

Here 120596br is the frequency of the breathing cracked beam Itis given by

120596br = 2120596119900120596119888120596119900 + 120596119888 (28)

42 Dynamic Stress Intensity Factor (DSIF) For a vibratingcracked beam the crack propagation speed relies on notonly the amplitude of dynamic stress response but also thecrack depth For an edge cracked beam as shown in Figure 1ignoring the shear force the beam endures bending loadmainlyThe crackwill open as type I crack In real engineeringpractice mode I crack maybe is the most common crackmodel and it is very sensitive to dynamic responseThereforethe mode I crack is considered in this article Based on thestrength theory and Hookersquos law the nominal dynamic bulk

stress at the top edge of the cracked beamunder pure bendingstates can be estimated approximately by

120590119889 = 119864ℎ2 1205972119908 (119909 119905)1205971199092100381610038161003816100381610038161003816100381610038161003816119909=119897 (29)

Plugging the dynamic displacement response into thestress equation we obtain

120590119889 = 119864120578119885stℎ2 1205972119882(119909)1205971199092

100381610038161003816100381610038161003816100381610038161003816119909=119897 (30)

Here 119885st is the static displacement it is the ratio of thegeneralized force and the stiffness of the cracked beam 120578 isthe amplify ratio of displacement response it is given by [26]and expressed by

120578 = 1radic(1 minus 1205822)2 + ((120574119898 + 120574119891) 120582)2

(31)

Here 120582 is the frequency ratio When 120582 is near to 1 the beamis near or at resonance the amplify ratio is very large and thedamping ratio plays important role in the resonance responseand crack growth

Based on the definition of stress intensity factor (SIF) infracture mechanics the SIF of the beam under pure bendingcan be defined by empirical formula

119870119868119889 = 120590119889radic120587119886119865119868 (119886ℎ) (32)

Here 119870119868119889 is SIF 120590119889 is the nominal dynamic bulk stress119865119868(119886ℎ) is crack shape function and it is given by manyauthors in approximate expression as [28]

119865119868 (119886ℎ) = 1122 minus 140 (119886ℎ) + 733 (119886ℎ)2

minus 1308 (119886ℎ)3 + 140 (119886ℎ)4 (33)

The shape function is claimed to be of engineeringaccuracy for any 119886ℎ lt 065 Crack Growth Analysis

51 The Modified Forman Equation The parameters such asthe crack closure effect stress ratio SIF range critical SIF andthe threshold of SIF are considered for the model of crack

Shock and Vibration 7

growth which is called themodified FormanModel [21]Thismodel is employed for crack growth analysis in this study andshown as

119889119886119889119873 = 119861(1 minus 1198911 minus 119903 Δ119870)119899 [1 minus Δ119870thΔ119870]119901[1 minus Δ119870 (1 minus 119903)119870119888]119902 (34)

Here 119903 is the stress ratio 119870119888 is the critical SIF Δ119870th isthe threshold SIF and Δ119870 is the stress intensity range 119873 isthe numbers of vibration cycle 119861 119899 119901 and 119902 are fatigueproperties of materials 119891 is closure function and given byNewman Jr [29]119891

=

max (1199031198980 + 1198981119903 + 11989821199032 + 11989831199033) 119903 ge 01198980 + 1198981119903 0 gt 119903 ge minus21198980 minus 21198981 119903 lt minus2

(35)

where

1198980 = (0825 minus 034120572 + 0051205722) [cos(120587119878max(21205900) )]1120572

1198981 = (0415 minus 0071120572) 119878max1205900 1198982 = 1 minus 1198980 minus 1198981 minus 11989831198983 = 21198980 + 1198981 minus 1

(36)

The threshold SIF is given by

Δ119870th = Δ119870th0 ( 119886119886 + 1198860)05 ( 4120587 tan (1 minus 119903)) (37)

Here Δ119870th0 is the threshold SIF when 119903 is 0 1198860 is the initialcrack depth

In these formulas 120572 is the constraint coefficient for planestress or plane strain problem 119878max1205900 is the ratio of themaximum stress and stress flow For the constant amplitudeloading the increment of crack growth can be calculated byintegrating (34) Because the stress relies on the crack depththe exciting frequency and the damping the amplitude ofdynamic stress response is assumed to be a constant valuewithin one cycle The incremental crack growth in each cyclecan be estimated by (34) Then the crack depth after 119894 timescycle can be calculated by the superposition method andexpressed as

119886119894 = 1198860 + 119894sum119895=1

Δ119886119895 (38)

Here 119886119894 is the total crack depth after 119894 times cycle and Δ119886119895 isthe crack incremental of the 119895th step vibration

The incremental of crack growth within one vibrationcycle is calculated by employing the modified Forman crackgrowth equation It is expressed by

Δ119886 = 119861(1 minus 1198911 minus 119903 Δ119870)119899 [1 minus Δ119870thΔ119870]119901[1 minus Δ119870 (1 minus 119903)119870119888]119902 (39)

Table 1 Material properties of 304SS

Youngrsquos modulus 119864 (MPa) 204 times 105

Poissonrsquos ratio ] 03Mass density 120588 (kgm3) 7860Yield stress 120590119904 (MPa) 2758Ultimate stress 120590119887 (MPa) 620Fracture toughness 119870119862 (MPasdotmm05) 7645Threshold value Δ119870th0 (MPasdotmm05) 1216

Table 2 Crack growth properties of 304SS

119861 6 times 10minus10119901 025119902 025119899 30119878max1205900 03120572 25

The coupling analysis can be done by the above math-ematical equations The stiffness is changed gradually withcrack growth first The damping is also introduced by thefriction energy dissipation from crack surfaces Both of themaffect the dynamic response greatly On the contrary thecrack propagation ratio is also affected by the change ofdynamic response

52 Failure Criteria If the crack depth extends to a certainvalue or the dynamic stress is too large alternative failuremodes may occur to the beam Therefore it is especiallyimportant to work out reasonable criteria to indicate thefailure Three failure criteria are considered and shown asfollows

(1) Geometrical Limit The failure occurs automatically if thecrack depth is larger than the half height of the beam becauseSIF is not suitable when used to estimate the fatigue life incrack growth equation

(2) Critical SIF The fracture occurs if the maximum SIF islarger than critical SIF (119870119868119889max ge 119870119862)(3) Fracture Failure The static fracture failure occurs if themaximum value of the bulk dynamic stress is larger than theultimate strength stress

6 Results and Discussion

A cantilever beam of 304SS is considered as 1m in length012m in width 015m in height and a straight-edge crackwith distance 119897 = 05m from the fixed end The force thatacts on the free end of the cracked beam is a sine excitationThe amplitude of the force is 1198650 = 500N and the stress ratio is119903 = minus1 The materials of the beam are shown in Table 1 Thecrack growth properties of the beam are shown in Table 2

Because the damping properties in the stainless steelsubjected to deformation and temperature the maximum

8 Shock and Vibration

07

075

08

085

09

095

1

Freq

uenc

y ra

tio

01 02 03 04 050Crack depth ratio ah

brc by ABAQUSbrc by Equation (28)oc by Equation (26)

Figure 7 Frequencies change with crack severity

damping loss factor is not bigger than 001 [30] and thedamping loss factor 120574119898 of 304SS is assumed to be 0005 Inorder to investigate the effects of breathing behaviour on thedynamic response and the crack growth themode frequencyfriction damping loss factor dynamic stress dynamic stressintensity and crack growth are analyzed

Figure 7 displays the relation of the mode frequency ratioand the crack depth ratio Results indicate that the predictedfrequency of breathing cracked beam by (28) is in goodagreement with the frequency calculated by ABAQUS Thedecrease ratio of the frequency of the breathing cracked beam120596br120596119888 is smaller than that of the open cracked beam120596119900120596119888 Itis implied that if the frequency of the force acted on the beamis near to the resonance frequency the breathing crackedbeam is subjected to a bigger response because the resonanceregion of the breathing cracked beam is wider than that ofthe open cracked beam Furthermore the crack growth speedof the breathing crack is faster than that of the open cracknear the resonance condition Therefore using a breathingcrack mode to carry out dynamic analysis or crack growthcalculation is more accurate than using an open crack model

Figure 8 investigates the friction damping loss factor byusing different friction coefficient Results indicate that thefriction damping loss factor varies with not only the crackseverity but also the friction coefficient With the increaseof the crack depth or the friction coefficient the frictiondamping loss factor increases That implied that the energydissipated by the friction force is increasing during crackgrowth It is very helpful to reduce the dynamic responseof the resonant beam Thus the crack growth ratio of thebreathing crack is reduced too

Figure 9 shows the maximum dynamic stress of thebreathing cracked beam It is compared with that of the opencracked beam near the resonance region without considering

0

0005

001

0015

002

0025

003

Fric

tion

dam

ping

loss

fact

or

01 02 03 04 050Crack depth ratio ah

= 01

= 03

= 05

f

Figure 8 Friction damping loss factor with crack severity changeunder different friction coefficients

Breathing crack c = 098

Open crack c = 098Breathing crack c = 099

Open crack c = 099Breathing crack c = 1

Open crack c = 1

01 02 03 04 050Crack depth ratio ah

0

20

40

60

80

100

120

140

160

180

Max

imum

dyn

amic

stre

ssd

max

(MPa

)

Figure 9 The maximum dynamic stress with crack severity change(120583 = 0)

the friction damping Results indicate that the stress responseamplitude varies with crack depth Resonance vibrationhappens when the crack reaches a certain depth with onefixed exciting frequencyThe maximum stress happens at theresonant point And themaximumdynamic stress of the opencracked beam is smaller than that of the breathing crackedbeam with the same exciting frequency This implies that the

Shock and Vibration 9

0

20

40

60

80

100

120

140

160

180

Max

imum

dyn

amic

stre

ssd

max

(MPa

)

01 02 03 04 050Crack depth ratio ah

c = 098 = 01

c = 098 = 03

c = 099 = 01

c = 099 = 03

c = 1 = 01

c = 1 = 03

c = 098 = 05

c = 099 = 05

c = 1 = 05

Figure 10Themaximumdynamic stress with crack severity changeof breathing cracked beam

closure effect changes the stiffness of the breathing crackedbeam so that the maximum stress response also changes

Figure 10 presents the effect of the friction damping on themaximum dynamic stress of a breathing cracked beam as thebeam working under resonance condition It can be shownthat the maximum stress of the beam is greatly reduced bythe friction damping This implied that the friction dampingis an important factor to reduce the crack growth ratio

Figure 11 discovers the relation of the maximum dynamicSIFs of the open cracked beam and that of the breathingcracked beam without friction damping It is shown that themaximum dynamic SIF of the breathing cracked beam islarger than that of the open cracked beam without frictiondamping considered under the same exciting frequency Buteach of them happens at different crack depths

Figure 12 studies the effects of friction damping loss factoron themaximum dynamic SIF of the breathing cracked beamat resonance It can be seen that dynamic SIFs are verysensitive to the friction damping Comparing Figure 10 withFigure 12 we can find that the change law of the dynamicstress is different from that of the dynamic SIF The reasonis because the crack depth also plays an important role in SIFduring crack growth

Figures 13 and 14 examine the relation of the crack depthwith the vibration cycles It can be seen that the crackgrowth ratio of the open cracked beam is slower than thatof the breathing cracked beam if the friction damping is notconsidered The reason is because the decrease ratio of the

Breathing crack c = 098

Open crack c = 098

Breathing crack c = 099

Open crack c = 099

Breathing crack c = 1

Open crack c = 1

0

200

400

600

800

1000

1200

1400

1600

1800

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

Figure 11 The maximum dynamic SIF with crack severity changefor different frequency (120583 = 0)

0

500

1000

1500

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 12 The maximum dynamic SIF with crack severity changeof breathing cracked beam

mode frequency of the breathing cracked beam is slower thanthat of the open cracked beam This means the breathingcracked beam has a wider resonance region Considering

10 Shock and Vibration

01 015 02 025 03 035 04 045 05

Breathing crack c = 098

Breathing crack c = 099

Breathing crack c = 1

Open crack c = 098

Open crack c = 099

Open crack c = 1

Crack depth ratio ah

times104

0

1

2

3

4

5

Num

bers

of v

ibra

tion

cycl

e N

Figure 13 Crack depth with vibration cycles for different frequency(120583 = 0)

the friction damping the crack growth ratio is reduced bythe friction damping for a breathing cracked beam obviouslywhen the beam comes into resonance region as shown inFigure 14

Figure 15 shows the relation of the decrease ratio offrequency ratio and the numbers of vibration cycles It can beseen that the decrease ratio of frequency of the open crackedbeam is faster than that of the breathing cracked beam at thefirst phase of the crack growth when the exciting frequency is098 times of themode frequency of the intact beam But aftercertain numbers of vibration cycle the decrease ratio of thefrequency ratio of the breathing cracked beam is faster thanthat of the open cracked beam

7 Conclusions

In this article the stiffness model and the friction dampingmodel of a breathing cracked beam are derived The firstmode frequency is analyzed by using a SDOF model whichis simplified by the Galerkin method The dynamic stressis derived and the crack growth is studied The effects ofthe breathing behaviour on the dynamic response and crackgrowth are revealed so as to analyze the coupling mechanismof the crack growth anddynamic response approximatelyThemain conclusions are obtained

(1) During vibration analysis of a cracked beam the crackclosure effect between opening and closing states must beconsidered A breathing crack model is better than the opencrackmodel for dynamic response and crack growth analysisA sinusoidal breathing function is useful to simulate the

0

2

4

6

8

10

Num

bers

of v

ibra

tion

cycl

e N

times104

015 02 025 03 035 04 045 0501Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 14 Breathing crack depth with vibration cycles for differentfriction coefficient and frequency

times104

088

09

092

094

096

098

1

Freq

uenc

y ra

tio

br

cor

o

c

1 2 3 4 50Numbers of vibration cycle N

oc

brc = 0brc = 02

brc = 04

Figure 15 Frequency ratio with vibration cycles (120596120596119888 = 098)

transition state between the fully open state and the fullyclosed state It meets the object facts and the physical naturalof crack It may be closer to the real motion of the breathingcrack than that of the squarewave function Using a breathingcrack model is more realistic than using an opening crackmodel

Shock and Vibration 11

(2) Based on energy principle and Coulomb frictionmodel the friction model is set up and friction damping lossfactor is derived Friction damping not only plays importantrole in the maximum dynamic response but also playsimportant role in the crack growth Especially to resonanceresponse and crack growth friction damping effect is veryobvious

(3)TheGalerkinmethod is used to simplify the breathingcracked beam into a SDOF systemThe first mode frequencyequation is derived approximately The first mode frequencychanges with the crack severity The frequency decrease ratioof the breathing cracked beam is slower than that of theopen cracked beam If the closure behaviour of the breathingcrack is not considered the degree of crack growth may beunderestimated

(4) The modified Forman equation is employed to simu-late the crack growth A crack growth life analytical methodis proposed with the coupling effect of vibration and fatiguecrack growth included Breathing behaviour of the crack isimportant to crack growthThe dynamical properties play animportant role in crack growthWhen the exciting frequencyis near to the first mode frequency the crack growth ratio isvery fastThe crack growth life of the breathing cracked beamis smaller than that of the open cracked beam

Nomenclature

119886 1198860 Crack depth initial crack depth119886119894 Δ119886119895 The 119894th step crack depth the 119895th step crackincremental119860 Area of cross section119887 Width of beam cross section119861 Fatigue properties of materials119888lowast Coefficient of modal damping119862 Flexibility of the intact beam119862119879 Flexibility due to the presence of the crack119862119900 Total flexibility of the open cracked beam119889 Sliding distance between crack surfaces119864 Youngrsquos modulus119891 Closure function of breathing crack119865 Harmonic load119865lowast 1198650 The generalized force loading amplitude119865119899 119865119891 Normal pressing force friction force119865av Average pressing force119865119868 Crack shape functionℎ Height of the beam cross section119868 Area moment of inertial of the beam crosssection119896lowast The generalized stiffness119896br Stiffness of the breathing cracked beam119896119888 Stiffness of the close cracked beam119896119900 Stiffness of the open cracked beam119896119879 Stiffness of the torsional spring119870119868119889 Dynamic stress intensity factor119870119868119889max The maximum dynamic stress intensity factor119870119862 Fracture toughnessΔ119870th The threshold of stress intensity factorΔ119870 The range of stress intensity factor119897 Distance of crack from the fixed end

119871 Length of the beam119898lowast The generalized mass119872 Moment119899 119901 119902 Fatigue properties of materials119873 Numbers of vibration cycle119876 Shear force119903 Stress ratio119905 Time119879 Amplitude-time function119881max The strain energy119882119862 Energy dissipation119908 Lateral deflection in 119911 direction of beam119882(119909) Modal function119909 119910 119911 The coordinate119885st Static displacement120588 ] Density of material Poissonrsquos ratio120583 Friction coefficient120596 Excitation frequency120596119888 Frequency of the close cracked beam120596119900 Frequency of the open cracked beam120574119898 Material damping120574119891 Frictional damping120578 Amplify ratio120582 Frequency ratio120573 The breathing function120575 The Dirac function120590119899 120590119889 Nominal stress normal bulk dynamic stress120590119887 120590119904 Ultimate stress yield stress

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research work is supported by the National NaturalScience Foundation of China (51565039)

References

[1] A R Biswal T Roy R K Behera S K Pradhan and P KParida ldquoFinite element based vibration analysis of a nonpris-matic Timoshenko beam with transverse open crackrdquo ProcediaEngineering vol 144 pp 226ndash233 2016

[2] MKrawczuk andWMOstachowicz ldquoModelling and vibrationanalysis of a cantilever composite beam with a transverse opencrackrdquo Journal of Sound and Vibration vol 183 no 1 pp 69ndash891995

[3] C A Papadopoulos and A D Dimarogonas ldquoCoupled longi-tudinal and bending vibrations of a rotating shaft with an opencrackrdquo Journal of Sound and Vibration vol 117 no 1 pp 81ndash931987

[4] W M Ostachowicz and M Krawczuk ldquoAnalysis of the effect ofcracks on the natural frequencies of a cantilever beamrdquo Journalof Sound and Vibration vol 150 no 2 pp 191ndash201 1991

[5] T G Chondros andA D Dimarogonas ldquoVibration of a crackedcantilever beamrdquo Journal of Vibration andAcoustics vol 120 no3 pp 742ndash746 1998

12 Shock and Vibration

[6] T G Chondros A D Dimarogonas and J Yao ldquoA continuouscracked beam vibration theoryrdquo Journal of Sound and Vibrationvol 215 no 1 pp 17ndash34 1998

[7] M Heydari A Ebrahimi and M Behzad ldquoContinuous modelfor flexural vibration analysis of Timoshenko beams with avertical edge crackrdquo Archive of Applied Mechanics vol 85 no5 pp 601ndash615 2015

[8] R K Behera A Pandey and D R Parhi ldquoNumerical andexperimental verification of a method for prognosis of inclinededge crack in cantilever beam based on synthesis of modeshapesrdquo Procedia Technology vol 14 pp 67ndash74 2014

[9] S M Cheng A S J Swamidas X J Wu and W WallaceldquoVibrational response of a beamwith a breathing crackrdquo Journalof Sound and Vibration vol 225 no 1 pp 201ndash208 1999

[10] T G Chondros A D Dimarogonas and J Yao ldquoVibration of abeam with a breathing crackrdquo Journal of Sound and Vibrationvol 239 no 1 pp 57ndash67 2001

[11] E Douka and L J Hadjileontiadis ldquoTime-frequency analysis ofthe free vibration response of a beam with a breathing crackrdquoNDT amp E International vol 38 no 1 pp 3ndash10 2005

[12] M Kisa and J Brandon ldquoEffects of closure of cracks on thedynamics of a cracked cantilever beamrdquo Journal of Sound andVibration vol 238 no 1 pp 1ndash18 2000

[13] C Surace R Ruotolo and D Storer ldquoDetecting nonlinearbehaviour using the volterra series to assess damage in beam-like structuresrdquo Journal of Theoretical and Applied Mechanicsvol 49 no 3 pp 905ndash926 2011

[14] AChatterjee ldquoNonlinear dynamics anddamage assessment of acantilever beamwith breathing edge crackrdquo Journal of Vibrationand Acoustics vol 133 no 5 Article ID 051004 2011

[15] R Ruotolo C Surace P Crespo and D Storer ldquoHarmonicanalysis of the vibrations of a cantilevered beam with a closingcrackrdquo Computers amp Structures vol 61 no 6 pp 1057ndash10741996

[16] A Nandi and S Neogy ldquoModelling of a beam with a breathingedge crack and some observations for crack detectionrdquo Journalof Vibration and Control vol 8 no 5 pp 673ndash693 2002

[17] W Liu and M E Barkey ldquoNonlinear vibrational response ofa single edge cracked beamrdquo Journal of Mechanical Science ampTechnology vol 31 no 11 pp 5231ndash5243 2017

[18] J Zeng H MaW Zhang and BWen ldquoDynamic characteristicanalysis of cracked cantilever beams under different cracktypesrdquo Engineering Failure Analysis vol 74 pp 80ndash94 2017

[19] W Zhang H Ma J Zeng S Wu and B Wen ldquoVibrationresponses analysis of an elastic-support cantilever beam withcrack and offset boundaryrdquo Mechanical Systems and SignalProcessing vol 95 pp 205ndash218 2017

[20] HMa J Zeng Z Lang L Zhang Y Guo and BWen ldquoAnalysisof the dynamic characteristics of a slant-cracked cantileverbeamrdquo Mechanical Systems and Signal Processing vol 75 pp261ndash279 2016

[21] J Liu W D Zhu P G Charalambides Y M Shao Y F Xuand X M Fang ldquoA dynamic model of a cantilever beam with aclosed embedded horizontal crack including local flexibilitiesat crack tipsrdquo Journal of Sound and Vibration vol 382 pp 274ndash290 2016

[22] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[23] A Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagationrdquo Engineering FractureMechanics vol17 no 4 pp 381ndash386 1983

[24] A J Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagation in a beam under longitudinal vibra-tionsrdquo International Journal of Fracture vol 23 no 1 pp 15ndash221983

[25] A J Dentsoras andADDimarogonas ldquoFatigue crack propaga-tion in resonating structuresrdquo Engineering Fracture Mechanicsvol 34 no 3 pp 721ndash728 1989

[26] M Lalanne P Berthier and J D HagopianMechanical Vibra-tions for Engineers John Wiley amp Sons New York NY USA1983

[27] A D Dimarogonas S A Paipetis and T G ChondrosAnalyticalMethods inRotorDynamics SpringerDordrechtTheNetherlands 2013

[28] H Tada P C Paris andG R IrwinTheStress Analysis of CracksHandbook The American Society of Mechanical Engineers2000

[29] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984

[30] R Mulyukov S Mikhailov R Zaripova and D SalimonenkoldquoDamping properties of 18Cr-10Ni stainless steel with submi-crocrystalline structurerdquoMaterials Research Bulletin vol 31 no6 pp 639ndash645 1996

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Shock and Vibration 7

growth which is called themodified FormanModel [21]Thismodel is employed for crack growth analysis in this study andshown as

119889119886119889119873 = 119861(1 minus 1198911 minus 119903 Δ119870)119899 [1 minus Δ119870thΔ119870]119901[1 minus Δ119870 (1 minus 119903)119870119888]119902 (34)

Here 119903 is the stress ratio 119870119888 is the critical SIF Δ119870th isthe threshold SIF and Δ119870 is the stress intensity range 119873 isthe numbers of vibration cycle 119861 119899 119901 and 119902 are fatigueproperties of materials 119891 is closure function and given byNewman Jr [29]119891

=

max (1199031198980 + 1198981119903 + 11989821199032 + 11989831199033) 119903 ge 01198980 + 1198981119903 0 gt 119903 ge minus21198980 minus 21198981 119903 lt minus2

(35)

where

1198980 = (0825 minus 034120572 + 0051205722) [cos(120587119878max(21205900) )]1120572

1198981 = (0415 minus 0071120572) 119878max1205900 1198982 = 1 minus 1198980 minus 1198981 minus 11989831198983 = 21198980 + 1198981 minus 1

(36)

The threshold SIF is given by

Δ119870th = Δ119870th0 ( 119886119886 + 1198860)05 ( 4120587 tan (1 minus 119903)) (37)

Here Δ119870th0 is the threshold SIF when 119903 is 0 1198860 is the initialcrack depth

In these formulas 120572 is the constraint coefficient for planestress or plane strain problem 119878max1205900 is the ratio of themaximum stress and stress flow For the constant amplitudeloading the increment of crack growth can be calculated byintegrating (34) Because the stress relies on the crack depththe exciting frequency and the damping the amplitude ofdynamic stress response is assumed to be a constant valuewithin one cycle The incremental crack growth in each cyclecan be estimated by (34) Then the crack depth after 119894 timescycle can be calculated by the superposition method andexpressed as

119886119894 = 1198860 + 119894sum119895=1

Δ119886119895 (38)

Here 119886119894 is the total crack depth after 119894 times cycle and Δ119886119895 isthe crack incremental of the 119895th step vibration

The incremental of crack growth within one vibrationcycle is calculated by employing the modified Forman crackgrowth equation It is expressed by

Δ119886 = 119861(1 minus 1198911 minus 119903 Δ119870)119899 [1 minus Δ119870thΔ119870]119901[1 minus Δ119870 (1 minus 119903)119870119888]119902 (39)

Table 1 Material properties of 304SS

Youngrsquos modulus 119864 (MPa) 204 times 105

Poissonrsquos ratio ] 03Mass density 120588 (kgm3) 7860Yield stress 120590119904 (MPa) 2758Ultimate stress 120590119887 (MPa) 620Fracture toughness 119870119862 (MPasdotmm05) 7645Threshold value Δ119870th0 (MPasdotmm05) 1216

Table 2 Crack growth properties of 304SS

119861 6 times 10minus10119901 025119902 025119899 30119878max1205900 03120572 25

The coupling analysis can be done by the above math-ematical equations The stiffness is changed gradually withcrack growth first The damping is also introduced by thefriction energy dissipation from crack surfaces Both of themaffect the dynamic response greatly On the contrary thecrack propagation ratio is also affected by the change ofdynamic response

52 Failure Criteria If the crack depth extends to a certainvalue or the dynamic stress is too large alternative failuremodes may occur to the beam Therefore it is especiallyimportant to work out reasonable criteria to indicate thefailure Three failure criteria are considered and shown asfollows

(1) Geometrical Limit The failure occurs automatically if thecrack depth is larger than the half height of the beam becauseSIF is not suitable when used to estimate the fatigue life incrack growth equation

(2) Critical SIF The fracture occurs if the maximum SIF islarger than critical SIF (119870119868119889max ge 119870119862)(3) Fracture Failure The static fracture failure occurs if themaximum value of the bulk dynamic stress is larger than theultimate strength stress

6 Results and Discussion

A cantilever beam of 304SS is considered as 1m in length012m in width 015m in height and a straight-edge crackwith distance 119897 = 05m from the fixed end The force thatacts on the free end of the cracked beam is a sine excitationThe amplitude of the force is 1198650 = 500N and the stress ratio is119903 = minus1 The materials of the beam are shown in Table 1 Thecrack growth properties of the beam are shown in Table 2

Because the damping properties in the stainless steelsubjected to deformation and temperature the maximum

8 Shock and Vibration

07

075

08

085

09

095

1

Freq

uenc

y ra

tio

01 02 03 04 050Crack depth ratio ah

brc by ABAQUSbrc by Equation (28)oc by Equation (26)

Figure 7 Frequencies change with crack severity

damping loss factor is not bigger than 001 [30] and thedamping loss factor 120574119898 of 304SS is assumed to be 0005 Inorder to investigate the effects of breathing behaviour on thedynamic response and the crack growth themode frequencyfriction damping loss factor dynamic stress dynamic stressintensity and crack growth are analyzed

Figure 7 displays the relation of the mode frequency ratioand the crack depth ratio Results indicate that the predictedfrequency of breathing cracked beam by (28) is in goodagreement with the frequency calculated by ABAQUS Thedecrease ratio of the frequency of the breathing cracked beam120596br120596119888 is smaller than that of the open cracked beam120596119900120596119888 Itis implied that if the frequency of the force acted on the beamis near to the resonance frequency the breathing crackedbeam is subjected to a bigger response because the resonanceregion of the breathing cracked beam is wider than that ofthe open cracked beam Furthermore the crack growth speedof the breathing crack is faster than that of the open cracknear the resonance condition Therefore using a breathingcrack mode to carry out dynamic analysis or crack growthcalculation is more accurate than using an open crack model

Figure 8 investigates the friction damping loss factor byusing different friction coefficient Results indicate that thefriction damping loss factor varies with not only the crackseverity but also the friction coefficient With the increaseof the crack depth or the friction coefficient the frictiondamping loss factor increases That implied that the energydissipated by the friction force is increasing during crackgrowth It is very helpful to reduce the dynamic responseof the resonant beam Thus the crack growth ratio of thebreathing crack is reduced too

Figure 9 shows the maximum dynamic stress of thebreathing cracked beam It is compared with that of the opencracked beam near the resonance region without considering

0

0005

001

0015

002

0025

003

Fric

tion

dam

ping

loss

fact

or

01 02 03 04 050Crack depth ratio ah

= 01

= 03

= 05

f

Figure 8 Friction damping loss factor with crack severity changeunder different friction coefficients

Breathing crack c = 098

Open crack c = 098Breathing crack c = 099

Open crack c = 099Breathing crack c = 1

Open crack c = 1

01 02 03 04 050Crack depth ratio ah

0

20

40

60

80

100

120

140

160

180

Max

imum

dyn

amic

stre

ssd

max

(MPa

)

Figure 9 The maximum dynamic stress with crack severity change(120583 = 0)

the friction damping Results indicate that the stress responseamplitude varies with crack depth Resonance vibrationhappens when the crack reaches a certain depth with onefixed exciting frequencyThe maximum stress happens at theresonant point And themaximumdynamic stress of the opencracked beam is smaller than that of the breathing crackedbeam with the same exciting frequency This implies that the

Shock and Vibration 9

0

20

40

60

80

100

120

140

160

180

Max

imum

dyn

amic

stre

ssd

max

(MPa

)

01 02 03 04 050Crack depth ratio ah

c = 098 = 01

c = 098 = 03

c = 099 = 01

c = 099 = 03

c = 1 = 01

c = 1 = 03

c = 098 = 05

c = 099 = 05

c = 1 = 05

Figure 10Themaximumdynamic stress with crack severity changeof breathing cracked beam

closure effect changes the stiffness of the breathing crackedbeam so that the maximum stress response also changes

Figure 10 presents the effect of the friction damping on themaximum dynamic stress of a breathing cracked beam as thebeam working under resonance condition It can be shownthat the maximum stress of the beam is greatly reduced bythe friction damping This implied that the friction dampingis an important factor to reduce the crack growth ratio

Figure 11 discovers the relation of the maximum dynamicSIFs of the open cracked beam and that of the breathingcracked beam without friction damping It is shown that themaximum dynamic SIF of the breathing cracked beam islarger than that of the open cracked beam without frictiondamping considered under the same exciting frequency Buteach of them happens at different crack depths

Figure 12 studies the effects of friction damping loss factoron themaximum dynamic SIF of the breathing cracked beamat resonance It can be seen that dynamic SIFs are verysensitive to the friction damping Comparing Figure 10 withFigure 12 we can find that the change law of the dynamicstress is different from that of the dynamic SIF The reasonis because the crack depth also plays an important role in SIFduring crack growth

Figures 13 and 14 examine the relation of the crack depthwith the vibration cycles It can be seen that the crackgrowth ratio of the open cracked beam is slower than thatof the breathing cracked beam if the friction damping is notconsidered The reason is because the decrease ratio of the

Breathing crack c = 098

Open crack c = 098

Breathing crack c = 099

Open crack c = 099

Breathing crack c = 1

Open crack c = 1

0

200

400

600

800

1000

1200

1400

1600

1800

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

Figure 11 The maximum dynamic SIF with crack severity changefor different frequency (120583 = 0)

0

500

1000

1500

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 12 The maximum dynamic SIF with crack severity changeof breathing cracked beam

mode frequency of the breathing cracked beam is slower thanthat of the open cracked beam This means the breathingcracked beam has a wider resonance region Considering

10 Shock and Vibration

01 015 02 025 03 035 04 045 05

Breathing crack c = 098

Breathing crack c = 099

Breathing crack c = 1

Open crack c = 098

Open crack c = 099

Open crack c = 1

Crack depth ratio ah

times104

0

1

2

3

4

5

Num

bers

of v

ibra

tion

cycl

e N

Figure 13 Crack depth with vibration cycles for different frequency(120583 = 0)

the friction damping the crack growth ratio is reduced bythe friction damping for a breathing cracked beam obviouslywhen the beam comes into resonance region as shown inFigure 14

Figure 15 shows the relation of the decrease ratio offrequency ratio and the numbers of vibration cycles It can beseen that the decrease ratio of frequency of the open crackedbeam is faster than that of the breathing cracked beam at thefirst phase of the crack growth when the exciting frequency is098 times of themode frequency of the intact beam But aftercertain numbers of vibration cycle the decrease ratio of thefrequency ratio of the breathing cracked beam is faster thanthat of the open cracked beam

7 Conclusions

In this article the stiffness model and the friction dampingmodel of a breathing cracked beam are derived The firstmode frequency is analyzed by using a SDOF model whichis simplified by the Galerkin method The dynamic stressis derived and the crack growth is studied The effects ofthe breathing behaviour on the dynamic response and crackgrowth are revealed so as to analyze the coupling mechanismof the crack growth anddynamic response approximatelyThemain conclusions are obtained

(1) During vibration analysis of a cracked beam the crackclosure effect between opening and closing states must beconsidered A breathing crack model is better than the opencrackmodel for dynamic response and crack growth analysisA sinusoidal breathing function is useful to simulate the

0

2

4

6

8

10

Num

bers

of v

ibra

tion

cycl

e N

times104

015 02 025 03 035 04 045 0501Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 14 Breathing crack depth with vibration cycles for differentfriction coefficient and frequency

times104

088

09

092

094

096

098

1

Freq

uenc

y ra

tio

br

cor

o

c

1 2 3 4 50Numbers of vibration cycle N

oc

brc = 0brc = 02

brc = 04

Figure 15 Frequency ratio with vibration cycles (120596120596119888 = 098)

transition state between the fully open state and the fullyclosed state It meets the object facts and the physical naturalof crack It may be closer to the real motion of the breathingcrack than that of the squarewave function Using a breathingcrack model is more realistic than using an opening crackmodel

Shock and Vibration 11

(2) Based on energy principle and Coulomb frictionmodel the friction model is set up and friction damping lossfactor is derived Friction damping not only plays importantrole in the maximum dynamic response but also playsimportant role in the crack growth Especially to resonanceresponse and crack growth friction damping effect is veryobvious

(3)TheGalerkinmethod is used to simplify the breathingcracked beam into a SDOF systemThe first mode frequencyequation is derived approximately The first mode frequencychanges with the crack severity The frequency decrease ratioof the breathing cracked beam is slower than that of theopen cracked beam If the closure behaviour of the breathingcrack is not considered the degree of crack growth may beunderestimated

(4) The modified Forman equation is employed to simu-late the crack growth A crack growth life analytical methodis proposed with the coupling effect of vibration and fatiguecrack growth included Breathing behaviour of the crack isimportant to crack growthThe dynamical properties play animportant role in crack growthWhen the exciting frequencyis near to the first mode frequency the crack growth ratio isvery fastThe crack growth life of the breathing cracked beamis smaller than that of the open cracked beam

Nomenclature

119886 1198860 Crack depth initial crack depth119886119894 Δ119886119895 The 119894th step crack depth the 119895th step crackincremental119860 Area of cross section119887 Width of beam cross section119861 Fatigue properties of materials119888lowast Coefficient of modal damping119862 Flexibility of the intact beam119862119879 Flexibility due to the presence of the crack119862119900 Total flexibility of the open cracked beam119889 Sliding distance between crack surfaces119864 Youngrsquos modulus119891 Closure function of breathing crack119865 Harmonic load119865lowast 1198650 The generalized force loading amplitude119865119899 119865119891 Normal pressing force friction force119865av Average pressing force119865119868 Crack shape functionℎ Height of the beam cross section119868 Area moment of inertial of the beam crosssection119896lowast The generalized stiffness119896br Stiffness of the breathing cracked beam119896119888 Stiffness of the close cracked beam119896119900 Stiffness of the open cracked beam119896119879 Stiffness of the torsional spring119870119868119889 Dynamic stress intensity factor119870119868119889max The maximum dynamic stress intensity factor119870119862 Fracture toughnessΔ119870th The threshold of stress intensity factorΔ119870 The range of stress intensity factor119897 Distance of crack from the fixed end

119871 Length of the beam119898lowast The generalized mass119872 Moment119899 119901 119902 Fatigue properties of materials119873 Numbers of vibration cycle119876 Shear force119903 Stress ratio119905 Time119879 Amplitude-time function119881max The strain energy119882119862 Energy dissipation119908 Lateral deflection in 119911 direction of beam119882(119909) Modal function119909 119910 119911 The coordinate119885st Static displacement120588 ] Density of material Poissonrsquos ratio120583 Friction coefficient120596 Excitation frequency120596119888 Frequency of the close cracked beam120596119900 Frequency of the open cracked beam120574119898 Material damping120574119891 Frictional damping120578 Amplify ratio120582 Frequency ratio120573 The breathing function120575 The Dirac function120590119899 120590119889 Nominal stress normal bulk dynamic stress120590119887 120590119904 Ultimate stress yield stress

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research work is supported by the National NaturalScience Foundation of China (51565039)

References

[1] A R Biswal T Roy R K Behera S K Pradhan and P KParida ldquoFinite element based vibration analysis of a nonpris-matic Timoshenko beam with transverse open crackrdquo ProcediaEngineering vol 144 pp 226ndash233 2016

[2] MKrawczuk andWMOstachowicz ldquoModelling and vibrationanalysis of a cantilever composite beam with a transverse opencrackrdquo Journal of Sound and Vibration vol 183 no 1 pp 69ndash891995

[3] C A Papadopoulos and A D Dimarogonas ldquoCoupled longi-tudinal and bending vibrations of a rotating shaft with an opencrackrdquo Journal of Sound and Vibration vol 117 no 1 pp 81ndash931987

[4] W M Ostachowicz and M Krawczuk ldquoAnalysis of the effect ofcracks on the natural frequencies of a cantilever beamrdquo Journalof Sound and Vibration vol 150 no 2 pp 191ndash201 1991

[5] T G Chondros andA D Dimarogonas ldquoVibration of a crackedcantilever beamrdquo Journal of Vibration andAcoustics vol 120 no3 pp 742ndash746 1998

12 Shock and Vibration

[6] T G Chondros A D Dimarogonas and J Yao ldquoA continuouscracked beam vibration theoryrdquo Journal of Sound and Vibrationvol 215 no 1 pp 17ndash34 1998

[7] M Heydari A Ebrahimi and M Behzad ldquoContinuous modelfor flexural vibration analysis of Timoshenko beams with avertical edge crackrdquo Archive of Applied Mechanics vol 85 no5 pp 601ndash615 2015

[8] R K Behera A Pandey and D R Parhi ldquoNumerical andexperimental verification of a method for prognosis of inclinededge crack in cantilever beam based on synthesis of modeshapesrdquo Procedia Technology vol 14 pp 67ndash74 2014

[9] S M Cheng A S J Swamidas X J Wu and W WallaceldquoVibrational response of a beamwith a breathing crackrdquo Journalof Sound and Vibration vol 225 no 1 pp 201ndash208 1999

[10] T G Chondros A D Dimarogonas and J Yao ldquoVibration of abeam with a breathing crackrdquo Journal of Sound and Vibrationvol 239 no 1 pp 57ndash67 2001

[11] E Douka and L J Hadjileontiadis ldquoTime-frequency analysis ofthe free vibration response of a beam with a breathing crackrdquoNDT amp E International vol 38 no 1 pp 3ndash10 2005

[12] M Kisa and J Brandon ldquoEffects of closure of cracks on thedynamics of a cracked cantilever beamrdquo Journal of Sound andVibration vol 238 no 1 pp 1ndash18 2000

[13] C Surace R Ruotolo and D Storer ldquoDetecting nonlinearbehaviour using the volterra series to assess damage in beam-like structuresrdquo Journal of Theoretical and Applied Mechanicsvol 49 no 3 pp 905ndash926 2011

[14] AChatterjee ldquoNonlinear dynamics anddamage assessment of acantilever beamwith breathing edge crackrdquo Journal of Vibrationand Acoustics vol 133 no 5 Article ID 051004 2011

[15] R Ruotolo C Surace P Crespo and D Storer ldquoHarmonicanalysis of the vibrations of a cantilevered beam with a closingcrackrdquo Computers amp Structures vol 61 no 6 pp 1057ndash10741996

[16] A Nandi and S Neogy ldquoModelling of a beam with a breathingedge crack and some observations for crack detectionrdquo Journalof Vibration and Control vol 8 no 5 pp 673ndash693 2002

[17] W Liu and M E Barkey ldquoNonlinear vibrational response ofa single edge cracked beamrdquo Journal of Mechanical Science ampTechnology vol 31 no 11 pp 5231ndash5243 2017

[18] J Zeng H MaW Zhang and BWen ldquoDynamic characteristicanalysis of cracked cantilever beams under different cracktypesrdquo Engineering Failure Analysis vol 74 pp 80ndash94 2017

[19] W Zhang H Ma J Zeng S Wu and B Wen ldquoVibrationresponses analysis of an elastic-support cantilever beam withcrack and offset boundaryrdquo Mechanical Systems and SignalProcessing vol 95 pp 205ndash218 2017

[20] HMa J Zeng Z Lang L Zhang Y Guo and BWen ldquoAnalysisof the dynamic characteristics of a slant-cracked cantileverbeamrdquo Mechanical Systems and Signal Processing vol 75 pp261ndash279 2016

[21] J Liu W D Zhu P G Charalambides Y M Shao Y F Xuand X M Fang ldquoA dynamic model of a cantilever beam with aclosed embedded horizontal crack including local flexibilitiesat crack tipsrdquo Journal of Sound and Vibration vol 382 pp 274ndash290 2016

[22] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[23] A Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagationrdquo Engineering FractureMechanics vol17 no 4 pp 381ndash386 1983

[24] A J Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagation in a beam under longitudinal vibra-tionsrdquo International Journal of Fracture vol 23 no 1 pp 15ndash221983

[25] A J Dentsoras andADDimarogonas ldquoFatigue crack propaga-tion in resonating structuresrdquo Engineering Fracture Mechanicsvol 34 no 3 pp 721ndash728 1989

[26] M Lalanne P Berthier and J D HagopianMechanical Vibra-tions for Engineers John Wiley amp Sons New York NY USA1983

[27] A D Dimarogonas S A Paipetis and T G ChondrosAnalyticalMethods inRotorDynamics SpringerDordrechtTheNetherlands 2013

[28] H Tada P C Paris andG R IrwinTheStress Analysis of CracksHandbook The American Society of Mechanical Engineers2000

[29] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984

[30] R Mulyukov S Mikhailov R Zaripova and D SalimonenkoldquoDamping properties of 18Cr-10Ni stainless steel with submi-crocrystalline structurerdquoMaterials Research Bulletin vol 31 no6 pp 639ndash645 1996

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Submit your manuscripts atwwwhindawicom

8 Shock and Vibration

07

075

08

085

09

095

1

Freq

uenc

y ra

tio

01 02 03 04 050Crack depth ratio ah

brc by ABAQUSbrc by Equation (28)oc by Equation (26)

Figure 7 Frequencies change with crack severity

damping loss factor is not bigger than 001 [30] and thedamping loss factor 120574119898 of 304SS is assumed to be 0005 Inorder to investigate the effects of breathing behaviour on thedynamic response and the crack growth themode frequencyfriction damping loss factor dynamic stress dynamic stressintensity and crack growth are analyzed

Figure 7 displays the relation of the mode frequency ratioand the crack depth ratio Results indicate that the predictedfrequency of breathing cracked beam by (28) is in goodagreement with the frequency calculated by ABAQUS Thedecrease ratio of the frequency of the breathing cracked beam120596br120596119888 is smaller than that of the open cracked beam120596119900120596119888 Itis implied that if the frequency of the force acted on the beamis near to the resonance frequency the breathing crackedbeam is subjected to a bigger response because the resonanceregion of the breathing cracked beam is wider than that ofthe open cracked beam Furthermore the crack growth speedof the breathing crack is faster than that of the open cracknear the resonance condition Therefore using a breathingcrack mode to carry out dynamic analysis or crack growthcalculation is more accurate than using an open crack model

Figure 8 investigates the friction damping loss factor byusing different friction coefficient Results indicate that thefriction damping loss factor varies with not only the crackseverity but also the friction coefficient With the increaseof the crack depth or the friction coefficient the frictiondamping loss factor increases That implied that the energydissipated by the friction force is increasing during crackgrowth It is very helpful to reduce the dynamic responseof the resonant beam Thus the crack growth ratio of thebreathing crack is reduced too

Figure 9 shows the maximum dynamic stress of thebreathing cracked beam It is compared with that of the opencracked beam near the resonance region without considering

0

0005

001

0015

002

0025

003

Fric

tion

dam

ping

loss

fact

or

01 02 03 04 050Crack depth ratio ah

= 01

= 03

= 05

f

Figure 8 Friction damping loss factor with crack severity changeunder different friction coefficients

Breathing crack c = 098

Open crack c = 098Breathing crack c = 099

Open crack c = 099Breathing crack c = 1

Open crack c = 1

01 02 03 04 050Crack depth ratio ah

0

20

40

60

80

100

120

140

160

180

Max

imum

dyn

amic

stre

ssd

max

(MPa

)

Figure 9 The maximum dynamic stress with crack severity change(120583 = 0)

the friction damping Results indicate that the stress responseamplitude varies with crack depth Resonance vibrationhappens when the crack reaches a certain depth with onefixed exciting frequencyThe maximum stress happens at theresonant point And themaximumdynamic stress of the opencracked beam is smaller than that of the breathing crackedbeam with the same exciting frequency This implies that the

Shock and Vibration 9

0

20

40

60

80

100

120

140

160

180

Max

imum

dyn

amic

stre

ssd

max

(MPa

)

01 02 03 04 050Crack depth ratio ah

c = 098 = 01

c = 098 = 03

c = 099 = 01

c = 099 = 03

c = 1 = 01

c = 1 = 03

c = 098 = 05

c = 099 = 05

c = 1 = 05

Figure 10Themaximumdynamic stress with crack severity changeof breathing cracked beam

closure effect changes the stiffness of the breathing crackedbeam so that the maximum stress response also changes

Figure 10 presents the effect of the friction damping on themaximum dynamic stress of a breathing cracked beam as thebeam working under resonance condition It can be shownthat the maximum stress of the beam is greatly reduced bythe friction damping This implied that the friction dampingis an important factor to reduce the crack growth ratio

Figure 11 discovers the relation of the maximum dynamicSIFs of the open cracked beam and that of the breathingcracked beam without friction damping It is shown that themaximum dynamic SIF of the breathing cracked beam islarger than that of the open cracked beam without frictiondamping considered under the same exciting frequency Buteach of them happens at different crack depths

Figure 12 studies the effects of friction damping loss factoron themaximum dynamic SIF of the breathing cracked beamat resonance It can be seen that dynamic SIFs are verysensitive to the friction damping Comparing Figure 10 withFigure 12 we can find that the change law of the dynamicstress is different from that of the dynamic SIF The reasonis because the crack depth also plays an important role in SIFduring crack growth

Figures 13 and 14 examine the relation of the crack depthwith the vibration cycles It can be seen that the crackgrowth ratio of the open cracked beam is slower than thatof the breathing cracked beam if the friction damping is notconsidered The reason is because the decrease ratio of the

Breathing crack c = 098

Open crack c = 098

Breathing crack c = 099

Open crack c = 099

Breathing crack c = 1

Open crack c = 1

0

200

400

600

800

1000

1200

1400

1600

1800

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

Figure 11 The maximum dynamic SIF with crack severity changefor different frequency (120583 = 0)

0

500

1000

1500

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 12 The maximum dynamic SIF with crack severity changeof breathing cracked beam

mode frequency of the breathing cracked beam is slower thanthat of the open cracked beam This means the breathingcracked beam has a wider resonance region Considering

10 Shock and Vibration

01 015 02 025 03 035 04 045 05

Breathing crack c = 098

Breathing crack c = 099

Breathing crack c = 1

Open crack c = 098

Open crack c = 099

Open crack c = 1

Crack depth ratio ah

times104

0

1

2

3

4

5

Num

bers

of v

ibra

tion

cycl

e N

Figure 13 Crack depth with vibration cycles for different frequency(120583 = 0)

the friction damping the crack growth ratio is reduced bythe friction damping for a breathing cracked beam obviouslywhen the beam comes into resonance region as shown inFigure 14

Figure 15 shows the relation of the decrease ratio offrequency ratio and the numbers of vibration cycles It can beseen that the decrease ratio of frequency of the open crackedbeam is faster than that of the breathing cracked beam at thefirst phase of the crack growth when the exciting frequency is098 times of themode frequency of the intact beam But aftercertain numbers of vibration cycle the decrease ratio of thefrequency ratio of the breathing cracked beam is faster thanthat of the open cracked beam

7 Conclusions

In this article the stiffness model and the friction dampingmodel of a breathing cracked beam are derived The firstmode frequency is analyzed by using a SDOF model whichis simplified by the Galerkin method The dynamic stressis derived and the crack growth is studied The effects ofthe breathing behaviour on the dynamic response and crackgrowth are revealed so as to analyze the coupling mechanismof the crack growth anddynamic response approximatelyThemain conclusions are obtained

(1) During vibration analysis of a cracked beam the crackclosure effect between opening and closing states must beconsidered A breathing crack model is better than the opencrackmodel for dynamic response and crack growth analysisA sinusoidal breathing function is useful to simulate the

0

2

4

6

8

10

Num

bers

of v

ibra

tion

cycl

e N

times104

015 02 025 03 035 04 045 0501Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 14 Breathing crack depth with vibration cycles for differentfriction coefficient and frequency

times104

088

09

092

094

096

098

1

Freq

uenc

y ra

tio

br

cor

o

c

1 2 3 4 50Numbers of vibration cycle N

oc

brc = 0brc = 02

brc = 04

Figure 15 Frequency ratio with vibration cycles (120596120596119888 = 098)

transition state between the fully open state and the fullyclosed state It meets the object facts and the physical naturalof crack It may be closer to the real motion of the breathingcrack than that of the squarewave function Using a breathingcrack model is more realistic than using an opening crackmodel

Shock and Vibration 11

(2) Based on energy principle and Coulomb frictionmodel the friction model is set up and friction damping lossfactor is derived Friction damping not only plays importantrole in the maximum dynamic response but also playsimportant role in the crack growth Especially to resonanceresponse and crack growth friction damping effect is veryobvious

(3)TheGalerkinmethod is used to simplify the breathingcracked beam into a SDOF systemThe first mode frequencyequation is derived approximately The first mode frequencychanges with the crack severity The frequency decrease ratioof the breathing cracked beam is slower than that of theopen cracked beam If the closure behaviour of the breathingcrack is not considered the degree of crack growth may beunderestimated

(4) The modified Forman equation is employed to simu-late the crack growth A crack growth life analytical methodis proposed with the coupling effect of vibration and fatiguecrack growth included Breathing behaviour of the crack isimportant to crack growthThe dynamical properties play animportant role in crack growthWhen the exciting frequencyis near to the first mode frequency the crack growth ratio isvery fastThe crack growth life of the breathing cracked beamis smaller than that of the open cracked beam

Nomenclature

119886 1198860 Crack depth initial crack depth119886119894 Δ119886119895 The 119894th step crack depth the 119895th step crackincremental119860 Area of cross section119887 Width of beam cross section119861 Fatigue properties of materials119888lowast Coefficient of modal damping119862 Flexibility of the intact beam119862119879 Flexibility due to the presence of the crack119862119900 Total flexibility of the open cracked beam119889 Sliding distance between crack surfaces119864 Youngrsquos modulus119891 Closure function of breathing crack119865 Harmonic load119865lowast 1198650 The generalized force loading amplitude119865119899 119865119891 Normal pressing force friction force119865av Average pressing force119865119868 Crack shape functionℎ Height of the beam cross section119868 Area moment of inertial of the beam crosssection119896lowast The generalized stiffness119896br Stiffness of the breathing cracked beam119896119888 Stiffness of the close cracked beam119896119900 Stiffness of the open cracked beam119896119879 Stiffness of the torsional spring119870119868119889 Dynamic stress intensity factor119870119868119889max The maximum dynamic stress intensity factor119870119862 Fracture toughnessΔ119870th The threshold of stress intensity factorΔ119870 The range of stress intensity factor119897 Distance of crack from the fixed end

119871 Length of the beam119898lowast The generalized mass119872 Moment119899 119901 119902 Fatigue properties of materials119873 Numbers of vibration cycle119876 Shear force119903 Stress ratio119905 Time119879 Amplitude-time function119881max The strain energy119882119862 Energy dissipation119908 Lateral deflection in 119911 direction of beam119882(119909) Modal function119909 119910 119911 The coordinate119885st Static displacement120588 ] Density of material Poissonrsquos ratio120583 Friction coefficient120596 Excitation frequency120596119888 Frequency of the close cracked beam120596119900 Frequency of the open cracked beam120574119898 Material damping120574119891 Frictional damping120578 Amplify ratio120582 Frequency ratio120573 The breathing function120575 The Dirac function120590119899 120590119889 Nominal stress normal bulk dynamic stress120590119887 120590119904 Ultimate stress yield stress

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research work is supported by the National NaturalScience Foundation of China (51565039)

References

[1] A R Biswal T Roy R K Behera S K Pradhan and P KParida ldquoFinite element based vibration analysis of a nonpris-matic Timoshenko beam with transverse open crackrdquo ProcediaEngineering vol 144 pp 226ndash233 2016

[2] MKrawczuk andWMOstachowicz ldquoModelling and vibrationanalysis of a cantilever composite beam with a transverse opencrackrdquo Journal of Sound and Vibration vol 183 no 1 pp 69ndash891995

[3] C A Papadopoulos and A D Dimarogonas ldquoCoupled longi-tudinal and bending vibrations of a rotating shaft with an opencrackrdquo Journal of Sound and Vibration vol 117 no 1 pp 81ndash931987

[4] W M Ostachowicz and M Krawczuk ldquoAnalysis of the effect ofcracks on the natural frequencies of a cantilever beamrdquo Journalof Sound and Vibration vol 150 no 2 pp 191ndash201 1991

[5] T G Chondros andA D Dimarogonas ldquoVibration of a crackedcantilever beamrdquo Journal of Vibration andAcoustics vol 120 no3 pp 742ndash746 1998

12 Shock and Vibration

[6] T G Chondros A D Dimarogonas and J Yao ldquoA continuouscracked beam vibration theoryrdquo Journal of Sound and Vibrationvol 215 no 1 pp 17ndash34 1998

[7] M Heydari A Ebrahimi and M Behzad ldquoContinuous modelfor flexural vibration analysis of Timoshenko beams with avertical edge crackrdquo Archive of Applied Mechanics vol 85 no5 pp 601ndash615 2015

[8] R K Behera A Pandey and D R Parhi ldquoNumerical andexperimental verification of a method for prognosis of inclinededge crack in cantilever beam based on synthesis of modeshapesrdquo Procedia Technology vol 14 pp 67ndash74 2014

[9] S M Cheng A S J Swamidas X J Wu and W WallaceldquoVibrational response of a beamwith a breathing crackrdquo Journalof Sound and Vibration vol 225 no 1 pp 201ndash208 1999

[10] T G Chondros A D Dimarogonas and J Yao ldquoVibration of abeam with a breathing crackrdquo Journal of Sound and Vibrationvol 239 no 1 pp 57ndash67 2001

[11] E Douka and L J Hadjileontiadis ldquoTime-frequency analysis ofthe free vibration response of a beam with a breathing crackrdquoNDT amp E International vol 38 no 1 pp 3ndash10 2005

[12] M Kisa and J Brandon ldquoEffects of closure of cracks on thedynamics of a cracked cantilever beamrdquo Journal of Sound andVibration vol 238 no 1 pp 1ndash18 2000

[13] C Surace R Ruotolo and D Storer ldquoDetecting nonlinearbehaviour using the volterra series to assess damage in beam-like structuresrdquo Journal of Theoretical and Applied Mechanicsvol 49 no 3 pp 905ndash926 2011

[14] AChatterjee ldquoNonlinear dynamics anddamage assessment of acantilever beamwith breathing edge crackrdquo Journal of Vibrationand Acoustics vol 133 no 5 Article ID 051004 2011

[15] R Ruotolo C Surace P Crespo and D Storer ldquoHarmonicanalysis of the vibrations of a cantilevered beam with a closingcrackrdquo Computers amp Structures vol 61 no 6 pp 1057ndash10741996

[16] A Nandi and S Neogy ldquoModelling of a beam with a breathingedge crack and some observations for crack detectionrdquo Journalof Vibration and Control vol 8 no 5 pp 673ndash693 2002

[17] W Liu and M E Barkey ldquoNonlinear vibrational response ofa single edge cracked beamrdquo Journal of Mechanical Science ampTechnology vol 31 no 11 pp 5231ndash5243 2017

[18] J Zeng H MaW Zhang and BWen ldquoDynamic characteristicanalysis of cracked cantilever beams under different cracktypesrdquo Engineering Failure Analysis vol 74 pp 80ndash94 2017

[19] W Zhang H Ma J Zeng S Wu and B Wen ldquoVibrationresponses analysis of an elastic-support cantilever beam withcrack and offset boundaryrdquo Mechanical Systems and SignalProcessing vol 95 pp 205ndash218 2017

[20] HMa J Zeng Z Lang L Zhang Y Guo and BWen ldquoAnalysisof the dynamic characteristics of a slant-cracked cantileverbeamrdquo Mechanical Systems and Signal Processing vol 75 pp261ndash279 2016

[21] J Liu W D Zhu P G Charalambides Y M Shao Y F Xuand X M Fang ldquoA dynamic model of a cantilever beam with aclosed embedded horizontal crack including local flexibilitiesat crack tipsrdquo Journal of Sound and Vibration vol 382 pp 274ndash290 2016

[22] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[23] A Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagationrdquo Engineering FractureMechanics vol17 no 4 pp 381ndash386 1983

[24] A J Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagation in a beam under longitudinal vibra-tionsrdquo International Journal of Fracture vol 23 no 1 pp 15ndash221983

[25] A J Dentsoras andADDimarogonas ldquoFatigue crack propaga-tion in resonating structuresrdquo Engineering Fracture Mechanicsvol 34 no 3 pp 721ndash728 1989

[26] M Lalanne P Berthier and J D HagopianMechanical Vibra-tions for Engineers John Wiley amp Sons New York NY USA1983

[27] A D Dimarogonas S A Paipetis and T G ChondrosAnalyticalMethods inRotorDynamics SpringerDordrechtTheNetherlands 2013

[28] H Tada P C Paris andG R IrwinTheStress Analysis of CracksHandbook The American Society of Mechanical Engineers2000

[29] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984

[30] R Mulyukov S Mikhailov R Zaripova and D SalimonenkoldquoDamping properties of 18Cr-10Ni stainless steel with submi-crocrystalline structurerdquoMaterials Research Bulletin vol 31 no6 pp 639ndash645 1996

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Shock and Vibration 9

0

20

40

60

80

100

120

140

160

180

Max

imum

dyn

amic

stre

ssd

max

(MPa

)

01 02 03 04 050Crack depth ratio ah

c = 098 = 01

c = 098 = 03

c = 099 = 01

c = 099 = 03

c = 1 = 01

c = 1 = 03

c = 098 = 05

c = 099 = 05

c = 1 = 05

Figure 10Themaximumdynamic stress with crack severity changeof breathing cracked beam

closure effect changes the stiffness of the breathing crackedbeam so that the maximum stress response also changes

Figure 10 presents the effect of the friction damping on themaximum dynamic stress of a breathing cracked beam as thebeam working under resonance condition It can be shownthat the maximum stress of the beam is greatly reduced bythe friction damping This implied that the friction dampingis an important factor to reduce the crack growth ratio

Figure 11 discovers the relation of the maximum dynamicSIFs of the open cracked beam and that of the breathingcracked beam without friction damping It is shown that themaximum dynamic SIF of the breathing cracked beam islarger than that of the open cracked beam without frictiondamping considered under the same exciting frequency Buteach of them happens at different crack depths

Figure 12 studies the effects of friction damping loss factoron themaximum dynamic SIF of the breathing cracked beamat resonance It can be seen that dynamic SIFs are verysensitive to the friction damping Comparing Figure 10 withFigure 12 we can find that the change law of the dynamicstress is different from that of the dynamic SIF The reasonis because the crack depth also plays an important role in SIFduring crack growth

Figures 13 and 14 examine the relation of the crack depthwith the vibration cycles It can be seen that the crackgrowth ratio of the open cracked beam is slower than thatof the breathing cracked beam if the friction damping is notconsidered The reason is because the decrease ratio of the

Breathing crack c = 098

Open crack c = 098

Breathing crack c = 099

Open crack c = 099

Breathing crack c = 1

Open crack c = 1

0

200

400

600

800

1000

1200

1400

1600

1800

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

Figure 11 The maximum dynamic SIF with crack severity changefor different frequency (120583 = 0)

0

500

1000

1500

Max

imum

dyn

amic

SIF

KId

max

(Mpa

middotmm

05)

01 02 03 04 050Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 12 The maximum dynamic SIF with crack severity changeof breathing cracked beam

mode frequency of the breathing cracked beam is slower thanthat of the open cracked beam This means the breathingcracked beam has a wider resonance region Considering

10 Shock and Vibration

01 015 02 025 03 035 04 045 05

Breathing crack c = 098

Breathing crack c = 099

Breathing crack c = 1

Open crack c = 098

Open crack c = 099

Open crack c = 1

Crack depth ratio ah

times104

0

1

2

3

4

5

Num

bers

of v

ibra

tion

cycl

e N

Figure 13 Crack depth with vibration cycles for different frequency(120583 = 0)

the friction damping the crack growth ratio is reduced bythe friction damping for a breathing cracked beam obviouslywhen the beam comes into resonance region as shown inFigure 14

Figure 15 shows the relation of the decrease ratio offrequency ratio and the numbers of vibration cycles It can beseen that the decrease ratio of frequency of the open crackedbeam is faster than that of the breathing cracked beam at thefirst phase of the crack growth when the exciting frequency is098 times of themode frequency of the intact beam But aftercertain numbers of vibration cycle the decrease ratio of thefrequency ratio of the breathing cracked beam is faster thanthat of the open cracked beam

7 Conclusions

In this article the stiffness model and the friction dampingmodel of a breathing cracked beam are derived The firstmode frequency is analyzed by using a SDOF model whichis simplified by the Galerkin method The dynamic stressis derived and the crack growth is studied The effects ofthe breathing behaviour on the dynamic response and crackgrowth are revealed so as to analyze the coupling mechanismof the crack growth anddynamic response approximatelyThemain conclusions are obtained

(1) During vibration analysis of a cracked beam the crackclosure effect between opening and closing states must beconsidered A breathing crack model is better than the opencrackmodel for dynamic response and crack growth analysisA sinusoidal breathing function is useful to simulate the

0

2

4

6

8

10

Num

bers

of v

ibra

tion

cycl

e N

times104

015 02 025 03 035 04 045 0501Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 14 Breathing crack depth with vibration cycles for differentfriction coefficient and frequency

times104

088

09

092

094

096

098

1

Freq

uenc

y ra

tio

br

cor

o

c

1 2 3 4 50Numbers of vibration cycle N

oc

brc = 0brc = 02

brc = 04

Figure 15 Frequency ratio with vibration cycles (120596120596119888 = 098)

transition state between the fully open state and the fullyclosed state It meets the object facts and the physical naturalof crack It may be closer to the real motion of the breathingcrack than that of the squarewave function Using a breathingcrack model is more realistic than using an opening crackmodel

Shock and Vibration 11

(2) Based on energy principle and Coulomb frictionmodel the friction model is set up and friction damping lossfactor is derived Friction damping not only plays importantrole in the maximum dynamic response but also playsimportant role in the crack growth Especially to resonanceresponse and crack growth friction damping effect is veryobvious

(3)TheGalerkinmethod is used to simplify the breathingcracked beam into a SDOF systemThe first mode frequencyequation is derived approximately The first mode frequencychanges with the crack severity The frequency decrease ratioof the breathing cracked beam is slower than that of theopen cracked beam If the closure behaviour of the breathingcrack is not considered the degree of crack growth may beunderestimated

(4) The modified Forman equation is employed to simu-late the crack growth A crack growth life analytical methodis proposed with the coupling effect of vibration and fatiguecrack growth included Breathing behaviour of the crack isimportant to crack growthThe dynamical properties play animportant role in crack growthWhen the exciting frequencyis near to the first mode frequency the crack growth ratio isvery fastThe crack growth life of the breathing cracked beamis smaller than that of the open cracked beam

Nomenclature

119886 1198860 Crack depth initial crack depth119886119894 Δ119886119895 The 119894th step crack depth the 119895th step crackincremental119860 Area of cross section119887 Width of beam cross section119861 Fatigue properties of materials119888lowast Coefficient of modal damping119862 Flexibility of the intact beam119862119879 Flexibility due to the presence of the crack119862119900 Total flexibility of the open cracked beam119889 Sliding distance between crack surfaces119864 Youngrsquos modulus119891 Closure function of breathing crack119865 Harmonic load119865lowast 1198650 The generalized force loading amplitude119865119899 119865119891 Normal pressing force friction force119865av Average pressing force119865119868 Crack shape functionℎ Height of the beam cross section119868 Area moment of inertial of the beam crosssection119896lowast The generalized stiffness119896br Stiffness of the breathing cracked beam119896119888 Stiffness of the close cracked beam119896119900 Stiffness of the open cracked beam119896119879 Stiffness of the torsional spring119870119868119889 Dynamic stress intensity factor119870119868119889max The maximum dynamic stress intensity factor119870119862 Fracture toughnessΔ119870th The threshold of stress intensity factorΔ119870 The range of stress intensity factor119897 Distance of crack from the fixed end

119871 Length of the beam119898lowast The generalized mass119872 Moment119899 119901 119902 Fatigue properties of materials119873 Numbers of vibration cycle119876 Shear force119903 Stress ratio119905 Time119879 Amplitude-time function119881max The strain energy119882119862 Energy dissipation119908 Lateral deflection in 119911 direction of beam119882(119909) Modal function119909 119910 119911 The coordinate119885st Static displacement120588 ] Density of material Poissonrsquos ratio120583 Friction coefficient120596 Excitation frequency120596119888 Frequency of the close cracked beam120596119900 Frequency of the open cracked beam120574119898 Material damping120574119891 Frictional damping120578 Amplify ratio120582 Frequency ratio120573 The breathing function120575 The Dirac function120590119899 120590119889 Nominal stress normal bulk dynamic stress120590119887 120590119904 Ultimate stress yield stress

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research work is supported by the National NaturalScience Foundation of China (51565039)

References

[1] A R Biswal T Roy R K Behera S K Pradhan and P KParida ldquoFinite element based vibration analysis of a nonpris-matic Timoshenko beam with transverse open crackrdquo ProcediaEngineering vol 144 pp 226ndash233 2016

[2] MKrawczuk andWMOstachowicz ldquoModelling and vibrationanalysis of a cantilever composite beam with a transverse opencrackrdquo Journal of Sound and Vibration vol 183 no 1 pp 69ndash891995

[3] C A Papadopoulos and A D Dimarogonas ldquoCoupled longi-tudinal and bending vibrations of a rotating shaft with an opencrackrdquo Journal of Sound and Vibration vol 117 no 1 pp 81ndash931987

[4] W M Ostachowicz and M Krawczuk ldquoAnalysis of the effect ofcracks on the natural frequencies of a cantilever beamrdquo Journalof Sound and Vibration vol 150 no 2 pp 191ndash201 1991

[5] T G Chondros andA D Dimarogonas ldquoVibration of a crackedcantilever beamrdquo Journal of Vibration andAcoustics vol 120 no3 pp 742ndash746 1998

12 Shock and Vibration

[6] T G Chondros A D Dimarogonas and J Yao ldquoA continuouscracked beam vibration theoryrdquo Journal of Sound and Vibrationvol 215 no 1 pp 17ndash34 1998

[7] M Heydari A Ebrahimi and M Behzad ldquoContinuous modelfor flexural vibration analysis of Timoshenko beams with avertical edge crackrdquo Archive of Applied Mechanics vol 85 no5 pp 601ndash615 2015

[8] R K Behera A Pandey and D R Parhi ldquoNumerical andexperimental verification of a method for prognosis of inclinededge crack in cantilever beam based on synthesis of modeshapesrdquo Procedia Technology vol 14 pp 67ndash74 2014

[9] S M Cheng A S J Swamidas X J Wu and W WallaceldquoVibrational response of a beamwith a breathing crackrdquo Journalof Sound and Vibration vol 225 no 1 pp 201ndash208 1999

[10] T G Chondros A D Dimarogonas and J Yao ldquoVibration of abeam with a breathing crackrdquo Journal of Sound and Vibrationvol 239 no 1 pp 57ndash67 2001

[11] E Douka and L J Hadjileontiadis ldquoTime-frequency analysis ofthe free vibration response of a beam with a breathing crackrdquoNDT amp E International vol 38 no 1 pp 3ndash10 2005

[12] M Kisa and J Brandon ldquoEffects of closure of cracks on thedynamics of a cracked cantilever beamrdquo Journal of Sound andVibration vol 238 no 1 pp 1ndash18 2000

[13] C Surace R Ruotolo and D Storer ldquoDetecting nonlinearbehaviour using the volterra series to assess damage in beam-like structuresrdquo Journal of Theoretical and Applied Mechanicsvol 49 no 3 pp 905ndash926 2011

[14] AChatterjee ldquoNonlinear dynamics anddamage assessment of acantilever beamwith breathing edge crackrdquo Journal of Vibrationand Acoustics vol 133 no 5 Article ID 051004 2011

[15] R Ruotolo C Surace P Crespo and D Storer ldquoHarmonicanalysis of the vibrations of a cantilevered beam with a closingcrackrdquo Computers amp Structures vol 61 no 6 pp 1057ndash10741996

[16] A Nandi and S Neogy ldquoModelling of a beam with a breathingedge crack and some observations for crack detectionrdquo Journalof Vibration and Control vol 8 no 5 pp 673ndash693 2002

[17] W Liu and M E Barkey ldquoNonlinear vibrational response ofa single edge cracked beamrdquo Journal of Mechanical Science ampTechnology vol 31 no 11 pp 5231ndash5243 2017

[18] J Zeng H MaW Zhang and BWen ldquoDynamic characteristicanalysis of cracked cantilever beams under different cracktypesrdquo Engineering Failure Analysis vol 74 pp 80ndash94 2017

[19] W Zhang H Ma J Zeng S Wu and B Wen ldquoVibrationresponses analysis of an elastic-support cantilever beam withcrack and offset boundaryrdquo Mechanical Systems and SignalProcessing vol 95 pp 205ndash218 2017

[20] HMa J Zeng Z Lang L Zhang Y Guo and BWen ldquoAnalysisof the dynamic characteristics of a slant-cracked cantileverbeamrdquo Mechanical Systems and Signal Processing vol 75 pp261ndash279 2016

[21] J Liu W D Zhu P G Charalambides Y M Shao Y F Xuand X M Fang ldquoA dynamic model of a cantilever beam with aclosed embedded horizontal crack including local flexibilitiesat crack tipsrdquo Journal of Sound and Vibration vol 382 pp 274ndash290 2016

[22] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[23] A Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagationrdquo Engineering FractureMechanics vol17 no 4 pp 381ndash386 1983

[24] A J Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagation in a beam under longitudinal vibra-tionsrdquo International Journal of Fracture vol 23 no 1 pp 15ndash221983

[25] A J Dentsoras andADDimarogonas ldquoFatigue crack propaga-tion in resonating structuresrdquo Engineering Fracture Mechanicsvol 34 no 3 pp 721ndash728 1989

[26] M Lalanne P Berthier and J D HagopianMechanical Vibra-tions for Engineers John Wiley amp Sons New York NY USA1983

[27] A D Dimarogonas S A Paipetis and T G ChondrosAnalyticalMethods inRotorDynamics SpringerDordrechtTheNetherlands 2013

[28] H Tada P C Paris andG R IrwinTheStress Analysis of CracksHandbook The American Society of Mechanical Engineers2000

[29] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984

[30] R Mulyukov S Mikhailov R Zaripova and D SalimonenkoldquoDamping properties of 18Cr-10Ni stainless steel with submi-crocrystalline structurerdquoMaterials Research Bulletin vol 31 no6 pp 639ndash645 1996

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

10 Shock and Vibration

01 015 02 025 03 035 04 045 05

Breathing crack c = 098

Breathing crack c = 099

Breathing crack c = 1

Open crack c = 098

Open crack c = 099

Open crack c = 1

Crack depth ratio ah

times104

0

1

2

3

4

5

Num

bers

of v

ibra

tion

cycl

e N

Figure 13 Crack depth with vibration cycles for different frequency(120583 = 0)

the friction damping the crack growth ratio is reduced bythe friction damping for a breathing cracked beam obviouslywhen the beam comes into resonance region as shown inFigure 14

Figure 15 shows the relation of the decrease ratio offrequency ratio and the numbers of vibration cycles It can beseen that the decrease ratio of frequency of the open crackedbeam is faster than that of the breathing cracked beam at thefirst phase of the crack growth when the exciting frequency is098 times of themode frequency of the intact beam But aftercertain numbers of vibration cycle the decrease ratio of thefrequency ratio of the breathing cracked beam is faster thanthat of the open cracked beam

7 Conclusions

In this article the stiffness model and the friction dampingmodel of a breathing cracked beam are derived The firstmode frequency is analyzed by using a SDOF model whichis simplified by the Galerkin method The dynamic stressis derived and the crack growth is studied The effects ofthe breathing behaviour on the dynamic response and crackgrowth are revealed so as to analyze the coupling mechanismof the crack growth anddynamic response approximatelyThemain conclusions are obtained

(1) During vibration analysis of a cracked beam the crackclosure effect between opening and closing states must beconsidered A breathing crack model is better than the opencrackmodel for dynamic response and crack growth analysisA sinusoidal breathing function is useful to simulate the

0

2

4

6

8

10

Num

bers

of v

ibra

tion

cycl

e N

times104

015 02 025 03 035 04 045 0501Crack depth ratio ah

c = 099 = 01

c = 098 = 01

c = 098 = 03

c = 099 = 03

c = 098 = 05

c = 099 = 05

c = 10 = 01

c = 10 = 03

c = 10 = 05

Figure 14 Breathing crack depth with vibration cycles for differentfriction coefficient and frequency

times104

088

09

092

094

096

098

1

Freq

uenc

y ra

tio

br

cor

o

c

1 2 3 4 50Numbers of vibration cycle N

oc

brc = 0brc = 02

brc = 04

Figure 15 Frequency ratio with vibration cycles (120596120596119888 = 098)

transition state between the fully open state and the fullyclosed state It meets the object facts and the physical naturalof crack It may be closer to the real motion of the breathingcrack than that of the squarewave function Using a breathingcrack model is more realistic than using an opening crackmodel

Shock and Vibration 11

(2) Based on energy principle and Coulomb frictionmodel the friction model is set up and friction damping lossfactor is derived Friction damping not only plays importantrole in the maximum dynamic response but also playsimportant role in the crack growth Especially to resonanceresponse and crack growth friction damping effect is veryobvious

(3)TheGalerkinmethod is used to simplify the breathingcracked beam into a SDOF systemThe first mode frequencyequation is derived approximately The first mode frequencychanges with the crack severity The frequency decrease ratioof the breathing cracked beam is slower than that of theopen cracked beam If the closure behaviour of the breathingcrack is not considered the degree of crack growth may beunderestimated

(4) The modified Forman equation is employed to simu-late the crack growth A crack growth life analytical methodis proposed with the coupling effect of vibration and fatiguecrack growth included Breathing behaviour of the crack isimportant to crack growthThe dynamical properties play animportant role in crack growthWhen the exciting frequencyis near to the first mode frequency the crack growth ratio isvery fastThe crack growth life of the breathing cracked beamis smaller than that of the open cracked beam

Nomenclature

119886 1198860 Crack depth initial crack depth119886119894 Δ119886119895 The 119894th step crack depth the 119895th step crackincremental119860 Area of cross section119887 Width of beam cross section119861 Fatigue properties of materials119888lowast Coefficient of modal damping119862 Flexibility of the intact beam119862119879 Flexibility due to the presence of the crack119862119900 Total flexibility of the open cracked beam119889 Sliding distance between crack surfaces119864 Youngrsquos modulus119891 Closure function of breathing crack119865 Harmonic load119865lowast 1198650 The generalized force loading amplitude119865119899 119865119891 Normal pressing force friction force119865av Average pressing force119865119868 Crack shape functionℎ Height of the beam cross section119868 Area moment of inertial of the beam crosssection119896lowast The generalized stiffness119896br Stiffness of the breathing cracked beam119896119888 Stiffness of the close cracked beam119896119900 Stiffness of the open cracked beam119896119879 Stiffness of the torsional spring119870119868119889 Dynamic stress intensity factor119870119868119889max The maximum dynamic stress intensity factor119870119862 Fracture toughnessΔ119870th The threshold of stress intensity factorΔ119870 The range of stress intensity factor119897 Distance of crack from the fixed end

119871 Length of the beam119898lowast The generalized mass119872 Moment119899 119901 119902 Fatigue properties of materials119873 Numbers of vibration cycle119876 Shear force119903 Stress ratio119905 Time119879 Amplitude-time function119881max The strain energy119882119862 Energy dissipation119908 Lateral deflection in 119911 direction of beam119882(119909) Modal function119909 119910 119911 The coordinate119885st Static displacement120588 ] Density of material Poissonrsquos ratio120583 Friction coefficient120596 Excitation frequency120596119888 Frequency of the close cracked beam120596119900 Frequency of the open cracked beam120574119898 Material damping120574119891 Frictional damping120578 Amplify ratio120582 Frequency ratio120573 The breathing function120575 The Dirac function120590119899 120590119889 Nominal stress normal bulk dynamic stress120590119887 120590119904 Ultimate stress yield stress

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research work is supported by the National NaturalScience Foundation of China (51565039)

References

[1] A R Biswal T Roy R K Behera S K Pradhan and P KParida ldquoFinite element based vibration analysis of a nonpris-matic Timoshenko beam with transverse open crackrdquo ProcediaEngineering vol 144 pp 226ndash233 2016

[2] MKrawczuk andWMOstachowicz ldquoModelling and vibrationanalysis of a cantilever composite beam with a transverse opencrackrdquo Journal of Sound and Vibration vol 183 no 1 pp 69ndash891995

[3] C A Papadopoulos and A D Dimarogonas ldquoCoupled longi-tudinal and bending vibrations of a rotating shaft with an opencrackrdquo Journal of Sound and Vibration vol 117 no 1 pp 81ndash931987

[4] W M Ostachowicz and M Krawczuk ldquoAnalysis of the effect ofcracks on the natural frequencies of a cantilever beamrdquo Journalof Sound and Vibration vol 150 no 2 pp 191ndash201 1991

[5] T G Chondros andA D Dimarogonas ldquoVibration of a crackedcantilever beamrdquo Journal of Vibration andAcoustics vol 120 no3 pp 742ndash746 1998

12 Shock and Vibration

[6] T G Chondros A D Dimarogonas and J Yao ldquoA continuouscracked beam vibration theoryrdquo Journal of Sound and Vibrationvol 215 no 1 pp 17ndash34 1998

[7] M Heydari A Ebrahimi and M Behzad ldquoContinuous modelfor flexural vibration analysis of Timoshenko beams with avertical edge crackrdquo Archive of Applied Mechanics vol 85 no5 pp 601ndash615 2015

[8] R K Behera A Pandey and D R Parhi ldquoNumerical andexperimental verification of a method for prognosis of inclinededge crack in cantilever beam based on synthesis of modeshapesrdquo Procedia Technology vol 14 pp 67ndash74 2014

[9] S M Cheng A S J Swamidas X J Wu and W WallaceldquoVibrational response of a beamwith a breathing crackrdquo Journalof Sound and Vibration vol 225 no 1 pp 201ndash208 1999

[10] T G Chondros A D Dimarogonas and J Yao ldquoVibration of abeam with a breathing crackrdquo Journal of Sound and Vibrationvol 239 no 1 pp 57ndash67 2001

[11] E Douka and L J Hadjileontiadis ldquoTime-frequency analysis ofthe free vibration response of a beam with a breathing crackrdquoNDT amp E International vol 38 no 1 pp 3ndash10 2005

[12] M Kisa and J Brandon ldquoEffects of closure of cracks on thedynamics of a cracked cantilever beamrdquo Journal of Sound andVibration vol 238 no 1 pp 1ndash18 2000

[13] C Surace R Ruotolo and D Storer ldquoDetecting nonlinearbehaviour using the volterra series to assess damage in beam-like structuresrdquo Journal of Theoretical and Applied Mechanicsvol 49 no 3 pp 905ndash926 2011

[14] AChatterjee ldquoNonlinear dynamics anddamage assessment of acantilever beamwith breathing edge crackrdquo Journal of Vibrationand Acoustics vol 133 no 5 Article ID 051004 2011

[15] R Ruotolo C Surace P Crespo and D Storer ldquoHarmonicanalysis of the vibrations of a cantilevered beam with a closingcrackrdquo Computers amp Structures vol 61 no 6 pp 1057ndash10741996

[16] A Nandi and S Neogy ldquoModelling of a beam with a breathingedge crack and some observations for crack detectionrdquo Journalof Vibration and Control vol 8 no 5 pp 673ndash693 2002

[17] W Liu and M E Barkey ldquoNonlinear vibrational response ofa single edge cracked beamrdquo Journal of Mechanical Science ampTechnology vol 31 no 11 pp 5231ndash5243 2017

[18] J Zeng H MaW Zhang and BWen ldquoDynamic characteristicanalysis of cracked cantilever beams under different cracktypesrdquo Engineering Failure Analysis vol 74 pp 80ndash94 2017

[19] W Zhang H Ma J Zeng S Wu and B Wen ldquoVibrationresponses analysis of an elastic-support cantilever beam withcrack and offset boundaryrdquo Mechanical Systems and SignalProcessing vol 95 pp 205ndash218 2017

[20] HMa J Zeng Z Lang L Zhang Y Guo and BWen ldquoAnalysisof the dynamic characteristics of a slant-cracked cantileverbeamrdquo Mechanical Systems and Signal Processing vol 75 pp261ndash279 2016

[21] J Liu W D Zhu P G Charalambides Y M Shao Y F Xuand X M Fang ldquoA dynamic model of a cantilever beam with aclosed embedded horizontal crack including local flexibilitiesat crack tipsrdquo Journal of Sound and Vibration vol 382 pp 274ndash290 2016

[22] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[23] A Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagationrdquo Engineering FractureMechanics vol17 no 4 pp 381ndash386 1983

[24] A J Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagation in a beam under longitudinal vibra-tionsrdquo International Journal of Fracture vol 23 no 1 pp 15ndash221983

[25] A J Dentsoras andADDimarogonas ldquoFatigue crack propaga-tion in resonating structuresrdquo Engineering Fracture Mechanicsvol 34 no 3 pp 721ndash728 1989

[26] M Lalanne P Berthier and J D HagopianMechanical Vibra-tions for Engineers John Wiley amp Sons New York NY USA1983

[27] A D Dimarogonas S A Paipetis and T G ChondrosAnalyticalMethods inRotorDynamics SpringerDordrechtTheNetherlands 2013

[28] H Tada P C Paris andG R IrwinTheStress Analysis of CracksHandbook The American Society of Mechanical Engineers2000

[29] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984

[30] R Mulyukov S Mikhailov R Zaripova and D SalimonenkoldquoDamping properties of 18Cr-10Ni stainless steel with submi-crocrystalline structurerdquoMaterials Research Bulletin vol 31 no6 pp 639ndash645 1996

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Shock and Vibration 11

(2) Based on energy principle and Coulomb frictionmodel the friction model is set up and friction damping lossfactor is derived Friction damping not only plays importantrole in the maximum dynamic response but also playsimportant role in the crack growth Especially to resonanceresponse and crack growth friction damping effect is veryobvious

(3)TheGalerkinmethod is used to simplify the breathingcracked beam into a SDOF systemThe first mode frequencyequation is derived approximately The first mode frequencychanges with the crack severity The frequency decrease ratioof the breathing cracked beam is slower than that of theopen cracked beam If the closure behaviour of the breathingcrack is not considered the degree of crack growth may beunderestimated

(4) The modified Forman equation is employed to simu-late the crack growth A crack growth life analytical methodis proposed with the coupling effect of vibration and fatiguecrack growth included Breathing behaviour of the crack isimportant to crack growthThe dynamical properties play animportant role in crack growthWhen the exciting frequencyis near to the first mode frequency the crack growth ratio isvery fastThe crack growth life of the breathing cracked beamis smaller than that of the open cracked beam

Nomenclature

119886 1198860 Crack depth initial crack depth119886119894 Δ119886119895 The 119894th step crack depth the 119895th step crackincremental119860 Area of cross section119887 Width of beam cross section119861 Fatigue properties of materials119888lowast Coefficient of modal damping119862 Flexibility of the intact beam119862119879 Flexibility due to the presence of the crack119862119900 Total flexibility of the open cracked beam119889 Sliding distance between crack surfaces119864 Youngrsquos modulus119891 Closure function of breathing crack119865 Harmonic load119865lowast 1198650 The generalized force loading amplitude119865119899 119865119891 Normal pressing force friction force119865av Average pressing force119865119868 Crack shape functionℎ Height of the beam cross section119868 Area moment of inertial of the beam crosssection119896lowast The generalized stiffness119896br Stiffness of the breathing cracked beam119896119888 Stiffness of the close cracked beam119896119900 Stiffness of the open cracked beam119896119879 Stiffness of the torsional spring119870119868119889 Dynamic stress intensity factor119870119868119889max The maximum dynamic stress intensity factor119870119862 Fracture toughnessΔ119870th The threshold of stress intensity factorΔ119870 The range of stress intensity factor119897 Distance of crack from the fixed end

119871 Length of the beam119898lowast The generalized mass119872 Moment119899 119901 119902 Fatigue properties of materials119873 Numbers of vibration cycle119876 Shear force119903 Stress ratio119905 Time119879 Amplitude-time function119881max The strain energy119882119862 Energy dissipation119908 Lateral deflection in 119911 direction of beam119882(119909) Modal function119909 119910 119911 The coordinate119885st Static displacement120588 ] Density of material Poissonrsquos ratio120583 Friction coefficient120596 Excitation frequency120596119888 Frequency of the close cracked beam120596119900 Frequency of the open cracked beam120574119898 Material damping120574119891 Frictional damping120578 Amplify ratio120582 Frequency ratio120573 The breathing function120575 The Dirac function120590119899 120590119889 Nominal stress normal bulk dynamic stress120590119887 120590119904 Ultimate stress yield stress

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research work is supported by the National NaturalScience Foundation of China (51565039)

References

[1] A R Biswal T Roy R K Behera S K Pradhan and P KParida ldquoFinite element based vibration analysis of a nonpris-matic Timoshenko beam with transverse open crackrdquo ProcediaEngineering vol 144 pp 226ndash233 2016

[2] MKrawczuk andWMOstachowicz ldquoModelling and vibrationanalysis of a cantilever composite beam with a transverse opencrackrdquo Journal of Sound and Vibration vol 183 no 1 pp 69ndash891995

[3] C A Papadopoulos and A D Dimarogonas ldquoCoupled longi-tudinal and bending vibrations of a rotating shaft with an opencrackrdquo Journal of Sound and Vibration vol 117 no 1 pp 81ndash931987

[4] W M Ostachowicz and M Krawczuk ldquoAnalysis of the effect ofcracks on the natural frequencies of a cantilever beamrdquo Journalof Sound and Vibration vol 150 no 2 pp 191ndash201 1991

[5] T G Chondros andA D Dimarogonas ldquoVibration of a crackedcantilever beamrdquo Journal of Vibration andAcoustics vol 120 no3 pp 742ndash746 1998

12 Shock and Vibration

[6] T G Chondros A D Dimarogonas and J Yao ldquoA continuouscracked beam vibration theoryrdquo Journal of Sound and Vibrationvol 215 no 1 pp 17ndash34 1998

[7] M Heydari A Ebrahimi and M Behzad ldquoContinuous modelfor flexural vibration analysis of Timoshenko beams with avertical edge crackrdquo Archive of Applied Mechanics vol 85 no5 pp 601ndash615 2015

[8] R K Behera A Pandey and D R Parhi ldquoNumerical andexperimental verification of a method for prognosis of inclinededge crack in cantilever beam based on synthesis of modeshapesrdquo Procedia Technology vol 14 pp 67ndash74 2014

[9] S M Cheng A S J Swamidas X J Wu and W WallaceldquoVibrational response of a beamwith a breathing crackrdquo Journalof Sound and Vibration vol 225 no 1 pp 201ndash208 1999

[10] T G Chondros A D Dimarogonas and J Yao ldquoVibration of abeam with a breathing crackrdquo Journal of Sound and Vibrationvol 239 no 1 pp 57ndash67 2001

[11] E Douka and L J Hadjileontiadis ldquoTime-frequency analysis ofthe free vibration response of a beam with a breathing crackrdquoNDT amp E International vol 38 no 1 pp 3ndash10 2005

[12] M Kisa and J Brandon ldquoEffects of closure of cracks on thedynamics of a cracked cantilever beamrdquo Journal of Sound andVibration vol 238 no 1 pp 1ndash18 2000

[13] C Surace R Ruotolo and D Storer ldquoDetecting nonlinearbehaviour using the volterra series to assess damage in beam-like structuresrdquo Journal of Theoretical and Applied Mechanicsvol 49 no 3 pp 905ndash926 2011

[14] AChatterjee ldquoNonlinear dynamics anddamage assessment of acantilever beamwith breathing edge crackrdquo Journal of Vibrationand Acoustics vol 133 no 5 Article ID 051004 2011

[15] R Ruotolo C Surace P Crespo and D Storer ldquoHarmonicanalysis of the vibrations of a cantilevered beam with a closingcrackrdquo Computers amp Structures vol 61 no 6 pp 1057ndash10741996

[16] A Nandi and S Neogy ldquoModelling of a beam with a breathingedge crack and some observations for crack detectionrdquo Journalof Vibration and Control vol 8 no 5 pp 673ndash693 2002

[17] W Liu and M E Barkey ldquoNonlinear vibrational response ofa single edge cracked beamrdquo Journal of Mechanical Science ampTechnology vol 31 no 11 pp 5231ndash5243 2017

[18] J Zeng H MaW Zhang and BWen ldquoDynamic characteristicanalysis of cracked cantilever beams under different cracktypesrdquo Engineering Failure Analysis vol 74 pp 80ndash94 2017

[19] W Zhang H Ma J Zeng S Wu and B Wen ldquoVibrationresponses analysis of an elastic-support cantilever beam withcrack and offset boundaryrdquo Mechanical Systems and SignalProcessing vol 95 pp 205ndash218 2017

[20] HMa J Zeng Z Lang L Zhang Y Guo and BWen ldquoAnalysisof the dynamic characteristics of a slant-cracked cantileverbeamrdquo Mechanical Systems and Signal Processing vol 75 pp261ndash279 2016

[21] J Liu W D Zhu P G Charalambides Y M Shao Y F Xuand X M Fang ldquoA dynamic model of a cantilever beam with aclosed embedded horizontal crack including local flexibilitiesat crack tipsrdquo Journal of Sound and Vibration vol 382 pp 274ndash290 2016

[22] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[23] A Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagationrdquo Engineering FractureMechanics vol17 no 4 pp 381ndash386 1983

[24] A J Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagation in a beam under longitudinal vibra-tionsrdquo International Journal of Fracture vol 23 no 1 pp 15ndash221983

[25] A J Dentsoras andADDimarogonas ldquoFatigue crack propaga-tion in resonating structuresrdquo Engineering Fracture Mechanicsvol 34 no 3 pp 721ndash728 1989

[26] M Lalanne P Berthier and J D HagopianMechanical Vibra-tions for Engineers John Wiley amp Sons New York NY USA1983

[27] A D Dimarogonas S A Paipetis and T G ChondrosAnalyticalMethods inRotorDynamics SpringerDordrechtTheNetherlands 2013

[28] H Tada P C Paris andG R IrwinTheStress Analysis of CracksHandbook The American Society of Mechanical Engineers2000

[29] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984

[30] R Mulyukov S Mikhailov R Zaripova and D SalimonenkoldquoDamping properties of 18Cr-10Ni stainless steel with submi-crocrystalline structurerdquoMaterials Research Bulletin vol 31 no6 pp 639ndash645 1996

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

12 Shock and Vibration

[6] T G Chondros A D Dimarogonas and J Yao ldquoA continuouscracked beam vibration theoryrdquo Journal of Sound and Vibrationvol 215 no 1 pp 17ndash34 1998

[7] M Heydari A Ebrahimi and M Behzad ldquoContinuous modelfor flexural vibration analysis of Timoshenko beams with avertical edge crackrdquo Archive of Applied Mechanics vol 85 no5 pp 601ndash615 2015

[8] R K Behera A Pandey and D R Parhi ldquoNumerical andexperimental verification of a method for prognosis of inclinededge crack in cantilever beam based on synthesis of modeshapesrdquo Procedia Technology vol 14 pp 67ndash74 2014

[9] S M Cheng A S J Swamidas X J Wu and W WallaceldquoVibrational response of a beamwith a breathing crackrdquo Journalof Sound and Vibration vol 225 no 1 pp 201ndash208 1999

[10] T G Chondros A D Dimarogonas and J Yao ldquoVibration of abeam with a breathing crackrdquo Journal of Sound and Vibrationvol 239 no 1 pp 57ndash67 2001

[11] E Douka and L J Hadjileontiadis ldquoTime-frequency analysis ofthe free vibration response of a beam with a breathing crackrdquoNDT amp E International vol 38 no 1 pp 3ndash10 2005

[12] M Kisa and J Brandon ldquoEffects of closure of cracks on thedynamics of a cracked cantilever beamrdquo Journal of Sound andVibration vol 238 no 1 pp 1ndash18 2000

[13] C Surace R Ruotolo and D Storer ldquoDetecting nonlinearbehaviour using the volterra series to assess damage in beam-like structuresrdquo Journal of Theoretical and Applied Mechanicsvol 49 no 3 pp 905ndash926 2011

[14] AChatterjee ldquoNonlinear dynamics anddamage assessment of acantilever beamwith breathing edge crackrdquo Journal of Vibrationand Acoustics vol 133 no 5 Article ID 051004 2011

[15] R Ruotolo C Surace P Crespo and D Storer ldquoHarmonicanalysis of the vibrations of a cantilevered beam with a closingcrackrdquo Computers amp Structures vol 61 no 6 pp 1057ndash10741996

[16] A Nandi and S Neogy ldquoModelling of a beam with a breathingedge crack and some observations for crack detectionrdquo Journalof Vibration and Control vol 8 no 5 pp 673ndash693 2002

[17] W Liu and M E Barkey ldquoNonlinear vibrational response ofa single edge cracked beamrdquo Journal of Mechanical Science ampTechnology vol 31 no 11 pp 5231ndash5243 2017

[18] J Zeng H MaW Zhang and BWen ldquoDynamic characteristicanalysis of cracked cantilever beams under different cracktypesrdquo Engineering Failure Analysis vol 74 pp 80ndash94 2017

[19] W Zhang H Ma J Zeng S Wu and B Wen ldquoVibrationresponses analysis of an elastic-support cantilever beam withcrack and offset boundaryrdquo Mechanical Systems and SignalProcessing vol 95 pp 205ndash218 2017

[20] HMa J Zeng Z Lang L Zhang Y Guo and BWen ldquoAnalysisof the dynamic characteristics of a slant-cracked cantileverbeamrdquo Mechanical Systems and Signal Processing vol 75 pp261ndash279 2016

[21] J Liu W D Zhu P G Charalambides Y M Shao Y F Xuand X M Fang ldquoA dynamic model of a cantilever beam with aclosed embedded horizontal crack including local flexibilitiesat crack tipsrdquo Journal of Sound and Vibration vol 382 pp 274ndash290 2016

[22] Y S Shih and J J Chen ldquoAnalysis of fatigue crack growth on acracked shaftrdquo International Journal of Fatigue vol 19 no 6 pp477ndash485 1997

[23] A Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagationrdquo Engineering FractureMechanics vol17 no 4 pp 381ndash386 1983

[24] A J Dentsoras and A D Dimarogonas ldquoResonance controlledfatigue crack propagation in a beam under longitudinal vibra-tionsrdquo International Journal of Fracture vol 23 no 1 pp 15ndash221983

[25] A J Dentsoras andADDimarogonas ldquoFatigue crack propaga-tion in resonating structuresrdquo Engineering Fracture Mechanicsvol 34 no 3 pp 721ndash728 1989

[26] M Lalanne P Berthier and J D HagopianMechanical Vibra-tions for Engineers John Wiley amp Sons New York NY USA1983

[27] A D Dimarogonas S A Paipetis and T G ChondrosAnalyticalMethods inRotorDynamics SpringerDordrechtTheNetherlands 2013

[28] H Tada P C Paris andG R IrwinTheStress Analysis of CracksHandbook The American Society of Mechanical Engineers2000

[29] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984

[30] R Mulyukov S Mikhailov R Zaripova and D SalimonenkoldquoDamping properties of 18Cr-10Ni stainless steel with submi-crocrystalline structurerdquoMaterials Research Bulletin vol 31 no6 pp 639ndash645 1996

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom