Analytical approach and numerical simulation to

Mechanics & Industry 20, 629 (2019)© AFM, EDP Sciences 2019https://doi.org/10.1051/meca/2019065

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Analytical approach and numerical simulation to investigatethe stress field and the dynamic stress intensity factorsof a cracked tooth subjected to a periodic loadingMohamed Boughazala*, Khemais Hamrouni, Abdessattar Aloui, and Mohamed Haddar

Mechanics, Modelling and Manufacturing Laboratory (LA2MP), Mechanical Engineering Department, National Engineers Schoolof Sfax, B.P. W3038, Sfax, Tunisia

* e-mail: m

Received: 3 March 2019 / Accepted: 24 September 2019

Abstract. A new analytical approach was developed in this paper to study the dynamic analysis of a crackedgear tooth subjected to a periodic loading. The finite elements method (FEM) based on the contour integraltechnique was used in order to validate the effectiveness and reliability of the analytical formulation. A three-dimensional (3D) model of a cracked tooth was designed where a refined mesh was applied in the crack region tobetter simulate the stress concentration. The main objective of this study was to investigate the influence of thecrack depth and the external load on the variation of the stress intensity factors (SIF)KI andKII associated withthe opening crack modes I and II, respectively, and on the stress field near the crack tip. The obtained resultsshow a significant agreement between the analytical results and the FEM findings.

Keywords: Cracked tooth / periodic loading / dynamic stress intensity factors / stress field /finite elements method

1 Introduction

Gears, as the most important parts of rotating machinery,are power transmission elements frequently used inindustries such as manufacturing, aerospace, automotive,aircraft engines, and so on. They often work in severeconditions and are therefore subject to progressivedeterioration especially in the teeth such as wear, cracks,etc. These defects cause a failure in the operation of themechanical system since they affect the reliability andsecurity of the gear system. Consequently, the study of thedynamic behavior of a cracked gear tooth has been thesubject of many investigations owing to its important rolein fault detection analysis. Chaari et al. [1] presented ananalytical modeling of tooth cracks to quantify thereduction of gear mesh stiffness when the crack is assumedas a straight line shape occurring in the tooth root circle. Inorder to validate this, a FEM is also computed incorporat-ing the tooth crack. In the same context, Mohammed andRantatalo [2] developed a dynamic model, including agyroscopic effect of a one-stage spur gear system, tocalculate the time varying mesh stiffness for different cracksizes. Remping et al. [3] investigated the influence of crackposition and length on the dynamic characteristics of acracked gear. Furthermore, the variation of the stress

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intensity factors related to changes of various parameterswas analyzed using the fracture mechanics theory and thefinite elements method by ANSYS software. Fakhfakh andChaari [4] developed a one-stage geared transmissionmodel with two degrees of freedom to study the dynamicresponse in both cases of healthy and defected teeth.Moreover, they presented a series of tests in theexperimental setup to observe and analyze the dynamicbehavior and signal responses of perfect and cracked teeth.Nicoletto [5] combined the weight function and thecomplex potential method in an approximate approachto determine the stress intensity factors of cracked gearstaking into account the influence of different parameterssuch as, loading type, crack propagation direction andteeth number. Themodes I and II SIFs were compared withthe finite element analysis results reached by reference [6].An experimental technique based on strain gauge wasproposed by Naresh and Anand [7] to measure the gearmesh stiffness of healthy spur gear pair system. Likewise,they calculated the mesh stiffness of cracked spur gear withthe help of the strain energy release and the SIFcalculation. Thus, the mesh stiffness reduction due tocrack have been observed. The experimental results havebeen validated by the analytical method of Wu et al. [8].Pehan et al. [9] considered both two- and three-dimensionalanalysis to evaluate the stress intensity factors in a toothroot as a function of crack depth, and determine the crackpropagation direction from the tip of the initial crack.

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Fig. 1. Gear tooth in the presence of a crack.

Fig. 3. Typical variation of the periodic load.

Fig. 2. Teeth engaged during meshing [15].

2 M. Boughazala et al.: Mechanics & Industry 20, 629 (2019)

Zouari et al. [10] presented a finite element method with athree-dimensional survey to study the influence of crackdimension and the direction of crack propagation on themesh stiffness for spur gear with a crack in the tooth foot.The same computer software (FORTRAN program andANSYS code) and the same gear geometrical parametershave been used in another work of Zouari et al. [11] in orderto estimate the variation of the stress intensity factorsaccording to crack depth and crack propagation angle andtrack the crack propagation in the tooth foot of a spur gear.Guagliano et al. [12] proposed a versatile approach thatconsiders the contact pressure distribution to evaluate thestress intensity factors along the front of any subsurfacecracks found in hypoid gear. Zhanewi et al. [13] establishedan analytical model based on the potential energy methodto investigate the effects on the (TVMS) under threedifferent gear crack types. A three-dimensional (3D) modelof spatial cracked gear pair was adopted to verify thereliability of the analytical method. Meanwhile, the effectof these three types of cracks on the equivalent stress, thetooth displacement and contact pressure were analyzed byusing the FE method. Recently, Yang et al. [14] proposedanalytical formulations, validated by a finite elementmodel, to investigate the influence of the tip fillet radius ofrack-type tools and tooth addendum modification on gearmesh stiffness for a spur gear pair.

In light of the review above, the SIF and the stress fieldaround the crack were not extensively studied when a toothcrack appeared and the stress concentration was observed.This needs to be deeply studied. To this end, this paperpresented a new analytical approach, solved by applyingthe Newmark iterative schema in a MATLAB computerprogram. The proposed approach was validated through acomparison with the FE results provided by a numericalsimulation using ABAQUS software. The effects of loadand crack depth were discussed, and some conclusions weredrawn in the conclusive section.

2 Analytical approach

2.1 Gear tooth modeling

A cracked gear tooth, with a width (b) and a crack depth(a), was modeled by a variable section beam fixed at itsend. The crack is located at a distance (xc) from the toothroot as shown in Figure 1.

The load F(t) applied on the tooth is a time varyingparameter that reflects gear mesh conditions as the numberof teeth in contact varies and as the line of contact of theengaged tooth varies consequently (see Fig. 2). Thus, thetooth is subjected to a periodic load spread over a period(T) and the loading cycle can be modeled by a trapezoidalforce (Fig. 3).

FðtÞ ¼

F

eTt 0 � t � eT

F eT < t � hT

�F

ð& � hÞT tþ F&

ð& � hÞ hT < t � &T

0 &T < t � T

:

8>>>>>><>>>>>>:

ð1Þ

2.2 Dynamic study of a gear tooth without crack

The tooth was modeled by a recessed beam of a variablesection S(x) and a quadratic momentI(x). The dynamicequation of the tooth without crack is written as follows:

rSðxÞ ∂2yðx; tÞ∂t2

þ ∂2

∂x2EIðxÞ ∂

2yðx; tÞ∂x2

� �¼ pðx; tÞdðx� xF Þ

ð2Þp(x, t) is a distributed load per unit length, such that:

FðtÞ ¼ �Z l

0

pðx; tÞdðx� xF Þdx:

The kinetic energy T and potential strain energy U ofthe beam can be expressed, respectively, as,

T ¼ 1

2

Z l

0

rSðxÞ ∂yðx; tÞ∂t

� �2dx ð3Þ

U ¼ 1

2

Z l

0

EIðxÞ ∂2yðx; tÞ∂x2

� �2dx: ð4Þ

M. Boughazala et al.: Mechanics & Industry 20, 629 (2019) 3

The work achieved by this force F(t) is:

W ¼Z

pðx; tÞdðx� xF Þyðx; tÞdx: ð5Þ

The solution to the dynamic equation (2) can be writtenin the following form:

yðx; tÞ ¼X∞i¼1

Y iðxÞqiðtÞ; ð6Þ

where,Y iðxÞ¼x sin ipx

2l is a function which checks the followingboundary conditions:

y(0, t)= 0,dyð0; tÞdx

¼ 0:

Then, we obtain:

T ¼ 1

2

Z l

0

rSðxÞX∞i¼1

Y iðxÞ _qiðtÞX∞j¼1

Y jðxÞ _qjðtÞ" #

dx

¼ 1

2

X∞i¼1

X∞j¼1

_qiðtÞmij _qjðtÞ ð7Þ

U ¼ 1

2

Z l

0

EIðxÞX∞i¼1

∂2Y iðxÞ∂x2

qiðtÞX∞j¼1

∂2Y jðxÞ∂x2

qjðtÞ !

dx

¼ 1

2

X∞i¼1

X∞j¼1

qiðtÞkijqjðtÞ ð8Þ

W ¼Z l

0

pðx; tÞdðx� xF ÞX∞i¼1

Y iðxÞ qiðtÞdx

¼X∞i¼1

FðtÞY iðxF Þ qiðtÞ ¼X∞i¼1

fiðtÞ qiðtÞ; ð9Þ

where,

mij ¼Z l

0

rSðxÞY iðxÞY jðxÞdx

kij ¼Z l

0

EIðxÞ ∂2Y iðxÞ∂x2

∂2Y jðxÞ∂x2

dx

fiðtÞ ¼ FðtÞY iðxF Þ:Using the Lagrange equation:

d

dt

∂T∂ _qi

� �� ∂T∂qi

þ ∂U∂qi

¼ ∂W∂qi

ð10Þ

we obtain the following equations system:X∞j¼1

mij€qjðtÞ þX∞j¼1

kijqjðtÞ ¼ fiðtÞ: ð11Þ

This can be represented in a matrix form as follows,

M½ � _q_ðtÞ þ K½ �qðtÞ ¼ fðtÞ: ð12Þ

If we limit to i=1,...., n and j=1,..., n, we get:

M½ � ¼ ⌊mij⌋; K½ � ¼ ⌊kij⌋;

qðtÞ ¼ q1ðtÞ; :::; qnðtÞf gt; fðtÞ ¼ f1ðtÞ; :::; fnðtÞf gt:

Applying the time integration schema developed by theNewmarkmethod, and adopting the notation: qðtÞ ¼ q

t, we

get:

qt¼ q

t�Dtþ Dt _q

t�Dtþ Dt2

2€qt�Dt

þ bDt3 ⃛qt�Dt

_qt¼ _q

t�Dtþ Dt€q

t�Dtþ g Dt2 q⃛

t�Dt :

8>>><>>>:

ð13Þ

Using the approximation q⃛t�Dt

¼ €qt�€q

t�Dt

Dt we can write:

qt¼ q

t�Dtþ Dt _q

t�Dtþ�1

2� b

�Dt2€q

t�Dtþ bDt2€q

t

_qt ¼ _q

t�Dt þ ð1� gÞDt€qt�Dt

þ g Dt€qt

8>>><>>>:

where b ¼ 14 and g ¼ 1

2.Or in the following form:

€qt¼ b1ðqt � q

t�DtÞ þ b2 _q t�Dt

þ b3€qt�Dt

_qt¼ b4ðqt � q

t�DtÞ þ b5 _q t�Dt

þ b6€qt�Dt

8><>: ð14Þ

b1 ¼ 1

bDt2, b2 ¼ �1

bDt, b3 ¼ 1� 1

2b, b4 ¼ g

bDt,

b5 ¼ 1� g

b, b6 ¼

�1� g

2b

�Dt.

The differential equation becomes:

b1M þK½ � qt¼ fðtÞ þ M½ � b1qt�Dt

� b2 _q t�Dt� b3€q t�Dt

� �:

It is an equation of the form:

K�

qt¼ F t; ð15Þ

where K� ¼ b1M þK½ � : the fictitious stiffness matrix.

F t ¼ fðtÞ þ M½ � ðb1qt�Dt� b2 _q t�Dt

� b3€q t�DtÞ: the ficti-

tious effort.The Newmark’s iterative schema is used with time

increments (t=Dt, 2Dt, 3Dt,…NDt) to determine the vectorqt¼ ðq1ðtÞ; :::; qiðtÞ; :::; qnðtÞÞ. Then, the function is deter-

mined using equation (6).

Fig. 5. Three-dimensional model with the boundary conditions.

Fig. 4. Polar coordinates at a crack tip.


2.3 Dynamic analysis of the cracked tooth

In the presence of a crack located orthogonally to the toothmedian line, the stress field near the tip of the crack can beexpressed in the polar coordinates (r, ’) (Fig. 4) as:

sij ¼ KIðtÞffiffiffiffiffiffiffiffiffi2p r

p gijð’Þ þKIIðtÞffiffiffiffiffiffiffiffiffi2p r

p hijð’Þ ð16Þ

KI(t) and KII(t) are the stress intensity factors associated,respectively, with the crackingmodes I and II, which can beexpressed according to reference [16] by the followingrelations:

KIðtÞ ¼ sxðtÞffiffiffiffiffiffiffip a

pY I ; ð17Þ

KIIðtÞ ¼ txyðtÞffiffiffiffiffiffiffip a

pY II ; ð18Þ

whereYI andYII are factors that depend on the ratio j ¼ ahc

between the crack depth and the height of the section.According to [17] we have:

Y I ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p

jtg

pj

2

� �s 0:923þ 0:199 1� sinpj

2

� �� 4

cospj

2

� � ; ð19Þ

Y II ¼ ð3j� 2j2Þ 1:222� 0:561jþ 0:085j2 þ 0:18j3ffiffiffiffiffiffiffiffiffiffiffi1� j

p ; ð20Þ

where sx(t) and txy(t) are, respectively, the nominal and thetangential stresses, determined in the case of the healthytooth and written as,

sxðtÞ ¼ �E∂2yðx; tÞ

∂x2

��x¼xc

hc

2ð21Þ

txyðtÞ ¼ 6FðtÞbhc

a

hc

� �� a

hc

� �2" #

: ð22Þ

The stress field near the crack tip is given by [18] asshown in the following formulas:

sxðr;’; tÞ ¼ KIðtÞffiffiffiffiffiffiffiffiffi2p r

p cos’

21þ sin

’

2sin

3’

2

� �

þKIIðtÞffiffiffiffiffiffiffiffiffi2p r

p sin’

2cos

’

2cos

3’

2ð23Þ

txyðr;’; tÞ ¼ KIðtÞffiffiffiffiffiffiffiffiffi2p r

p sin’

2cos

’

2cos

3’

2

þKIIðtÞffiffiffiffiffiffiffiffiffi2p r

p cos’

21� sin

’

2sin

3’

2

� �: ð24Þ

3 The finite elements method

In order to numerically analyze the dynamic behavior of acracked tooth and to study the influence of someparameters, namely, the crack depth and the applied load,a three-dimensional model was created and simulated bythe ABAQUS software. The cracked tooth was modeled asa non uniform cantilever beam. Assuming that the crackposition was set at a distance (xc) from the tooth root andthat this crack extends along the whole tooth width with auniform depth distribution, the tooth with this type ofcrack is still considered as a cantilever beam. The boundaryconditions of the cracked tooth can be set so that the toothroot is fixed and the top is free. The external load isconsidered uniformly distributed along the entire toothwidth and is located at a distance (xF) from the tooth root(Fig. 5).

The model, for which the geometrical parameters arelisted in Table 1, is selected to check the validity of theanalytical results.

The finite elements model consists of 17 501 nodes and15 768 linear hexahedral elements of type C3D8R. Tobetter simulate the stress singularity and obtain accurateresults in this paper, the mesh of this model was designed

Table 1. Cracked tooth parameters.

Symbol Value

Tooth material 42CrMo4Young modulus (GPa) E 210Width of the tooth (mm) b 20Module (mm) m 2Density (kg/m3) r 7800Poisson’s ratio n 0.3Load position (mm) xF 3Crack position (mm) xc 1.4

Fig. 6. Finite elements model of the cracked tooth.

Fig. 7. Numerical method procedure.



using local mesh refinements around crack tip and the restwas designed so as not to be too dense and the elements arenot too distorted. Figure 6 illustrates the tooth mesh andthe refined mesh around crack.

Fig. 8. Stress concentration near the crack tip.

Fig. 9. KI varia

Fig. 10. KII vari

The corresponding steps of the numerical method areillustrated in Figure 7.

4 Results and discussion

The Contour Integral Method was used in the dynamicimplicit simulation to extract the stress field and the stressintensity factors. This method was applied also to clearlyshow the stress concentration near the crack tip as shown inFigure 8.

The stress intensity factor is a failure criterionthat depends on the geometry of the cracked tooth, thesize of the crack, as well as the external loading. Thedynamic SIF for both mode one and mode two wasinvestigated according to changes in crack length andapplied load.

tion with load.

ation with load.

Fig. 11. KI variation with the crack depth.

Fig. 12. KII variation with the crack depth.


Considering a contact ratio c=1.6 and rotational speedN=400 rpm of the pinion which has Z=15 teeth, the gearmesh period (in second) is defined by [19]:

zT ¼ 60

NZ¼ 0:01 s:

Assuming a crack depth (a=0.5mm), while the otherparameters are fixed and only the magnitude of the load isvaried, the SIFs variation depending on different forcevalues are presented in Figures 9 and 10.

Figures 11 and 12 illustrate the SIFs variation for aconstant load (F=500N), and for various crack depthvalues with respect to the position of the crack from thetooth root.

From Figures 9–12, it can be observed that thevariation of the SIF’s (KI and KII) according to time has

a trapezoidal form. This is explained by the periodic force,exerted on the tooth, which varies with time depending onthe conditions of the gear and the contact between theteeth.

For the case of low contact ratio (c< 2), a fluctuation ofone pair-two pairs of teeth in contact is observed whichyields to a time varying stress intensity factor. As it ispresented in Figures 9–12, the maximum values of SIF arecorresponding to one pair in contact and are observedduring (2-c)&T.

The mixed mode of the loading near the crack tip isillustrated with the dynamic SIFs associated with modes Iand II. The evolution of stress intensity factors according tothe applied load and the crack depth obtained by thenumerical method is presented in Figures 13 and 14. It canbe observed from these figures that the magnitudes of both

Fig. 15. Crack depth influence on sxx.

Fig. 13. Evolution of stress intensity factors according to theapplied load for a=0.5mm.

Fig. 14. Evolution of stress intensity factors according to thecrack depth for F=500N.


Fig. 16. Crack depth influence on txy.


SIFs KI and KII are enhanced with the increasing load andcrack depth.

In Figure 14, different crack sizes are selected to studythe effect of crack depth on the SIF. As well shown in thisfigure the SIF’s values of both modes I and IIwere influenced by the growth of crack’s size. Italso showed that this effect becomes more pronouncedwhen the crack depth exceeds 0.6mm. Moreover, it can be

observed that although KII value increases obviously, itseems always very negligible compared to KI value, and thevariation (KI � KII) is still growing.

Furthermore, when a crack appears, it can be seen thatthe influence of the crack depth on the stress intensityfactors is bigger than the effect of the applied load. This isbecause the cracked tooth loses much rigidity and becomesmore and more flexible with the crack depth increase.

Fig. 17. Load influence on sxx.


Figures 15 and 16 present, under five cases, the effect ofcrack depth on the stress field near the crack by numericaland analytical methods in order to clearly prove that thestress values will decrease when we are far from the crack.This explains that the trends of the nominal and tangentialstresses are parabolic. From these figures, it is clear thatthe longer the crack is, the more severe the stressconcentration becomes. Also, it can be seen that in case 5,

which represents a deep crack (a=1mm), the levels of sxxand txy increase significantly compared to that of case 1which is a smaller crack depth (a=0.2mm).

In order to analyze the influence of applied load on thecracked tooth, by two methods, five cases were considered.Each of these cases is related to a different value of force fora constant crack depth (a=0.5mm) as shown in Figures 17and 18.

Fig. 18. Load influence on txy.


Through Figures 15–18, it can be determined thatthe magnitudes of the nominal stress (sxx) and thetangential stress (txy) rise with both the load andcrack depth increase. Furthermore, the applied load playsa less important role than the crack depth in the growth ofstress concentration near the crack. This clearly provesthat the stress field changes according to the crack depthrather than to the applied load.

The results of the stress field at the crack tippresented in Figures 15–18 have shown significantagreement between the analytical approach and thenumerical findings. These observations demonstrate

that this analytical approach will be a major step tostudy the dynamic behavior for more realistic industrialsystems.

5 Conclusion

Analytical approach and FEM were used to determine thedynamic stress intensity factors for mode I and mode II aswell as the stress field in the vicinity of crack, taking intoconsideration the influence of the crack depth and thevariation of applied load. Obviously, the results from


the two methods have shown a good agreement and thefollowing conclusions can be forwarded:
– The SIF increases with the increase of both the appliedload and the crack depth.
–
The influence of the applied load on the SIF appeared tobe smaller than that of the crack depth.
–
The finite elements analysis, using ABAQUS softwareand based on the dynamic implicit method, isconsidered as an efficient and reliable tool to solvedynamic problems and validate the theoreticalresults.
–
This analytical approach plays an important role inconnecting the fundamental research and the practicalusage, and it may be extended in future research studiesto describe three-dimensional dynamic problems arisingin other types of gearing and defects.
This work is supported by the Mechanics, Modeling andProduction Laboratory (LA2MP) of the National EngineeringSchool of Sfax (Tunisia) within a research project entitled“Study and dynamic analysis of a shafts line in the presence ofcracks”.

References

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Cite this article as: M. Boughazala, K. Hamrouni, A. Aloui, M. Haddar, Analytical approach and numerical simulation toinvestigate the stress field and the dynamic stress intensity factors of a cracked tooth subjected to a periodic loading, Mechanics &Industry 20, 629 (2019)

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