NumericalSimulationofMixed-ModeFatigueCrackGrowthfor ...the present study show consistency with the...
14
Research Article Numerical Simulation of Mixed-Mode Fatigue Crack Growth for Compact Tension Shear Specimen Yahya Ali Fageehi and Abdulnaser M. Alshoaibi Department of Mechanical Engineering, Jazan University, P. O. Box 706, Jazan 45142, Saudi Arabia Correspondence should be addressed to Abdulnaser M. Alshoaibi; [email protected] Received 30 January 2020; Revised 3 March 2020; Accepted 20 March 2020; Published 23 April 2020 Academic Editor: Davide Palumbo Copyright © 2020 Yahya Ali Fageehi and Abdulnaser M. Alshoaibi. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is work concentrates on the fracture behaviour of the compact tension specimen under mixed-mode loading, and numerical investigation using ANSYS Mechanical APDL 19.2 finite element program with different modes of mix angles is carried out. e prediction of mixed-mode fatigue life under constant amplitude fatigue loading for the compact tension shear specimen (CTS) is employed using Paris’ law model for two different loading angles with agreement to the experimental results. e predicted values of ΔK eq were compared with the experimental and analytical data for various models. Depending on the analysis, the findings of the present study show consistency with the results achieved with similar models of predicting the equivalent stress intensity factor. In addition, the direction of crack growth derived from the analysis was observed to follow the same trend of the literature experimental results. 1. Introduction is work describes the use of ANSYS Mechanical APDL 19.2 finite element program in crack analysis of the engi- neering structures containing discontinuities and holes. In the proposed study of the cracks throughout elastic mate- rials, linear elastic fracture mechanics are employed where the crack driving force called stress intensity factor (SIF) is used as a measure of fracture associated with the materials threshold SIFs (K th ) for dynamic loading. e accuracy of a crack propagation simulation depends highly on the accu- racy achieved for the calculated SIFs. Engineering structures are generally used under periodic and cyclic loading con- ditions [1] [2–7] [6]. e periodic as well as cyclic loading decreases the load limit required to initiate deformation in the material under static loading and results in progressive fracture leading to catastrophic failures of the material known as fatigue failures [6]. It has been estimated that 60% to 80% of the failures of mechanical components are as- sociated with fatigue [8]. Various problems concerning fatigue crack growth can be found in the literature using different approaches for simple and complex geometries in two and three dimensions [9–13]. Combined action of tensile stress, cyclic stress, and plastic strain initiates failure due to fatigue phenomena. If any of these three does not exist, it will not trigger and spread a fatigue crack growth [14]. As environmental loads are also multidirectional in nature, the fatigue cracks may propagate under mixed-mode conditions [15–18]. e majority of failures in service is of mixed-mode type, where the cracks do not propagate in the direction normal to the applied load. However, the majority of fatigue crack growth behavior studies focus on mode I cracks, whereas the cracks and defects in actual engineering components (e.g., pressure vessels, pipelines, and fan blades) tend to be plane-stress mixed-modes I–II cracks [19]. In reality also mixed mode can be observed, for example, in turbine shafts or railway tracks by a sudden change of the loading direction. e crack will initiate due to the plastic strain occasioning from the cyclic stress, whereas the propagation steps of the cracks result from the tensile stress. Furthermore, the cause of local tensile stresses happens to be the fact that compressive loads may not lead to the initiation of fatigue crack [20]. To prevent failure due to fatigue, ex- tensive research has been performed to get developed and Hindawi Advances in Materials Science and Engineering Volume 2020, Article ID 5426831, 14 pages https://doi.org/10.1155/2020/5426831
Transcript of NumericalSimulationofMixed-ModeFatigueCrackGrowthfor ...the present study show consistency with the...
Research ArticleNumerical Simulation of Mixed-Mode Fatigue Crack Growth forCompact Tension Shear Specimen
Yahya Ali Fageehi and Abdulnaser M Alshoaibi
Department of Mechanical Engineering Jazan University P O Box 706 Jazan 45142 Saudi Arabia
Correspondence should be addressed to Abdulnaser M Alshoaibi alshoaibigmailcom
Received 30 January 2020 Revised 3 March 2020 Accepted 20 March 2020 Published 23 April 2020
Academic Editor Davide Palumbo
Copyright copy 2020 Yahya Ali Fageehi and Abdulnaser M Alshoaibi ampis is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited
ampis work concentrates on the fracture behaviour of the compact tension specimen under mixed-mode loading and numericalinvestigation using ANSYS Mechanical APDL 192 finite element program with different modes of mix angles is carried out ampeprediction of mixed-mode fatigue life under constant amplitude fatigue loading for the compact tension shear specimen (CTS) isemployed using Parisrsquo law model for two different loading angles with agreement to the experimental results ampe predicted valuesof ΔKeq were compared with the experimental and analytical data for various models Depending on the analysis the findings ofthe present study show consistency with the results achieved with similar models of predicting the equivalent stress intensityfactor In addition the direction of crack growth derived from the analysis was observed to follow the same trend of the literatureexperimental results
1 Introduction
ampis work describes the use of ANSYS Mechanical APDL192 finite element program in crack analysis of the engi-neering structures containing discontinuities and holes Inthe proposed study of the cracks throughout elastic mate-rials linear elastic fracture mechanics are employed wherethe crack driving force called stress intensity factor (SIF) isused as a measure of fracture associated with the materialsthreshold SIFs (Kth) for dynamic loading ampe accuracy of acrack propagation simulation depends highly on the accu-racy achieved for the calculated SIFs Engineering structuresare generally used under periodic and cyclic loading con-ditions [1] [2ndash7] [6] ampe periodic as well as cyclic loadingdecreases the load limit required to initiate deformation inthe material under static loading and results in progressivefracture leading to catastrophic failures of the materialknown as fatigue failures [6] It has been estimated that 60to 80 of the failures of mechanical components are as-sociated with fatigue [8] Various problems concerningfatigue crack growth can be found in the literature usingdifferent approaches for simple and complex geometries in
two and three dimensions [9ndash13] Combined action oftensile stress cyclic stress and plastic strain initiates failuredue to fatigue phenomena If any of these three does notexist it will not trigger and spread a fatigue crack growth[14] As environmental loads are also multidirectional innature the fatigue cracks may propagate under mixed-modeconditions [15ndash18] ampe majority of failures in service is ofmixed-mode type where the cracks do not propagate in thedirection normal to the applied load However the majorityof fatigue crack growth behavior studies focus on mode Icracks whereas the cracks and defects in actual engineeringcomponents (eg pressure vessels pipelines and fan blades)tend to be plane-stress mixed-modes IndashII cracks [19] Inreality also mixed mode can be observed for example inturbine shafts or railway tracks by a sudden change of theloading direction ampe crack will initiate due to the plasticstrain occasioning from the cyclic stress whereas thepropagation steps of the cracks result from the tensile stressFurthermore the cause of local tensile stresses happens to bethe fact that compressive loads may not lead to the initiationof fatigue crack [20] To prevent failure due to fatigue ex-tensive research has been performed to get developed and
HindawiAdvances in Materials Science and EngineeringVolume 2020 Article ID 5426831 14 pageshttpsdoiorg10115520205426831
effective models to predict fatigue crack propagation andfatigue lifetime Several experimental models have beenproposed but carrying out the experiment is generallyexpensive and time-consuming Numerical simulationusing the extended finite element approach is a suitableway to minimize the time and cost associated with ex-perimental work Under linear elastic fracture mechanics(LEFM) analysis the SIFs are used to describe displace-ments and stresses around the crack front as well as theanalysis of fatigue crack propagation As the crackpropagates larger the SIF increases until it reaches acritical range where the assembly might start to deformby initiating fracture As a result of the complex mode ofapplied loads the geometry specification mixed modes (III) are regularly the general types of loads dependent onΔKeq used in the fatigue life prediction [21ndash24] ampe aim ofthis paper is to use the extended finite element methodimplemented by ANSYS APDL 192 to investigate theeffect of the loading angle for the CTS specimen withdifferent initial cracks
2 Numerical Predication of Mixed-ModeFatigue Life
Mainly three approaches have been commonly used forillustration of the fatigue assessment of materials which arethe fracture mechanics method established by [25] thestrain-life method proposed independently by [26] and thestress-life method proposed by [27] In the present study thefirst method was used for fatigue life prediction by which thecrack tip can be individually defined by the SIFsamperefore itis essential to estimate the fatigue crack path accurately inorder to predict the fatigue life assessment To this end themaximum tangential stress criterion theory was used topredict the crack deflection angle [28ndash30] as
θ 2 arctan14
KI
KII
+14
KI
KII
1113888 1113889
2
+ 8
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ forKII lt 0
(1)
θ 2 arctan14
KI
KII
minus14
KI
KII
1113888 1113889
2
+ 8
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ forKII gt 0
(2)
ampe two possibilities for the crack growth direction areshown in Figure 1
ampe calculation of fatigue crack growth using the cor-responding stress intensity factor is the most widely usedmethod for structures under mixed-mode dynamic loadingUsing a modified formula of Paris law a researcher [24]proposed a power law for the fatigue crack growth rela-tionship with ΔKeq which is specified as
da
dN C ΔKeq1113872 1113873
m(3)
From equation (3) for a crack increment the number oflife cycles of fatigue may be predicted as
1113946
Δa
0
da
C ΔKeq1113872 1113873m 1113946
ΔN
0
dN ΔN (4)
Several of the commonly used ΔKeq formulas accom-panied by the proposed authors are listed in Table 1
21 Numerical Results and Discussion Finite element codeANSYS Mechanical APDL 192 is implemented to simulatethe different types of loading angle of the compact tensionspecimen (CTS) ampe CTS geometry is shown in Figure 2(a)and its suggested loading angles are shown in Figure 2(b)ampere are three types of cracks that can be introduced inANSYS which are arbitrary semielliptical and premeshedcrack ampe premeshed crack method involves the crack frontthat is used by the Smart Crack Growth analysis enginewhere the failure criterion is the stress intensity factor ampepreviously created node sets are allocated to the crack frontwithin the premesh crack unit and the top and bottom facesof the crack ampe crack coordinate system is referenced ampenumber of contours for solution is set to 5 ampese are theldquoloopsrdquo through the mesh around the crack tip which areused to evaluate the stress intensity factor by integrating thecrack tip region strain energy In the analysis the fracturemechanics method avoids the stress singularities at the cracktip ampe process is repeated for the top and bottom crackfaces ampe new implemented feature in ANSYS is the ldquoSmartCrack Growthrdquo with tetrahedron mesh added after finishingthe requirements of the ldquopremeshed crackrdquo in which theuser can select the type of crack growth option
KI
KI
InitialCrack
ndashKII
ndashKII+θ
Keq
Keq
Crack
Propagation
(a)
KI
KI
KII
KIIndashθ
Keq
KeqInitialCrack
Crack
Propagation
(b)
Figure 1 Crack growth angle (a) KII rang 0 (b) KII lang 0
2 Advances in Materials Science and Engineering
Table 1 Commonly applied ΔKeq formulas
Model provided by authors ΔKeq expression
[24] ΔKeq (ΔK2I + 2ΔK2
II)12
[24] ΔKeq (ΔK4I + 8ΔK4
II)14
[31] ΔKeq ΔK2
1 + ΔK2II
1113969
[32 33] ΔKeq ΔKI
2+12
ΔK2I + 4(1155ΔK2
II)
1113969
[21] ΔKeq (10519 times K4I minus 0035 times K4
II + 23056 times K2I times K2
II)12
[34] ΔKeq cosθ2
[KIcos2θ2
minus32KII sin θ]
Figure 3 3D ANSYS model for CTS specimen (no of nodes 201317 and no of elements 132886)
10W = 9018 27 27
54
425a
Oslash17
154
(a)
F
F
α
0deg 90deg15deg 75deg
30deg 60deg
(b)
Figure 2 CTS geometry (a) dimensions (in mm) and (b) loading angles
Advances in Materials Science and Engineering 3
Figure 3 displays the 3D finite element layout for thegeometry dimensions of the CTS with number ofnodes 201317 and number of elements 132886 Based onthe values of the loading angle (α) which can be changed instages of 15deg from 0 to 90deg the modes of loading will changeaccording to the value of the angle from mixed-modeloading (α15deg 30deg 45deg 60deg and 75deg) pure mode II (α 0deg)and pure mode I (α 90deg) ampe studied material was alu-minium alloy having an elasticity module of E 74GPav 033 σY 517MPa tensile strength 579MPa fracturetoughness KIC 3295MPa
m
radic threshold SIF
Kth 315MPam
radic Paris constant C 43378times10minus 7 7 (mm
cycle)(MPam
radic) and Paris exponent m 26183 [35]
ampe geometryrsquos thickness is 5mmwhile the ending partsrsquothicknesses are 10mm which can protect the specimenrsquoslinking holes from failing ampe plane-stress conditions areassumed in the simulations according to the thickness valuewhich is small compared to other dimensions as well as forprecise comparison with other researchers who assumedthat for the same geometry [35] ampe value of load isP 16 kN and the precrack length is a 45mm
ampe finite element model of ANSYS shown in Figure 3 iscompletely consistent with the state of experimental testsperformed by [21 35ndash37] In the present simulation the CTSgeometries are loaded in various angles 30deg 45deg 60deg and 75degin the original direction of cracking for the comparison ofΔKeq and crack paths directionsampere are also other loadingangles 0deg 30deg 60deg and 90deg to compare the direction of crackpropagation and assessment of fatigue life In the last casethree different crack length geometries of the CTS geome-tries have been simulated according to the initial cracklength of aW 03 05 and 07
Figure 4 shows the distribution of the equivalent loadingused in ANSYS analysis of the present work ampe uniaxialload F is correlated to the equivalent loads acting on differentholes of the six holes based on the following formulas [38]
F1 F6 F 05 cos α +c
bsin α1113874 1113875
F2 F5 F sin α
F3 F4 F 05lowast cos α minus (cb)sin α( 1113857
(5)
where α is the angle of load and c and b are length parametershown by Figure 4 (c b 54mm) ampe final values of allloading forces with different loading angles are shown inTable 2
ampe analytical solution for KI of this specimen is pro-vided by [23] as
KI F
Wt
πa
radic cos α(1 minus aW)
026 + 265α(W minus a)
1 + 055α(W minus 5) minus 008(a(W minus a))2
1113971
(6)
where F is the uniaxial load (Figure 2(b)) a is a crackrsquoslength W is the width of geometry and t is the geometrythickness It is worth comparing the SIFs for this geometrywith the analytical solution ofKI (equation (6)) for differentthicknesses (3 6 9 and 12mm) as represented in Figure 5
c
F1F2
F3
c
F4
F5
F6b
Figure 4 Loading and boundary condition for CTS geometry
Table 2 Values of forces F1 to F6 according to load angle α
α F2 F5 F1 F6 F3 F415 0259 F 0742 F 0224 F30 05 F 0933 F minus 0067 F45 0707 F 1061 F minus 0354 F60 0866 F 1116 F minus 0616 F75 0966 F 1095 F minus 0837 F90 F F minus F
800
700
600
500
400
300
200
100
0
K I (M
Pa m
m1
2 )
40 45 50 55 60 65 70 75 80Crack length a (mm)
t = 3mm
t = 6mm
t = 9mm
t = 12mm
ANSYSRichard criterion
Figure 5 Comparison for KI with analytical solution for differentspecimen thicknesses 3mm 6mm 9mm and 12mm
4 Advances in Materials Science and Engineering
As can be seen in this figure the values of SIFs are decreasingas the thickness increasing and the agreement with theanalytical solution obtained by equation (2) is obvious
ampe predicted values of ΔKeq were compared to theexperimental values performed by [37] as well as the ana-lytical expression given by [24 33 34 37] for different valuesof loading angles 30deg 45deg 60deg and 75degampese comparisons areshown in Figures 6ndash9
Figure 6 demonstrates the comparisons for the variationof the crack length as a function of the corresponding ΔKeq
with load angle of 30deg and the amount of applied loading is88 kN (loading ratio R 01) As illustrated in this figure thefour criteria of ΔKeq have around the same tendency as dataobtained from experimental tests by [37] For this loadingangle even there is a slight divergence for the [34] criterionespecially after the mid of the crack length In Figure 7 theloading angle is 45deg and the amount of the load is 114 kNwith R 01 ampe agreement for the ANSYS results with theexperimental results is clear as well as with other analyticalcriteria
Figure 8 shows the comparisons for the equivalent SIFfor 60deg loading angle and the amount of load is 1365 kNwithload ratio R 01 ampere is clearly divergence of the initialvalues of the equivalent SIF for all criteria which is de-pendent in the initial step of the crack growth ampis di-vergence is increased for the loading angle of 75deg and amountof load is 13 kN (R 01) as shown in Figure 9 ampough all ofthe four criteria almost provide close predictions to theexperimental results obtained by [37]
In addition the paths of crack growth of load angles 30deg45deg 60deg and 75deg were validated with comparison to practicalcrack paths tested by [36] as shown in Figure 10 with goodagreement
ampe predicted crack growth path has been compared tothe experimental test performed by [39] with different cracklengths (aw 03 05 and 07) and different load angles 030 60 and 90 as shown in Table 3
For pure loading of mode II (α 0deg) the direction of thecrack growth is controlled by the second mode of SIF (KII)
47 49 51 53 5545Crack length (mm)
K ep (M
Pa m
12 )
Experimental(Demir et al 2018)Richard criterion
Tanaka criterionDemir et al (2017)criterion
Erdogan and Sih ANSYS
201816141210
86420
Figure 7 Equivalent SIF with loading angle 45deg
47 49 51 53 5545Crack length (mm)
0
5
10
15
20
25
K ep (M
Pa m
12 )
Experimental(Demir et al 2018)Richard criterion
Tanaka criterionDemir et al (2017)criterion
Erdogan and Sih ANSYS
Figure 8 Equivalent SIF with loading angle 60deg
0
5
10
15
20
25
45 47 49 51 53 55
Keq
(MPa
m1
2 )
Crack length (mm)
Experimental(Demir et al 2018)Richard criterionErdogan and Sih
Tanaka criterionDemir et al (2017)criterionAnsys
Figure 9 Equivalent SIF with loading angle 75deg
Experimental(Demir et al 2018)Richard criterion
Tanaka criterionDemir et al (2017)criterion
Erdogan and Sih ANSYS
0
5
10
15
20
25
K eq (M
Pa m
12 )
47 49 51 53 55 57 59 61 63 6545Crack length (mm)
Figure 6 Equivalent SIF with loading angle 30deg
Advances in Materials Science and Engineering 5
4250 4450 4650 4850 5050 5250 5450 5650 5850 6050 6250 6450X (mm)
Y (m
m)
1800
1600
1400
1200
1000
800
600
400
200
000
75 deg (Antunes et al 2019)75 deg ANSYS60 deg Exp (Antunes et al 2019)60 deg ANSYS
45 deg Exp (Antunes et al 2019)45 deg ANSYS30 deg Exp (Antunes et al 2019)30 deg ANSYS
Figure 10 Comparison for crack growth paths between present study results and experimental results of [36]
Table 3 Comparison of paths of crack propagation among ANSYSrsquos results and experimental results [39]
ANSYS Experimental test [39]
(aw= 03) α 0deg
6 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 03) α 30deg
(aw= 03) α 60deg
Advances in Materials Science and Engineering 7
Table 3 Continued
ANSYS Experimental test [39]
(aw= 03) α 90deg
(aw= 05) α 0deg
8 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 30deg
(aw= 05) α 60deg
Advances in Materials Science and Engineering 9
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 90deg
(aw= 07) α 0deg
10 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 30deg
(aw= 07) α 60deg
Advances in Materials Science and Engineering 11
and the cracks all grow with different angles based onequation (2) as the values of KII are higher whereas forcomplete mode I loading (α 90deg) crack paths are domi-nated by the first mode KI and the crack is going to growstraight ampe influence of the first mode (KI) decreases as αincreases and vice versa for mode II
ampe crack direction angles of three geometries withdifferent initial length of cracks (aW 03 05 and 07)are 6deg 10deg and 12deg respectively which are identical to the
experimental angles tested by [39] Furthermore formixed-mode loading in the CTS geometries with loadingangle α 30deg the cracks are propagated to some extentdiffering to the original crack front direction For thesethree geometries the mixed-mode crack propagationangles (aW 03 05 and 07) are about 55deg 85deg and85deg respectively which are almost in agreement withexperimental angles of [39]
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 90deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 11 Comparison of simulated and experimental [35] fatiguelife for loading angle 30deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350 400 450 500000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 12 Comparison of simulated and experimental [35] fatiguelife for loading angle 60deg
12 Advances in Materials Science and Engineering
For the above three loading angles situations (α 90degα 0deg and α 30deg) regardless of the length of the initialcrack for the same load angle (α) the angles of the crackgrowth in each case are close to each other In contrast forcombined mode (III) loading with loading angle α 60deg thedirections of crack are highly dependent on the initial lengthof the crack On the other hand for small crack length (aW 03 and 05) the cracks also grow varying from theinitial crack tip against the clockwise direction with crackpropagation angles of 25deg and 30deg respectively which areidentical to those angles in the experimental test [39]Furthermore if the initial crack length is longer than theprevious cases (aW 07) the direction of the crack growthand the direction of the initial crack happen to be identicaland are presented in Table 3
In order to assess the fatigue life the number of cycles fordifferent loading angles 30 and 60 calculated in this study iscompared with the experimental results observed by [35]under mixed loading mode (III) resulting in clear agree-ments as can be observed in Figures 11 and 12 ampe amountof the applied load is F 16 kN with load ratio R 01whereas the Paris constant as mentioned previously isC 43378times10minus 7 (mmcycle)(MPa
m
radic) and m 26183
[40]
3 Conclusions
For the CTS specimen the finite element analysis of themixed-mode fatigue crack propagation was carried out usingANSYS Mechanical APDL and compared with experimentaldata for various angles of loading ampe predicted values ofΔKeq by the experimental and numerical methods are ob-served to be in good agreement with the present study re-sults ampe mixed-mode fatigue life is predicted andcompared with experimental data for two different loadingangles of 30deg and 60deg Interestingly the theoretically pre-dicted trajectories of the crack growth for all specimens areidentical to the experimental determined paths for differentloading angles Furthermore the predicted values of SIFs (KIand KII) are in good agreement with analytical solutions Itcan be stated that the structure of specimen geometry and itsconfiguration play an important role in obtaining highervalues of mixed-mode SIFs values ampis happens to besignificant in terms of applied loads particularly for highervalues of KII
Data Availability
ampe data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
References
[1] R I Stephens A Fatemi R R Stephens and H O FuchsMetal Fatigue in Engineering John Wiley amp Sons HobokenNJ USA 2000
[2] V V Bolotin Mechanics of Fatigue CRC Press Boca RatonFL USA 1999
[3] C A Brebbia and A V Farahani Fatigue Damage of Ma-terials Experiment and Analysis WIT Press SouthamptonUK 2003
[4] N E DowlingMechanical Behavior of Materials EngineeringMethods for Deformation Fracture and Fatigue PearsonLondon UK 2012
[5] G R Halford Fatigue and Durability of Structural MaterialsASM International London UK 2006
[6] R O Ritchie ldquoMechanisms of fatigue-crack propagation inductile and brittle solidsrdquo International Journal of Fracturevol 100 no 1 pp 55ndash83 1999
[7] S Suresh Fatigue of Materials Cambridge University PressCambridge UK 1998
[8] K N Solanki Two and ree-Dimensional Finite ElementAnalysis of Plasticity-Induced Fatigue Crack Closure AComprehensive Parametric Study Mississippi State Univer-sity Starkville MS USA 2002
[9] A M Alshoaibi and A K Ariffin ldquoFinite element modeling offatigue crack propagation using a self adaptive mesh strategyrdquoInternational Review of Aerospace Engineering (IREASE)vol 8 no 6 pp 209ndash215 2015
[10] A M Alshoaibi ldquoFinite element procedures for the numericalsimulation of fatigue crack propagation under mixed modeloadingrdquo Structural Engineering and Mechanics vol 35 no 3pp 283ndash299 2010
[11] A M Alshoaibi ldquoAdaptive finite element modeling of fatiguecrack propagationrdquo International Journal of Materials Scienceand Applications vol 2 no 3 pp 104ndash108 2013
[12] A M Alshoaibi ldquoAn adaptive finite element framework forfatigue crack propagation under constant amplitude loadingrdquoInternational Journal of Applied Science and Engineeringvol 13 no 3 pp 261ndash270 2015
[13] A M Alshoaibi ldquoComprehensive comparisons of two andthree dimensional numerical estimation of stress intensityfactors and crack propagation in linear elastic analysisrdquo In-ternational Journal of Integrated Engineering vol 11 no 6pp 45ndash52 2019
[14] A Handbook Fatigue and Fracture American Society ofMaterials vol 19 1996
[15] W He J Liu and D Xie ldquoNumerical study on fatigue crackgrowth at a web-stiffener of ship structural details by anobjected-oriented approach in conjunction with ABAQUSrdquoMarine Structures vol 35 pp 45ndash69 2014
[16] K Rege and D G Pavlou ldquoEffect of stop holes on structuralintegrity of offshore structures a numerical modelrdquo in Pro-ceedings of the Institution of Civil Engineers-Maritime Engi-neering ampomas Telford Ltd London UK 2019
[17] H Rhee ldquoFatigue crack growth analyses of offshore structuraltubular jointrdquo Journal of Offshore Mechanics and Arctic En-gineering vol 111 no 1 pp 49ndash55 1989
[18] H Riahi P Bressolette A Chateauneuf C Bouraoui andR Fathallah ldquoReliability analysis and inspection updating bystochastic response surface of fatigue cracks in mixed moderdquoEngineering Structures vol 33 no 12 pp 3392ndash3401 2011
[19] S Qi L X Cai C Bao H Chen K K Shi and H L WuldquoAnalytical theory for fatigue crack propagation rates ofmixed-mode I-II cracks and its applicationrdquo InternationalJournal of Fatigue vol 119 pp 150ndash159 2019
[20] A F Liu Mechanics and Mechanisms of Fracture An In-troduction ASM International London UK 2005
[21] O Demir A O Ayhan and S Iriccedil ldquoA new specimen formixed mode-III fracture tests modeling experiments and
Advances in Materials Science and Engineering 13
criteria developmentrdquo Engineering Fracture Mechanicsvol 178 pp 457ndash476 2017
[22] M Hussain S Pu and J Underwood ldquoStrain energy releaserate for a crack under combined mode I and mode II FractureAnalysisrdquo in Proceedings of the 1973 National Symposium onFracture Mechanics Part II ASTM International WestConshohocken PA USA 1974
[23] H Richard M Sander M Fulland and G Kullmer ldquoDe-velopment of fatigue crack growth in real structuresrdquo Engi-neering Fracture Mechanics vol 75 no 3-4 pp 331ndash3402008
[24] K Tanaka ldquoFatigue crack propagation from a crack inclinedto the cyclic tensile axisrdquo Engineering Fracture Mechanicsvol 6 no 3 pp 493ndash507 1974
[25] P Paris and F Erdogan ldquoA critical analysis of crack prop-agation lawsrdquo Journal of Basic Engineering vol 85 no 4pp 528ndash533 1963
[26] L Coffin ldquoCyclic deformation and fatigue of metalsrdquo eBook Fatigue and Staying Power of Metals Izo IL USSRMoscow Russia 1963
[27] A Wohler ldquoVersuche zur Ermittlung der auf die Eisen-bahnwagenachsen einwirkenden Krafte und die Wider-standsfahigkeit der Wagen-Achsenrdquo Zeitschrift fur Bauwesenvol 10 no 1860 pp 583ndash614 1860
[28] J M Alegre M Preciado and D Ferrentildeo ldquoStudy of thefatigue failure of an anti-return valve of a high pressuremachinerdquo Engineering Failure Analysis vol 14 no 2pp 408ndash416 2007
[29] H Chen Q Wang W Zeng et al ldquoDynamic brittle crackpropagation modeling using singular edge-based smoothedfinite element method with local mesh rezoningrdquo EuropeanJournal of MechanicsmdashASolids vol 76 pp 208ndash223 2019
[30] S Sajith K S R Krishna Murthy and P Robi ldquoPrediction ofAccurate Mixed Mode Fatigue Crack Growth Curves usingthe Parisrsquo Lawrdquo Journal of e Institution of Engineers (India)Series C vol 100 no 1 pp 165ndash174 2019
[31] G R Irwin ldquoAnalysis of stresses and strains near the end of acrack transversing a platerdquo Journal of Applied Mechanicsvol 24 pp 361ndash364 1957
[32] H A Richard F G Buchholz G Kullmer andM Schollmann 2D-and 3D-mixed mode fracture criteria KeyEngineering Materials Trans Tech Publication Stafa-ZurichSwitzerland 2003
[33] H A Richard B Schramm and N-H Schirmeisen ldquoCrackson Mixed Mode loading - ampeories experiments simula-tionsrdquo International Journal of Fatigue vol 62 pp 93ndash1032014
[34] F Erdogan and G C Sih ldquoOn the crack extension in platesunder plane loading and transverse shearrdquo Journal of BasicEngineering vol 85 no 4 pp 519ndash525 1963
[35] S Sajith K S R K Murthy and P S Robi ldquoExperimental andnumerical investigation of mixed mode fatigue crack growthmodels in aluminum 6061-T6rdquo International Journal of Fa-tigue vol 130 p 105285 2020
[36] F Antunes R Branco J Ferreira and L Borrego ldquoStressintensity factor solutions for CTS mixed mode specimenrdquoFrattura ed Integrita Strutturale vol 13 no 48 pp 676ndash6922019
[37] O Demir A O Ayhan S Iric and H Lekesiz ldquoEvaluation ofmixed mode-III criteria for fatigue crack propagation usingexperiments and modelingrdquo Chinese Journal of Aeronauticsvol 31 no 7 pp 1525ndash1534 2018
[38] H Richard Fracture predictions for cracks exposed to super-imposed normal and shear stresses International NuclearInformation System Vienna Austria 1985
[39] X-T Miao Q Yu C-Y Zhou J Li Y-Z Wang andX-H He ldquoExperimental and numerical investigation onfracture behavior of CTS specimen under I-II mixed modeloadingrdquo European Journal of MechanicsmdashASolids vol 72pp 235ndash244 2018
[40] S Sajith S S Shukla K S R K Murthy and P S RobildquoMixed mode fatigue crack growth studies in AISI 316stainless steelrdquo European Journal of MechanicsmdashASolidsvol 80 p 103898 2020
14 Advances in Materials Science and Engineering
effective models to predict fatigue crack propagation andfatigue lifetime Several experimental models have beenproposed but carrying out the experiment is generallyexpensive and time-consuming Numerical simulationusing the extended finite element approach is a suitableway to minimize the time and cost associated with ex-perimental work Under linear elastic fracture mechanics(LEFM) analysis the SIFs are used to describe displace-ments and stresses around the crack front as well as theanalysis of fatigue crack propagation As the crackpropagates larger the SIF increases until it reaches acritical range where the assembly might start to deformby initiating fracture As a result of the complex mode ofapplied loads the geometry specification mixed modes (III) are regularly the general types of loads dependent onΔKeq used in the fatigue life prediction [21ndash24] ampe aim ofthis paper is to use the extended finite element methodimplemented by ANSYS APDL 192 to investigate theeffect of the loading angle for the CTS specimen withdifferent initial cracks
2 Numerical Predication of Mixed-ModeFatigue Life
Mainly three approaches have been commonly used forillustration of the fatigue assessment of materials which arethe fracture mechanics method established by [25] thestrain-life method proposed independently by [26] and thestress-life method proposed by [27] In the present study thefirst method was used for fatigue life prediction by which thecrack tip can be individually defined by the SIFsamperefore itis essential to estimate the fatigue crack path accurately inorder to predict the fatigue life assessment To this end themaximum tangential stress criterion theory was used topredict the crack deflection angle [28ndash30] as
θ 2 arctan14
KI
KII
+14
KI
KII
1113888 1113889
2
+ 8
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ forKII lt 0
(1)
θ 2 arctan14
KI
KII
minus14
KI
KII
1113888 1113889
2
+ 8
11139741113972
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ forKII gt 0
(2)
ampe two possibilities for the crack growth direction areshown in Figure 1
ampe calculation of fatigue crack growth using the cor-responding stress intensity factor is the most widely usedmethod for structures under mixed-mode dynamic loadingUsing a modified formula of Paris law a researcher [24]proposed a power law for the fatigue crack growth rela-tionship with ΔKeq which is specified as
da
dN C ΔKeq1113872 1113873
m(3)
From equation (3) for a crack increment the number oflife cycles of fatigue may be predicted as
1113946
Δa
0
da
C ΔKeq1113872 1113873m 1113946
ΔN
0
dN ΔN (4)
Several of the commonly used ΔKeq formulas accom-panied by the proposed authors are listed in Table 1
21 Numerical Results and Discussion Finite element codeANSYS Mechanical APDL 192 is implemented to simulatethe different types of loading angle of the compact tensionspecimen (CTS) ampe CTS geometry is shown in Figure 2(a)and its suggested loading angles are shown in Figure 2(b)ampere are three types of cracks that can be introduced inANSYS which are arbitrary semielliptical and premeshedcrack ampe premeshed crack method involves the crack frontthat is used by the Smart Crack Growth analysis enginewhere the failure criterion is the stress intensity factor ampepreviously created node sets are allocated to the crack frontwithin the premesh crack unit and the top and bottom facesof the crack ampe crack coordinate system is referenced ampenumber of contours for solution is set to 5 ampese are theldquoloopsrdquo through the mesh around the crack tip which areused to evaluate the stress intensity factor by integrating thecrack tip region strain energy In the analysis the fracturemechanics method avoids the stress singularities at the cracktip ampe process is repeated for the top and bottom crackfaces ampe new implemented feature in ANSYS is the ldquoSmartCrack Growthrdquo with tetrahedron mesh added after finishingthe requirements of the ldquopremeshed crackrdquo in which theuser can select the type of crack growth option
KI
KI
InitialCrack
ndashKII
ndashKII+θ
Keq
Keq
Crack
Propagation
(a)
KI
KI
KII
KIIndashθ
Keq
KeqInitialCrack
Crack
Propagation
(b)
Figure 1 Crack growth angle (a) KII rang 0 (b) KII lang 0
2 Advances in Materials Science and Engineering
Table 1 Commonly applied ΔKeq formulas
Model provided by authors ΔKeq expression
[24] ΔKeq (ΔK2I + 2ΔK2
II)12
[24] ΔKeq (ΔK4I + 8ΔK4
II)14
[31] ΔKeq ΔK2
1 + ΔK2II
1113969
[32 33] ΔKeq ΔKI
2+12
ΔK2I + 4(1155ΔK2
II)
1113969
[21] ΔKeq (10519 times K4I minus 0035 times K4
II + 23056 times K2I times K2
II)12
[34] ΔKeq cosθ2
[KIcos2θ2
minus32KII sin θ]
Figure 3 3D ANSYS model for CTS specimen (no of nodes 201317 and no of elements 132886)
10W = 9018 27 27
54
425a
Oslash17
154
(a)
F
F
α
0deg 90deg15deg 75deg
30deg 60deg
(b)
Figure 2 CTS geometry (a) dimensions (in mm) and (b) loading angles
Advances in Materials Science and Engineering 3
Figure 3 displays the 3D finite element layout for thegeometry dimensions of the CTS with number ofnodes 201317 and number of elements 132886 Based onthe values of the loading angle (α) which can be changed instages of 15deg from 0 to 90deg the modes of loading will changeaccording to the value of the angle from mixed-modeloading (α15deg 30deg 45deg 60deg and 75deg) pure mode II (α 0deg)and pure mode I (α 90deg) ampe studied material was alu-minium alloy having an elasticity module of E 74GPav 033 σY 517MPa tensile strength 579MPa fracturetoughness KIC 3295MPa
m
radic threshold SIF
Kth 315MPam
radic Paris constant C 43378times10minus 7 7 (mm
cycle)(MPam
radic) and Paris exponent m 26183 [35]
ampe geometryrsquos thickness is 5mmwhile the ending partsrsquothicknesses are 10mm which can protect the specimenrsquoslinking holes from failing ampe plane-stress conditions areassumed in the simulations according to the thickness valuewhich is small compared to other dimensions as well as forprecise comparison with other researchers who assumedthat for the same geometry [35] ampe value of load isP 16 kN and the precrack length is a 45mm
ampe finite element model of ANSYS shown in Figure 3 iscompletely consistent with the state of experimental testsperformed by [21 35ndash37] In the present simulation the CTSgeometries are loaded in various angles 30deg 45deg 60deg and 75degin the original direction of cracking for the comparison ofΔKeq and crack paths directionsampere are also other loadingangles 0deg 30deg 60deg and 90deg to compare the direction of crackpropagation and assessment of fatigue life In the last casethree different crack length geometries of the CTS geome-tries have been simulated according to the initial cracklength of aW 03 05 and 07
Figure 4 shows the distribution of the equivalent loadingused in ANSYS analysis of the present work ampe uniaxialload F is correlated to the equivalent loads acting on differentholes of the six holes based on the following formulas [38]
F1 F6 F 05 cos α +c
bsin α1113874 1113875
F2 F5 F sin α
F3 F4 F 05lowast cos α minus (cb)sin α( 1113857
(5)
where α is the angle of load and c and b are length parametershown by Figure 4 (c b 54mm) ampe final values of allloading forces with different loading angles are shown inTable 2
ampe analytical solution for KI of this specimen is pro-vided by [23] as
KI F
Wt
πa
radic cos α(1 minus aW)
026 + 265α(W minus a)
1 + 055α(W minus 5) minus 008(a(W minus a))2
1113971
(6)
where F is the uniaxial load (Figure 2(b)) a is a crackrsquoslength W is the width of geometry and t is the geometrythickness It is worth comparing the SIFs for this geometrywith the analytical solution ofKI (equation (6)) for differentthicknesses (3 6 9 and 12mm) as represented in Figure 5
c
F1F2
F3
c
F4
F5
F6b
Figure 4 Loading and boundary condition for CTS geometry
Table 2 Values of forces F1 to F6 according to load angle α
α F2 F5 F1 F6 F3 F415 0259 F 0742 F 0224 F30 05 F 0933 F minus 0067 F45 0707 F 1061 F minus 0354 F60 0866 F 1116 F minus 0616 F75 0966 F 1095 F minus 0837 F90 F F minus F
800
700
600
500
400
300
200
100
0
K I (M
Pa m
m1
2 )
40 45 50 55 60 65 70 75 80Crack length a (mm)
t = 3mm
t = 6mm
t = 9mm
t = 12mm
ANSYSRichard criterion
Figure 5 Comparison for KI with analytical solution for differentspecimen thicknesses 3mm 6mm 9mm and 12mm
4 Advances in Materials Science and Engineering
As can be seen in this figure the values of SIFs are decreasingas the thickness increasing and the agreement with theanalytical solution obtained by equation (2) is obvious
ampe predicted values of ΔKeq were compared to theexperimental values performed by [37] as well as the ana-lytical expression given by [24 33 34 37] for different valuesof loading angles 30deg 45deg 60deg and 75degampese comparisons areshown in Figures 6ndash9
Figure 6 demonstrates the comparisons for the variationof the crack length as a function of the corresponding ΔKeq
with load angle of 30deg and the amount of applied loading is88 kN (loading ratio R 01) As illustrated in this figure thefour criteria of ΔKeq have around the same tendency as dataobtained from experimental tests by [37] For this loadingangle even there is a slight divergence for the [34] criterionespecially after the mid of the crack length In Figure 7 theloading angle is 45deg and the amount of the load is 114 kNwith R 01 ampe agreement for the ANSYS results with theexperimental results is clear as well as with other analyticalcriteria
Figure 8 shows the comparisons for the equivalent SIFfor 60deg loading angle and the amount of load is 1365 kNwithload ratio R 01 ampere is clearly divergence of the initialvalues of the equivalent SIF for all criteria which is de-pendent in the initial step of the crack growth ampis di-vergence is increased for the loading angle of 75deg and amountof load is 13 kN (R 01) as shown in Figure 9 ampough all ofthe four criteria almost provide close predictions to theexperimental results obtained by [37]
In addition the paths of crack growth of load angles 30deg45deg 60deg and 75deg were validated with comparison to practicalcrack paths tested by [36] as shown in Figure 10 with goodagreement
ampe predicted crack growth path has been compared tothe experimental test performed by [39] with different cracklengths (aw 03 05 and 07) and different load angles 030 60 and 90 as shown in Table 3
For pure loading of mode II (α 0deg) the direction of thecrack growth is controlled by the second mode of SIF (KII)
47 49 51 53 5545Crack length (mm)
K ep (M
Pa m
12 )
Experimental(Demir et al 2018)Richard criterion
Tanaka criterionDemir et al (2017)criterion
Erdogan and Sih ANSYS
201816141210
86420
Figure 7 Equivalent SIF with loading angle 45deg
47 49 51 53 5545Crack length (mm)
0
5
10
15
20
25
K ep (M
Pa m
12 )
Experimental(Demir et al 2018)Richard criterion
Tanaka criterionDemir et al (2017)criterion
Erdogan and Sih ANSYS
Figure 8 Equivalent SIF with loading angle 60deg
0
5
10
15
20
25
45 47 49 51 53 55
Keq
(MPa
m1
2 )
Crack length (mm)
Experimental(Demir et al 2018)Richard criterionErdogan and Sih
Tanaka criterionDemir et al (2017)criterionAnsys
Figure 9 Equivalent SIF with loading angle 75deg
Experimental(Demir et al 2018)Richard criterion
Tanaka criterionDemir et al (2017)criterion
Erdogan and Sih ANSYS
0
5
10
15
20
25
K eq (M
Pa m
12 )
47 49 51 53 55 57 59 61 63 6545Crack length (mm)
Figure 6 Equivalent SIF with loading angle 30deg
Advances in Materials Science and Engineering 5
4250 4450 4650 4850 5050 5250 5450 5650 5850 6050 6250 6450X (mm)
Y (m
m)
1800
1600
1400
1200
1000
800
600
400
200
000
75 deg (Antunes et al 2019)75 deg ANSYS60 deg Exp (Antunes et al 2019)60 deg ANSYS
45 deg Exp (Antunes et al 2019)45 deg ANSYS30 deg Exp (Antunes et al 2019)30 deg ANSYS
Figure 10 Comparison for crack growth paths between present study results and experimental results of [36]
Table 3 Comparison of paths of crack propagation among ANSYSrsquos results and experimental results [39]
ANSYS Experimental test [39]
(aw= 03) α 0deg
6 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 03) α 30deg
(aw= 03) α 60deg
Advances in Materials Science and Engineering 7
Table 3 Continued
ANSYS Experimental test [39]
(aw= 03) α 90deg
(aw= 05) α 0deg
8 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 30deg
(aw= 05) α 60deg
Advances in Materials Science and Engineering 9
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 90deg
(aw= 07) α 0deg
10 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 30deg
(aw= 07) α 60deg
Advances in Materials Science and Engineering 11
and the cracks all grow with different angles based onequation (2) as the values of KII are higher whereas forcomplete mode I loading (α 90deg) crack paths are domi-nated by the first mode KI and the crack is going to growstraight ampe influence of the first mode (KI) decreases as αincreases and vice versa for mode II
ampe crack direction angles of three geometries withdifferent initial length of cracks (aW 03 05 and 07)are 6deg 10deg and 12deg respectively which are identical to the
experimental angles tested by [39] Furthermore formixed-mode loading in the CTS geometries with loadingangle α 30deg the cracks are propagated to some extentdiffering to the original crack front direction For thesethree geometries the mixed-mode crack propagationangles (aW 03 05 and 07) are about 55deg 85deg and85deg respectively which are almost in agreement withexperimental angles of [39]
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 90deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 11 Comparison of simulated and experimental [35] fatiguelife for loading angle 30deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350 400 450 500000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 12 Comparison of simulated and experimental [35] fatiguelife for loading angle 60deg
12 Advances in Materials Science and Engineering
For the above three loading angles situations (α 90degα 0deg and α 30deg) regardless of the length of the initialcrack for the same load angle (α) the angles of the crackgrowth in each case are close to each other In contrast forcombined mode (III) loading with loading angle α 60deg thedirections of crack are highly dependent on the initial lengthof the crack On the other hand for small crack length (aW 03 and 05) the cracks also grow varying from theinitial crack tip against the clockwise direction with crackpropagation angles of 25deg and 30deg respectively which areidentical to those angles in the experimental test [39]Furthermore if the initial crack length is longer than theprevious cases (aW 07) the direction of the crack growthand the direction of the initial crack happen to be identicaland are presented in Table 3
In order to assess the fatigue life the number of cycles fordifferent loading angles 30 and 60 calculated in this study iscompared with the experimental results observed by [35]under mixed loading mode (III) resulting in clear agree-ments as can be observed in Figures 11 and 12 ampe amountof the applied load is F 16 kN with load ratio R 01whereas the Paris constant as mentioned previously isC 43378times10minus 7 (mmcycle)(MPa
m
radic) and m 26183
[40]
3 Conclusions
For the CTS specimen the finite element analysis of themixed-mode fatigue crack propagation was carried out usingANSYS Mechanical APDL and compared with experimentaldata for various angles of loading ampe predicted values ofΔKeq by the experimental and numerical methods are ob-served to be in good agreement with the present study re-sults ampe mixed-mode fatigue life is predicted andcompared with experimental data for two different loadingangles of 30deg and 60deg Interestingly the theoretically pre-dicted trajectories of the crack growth for all specimens areidentical to the experimental determined paths for differentloading angles Furthermore the predicted values of SIFs (KIand KII) are in good agreement with analytical solutions Itcan be stated that the structure of specimen geometry and itsconfiguration play an important role in obtaining highervalues of mixed-mode SIFs values ampis happens to besignificant in terms of applied loads particularly for highervalues of KII
Data Availability
ampe data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
References
[1] R I Stephens A Fatemi R R Stephens and H O FuchsMetal Fatigue in Engineering John Wiley amp Sons HobokenNJ USA 2000
[2] V V Bolotin Mechanics of Fatigue CRC Press Boca RatonFL USA 1999
[3] C A Brebbia and A V Farahani Fatigue Damage of Ma-terials Experiment and Analysis WIT Press SouthamptonUK 2003
[4] N E DowlingMechanical Behavior of Materials EngineeringMethods for Deformation Fracture and Fatigue PearsonLondon UK 2012
[5] G R Halford Fatigue and Durability of Structural MaterialsASM International London UK 2006
[6] R O Ritchie ldquoMechanisms of fatigue-crack propagation inductile and brittle solidsrdquo International Journal of Fracturevol 100 no 1 pp 55ndash83 1999
[7] S Suresh Fatigue of Materials Cambridge University PressCambridge UK 1998
[8] K N Solanki Two and ree-Dimensional Finite ElementAnalysis of Plasticity-Induced Fatigue Crack Closure AComprehensive Parametric Study Mississippi State Univer-sity Starkville MS USA 2002
[9] A M Alshoaibi and A K Ariffin ldquoFinite element modeling offatigue crack propagation using a self adaptive mesh strategyrdquoInternational Review of Aerospace Engineering (IREASE)vol 8 no 6 pp 209ndash215 2015
[10] A M Alshoaibi ldquoFinite element procedures for the numericalsimulation of fatigue crack propagation under mixed modeloadingrdquo Structural Engineering and Mechanics vol 35 no 3pp 283ndash299 2010
[11] A M Alshoaibi ldquoAdaptive finite element modeling of fatiguecrack propagationrdquo International Journal of Materials Scienceand Applications vol 2 no 3 pp 104ndash108 2013
[12] A M Alshoaibi ldquoAn adaptive finite element framework forfatigue crack propagation under constant amplitude loadingrdquoInternational Journal of Applied Science and Engineeringvol 13 no 3 pp 261ndash270 2015
[13] A M Alshoaibi ldquoComprehensive comparisons of two andthree dimensional numerical estimation of stress intensityfactors and crack propagation in linear elastic analysisrdquo In-ternational Journal of Integrated Engineering vol 11 no 6pp 45ndash52 2019
[14] A Handbook Fatigue and Fracture American Society ofMaterials vol 19 1996
[15] W He J Liu and D Xie ldquoNumerical study on fatigue crackgrowth at a web-stiffener of ship structural details by anobjected-oriented approach in conjunction with ABAQUSrdquoMarine Structures vol 35 pp 45ndash69 2014
[16] K Rege and D G Pavlou ldquoEffect of stop holes on structuralintegrity of offshore structures a numerical modelrdquo in Pro-ceedings of the Institution of Civil Engineers-Maritime Engi-neering ampomas Telford Ltd London UK 2019
[17] H Rhee ldquoFatigue crack growth analyses of offshore structuraltubular jointrdquo Journal of Offshore Mechanics and Arctic En-gineering vol 111 no 1 pp 49ndash55 1989
[18] H Riahi P Bressolette A Chateauneuf C Bouraoui andR Fathallah ldquoReliability analysis and inspection updating bystochastic response surface of fatigue cracks in mixed moderdquoEngineering Structures vol 33 no 12 pp 3392ndash3401 2011
[19] S Qi L X Cai C Bao H Chen K K Shi and H L WuldquoAnalytical theory for fatigue crack propagation rates ofmixed-mode I-II cracks and its applicationrdquo InternationalJournal of Fatigue vol 119 pp 150ndash159 2019
[20] A F Liu Mechanics and Mechanisms of Fracture An In-troduction ASM International London UK 2005
[21] O Demir A O Ayhan and S Iriccedil ldquoA new specimen formixed mode-III fracture tests modeling experiments and
Advances in Materials Science and Engineering 13
criteria developmentrdquo Engineering Fracture Mechanicsvol 178 pp 457ndash476 2017
[22] M Hussain S Pu and J Underwood ldquoStrain energy releaserate for a crack under combined mode I and mode II FractureAnalysisrdquo in Proceedings of the 1973 National Symposium onFracture Mechanics Part II ASTM International WestConshohocken PA USA 1974
[23] H Richard M Sander M Fulland and G Kullmer ldquoDe-velopment of fatigue crack growth in real structuresrdquo Engi-neering Fracture Mechanics vol 75 no 3-4 pp 331ndash3402008
[24] K Tanaka ldquoFatigue crack propagation from a crack inclinedto the cyclic tensile axisrdquo Engineering Fracture Mechanicsvol 6 no 3 pp 493ndash507 1974
[25] P Paris and F Erdogan ldquoA critical analysis of crack prop-agation lawsrdquo Journal of Basic Engineering vol 85 no 4pp 528ndash533 1963
[26] L Coffin ldquoCyclic deformation and fatigue of metalsrdquo eBook Fatigue and Staying Power of Metals Izo IL USSRMoscow Russia 1963
[27] A Wohler ldquoVersuche zur Ermittlung der auf die Eisen-bahnwagenachsen einwirkenden Krafte und die Wider-standsfahigkeit der Wagen-Achsenrdquo Zeitschrift fur Bauwesenvol 10 no 1860 pp 583ndash614 1860
[28] J M Alegre M Preciado and D Ferrentildeo ldquoStudy of thefatigue failure of an anti-return valve of a high pressuremachinerdquo Engineering Failure Analysis vol 14 no 2pp 408ndash416 2007
[29] H Chen Q Wang W Zeng et al ldquoDynamic brittle crackpropagation modeling using singular edge-based smoothedfinite element method with local mesh rezoningrdquo EuropeanJournal of MechanicsmdashASolids vol 76 pp 208ndash223 2019
[30] S Sajith K S R Krishna Murthy and P Robi ldquoPrediction ofAccurate Mixed Mode Fatigue Crack Growth Curves usingthe Parisrsquo Lawrdquo Journal of e Institution of Engineers (India)Series C vol 100 no 1 pp 165ndash174 2019
[31] G R Irwin ldquoAnalysis of stresses and strains near the end of acrack transversing a platerdquo Journal of Applied Mechanicsvol 24 pp 361ndash364 1957
[32] H A Richard F G Buchholz G Kullmer andM Schollmann 2D-and 3D-mixed mode fracture criteria KeyEngineering Materials Trans Tech Publication Stafa-ZurichSwitzerland 2003
[33] H A Richard B Schramm and N-H Schirmeisen ldquoCrackson Mixed Mode loading - ampeories experiments simula-tionsrdquo International Journal of Fatigue vol 62 pp 93ndash1032014
[34] F Erdogan and G C Sih ldquoOn the crack extension in platesunder plane loading and transverse shearrdquo Journal of BasicEngineering vol 85 no 4 pp 519ndash525 1963
[35] S Sajith K S R K Murthy and P S Robi ldquoExperimental andnumerical investigation of mixed mode fatigue crack growthmodels in aluminum 6061-T6rdquo International Journal of Fa-tigue vol 130 p 105285 2020
[36] F Antunes R Branco J Ferreira and L Borrego ldquoStressintensity factor solutions for CTS mixed mode specimenrdquoFrattura ed Integrita Strutturale vol 13 no 48 pp 676ndash6922019
[37] O Demir A O Ayhan S Iric and H Lekesiz ldquoEvaluation ofmixed mode-III criteria for fatigue crack propagation usingexperiments and modelingrdquo Chinese Journal of Aeronauticsvol 31 no 7 pp 1525ndash1534 2018
[38] H Richard Fracture predictions for cracks exposed to super-imposed normal and shear stresses International NuclearInformation System Vienna Austria 1985
[39] X-T Miao Q Yu C-Y Zhou J Li Y-Z Wang andX-H He ldquoExperimental and numerical investigation onfracture behavior of CTS specimen under I-II mixed modeloadingrdquo European Journal of MechanicsmdashASolids vol 72pp 235ndash244 2018
[40] S Sajith S S Shukla K S R K Murthy and P S RobildquoMixed mode fatigue crack growth studies in AISI 316stainless steelrdquo European Journal of MechanicsmdashASolidsvol 80 p 103898 2020
14 Advances in Materials Science and Engineering
Table 1 Commonly applied ΔKeq formulas
Model provided by authors ΔKeq expression
[24] ΔKeq (ΔK2I + 2ΔK2
II)12
[24] ΔKeq (ΔK4I + 8ΔK4
II)14
[31] ΔKeq ΔK2
1 + ΔK2II
1113969
[32 33] ΔKeq ΔKI
2+12
ΔK2I + 4(1155ΔK2
II)
1113969
[21] ΔKeq (10519 times K4I minus 0035 times K4
II + 23056 times K2I times K2
II)12
[34] ΔKeq cosθ2
[KIcos2θ2
minus32KII sin θ]
Figure 3 3D ANSYS model for CTS specimen (no of nodes 201317 and no of elements 132886)
10W = 9018 27 27
54
425a
Oslash17
154
(a)
F
F
α
0deg 90deg15deg 75deg
30deg 60deg
(b)
Figure 2 CTS geometry (a) dimensions (in mm) and (b) loading angles
Advances in Materials Science and Engineering 3
Figure 3 displays the 3D finite element layout for thegeometry dimensions of the CTS with number ofnodes 201317 and number of elements 132886 Based onthe values of the loading angle (α) which can be changed instages of 15deg from 0 to 90deg the modes of loading will changeaccording to the value of the angle from mixed-modeloading (α15deg 30deg 45deg 60deg and 75deg) pure mode II (α 0deg)and pure mode I (α 90deg) ampe studied material was alu-minium alloy having an elasticity module of E 74GPav 033 σY 517MPa tensile strength 579MPa fracturetoughness KIC 3295MPa
m
radic threshold SIF
Kth 315MPam
radic Paris constant C 43378times10minus 7 7 (mm
cycle)(MPam
radic) and Paris exponent m 26183 [35]
ampe geometryrsquos thickness is 5mmwhile the ending partsrsquothicknesses are 10mm which can protect the specimenrsquoslinking holes from failing ampe plane-stress conditions areassumed in the simulations according to the thickness valuewhich is small compared to other dimensions as well as forprecise comparison with other researchers who assumedthat for the same geometry [35] ampe value of load isP 16 kN and the precrack length is a 45mm
ampe finite element model of ANSYS shown in Figure 3 iscompletely consistent with the state of experimental testsperformed by [21 35ndash37] In the present simulation the CTSgeometries are loaded in various angles 30deg 45deg 60deg and 75degin the original direction of cracking for the comparison ofΔKeq and crack paths directionsampere are also other loadingangles 0deg 30deg 60deg and 90deg to compare the direction of crackpropagation and assessment of fatigue life In the last casethree different crack length geometries of the CTS geome-tries have been simulated according to the initial cracklength of aW 03 05 and 07
Figure 4 shows the distribution of the equivalent loadingused in ANSYS analysis of the present work ampe uniaxialload F is correlated to the equivalent loads acting on differentholes of the six holes based on the following formulas [38]
F1 F6 F 05 cos α +c
bsin α1113874 1113875
F2 F5 F sin α
F3 F4 F 05lowast cos α minus (cb)sin α( 1113857
(5)
where α is the angle of load and c and b are length parametershown by Figure 4 (c b 54mm) ampe final values of allloading forces with different loading angles are shown inTable 2
ampe analytical solution for KI of this specimen is pro-vided by [23] as
KI F
Wt
πa
radic cos α(1 minus aW)
026 + 265α(W minus a)
1 + 055α(W minus 5) minus 008(a(W minus a))2
1113971
(6)
where F is the uniaxial load (Figure 2(b)) a is a crackrsquoslength W is the width of geometry and t is the geometrythickness It is worth comparing the SIFs for this geometrywith the analytical solution ofKI (equation (6)) for differentthicknesses (3 6 9 and 12mm) as represented in Figure 5
c
F1F2
F3
c
F4
F5
F6b
Figure 4 Loading and boundary condition for CTS geometry
Table 2 Values of forces F1 to F6 according to load angle α
α F2 F5 F1 F6 F3 F415 0259 F 0742 F 0224 F30 05 F 0933 F minus 0067 F45 0707 F 1061 F minus 0354 F60 0866 F 1116 F minus 0616 F75 0966 F 1095 F minus 0837 F90 F F minus F
800
700
600
500
400
300
200
100
0
K I (M
Pa m
m1
2 )
40 45 50 55 60 65 70 75 80Crack length a (mm)
t = 3mm
t = 6mm
t = 9mm
t = 12mm
ANSYSRichard criterion
Figure 5 Comparison for KI with analytical solution for differentspecimen thicknesses 3mm 6mm 9mm and 12mm
4 Advances in Materials Science and Engineering
As can be seen in this figure the values of SIFs are decreasingas the thickness increasing and the agreement with theanalytical solution obtained by equation (2) is obvious
ampe predicted values of ΔKeq were compared to theexperimental values performed by [37] as well as the ana-lytical expression given by [24 33 34 37] for different valuesof loading angles 30deg 45deg 60deg and 75degampese comparisons areshown in Figures 6ndash9
Figure 6 demonstrates the comparisons for the variationof the crack length as a function of the corresponding ΔKeq
with load angle of 30deg and the amount of applied loading is88 kN (loading ratio R 01) As illustrated in this figure thefour criteria of ΔKeq have around the same tendency as dataobtained from experimental tests by [37] For this loadingangle even there is a slight divergence for the [34] criterionespecially after the mid of the crack length In Figure 7 theloading angle is 45deg and the amount of the load is 114 kNwith R 01 ampe agreement for the ANSYS results with theexperimental results is clear as well as with other analyticalcriteria
Figure 8 shows the comparisons for the equivalent SIFfor 60deg loading angle and the amount of load is 1365 kNwithload ratio R 01 ampere is clearly divergence of the initialvalues of the equivalent SIF for all criteria which is de-pendent in the initial step of the crack growth ampis di-vergence is increased for the loading angle of 75deg and amountof load is 13 kN (R 01) as shown in Figure 9 ampough all ofthe four criteria almost provide close predictions to theexperimental results obtained by [37]
In addition the paths of crack growth of load angles 30deg45deg 60deg and 75deg were validated with comparison to practicalcrack paths tested by [36] as shown in Figure 10 with goodagreement
ampe predicted crack growth path has been compared tothe experimental test performed by [39] with different cracklengths (aw 03 05 and 07) and different load angles 030 60 and 90 as shown in Table 3
For pure loading of mode II (α 0deg) the direction of thecrack growth is controlled by the second mode of SIF (KII)
47 49 51 53 5545Crack length (mm)
K ep (M
Pa m
12 )
Experimental(Demir et al 2018)Richard criterion
Tanaka criterionDemir et al (2017)criterion
Erdogan and Sih ANSYS
201816141210
86420
Figure 7 Equivalent SIF with loading angle 45deg
47 49 51 53 5545Crack length (mm)
0
5
10
15
20
25
K ep (M
Pa m
12 )
Experimental(Demir et al 2018)Richard criterion
Tanaka criterionDemir et al (2017)criterion
Erdogan and Sih ANSYS
Figure 8 Equivalent SIF with loading angle 60deg
0
5
10
15
20
25
45 47 49 51 53 55
Keq
(MPa
m1
2 )
Crack length (mm)
Experimental(Demir et al 2018)Richard criterionErdogan and Sih
Tanaka criterionDemir et al (2017)criterionAnsys
Figure 9 Equivalent SIF with loading angle 75deg
Experimental(Demir et al 2018)Richard criterion
Tanaka criterionDemir et al (2017)criterion
Erdogan and Sih ANSYS
0
5
10
15
20
25
K eq (M
Pa m
12 )
47 49 51 53 55 57 59 61 63 6545Crack length (mm)
Figure 6 Equivalent SIF with loading angle 30deg
Advances in Materials Science and Engineering 5
4250 4450 4650 4850 5050 5250 5450 5650 5850 6050 6250 6450X (mm)
Y (m
m)
1800
1600
1400
1200
1000
800
600
400
200
000
75 deg (Antunes et al 2019)75 deg ANSYS60 deg Exp (Antunes et al 2019)60 deg ANSYS
45 deg Exp (Antunes et al 2019)45 deg ANSYS30 deg Exp (Antunes et al 2019)30 deg ANSYS
Figure 10 Comparison for crack growth paths between present study results and experimental results of [36]
Table 3 Comparison of paths of crack propagation among ANSYSrsquos results and experimental results [39]
ANSYS Experimental test [39]
(aw= 03) α 0deg
6 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 03) α 30deg
(aw= 03) α 60deg
Advances in Materials Science and Engineering 7
Table 3 Continued
ANSYS Experimental test [39]
(aw= 03) α 90deg
(aw= 05) α 0deg
8 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 30deg
(aw= 05) α 60deg
Advances in Materials Science and Engineering 9
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 90deg
(aw= 07) α 0deg
10 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 30deg
(aw= 07) α 60deg
Advances in Materials Science and Engineering 11
and the cracks all grow with different angles based onequation (2) as the values of KII are higher whereas forcomplete mode I loading (α 90deg) crack paths are domi-nated by the first mode KI and the crack is going to growstraight ampe influence of the first mode (KI) decreases as αincreases and vice versa for mode II
ampe crack direction angles of three geometries withdifferent initial length of cracks (aW 03 05 and 07)are 6deg 10deg and 12deg respectively which are identical to the
experimental angles tested by [39] Furthermore formixed-mode loading in the CTS geometries with loadingangle α 30deg the cracks are propagated to some extentdiffering to the original crack front direction For thesethree geometries the mixed-mode crack propagationangles (aW 03 05 and 07) are about 55deg 85deg and85deg respectively which are almost in agreement withexperimental angles of [39]
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 90deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 11 Comparison of simulated and experimental [35] fatiguelife for loading angle 30deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350 400 450 500000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 12 Comparison of simulated and experimental [35] fatiguelife for loading angle 60deg
12 Advances in Materials Science and Engineering
For the above three loading angles situations (α 90degα 0deg and α 30deg) regardless of the length of the initialcrack for the same load angle (α) the angles of the crackgrowth in each case are close to each other In contrast forcombined mode (III) loading with loading angle α 60deg thedirections of crack are highly dependent on the initial lengthof the crack On the other hand for small crack length (aW 03 and 05) the cracks also grow varying from theinitial crack tip against the clockwise direction with crackpropagation angles of 25deg and 30deg respectively which areidentical to those angles in the experimental test [39]Furthermore if the initial crack length is longer than theprevious cases (aW 07) the direction of the crack growthand the direction of the initial crack happen to be identicaland are presented in Table 3
In order to assess the fatigue life the number of cycles fordifferent loading angles 30 and 60 calculated in this study iscompared with the experimental results observed by [35]under mixed loading mode (III) resulting in clear agree-ments as can be observed in Figures 11 and 12 ampe amountof the applied load is F 16 kN with load ratio R 01whereas the Paris constant as mentioned previously isC 43378times10minus 7 (mmcycle)(MPa
m
radic) and m 26183
[40]
3 Conclusions
For the CTS specimen the finite element analysis of themixed-mode fatigue crack propagation was carried out usingANSYS Mechanical APDL and compared with experimentaldata for various angles of loading ampe predicted values ofΔKeq by the experimental and numerical methods are ob-served to be in good agreement with the present study re-sults ampe mixed-mode fatigue life is predicted andcompared with experimental data for two different loadingangles of 30deg and 60deg Interestingly the theoretically pre-dicted trajectories of the crack growth for all specimens areidentical to the experimental determined paths for differentloading angles Furthermore the predicted values of SIFs (KIand KII) are in good agreement with analytical solutions Itcan be stated that the structure of specimen geometry and itsconfiguration play an important role in obtaining highervalues of mixed-mode SIFs values ampis happens to besignificant in terms of applied loads particularly for highervalues of KII
Data Availability
ampe data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
References
[1] R I Stephens A Fatemi R R Stephens and H O FuchsMetal Fatigue in Engineering John Wiley amp Sons HobokenNJ USA 2000
[2] V V Bolotin Mechanics of Fatigue CRC Press Boca RatonFL USA 1999
[3] C A Brebbia and A V Farahani Fatigue Damage of Ma-terials Experiment and Analysis WIT Press SouthamptonUK 2003
[4] N E DowlingMechanical Behavior of Materials EngineeringMethods for Deformation Fracture and Fatigue PearsonLondon UK 2012
[5] G R Halford Fatigue and Durability of Structural MaterialsASM International London UK 2006
[6] R O Ritchie ldquoMechanisms of fatigue-crack propagation inductile and brittle solidsrdquo International Journal of Fracturevol 100 no 1 pp 55ndash83 1999
[7] S Suresh Fatigue of Materials Cambridge University PressCambridge UK 1998
[8] K N Solanki Two and ree-Dimensional Finite ElementAnalysis of Plasticity-Induced Fatigue Crack Closure AComprehensive Parametric Study Mississippi State Univer-sity Starkville MS USA 2002
[9] A M Alshoaibi and A K Ariffin ldquoFinite element modeling offatigue crack propagation using a self adaptive mesh strategyrdquoInternational Review of Aerospace Engineering (IREASE)vol 8 no 6 pp 209ndash215 2015
[10] A M Alshoaibi ldquoFinite element procedures for the numericalsimulation of fatigue crack propagation under mixed modeloadingrdquo Structural Engineering and Mechanics vol 35 no 3pp 283ndash299 2010
[11] A M Alshoaibi ldquoAdaptive finite element modeling of fatiguecrack propagationrdquo International Journal of Materials Scienceand Applications vol 2 no 3 pp 104ndash108 2013
[12] A M Alshoaibi ldquoAn adaptive finite element framework forfatigue crack propagation under constant amplitude loadingrdquoInternational Journal of Applied Science and Engineeringvol 13 no 3 pp 261ndash270 2015
[13] A M Alshoaibi ldquoComprehensive comparisons of two andthree dimensional numerical estimation of stress intensityfactors and crack propagation in linear elastic analysisrdquo In-ternational Journal of Integrated Engineering vol 11 no 6pp 45ndash52 2019
[14] A Handbook Fatigue and Fracture American Society ofMaterials vol 19 1996
[15] W He J Liu and D Xie ldquoNumerical study on fatigue crackgrowth at a web-stiffener of ship structural details by anobjected-oriented approach in conjunction with ABAQUSrdquoMarine Structures vol 35 pp 45ndash69 2014
[16] K Rege and D G Pavlou ldquoEffect of stop holes on structuralintegrity of offshore structures a numerical modelrdquo in Pro-ceedings of the Institution of Civil Engineers-Maritime Engi-neering ampomas Telford Ltd London UK 2019
[17] H Rhee ldquoFatigue crack growth analyses of offshore structuraltubular jointrdquo Journal of Offshore Mechanics and Arctic En-gineering vol 111 no 1 pp 49ndash55 1989
[18] H Riahi P Bressolette A Chateauneuf C Bouraoui andR Fathallah ldquoReliability analysis and inspection updating bystochastic response surface of fatigue cracks in mixed moderdquoEngineering Structures vol 33 no 12 pp 3392ndash3401 2011
[19] S Qi L X Cai C Bao H Chen K K Shi and H L WuldquoAnalytical theory for fatigue crack propagation rates ofmixed-mode I-II cracks and its applicationrdquo InternationalJournal of Fatigue vol 119 pp 150ndash159 2019
[20] A F Liu Mechanics and Mechanisms of Fracture An In-troduction ASM International London UK 2005
[21] O Demir A O Ayhan and S Iriccedil ldquoA new specimen formixed mode-III fracture tests modeling experiments and
Advances in Materials Science and Engineering 13
criteria developmentrdquo Engineering Fracture Mechanicsvol 178 pp 457ndash476 2017
[22] M Hussain S Pu and J Underwood ldquoStrain energy releaserate for a crack under combined mode I and mode II FractureAnalysisrdquo in Proceedings of the 1973 National Symposium onFracture Mechanics Part II ASTM International WestConshohocken PA USA 1974
[23] H Richard M Sander M Fulland and G Kullmer ldquoDe-velopment of fatigue crack growth in real structuresrdquo Engi-neering Fracture Mechanics vol 75 no 3-4 pp 331ndash3402008
[24] K Tanaka ldquoFatigue crack propagation from a crack inclinedto the cyclic tensile axisrdquo Engineering Fracture Mechanicsvol 6 no 3 pp 493ndash507 1974
[25] P Paris and F Erdogan ldquoA critical analysis of crack prop-agation lawsrdquo Journal of Basic Engineering vol 85 no 4pp 528ndash533 1963
[26] L Coffin ldquoCyclic deformation and fatigue of metalsrdquo eBook Fatigue and Staying Power of Metals Izo IL USSRMoscow Russia 1963
[27] A Wohler ldquoVersuche zur Ermittlung der auf die Eisen-bahnwagenachsen einwirkenden Krafte und die Wider-standsfahigkeit der Wagen-Achsenrdquo Zeitschrift fur Bauwesenvol 10 no 1860 pp 583ndash614 1860
[28] J M Alegre M Preciado and D Ferrentildeo ldquoStudy of thefatigue failure of an anti-return valve of a high pressuremachinerdquo Engineering Failure Analysis vol 14 no 2pp 408ndash416 2007
[29] H Chen Q Wang W Zeng et al ldquoDynamic brittle crackpropagation modeling using singular edge-based smoothedfinite element method with local mesh rezoningrdquo EuropeanJournal of MechanicsmdashASolids vol 76 pp 208ndash223 2019
[30] S Sajith K S R Krishna Murthy and P Robi ldquoPrediction ofAccurate Mixed Mode Fatigue Crack Growth Curves usingthe Parisrsquo Lawrdquo Journal of e Institution of Engineers (India)Series C vol 100 no 1 pp 165ndash174 2019
[31] G R Irwin ldquoAnalysis of stresses and strains near the end of acrack transversing a platerdquo Journal of Applied Mechanicsvol 24 pp 361ndash364 1957
[32] H A Richard F G Buchholz G Kullmer andM Schollmann 2D-and 3D-mixed mode fracture criteria KeyEngineering Materials Trans Tech Publication Stafa-ZurichSwitzerland 2003
[33] H A Richard B Schramm and N-H Schirmeisen ldquoCrackson Mixed Mode loading - ampeories experiments simula-tionsrdquo International Journal of Fatigue vol 62 pp 93ndash1032014
[34] F Erdogan and G C Sih ldquoOn the crack extension in platesunder plane loading and transverse shearrdquo Journal of BasicEngineering vol 85 no 4 pp 519ndash525 1963
[35] S Sajith K S R K Murthy and P S Robi ldquoExperimental andnumerical investigation of mixed mode fatigue crack growthmodels in aluminum 6061-T6rdquo International Journal of Fa-tigue vol 130 p 105285 2020
[36] F Antunes R Branco J Ferreira and L Borrego ldquoStressintensity factor solutions for CTS mixed mode specimenrdquoFrattura ed Integrita Strutturale vol 13 no 48 pp 676ndash6922019
[37] O Demir A O Ayhan S Iric and H Lekesiz ldquoEvaluation ofmixed mode-III criteria for fatigue crack propagation usingexperiments and modelingrdquo Chinese Journal of Aeronauticsvol 31 no 7 pp 1525ndash1534 2018
[38] H Richard Fracture predictions for cracks exposed to super-imposed normal and shear stresses International NuclearInformation System Vienna Austria 1985
[39] X-T Miao Q Yu C-Y Zhou J Li Y-Z Wang andX-H He ldquoExperimental and numerical investigation onfracture behavior of CTS specimen under I-II mixed modeloadingrdquo European Journal of MechanicsmdashASolids vol 72pp 235ndash244 2018
[40] S Sajith S S Shukla K S R K Murthy and P S RobildquoMixed mode fatigue crack growth studies in AISI 316stainless steelrdquo European Journal of MechanicsmdashASolidsvol 80 p 103898 2020
14 Advances in Materials Science and Engineering
Figure 3 displays the 3D finite element layout for thegeometry dimensions of the CTS with number ofnodes 201317 and number of elements 132886 Based onthe values of the loading angle (α) which can be changed instages of 15deg from 0 to 90deg the modes of loading will changeaccording to the value of the angle from mixed-modeloading (α15deg 30deg 45deg 60deg and 75deg) pure mode II (α 0deg)and pure mode I (α 90deg) ampe studied material was alu-minium alloy having an elasticity module of E 74GPav 033 σY 517MPa tensile strength 579MPa fracturetoughness KIC 3295MPa
m
radic threshold SIF
Kth 315MPam
radic Paris constant C 43378times10minus 7 7 (mm
cycle)(MPam
radic) and Paris exponent m 26183 [35]
ampe geometryrsquos thickness is 5mmwhile the ending partsrsquothicknesses are 10mm which can protect the specimenrsquoslinking holes from failing ampe plane-stress conditions areassumed in the simulations according to the thickness valuewhich is small compared to other dimensions as well as forprecise comparison with other researchers who assumedthat for the same geometry [35] ampe value of load isP 16 kN and the precrack length is a 45mm
ampe finite element model of ANSYS shown in Figure 3 iscompletely consistent with the state of experimental testsperformed by [21 35ndash37] In the present simulation the CTSgeometries are loaded in various angles 30deg 45deg 60deg and 75degin the original direction of cracking for the comparison ofΔKeq and crack paths directionsampere are also other loadingangles 0deg 30deg 60deg and 90deg to compare the direction of crackpropagation and assessment of fatigue life In the last casethree different crack length geometries of the CTS geome-tries have been simulated according to the initial cracklength of aW 03 05 and 07
Figure 4 shows the distribution of the equivalent loadingused in ANSYS analysis of the present work ampe uniaxialload F is correlated to the equivalent loads acting on differentholes of the six holes based on the following formulas [38]
F1 F6 F 05 cos α +c
bsin α1113874 1113875
F2 F5 F sin α
F3 F4 F 05lowast cos α minus (cb)sin α( 1113857
(5)
where α is the angle of load and c and b are length parametershown by Figure 4 (c b 54mm) ampe final values of allloading forces with different loading angles are shown inTable 2
ampe analytical solution for KI of this specimen is pro-vided by [23] as
KI F
Wt
πa
radic cos α(1 minus aW)
026 + 265α(W minus a)
1 + 055α(W minus 5) minus 008(a(W minus a))2
1113971
(6)
where F is the uniaxial load (Figure 2(b)) a is a crackrsquoslength W is the width of geometry and t is the geometrythickness It is worth comparing the SIFs for this geometrywith the analytical solution ofKI (equation (6)) for differentthicknesses (3 6 9 and 12mm) as represented in Figure 5
c
F1F2
F3
c
F4
F5
F6b
Figure 4 Loading and boundary condition for CTS geometry
Table 2 Values of forces F1 to F6 according to load angle α
α F2 F5 F1 F6 F3 F415 0259 F 0742 F 0224 F30 05 F 0933 F minus 0067 F45 0707 F 1061 F minus 0354 F60 0866 F 1116 F minus 0616 F75 0966 F 1095 F minus 0837 F90 F F minus F
800
700
600
500
400
300
200
100
0
K I (M
Pa m
m1
2 )
40 45 50 55 60 65 70 75 80Crack length a (mm)
t = 3mm
t = 6mm
t = 9mm
t = 12mm
ANSYSRichard criterion
Figure 5 Comparison for KI with analytical solution for differentspecimen thicknesses 3mm 6mm 9mm and 12mm
4 Advances in Materials Science and Engineering
As can be seen in this figure the values of SIFs are decreasingas the thickness increasing and the agreement with theanalytical solution obtained by equation (2) is obvious
ampe predicted values of ΔKeq were compared to theexperimental values performed by [37] as well as the ana-lytical expression given by [24 33 34 37] for different valuesof loading angles 30deg 45deg 60deg and 75degampese comparisons areshown in Figures 6ndash9
Figure 6 demonstrates the comparisons for the variationof the crack length as a function of the corresponding ΔKeq
with load angle of 30deg and the amount of applied loading is88 kN (loading ratio R 01) As illustrated in this figure thefour criteria of ΔKeq have around the same tendency as dataobtained from experimental tests by [37] For this loadingangle even there is a slight divergence for the [34] criterionespecially after the mid of the crack length In Figure 7 theloading angle is 45deg and the amount of the load is 114 kNwith R 01 ampe agreement for the ANSYS results with theexperimental results is clear as well as with other analyticalcriteria
Figure 8 shows the comparisons for the equivalent SIFfor 60deg loading angle and the amount of load is 1365 kNwithload ratio R 01 ampere is clearly divergence of the initialvalues of the equivalent SIF for all criteria which is de-pendent in the initial step of the crack growth ampis di-vergence is increased for the loading angle of 75deg and amountof load is 13 kN (R 01) as shown in Figure 9 ampough all ofthe four criteria almost provide close predictions to theexperimental results obtained by [37]
In addition the paths of crack growth of load angles 30deg45deg 60deg and 75deg were validated with comparison to practicalcrack paths tested by [36] as shown in Figure 10 with goodagreement
ampe predicted crack growth path has been compared tothe experimental test performed by [39] with different cracklengths (aw 03 05 and 07) and different load angles 030 60 and 90 as shown in Table 3
For pure loading of mode II (α 0deg) the direction of thecrack growth is controlled by the second mode of SIF (KII)
47 49 51 53 5545Crack length (mm)
K ep (M
Pa m
12 )
Experimental(Demir et al 2018)Richard criterion
Tanaka criterionDemir et al (2017)criterion
Erdogan and Sih ANSYS
201816141210
86420
Figure 7 Equivalent SIF with loading angle 45deg
47 49 51 53 5545Crack length (mm)
0
5
10
15
20
25
K ep (M
Pa m
12 )
Experimental(Demir et al 2018)Richard criterion
Tanaka criterionDemir et al (2017)criterion
Erdogan and Sih ANSYS
Figure 8 Equivalent SIF with loading angle 60deg
0
5
10
15
20
25
45 47 49 51 53 55
Keq
(MPa
m1
2 )
Crack length (mm)
Experimental(Demir et al 2018)Richard criterionErdogan and Sih
Tanaka criterionDemir et al (2017)criterionAnsys
Figure 9 Equivalent SIF with loading angle 75deg
Experimental(Demir et al 2018)Richard criterion
Tanaka criterionDemir et al (2017)criterion
Erdogan and Sih ANSYS
0
5
10
15
20
25
K eq (M
Pa m
12 )
47 49 51 53 55 57 59 61 63 6545Crack length (mm)
Figure 6 Equivalent SIF with loading angle 30deg
Advances in Materials Science and Engineering 5
4250 4450 4650 4850 5050 5250 5450 5650 5850 6050 6250 6450X (mm)
Y (m
m)
1800
1600
1400
1200
1000
800
600
400
200
000
75 deg (Antunes et al 2019)75 deg ANSYS60 deg Exp (Antunes et al 2019)60 deg ANSYS
45 deg Exp (Antunes et al 2019)45 deg ANSYS30 deg Exp (Antunes et al 2019)30 deg ANSYS
Figure 10 Comparison for crack growth paths between present study results and experimental results of [36]
Table 3 Comparison of paths of crack propagation among ANSYSrsquos results and experimental results [39]
ANSYS Experimental test [39]
(aw= 03) α 0deg
6 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 03) α 30deg
(aw= 03) α 60deg
Advances in Materials Science and Engineering 7
Table 3 Continued
ANSYS Experimental test [39]
(aw= 03) α 90deg
(aw= 05) α 0deg
8 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 30deg
(aw= 05) α 60deg
Advances in Materials Science and Engineering 9
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 90deg
(aw= 07) α 0deg
10 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 30deg
(aw= 07) α 60deg
Advances in Materials Science and Engineering 11
and the cracks all grow with different angles based onequation (2) as the values of KII are higher whereas forcomplete mode I loading (α 90deg) crack paths are domi-nated by the first mode KI and the crack is going to growstraight ampe influence of the first mode (KI) decreases as αincreases and vice versa for mode II
ampe crack direction angles of three geometries withdifferent initial length of cracks (aW 03 05 and 07)are 6deg 10deg and 12deg respectively which are identical to the
experimental angles tested by [39] Furthermore formixed-mode loading in the CTS geometries with loadingangle α 30deg the cracks are propagated to some extentdiffering to the original crack front direction For thesethree geometries the mixed-mode crack propagationangles (aW 03 05 and 07) are about 55deg 85deg and85deg respectively which are almost in agreement withexperimental angles of [39]
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 90deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 11 Comparison of simulated and experimental [35] fatiguelife for loading angle 30deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350 400 450 500000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 12 Comparison of simulated and experimental [35] fatiguelife for loading angle 60deg
12 Advances in Materials Science and Engineering
For the above three loading angles situations (α 90degα 0deg and α 30deg) regardless of the length of the initialcrack for the same load angle (α) the angles of the crackgrowth in each case are close to each other In contrast forcombined mode (III) loading with loading angle α 60deg thedirections of crack are highly dependent on the initial lengthof the crack On the other hand for small crack length (aW 03 and 05) the cracks also grow varying from theinitial crack tip against the clockwise direction with crackpropagation angles of 25deg and 30deg respectively which areidentical to those angles in the experimental test [39]Furthermore if the initial crack length is longer than theprevious cases (aW 07) the direction of the crack growthand the direction of the initial crack happen to be identicaland are presented in Table 3
In order to assess the fatigue life the number of cycles fordifferent loading angles 30 and 60 calculated in this study iscompared with the experimental results observed by [35]under mixed loading mode (III) resulting in clear agree-ments as can be observed in Figures 11 and 12 ampe amountof the applied load is F 16 kN with load ratio R 01whereas the Paris constant as mentioned previously isC 43378times10minus 7 (mmcycle)(MPa
m
radic) and m 26183
[40]
3 Conclusions
For the CTS specimen the finite element analysis of themixed-mode fatigue crack propagation was carried out usingANSYS Mechanical APDL and compared with experimentaldata for various angles of loading ampe predicted values ofΔKeq by the experimental and numerical methods are ob-served to be in good agreement with the present study re-sults ampe mixed-mode fatigue life is predicted andcompared with experimental data for two different loadingangles of 30deg and 60deg Interestingly the theoretically pre-dicted trajectories of the crack growth for all specimens areidentical to the experimental determined paths for differentloading angles Furthermore the predicted values of SIFs (KIand KII) are in good agreement with analytical solutions Itcan be stated that the structure of specimen geometry and itsconfiguration play an important role in obtaining highervalues of mixed-mode SIFs values ampis happens to besignificant in terms of applied loads particularly for highervalues of KII
Data Availability
ampe data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
References
[1] R I Stephens A Fatemi R R Stephens and H O FuchsMetal Fatigue in Engineering John Wiley amp Sons HobokenNJ USA 2000
[2] V V Bolotin Mechanics of Fatigue CRC Press Boca RatonFL USA 1999
[3] C A Brebbia and A V Farahani Fatigue Damage of Ma-terials Experiment and Analysis WIT Press SouthamptonUK 2003
[4] N E DowlingMechanical Behavior of Materials EngineeringMethods for Deformation Fracture and Fatigue PearsonLondon UK 2012
[5] G R Halford Fatigue and Durability of Structural MaterialsASM International London UK 2006
[6] R O Ritchie ldquoMechanisms of fatigue-crack propagation inductile and brittle solidsrdquo International Journal of Fracturevol 100 no 1 pp 55ndash83 1999
[7] S Suresh Fatigue of Materials Cambridge University PressCambridge UK 1998
[8] K N Solanki Two and ree-Dimensional Finite ElementAnalysis of Plasticity-Induced Fatigue Crack Closure AComprehensive Parametric Study Mississippi State Univer-sity Starkville MS USA 2002
[9] A M Alshoaibi and A K Ariffin ldquoFinite element modeling offatigue crack propagation using a self adaptive mesh strategyrdquoInternational Review of Aerospace Engineering (IREASE)vol 8 no 6 pp 209ndash215 2015
[10] A M Alshoaibi ldquoFinite element procedures for the numericalsimulation of fatigue crack propagation under mixed modeloadingrdquo Structural Engineering and Mechanics vol 35 no 3pp 283ndash299 2010
[11] A M Alshoaibi ldquoAdaptive finite element modeling of fatiguecrack propagationrdquo International Journal of Materials Scienceand Applications vol 2 no 3 pp 104ndash108 2013
[12] A M Alshoaibi ldquoAn adaptive finite element framework forfatigue crack propagation under constant amplitude loadingrdquoInternational Journal of Applied Science and Engineeringvol 13 no 3 pp 261ndash270 2015
[13] A M Alshoaibi ldquoComprehensive comparisons of two andthree dimensional numerical estimation of stress intensityfactors and crack propagation in linear elastic analysisrdquo In-ternational Journal of Integrated Engineering vol 11 no 6pp 45ndash52 2019
[14] A Handbook Fatigue and Fracture American Society ofMaterials vol 19 1996
[15] W He J Liu and D Xie ldquoNumerical study on fatigue crackgrowth at a web-stiffener of ship structural details by anobjected-oriented approach in conjunction with ABAQUSrdquoMarine Structures vol 35 pp 45ndash69 2014
[16] K Rege and D G Pavlou ldquoEffect of stop holes on structuralintegrity of offshore structures a numerical modelrdquo in Pro-ceedings of the Institution of Civil Engineers-Maritime Engi-neering ampomas Telford Ltd London UK 2019
[17] H Rhee ldquoFatigue crack growth analyses of offshore structuraltubular jointrdquo Journal of Offshore Mechanics and Arctic En-gineering vol 111 no 1 pp 49ndash55 1989
[18] H Riahi P Bressolette A Chateauneuf C Bouraoui andR Fathallah ldquoReliability analysis and inspection updating bystochastic response surface of fatigue cracks in mixed moderdquoEngineering Structures vol 33 no 12 pp 3392ndash3401 2011
[19] S Qi L X Cai C Bao H Chen K K Shi and H L WuldquoAnalytical theory for fatigue crack propagation rates ofmixed-mode I-II cracks and its applicationrdquo InternationalJournal of Fatigue vol 119 pp 150ndash159 2019
[20] A F Liu Mechanics and Mechanisms of Fracture An In-troduction ASM International London UK 2005
[21] O Demir A O Ayhan and S Iriccedil ldquoA new specimen formixed mode-III fracture tests modeling experiments and
Advances in Materials Science and Engineering 13
criteria developmentrdquo Engineering Fracture Mechanicsvol 178 pp 457ndash476 2017
[22] M Hussain S Pu and J Underwood ldquoStrain energy releaserate for a crack under combined mode I and mode II FractureAnalysisrdquo in Proceedings of the 1973 National Symposium onFracture Mechanics Part II ASTM International WestConshohocken PA USA 1974
[23] H Richard M Sander M Fulland and G Kullmer ldquoDe-velopment of fatigue crack growth in real structuresrdquo Engi-neering Fracture Mechanics vol 75 no 3-4 pp 331ndash3402008
[24] K Tanaka ldquoFatigue crack propagation from a crack inclinedto the cyclic tensile axisrdquo Engineering Fracture Mechanicsvol 6 no 3 pp 493ndash507 1974
[25] P Paris and F Erdogan ldquoA critical analysis of crack prop-agation lawsrdquo Journal of Basic Engineering vol 85 no 4pp 528ndash533 1963
[26] L Coffin ldquoCyclic deformation and fatigue of metalsrdquo eBook Fatigue and Staying Power of Metals Izo IL USSRMoscow Russia 1963
[27] A Wohler ldquoVersuche zur Ermittlung der auf die Eisen-bahnwagenachsen einwirkenden Krafte und die Wider-standsfahigkeit der Wagen-Achsenrdquo Zeitschrift fur Bauwesenvol 10 no 1860 pp 583ndash614 1860
[28] J M Alegre M Preciado and D Ferrentildeo ldquoStudy of thefatigue failure of an anti-return valve of a high pressuremachinerdquo Engineering Failure Analysis vol 14 no 2pp 408ndash416 2007
[29] H Chen Q Wang W Zeng et al ldquoDynamic brittle crackpropagation modeling using singular edge-based smoothedfinite element method with local mesh rezoningrdquo EuropeanJournal of MechanicsmdashASolids vol 76 pp 208ndash223 2019
[30] S Sajith K S R Krishna Murthy and P Robi ldquoPrediction ofAccurate Mixed Mode Fatigue Crack Growth Curves usingthe Parisrsquo Lawrdquo Journal of e Institution of Engineers (India)Series C vol 100 no 1 pp 165ndash174 2019
[31] G R Irwin ldquoAnalysis of stresses and strains near the end of acrack transversing a platerdquo Journal of Applied Mechanicsvol 24 pp 361ndash364 1957
[32] H A Richard F G Buchholz G Kullmer andM Schollmann 2D-and 3D-mixed mode fracture criteria KeyEngineering Materials Trans Tech Publication Stafa-ZurichSwitzerland 2003
[33] H A Richard B Schramm and N-H Schirmeisen ldquoCrackson Mixed Mode loading - ampeories experiments simula-tionsrdquo International Journal of Fatigue vol 62 pp 93ndash1032014
[34] F Erdogan and G C Sih ldquoOn the crack extension in platesunder plane loading and transverse shearrdquo Journal of BasicEngineering vol 85 no 4 pp 519ndash525 1963
[35] S Sajith K S R K Murthy and P S Robi ldquoExperimental andnumerical investigation of mixed mode fatigue crack growthmodels in aluminum 6061-T6rdquo International Journal of Fa-tigue vol 130 p 105285 2020
[36] F Antunes R Branco J Ferreira and L Borrego ldquoStressintensity factor solutions for CTS mixed mode specimenrdquoFrattura ed Integrita Strutturale vol 13 no 48 pp 676ndash6922019
[37] O Demir A O Ayhan S Iric and H Lekesiz ldquoEvaluation ofmixed mode-III criteria for fatigue crack propagation usingexperiments and modelingrdquo Chinese Journal of Aeronauticsvol 31 no 7 pp 1525ndash1534 2018
[38] H Richard Fracture predictions for cracks exposed to super-imposed normal and shear stresses International NuclearInformation System Vienna Austria 1985
[39] X-T Miao Q Yu C-Y Zhou J Li Y-Z Wang andX-H He ldquoExperimental and numerical investigation onfracture behavior of CTS specimen under I-II mixed modeloadingrdquo European Journal of MechanicsmdashASolids vol 72pp 235ndash244 2018
[40] S Sajith S S Shukla K S R K Murthy and P S RobildquoMixed mode fatigue crack growth studies in AISI 316stainless steelrdquo European Journal of MechanicsmdashASolidsvol 80 p 103898 2020
14 Advances in Materials Science and Engineering
As can be seen in this figure the values of SIFs are decreasingas the thickness increasing and the agreement with theanalytical solution obtained by equation (2) is obvious
ampe predicted values of ΔKeq were compared to theexperimental values performed by [37] as well as the ana-lytical expression given by [24 33 34 37] for different valuesof loading angles 30deg 45deg 60deg and 75degampese comparisons areshown in Figures 6ndash9
Figure 6 demonstrates the comparisons for the variationof the crack length as a function of the corresponding ΔKeq
with load angle of 30deg and the amount of applied loading is88 kN (loading ratio R 01) As illustrated in this figure thefour criteria of ΔKeq have around the same tendency as dataobtained from experimental tests by [37] For this loadingangle even there is a slight divergence for the [34] criterionespecially after the mid of the crack length In Figure 7 theloading angle is 45deg and the amount of the load is 114 kNwith R 01 ampe agreement for the ANSYS results with theexperimental results is clear as well as with other analyticalcriteria
Figure 8 shows the comparisons for the equivalent SIFfor 60deg loading angle and the amount of load is 1365 kNwithload ratio R 01 ampere is clearly divergence of the initialvalues of the equivalent SIF for all criteria which is de-pendent in the initial step of the crack growth ampis di-vergence is increased for the loading angle of 75deg and amountof load is 13 kN (R 01) as shown in Figure 9 ampough all ofthe four criteria almost provide close predictions to theexperimental results obtained by [37]
In addition the paths of crack growth of load angles 30deg45deg 60deg and 75deg were validated with comparison to practicalcrack paths tested by [36] as shown in Figure 10 with goodagreement
ampe predicted crack growth path has been compared tothe experimental test performed by [39] with different cracklengths (aw 03 05 and 07) and different load angles 030 60 and 90 as shown in Table 3
For pure loading of mode II (α 0deg) the direction of thecrack growth is controlled by the second mode of SIF (KII)
47 49 51 53 5545Crack length (mm)
K ep (M
Pa m
12 )
Experimental(Demir et al 2018)Richard criterion
Tanaka criterionDemir et al (2017)criterion
Erdogan and Sih ANSYS
201816141210
86420
Figure 7 Equivalent SIF with loading angle 45deg
47 49 51 53 5545Crack length (mm)
0
5
10
15
20
25
K ep (M
Pa m
12 )
Experimental(Demir et al 2018)Richard criterion
Tanaka criterionDemir et al (2017)criterion
Erdogan and Sih ANSYS
Figure 8 Equivalent SIF with loading angle 60deg
0
5
10
15
20
25
45 47 49 51 53 55
Keq
(MPa
m1
2 )
Crack length (mm)
Experimental(Demir et al 2018)Richard criterionErdogan and Sih
Tanaka criterionDemir et al (2017)criterionAnsys
Figure 9 Equivalent SIF with loading angle 75deg
Experimental(Demir et al 2018)Richard criterion
Tanaka criterionDemir et al (2017)criterion
Erdogan and Sih ANSYS
0
5
10
15
20
25
K eq (M
Pa m
12 )
47 49 51 53 55 57 59 61 63 6545Crack length (mm)
Figure 6 Equivalent SIF with loading angle 30deg
Advances in Materials Science and Engineering 5
4250 4450 4650 4850 5050 5250 5450 5650 5850 6050 6250 6450X (mm)
Y (m
m)
1800
1600
1400
1200
1000
800
600
400
200
000
75 deg (Antunes et al 2019)75 deg ANSYS60 deg Exp (Antunes et al 2019)60 deg ANSYS
45 deg Exp (Antunes et al 2019)45 deg ANSYS30 deg Exp (Antunes et al 2019)30 deg ANSYS
Figure 10 Comparison for crack growth paths between present study results and experimental results of [36]
Table 3 Comparison of paths of crack propagation among ANSYSrsquos results and experimental results [39]
ANSYS Experimental test [39]
(aw= 03) α 0deg
6 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 03) α 30deg
(aw= 03) α 60deg
Advances in Materials Science and Engineering 7
Table 3 Continued
ANSYS Experimental test [39]
(aw= 03) α 90deg
(aw= 05) α 0deg
8 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 30deg
(aw= 05) α 60deg
Advances in Materials Science and Engineering 9
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 90deg
(aw= 07) α 0deg
10 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 30deg
(aw= 07) α 60deg
Advances in Materials Science and Engineering 11
and the cracks all grow with different angles based onequation (2) as the values of KII are higher whereas forcomplete mode I loading (α 90deg) crack paths are domi-nated by the first mode KI and the crack is going to growstraight ampe influence of the first mode (KI) decreases as αincreases and vice versa for mode II
ampe crack direction angles of three geometries withdifferent initial length of cracks (aW 03 05 and 07)are 6deg 10deg and 12deg respectively which are identical to the
experimental angles tested by [39] Furthermore formixed-mode loading in the CTS geometries with loadingangle α 30deg the cracks are propagated to some extentdiffering to the original crack front direction For thesethree geometries the mixed-mode crack propagationangles (aW 03 05 and 07) are about 55deg 85deg and85deg respectively which are almost in agreement withexperimental angles of [39]
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 90deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 11 Comparison of simulated and experimental [35] fatiguelife for loading angle 30deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350 400 450 500000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 12 Comparison of simulated and experimental [35] fatiguelife for loading angle 60deg
12 Advances in Materials Science and Engineering
For the above three loading angles situations (α 90degα 0deg and α 30deg) regardless of the length of the initialcrack for the same load angle (α) the angles of the crackgrowth in each case are close to each other In contrast forcombined mode (III) loading with loading angle α 60deg thedirections of crack are highly dependent on the initial lengthof the crack On the other hand for small crack length (aW 03 and 05) the cracks also grow varying from theinitial crack tip against the clockwise direction with crackpropagation angles of 25deg and 30deg respectively which areidentical to those angles in the experimental test [39]Furthermore if the initial crack length is longer than theprevious cases (aW 07) the direction of the crack growthand the direction of the initial crack happen to be identicaland are presented in Table 3
In order to assess the fatigue life the number of cycles fordifferent loading angles 30 and 60 calculated in this study iscompared with the experimental results observed by [35]under mixed loading mode (III) resulting in clear agree-ments as can be observed in Figures 11 and 12 ampe amountof the applied load is F 16 kN with load ratio R 01whereas the Paris constant as mentioned previously isC 43378times10minus 7 (mmcycle)(MPa
m
radic) and m 26183
[40]
3 Conclusions
For the CTS specimen the finite element analysis of themixed-mode fatigue crack propagation was carried out usingANSYS Mechanical APDL and compared with experimentaldata for various angles of loading ampe predicted values ofΔKeq by the experimental and numerical methods are ob-served to be in good agreement with the present study re-sults ampe mixed-mode fatigue life is predicted andcompared with experimental data for two different loadingangles of 30deg and 60deg Interestingly the theoretically pre-dicted trajectories of the crack growth for all specimens areidentical to the experimental determined paths for differentloading angles Furthermore the predicted values of SIFs (KIand KII) are in good agreement with analytical solutions Itcan be stated that the structure of specimen geometry and itsconfiguration play an important role in obtaining highervalues of mixed-mode SIFs values ampis happens to besignificant in terms of applied loads particularly for highervalues of KII
Data Availability
ampe data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
References
[1] R I Stephens A Fatemi R R Stephens and H O FuchsMetal Fatigue in Engineering John Wiley amp Sons HobokenNJ USA 2000
[2] V V Bolotin Mechanics of Fatigue CRC Press Boca RatonFL USA 1999
[3] C A Brebbia and A V Farahani Fatigue Damage of Ma-terials Experiment and Analysis WIT Press SouthamptonUK 2003
[4] N E DowlingMechanical Behavior of Materials EngineeringMethods for Deformation Fracture and Fatigue PearsonLondon UK 2012
[5] G R Halford Fatigue and Durability of Structural MaterialsASM International London UK 2006
[6] R O Ritchie ldquoMechanisms of fatigue-crack propagation inductile and brittle solidsrdquo International Journal of Fracturevol 100 no 1 pp 55ndash83 1999
[7] S Suresh Fatigue of Materials Cambridge University PressCambridge UK 1998
[8] K N Solanki Two and ree-Dimensional Finite ElementAnalysis of Plasticity-Induced Fatigue Crack Closure AComprehensive Parametric Study Mississippi State Univer-sity Starkville MS USA 2002
[9] A M Alshoaibi and A K Ariffin ldquoFinite element modeling offatigue crack propagation using a self adaptive mesh strategyrdquoInternational Review of Aerospace Engineering (IREASE)vol 8 no 6 pp 209ndash215 2015
[10] A M Alshoaibi ldquoFinite element procedures for the numericalsimulation of fatigue crack propagation under mixed modeloadingrdquo Structural Engineering and Mechanics vol 35 no 3pp 283ndash299 2010
[11] A M Alshoaibi ldquoAdaptive finite element modeling of fatiguecrack propagationrdquo International Journal of Materials Scienceand Applications vol 2 no 3 pp 104ndash108 2013
[12] A M Alshoaibi ldquoAn adaptive finite element framework forfatigue crack propagation under constant amplitude loadingrdquoInternational Journal of Applied Science and Engineeringvol 13 no 3 pp 261ndash270 2015
[13] A M Alshoaibi ldquoComprehensive comparisons of two andthree dimensional numerical estimation of stress intensityfactors and crack propagation in linear elastic analysisrdquo In-ternational Journal of Integrated Engineering vol 11 no 6pp 45ndash52 2019
[14] A Handbook Fatigue and Fracture American Society ofMaterials vol 19 1996
[15] W He J Liu and D Xie ldquoNumerical study on fatigue crackgrowth at a web-stiffener of ship structural details by anobjected-oriented approach in conjunction with ABAQUSrdquoMarine Structures vol 35 pp 45ndash69 2014
[16] K Rege and D G Pavlou ldquoEffect of stop holes on structuralintegrity of offshore structures a numerical modelrdquo in Pro-ceedings of the Institution of Civil Engineers-Maritime Engi-neering ampomas Telford Ltd London UK 2019
[17] H Rhee ldquoFatigue crack growth analyses of offshore structuraltubular jointrdquo Journal of Offshore Mechanics and Arctic En-gineering vol 111 no 1 pp 49ndash55 1989
[18] H Riahi P Bressolette A Chateauneuf C Bouraoui andR Fathallah ldquoReliability analysis and inspection updating bystochastic response surface of fatigue cracks in mixed moderdquoEngineering Structures vol 33 no 12 pp 3392ndash3401 2011
[19] S Qi L X Cai C Bao H Chen K K Shi and H L WuldquoAnalytical theory for fatigue crack propagation rates ofmixed-mode I-II cracks and its applicationrdquo InternationalJournal of Fatigue vol 119 pp 150ndash159 2019
[20] A F Liu Mechanics and Mechanisms of Fracture An In-troduction ASM International London UK 2005
[21] O Demir A O Ayhan and S Iriccedil ldquoA new specimen formixed mode-III fracture tests modeling experiments and
Advances in Materials Science and Engineering 13
criteria developmentrdquo Engineering Fracture Mechanicsvol 178 pp 457ndash476 2017
[22] M Hussain S Pu and J Underwood ldquoStrain energy releaserate for a crack under combined mode I and mode II FractureAnalysisrdquo in Proceedings of the 1973 National Symposium onFracture Mechanics Part II ASTM International WestConshohocken PA USA 1974
[23] H Richard M Sander M Fulland and G Kullmer ldquoDe-velopment of fatigue crack growth in real structuresrdquo Engi-neering Fracture Mechanics vol 75 no 3-4 pp 331ndash3402008
[24] K Tanaka ldquoFatigue crack propagation from a crack inclinedto the cyclic tensile axisrdquo Engineering Fracture Mechanicsvol 6 no 3 pp 493ndash507 1974
[25] P Paris and F Erdogan ldquoA critical analysis of crack prop-agation lawsrdquo Journal of Basic Engineering vol 85 no 4pp 528ndash533 1963
[26] L Coffin ldquoCyclic deformation and fatigue of metalsrdquo eBook Fatigue and Staying Power of Metals Izo IL USSRMoscow Russia 1963
[27] A Wohler ldquoVersuche zur Ermittlung der auf die Eisen-bahnwagenachsen einwirkenden Krafte und die Wider-standsfahigkeit der Wagen-Achsenrdquo Zeitschrift fur Bauwesenvol 10 no 1860 pp 583ndash614 1860
[28] J M Alegre M Preciado and D Ferrentildeo ldquoStudy of thefatigue failure of an anti-return valve of a high pressuremachinerdquo Engineering Failure Analysis vol 14 no 2pp 408ndash416 2007
[29] H Chen Q Wang W Zeng et al ldquoDynamic brittle crackpropagation modeling using singular edge-based smoothedfinite element method with local mesh rezoningrdquo EuropeanJournal of MechanicsmdashASolids vol 76 pp 208ndash223 2019
[30] S Sajith K S R Krishna Murthy and P Robi ldquoPrediction ofAccurate Mixed Mode Fatigue Crack Growth Curves usingthe Parisrsquo Lawrdquo Journal of e Institution of Engineers (India)Series C vol 100 no 1 pp 165ndash174 2019
[31] G R Irwin ldquoAnalysis of stresses and strains near the end of acrack transversing a platerdquo Journal of Applied Mechanicsvol 24 pp 361ndash364 1957
[32] H A Richard F G Buchholz G Kullmer andM Schollmann 2D-and 3D-mixed mode fracture criteria KeyEngineering Materials Trans Tech Publication Stafa-ZurichSwitzerland 2003
[33] H A Richard B Schramm and N-H Schirmeisen ldquoCrackson Mixed Mode loading - ampeories experiments simula-tionsrdquo International Journal of Fatigue vol 62 pp 93ndash1032014
[34] F Erdogan and G C Sih ldquoOn the crack extension in platesunder plane loading and transverse shearrdquo Journal of BasicEngineering vol 85 no 4 pp 519ndash525 1963
[35] S Sajith K S R K Murthy and P S Robi ldquoExperimental andnumerical investigation of mixed mode fatigue crack growthmodels in aluminum 6061-T6rdquo International Journal of Fa-tigue vol 130 p 105285 2020
[36] F Antunes R Branco J Ferreira and L Borrego ldquoStressintensity factor solutions for CTS mixed mode specimenrdquoFrattura ed Integrita Strutturale vol 13 no 48 pp 676ndash6922019
[37] O Demir A O Ayhan S Iric and H Lekesiz ldquoEvaluation ofmixed mode-III criteria for fatigue crack propagation usingexperiments and modelingrdquo Chinese Journal of Aeronauticsvol 31 no 7 pp 1525ndash1534 2018
[38] H Richard Fracture predictions for cracks exposed to super-imposed normal and shear stresses International NuclearInformation System Vienna Austria 1985
[39] X-T Miao Q Yu C-Y Zhou J Li Y-Z Wang andX-H He ldquoExperimental and numerical investigation onfracture behavior of CTS specimen under I-II mixed modeloadingrdquo European Journal of MechanicsmdashASolids vol 72pp 235ndash244 2018
[40] S Sajith S S Shukla K S R K Murthy and P S RobildquoMixed mode fatigue crack growth studies in AISI 316stainless steelrdquo European Journal of MechanicsmdashASolidsvol 80 p 103898 2020
14 Advances in Materials Science and Engineering
4250 4450 4650 4850 5050 5250 5450 5650 5850 6050 6250 6450X (mm)
Y (m
m)
1800
1600
1400
1200
1000
800
600
400
200
000
75 deg (Antunes et al 2019)75 deg ANSYS60 deg Exp (Antunes et al 2019)60 deg ANSYS
45 deg Exp (Antunes et al 2019)45 deg ANSYS30 deg Exp (Antunes et al 2019)30 deg ANSYS
Figure 10 Comparison for crack growth paths between present study results and experimental results of [36]
Table 3 Comparison of paths of crack propagation among ANSYSrsquos results and experimental results [39]
ANSYS Experimental test [39]
(aw= 03) α 0deg
6 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 03) α 30deg
(aw= 03) α 60deg
Advances in Materials Science and Engineering 7
Table 3 Continued
ANSYS Experimental test [39]
(aw= 03) α 90deg
(aw= 05) α 0deg
8 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 30deg
(aw= 05) α 60deg
Advances in Materials Science and Engineering 9
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 90deg
(aw= 07) α 0deg
10 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 30deg
(aw= 07) α 60deg
Advances in Materials Science and Engineering 11
and the cracks all grow with different angles based onequation (2) as the values of KII are higher whereas forcomplete mode I loading (α 90deg) crack paths are domi-nated by the first mode KI and the crack is going to growstraight ampe influence of the first mode (KI) decreases as αincreases and vice versa for mode II
ampe crack direction angles of three geometries withdifferent initial length of cracks (aW 03 05 and 07)are 6deg 10deg and 12deg respectively which are identical to the
experimental angles tested by [39] Furthermore formixed-mode loading in the CTS geometries with loadingangle α 30deg the cracks are propagated to some extentdiffering to the original crack front direction For thesethree geometries the mixed-mode crack propagationangles (aW 03 05 and 07) are about 55deg 85deg and85deg respectively which are almost in agreement withexperimental angles of [39]
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 90deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 11 Comparison of simulated and experimental [35] fatiguelife for loading angle 30deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350 400 450 500000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 12 Comparison of simulated and experimental [35] fatiguelife for loading angle 60deg
12 Advances in Materials Science and Engineering
For the above three loading angles situations (α 90degα 0deg and α 30deg) regardless of the length of the initialcrack for the same load angle (α) the angles of the crackgrowth in each case are close to each other In contrast forcombined mode (III) loading with loading angle α 60deg thedirections of crack are highly dependent on the initial lengthof the crack On the other hand for small crack length (aW 03 and 05) the cracks also grow varying from theinitial crack tip against the clockwise direction with crackpropagation angles of 25deg and 30deg respectively which areidentical to those angles in the experimental test [39]Furthermore if the initial crack length is longer than theprevious cases (aW 07) the direction of the crack growthand the direction of the initial crack happen to be identicaland are presented in Table 3
In order to assess the fatigue life the number of cycles fordifferent loading angles 30 and 60 calculated in this study iscompared with the experimental results observed by [35]under mixed loading mode (III) resulting in clear agree-ments as can be observed in Figures 11 and 12 ampe amountof the applied load is F 16 kN with load ratio R 01whereas the Paris constant as mentioned previously isC 43378times10minus 7 (mmcycle)(MPa
m
radic) and m 26183
[40]
3 Conclusions
For the CTS specimen the finite element analysis of themixed-mode fatigue crack propagation was carried out usingANSYS Mechanical APDL and compared with experimentaldata for various angles of loading ampe predicted values ofΔKeq by the experimental and numerical methods are ob-served to be in good agreement with the present study re-sults ampe mixed-mode fatigue life is predicted andcompared with experimental data for two different loadingangles of 30deg and 60deg Interestingly the theoretically pre-dicted trajectories of the crack growth for all specimens areidentical to the experimental determined paths for differentloading angles Furthermore the predicted values of SIFs (KIand KII) are in good agreement with analytical solutions Itcan be stated that the structure of specimen geometry and itsconfiguration play an important role in obtaining highervalues of mixed-mode SIFs values ampis happens to besignificant in terms of applied loads particularly for highervalues of KII
Data Availability
ampe data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
References
[1] R I Stephens A Fatemi R R Stephens and H O FuchsMetal Fatigue in Engineering John Wiley amp Sons HobokenNJ USA 2000
[2] V V Bolotin Mechanics of Fatigue CRC Press Boca RatonFL USA 1999
[3] C A Brebbia and A V Farahani Fatigue Damage of Ma-terials Experiment and Analysis WIT Press SouthamptonUK 2003
[4] N E DowlingMechanical Behavior of Materials EngineeringMethods for Deformation Fracture and Fatigue PearsonLondon UK 2012
[5] G R Halford Fatigue and Durability of Structural MaterialsASM International London UK 2006
[6] R O Ritchie ldquoMechanisms of fatigue-crack propagation inductile and brittle solidsrdquo International Journal of Fracturevol 100 no 1 pp 55ndash83 1999
[7] S Suresh Fatigue of Materials Cambridge University PressCambridge UK 1998
[8] K N Solanki Two and ree-Dimensional Finite ElementAnalysis of Plasticity-Induced Fatigue Crack Closure AComprehensive Parametric Study Mississippi State Univer-sity Starkville MS USA 2002
[9] A M Alshoaibi and A K Ariffin ldquoFinite element modeling offatigue crack propagation using a self adaptive mesh strategyrdquoInternational Review of Aerospace Engineering (IREASE)vol 8 no 6 pp 209ndash215 2015
[10] A M Alshoaibi ldquoFinite element procedures for the numericalsimulation of fatigue crack propagation under mixed modeloadingrdquo Structural Engineering and Mechanics vol 35 no 3pp 283ndash299 2010
[11] A M Alshoaibi ldquoAdaptive finite element modeling of fatiguecrack propagationrdquo International Journal of Materials Scienceand Applications vol 2 no 3 pp 104ndash108 2013
[12] A M Alshoaibi ldquoAn adaptive finite element framework forfatigue crack propagation under constant amplitude loadingrdquoInternational Journal of Applied Science and Engineeringvol 13 no 3 pp 261ndash270 2015
[13] A M Alshoaibi ldquoComprehensive comparisons of two andthree dimensional numerical estimation of stress intensityfactors and crack propagation in linear elastic analysisrdquo In-ternational Journal of Integrated Engineering vol 11 no 6pp 45ndash52 2019
[14] A Handbook Fatigue and Fracture American Society ofMaterials vol 19 1996
[15] W He J Liu and D Xie ldquoNumerical study on fatigue crackgrowth at a web-stiffener of ship structural details by anobjected-oriented approach in conjunction with ABAQUSrdquoMarine Structures vol 35 pp 45ndash69 2014
[16] K Rege and D G Pavlou ldquoEffect of stop holes on structuralintegrity of offshore structures a numerical modelrdquo in Pro-ceedings of the Institution of Civil Engineers-Maritime Engi-neering ampomas Telford Ltd London UK 2019
[17] H Rhee ldquoFatigue crack growth analyses of offshore structuraltubular jointrdquo Journal of Offshore Mechanics and Arctic En-gineering vol 111 no 1 pp 49ndash55 1989
[18] H Riahi P Bressolette A Chateauneuf C Bouraoui andR Fathallah ldquoReliability analysis and inspection updating bystochastic response surface of fatigue cracks in mixed moderdquoEngineering Structures vol 33 no 12 pp 3392ndash3401 2011
[19] S Qi L X Cai C Bao H Chen K K Shi and H L WuldquoAnalytical theory for fatigue crack propagation rates ofmixed-mode I-II cracks and its applicationrdquo InternationalJournal of Fatigue vol 119 pp 150ndash159 2019
[20] A F Liu Mechanics and Mechanisms of Fracture An In-troduction ASM International London UK 2005
[21] O Demir A O Ayhan and S Iriccedil ldquoA new specimen formixed mode-III fracture tests modeling experiments and
Advances in Materials Science and Engineering 13
criteria developmentrdquo Engineering Fracture Mechanicsvol 178 pp 457ndash476 2017
[22] M Hussain S Pu and J Underwood ldquoStrain energy releaserate for a crack under combined mode I and mode II FractureAnalysisrdquo in Proceedings of the 1973 National Symposium onFracture Mechanics Part II ASTM International WestConshohocken PA USA 1974
[23] H Richard M Sander M Fulland and G Kullmer ldquoDe-velopment of fatigue crack growth in real structuresrdquo Engi-neering Fracture Mechanics vol 75 no 3-4 pp 331ndash3402008
[24] K Tanaka ldquoFatigue crack propagation from a crack inclinedto the cyclic tensile axisrdquo Engineering Fracture Mechanicsvol 6 no 3 pp 493ndash507 1974
[25] P Paris and F Erdogan ldquoA critical analysis of crack prop-agation lawsrdquo Journal of Basic Engineering vol 85 no 4pp 528ndash533 1963
[26] L Coffin ldquoCyclic deformation and fatigue of metalsrdquo eBook Fatigue and Staying Power of Metals Izo IL USSRMoscow Russia 1963
[27] A Wohler ldquoVersuche zur Ermittlung der auf die Eisen-bahnwagenachsen einwirkenden Krafte und die Wider-standsfahigkeit der Wagen-Achsenrdquo Zeitschrift fur Bauwesenvol 10 no 1860 pp 583ndash614 1860
[28] J M Alegre M Preciado and D Ferrentildeo ldquoStudy of thefatigue failure of an anti-return valve of a high pressuremachinerdquo Engineering Failure Analysis vol 14 no 2pp 408ndash416 2007
[29] H Chen Q Wang W Zeng et al ldquoDynamic brittle crackpropagation modeling using singular edge-based smoothedfinite element method with local mesh rezoningrdquo EuropeanJournal of MechanicsmdashASolids vol 76 pp 208ndash223 2019
[30] S Sajith K S R Krishna Murthy and P Robi ldquoPrediction ofAccurate Mixed Mode Fatigue Crack Growth Curves usingthe Parisrsquo Lawrdquo Journal of e Institution of Engineers (India)Series C vol 100 no 1 pp 165ndash174 2019
[31] G R Irwin ldquoAnalysis of stresses and strains near the end of acrack transversing a platerdquo Journal of Applied Mechanicsvol 24 pp 361ndash364 1957
[32] H A Richard F G Buchholz G Kullmer andM Schollmann 2D-and 3D-mixed mode fracture criteria KeyEngineering Materials Trans Tech Publication Stafa-ZurichSwitzerland 2003
[33] H A Richard B Schramm and N-H Schirmeisen ldquoCrackson Mixed Mode loading - ampeories experiments simula-tionsrdquo International Journal of Fatigue vol 62 pp 93ndash1032014
[34] F Erdogan and G C Sih ldquoOn the crack extension in platesunder plane loading and transverse shearrdquo Journal of BasicEngineering vol 85 no 4 pp 519ndash525 1963
[35] S Sajith K S R K Murthy and P S Robi ldquoExperimental andnumerical investigation of mixed mode fatigue crack growthmodels in aluminum 6061-T6rdquo International Journal of Fa-tigue vol 130 p 105285 2020
[36] F Antunes R Branco J Ferreira and L Borrego ldquoStressintensity factor solutions for CTS mixed mode specimenrdquoFrattura ed Integrita Strutturale vol 13 no 48 pp 676ndash6922019
[37] O Demir A O Ayhan S Iric and H Lekesiz ldquoEvaluation ofmixed mode-III criteria for fatigue crack propagation usingexperiments and modelingrdquo Chinese Journal of Aeronauticsvol 31 no 7 pp 1525ndash1534 2018
[38] H Richard Fracture predictions for cracks exposed to super-imposed normal and shear stresses International NuclearInformation System Vienna Austria 1985
[39] X-T Miao Q Yu C-Y Zhou J Li Y-Z Wang andX-H He ldquoExperimental and numerical investigation onfracture behavior of CTS specimen under I-II mixed modeloadingrdquo European Journal of MechanicsmdashASolids vol 72pp 235ndash244 2018
[40] S Sajith S S Shukla K S R K Murthy and P S RobildquoMixed mode fatigue crack growth studies in AISI 316stainless steelrdquo European Journal of MechanicsmdashASolidsvol 80 p 103898 2020
14 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 03) α 30deg
(aw= 03) α 60deg
Advances in Materials Science and Engineering 7
Table 3 Continued
ANSYS Experimental test [39]
(aw= 03) α 90deg
(aw= 05) α 0deg
8 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 30deg
(aw= 05) α 60deg
Advances in Materials Science and Engineering 9
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 90deg
(aw= 07) α 0deg
10 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 30deg
(aw= 07) α 60deg
Advances in Materials Science and Engineering 11
and the cracks all grow with different angles based onequation (2) as the values of KII are higher whereas forcomplete mode I loading (α 90deg) crack paths are domi-nated by the first mode KI and the crack is going to growstraight ampe influence of the first mode (KI) decreases as αincreases and vice versa for mode II
ampe crack direction angles of three geometries withdifferent initial length of cracks (aW 03 05 and 07)are 6deg 10deg and 12deg respectively which are identical to the
experimental angles tested by [39] Furthermore formixed-mode loading in the CTS geometries with loadingangle α 30deg the cracks are propagated to some extentdiffering to the original crack front direction For thesethree geometries the mixed-mode crack propagationangles (aW 03 05 and 07) are about 55deg 85deg and85deg respectively which are almost in agreement withexperimental angles of [39]
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 90deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 11 Comparison of simulated and experimental [35] fatiguelife for loading angle 30deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350 400 450 500000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 12 Comparison of simulated and experimental [35] fatiguelife for loading angle 60deg
12 Advances in Materials Science and Engineering
For the above three loading angles situations (α 90degα 0deg and α 30deg) regardless of the length of the initialcrack for the same load angle (α) the angles of the crackgrowth in each case are close to each other In contrast forcombined mode (III) loading with loading angle α 60deg thedirections of crack are highly dependent on the initial lengthof the crack On the other hand for small crack length (aW 03 and 05) the cracks also grow varying from theinitial crack tip against the clockwise direction with crackpropagation angles of 25deg and 30deg respectively which areidentical to those angles in the experimental test [39]Furthermore if the initial crack length is longer than theprevious cases (aW 07) the direction of the crack growthand the direction of the initial crack happen to be identicaland are presented in Table 3
In order to assess the fatigue life the number of cycles fordifferent loading angles 30 and 60 calculated in this study iscompared with the experimental results observed by [35]under mixed loading mode (III) resulting in clear agree-ments as can be observed in Figures 11 and 12 ampe amountof the applied load is F 16 kN with load ratio R 01whereas the Paris constant as mentioned previously isC 43378times10minus 7 (mmcycle)(MPa
m
radic) and m 26183
[40]
3 Conclusions
For the CTS specimen the finite element analysis of themixed-mode fatigue crack propagation was carried out usingANSYS Mechanical APDL and compared with experimentaldata for various angles of loading ampe predicted values ofΔKeq by the experimental and numerical methods are ob-served to be in good agreement with the present study re-sults ampe mixed-mode fatigue life is predicted andcompared with experimental data for two different loadingangles of 30deg and 60deg Interestingly the theoretically pre-dicted trajectories of the crack growth for all specimens areidentical to the experimental determined paths for differentloading angles Furthermore the predicted values of SIFs (KIand KII) are in good agreement with analytical solutions Itcan be stated that the structure of specimen geometry and itsconfiguration play an important role in obtaining highervalues of mixed-mode SIFs values ampis happens to besignificant in terms of applied loads particularly for highervalues of KII
Data Availability
ampe data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
References
[1] R I Stephens A Fatemi R R Stephens and H O FuchsMetal Fatigue in Engineering John Wiley amp Sons HobokenNJ USA 2000
[2] V V Bolotin Mechanics of Fatigue CRC Press Boca RatonFL USA 1999
[3] C A Brebbia and A V Farahani Fatigue Damage of Ma-terials Experiment and Analysis WIT Press SouthamptonUK 2003
[4] N E DowlingMechanical Behavior of Materials EngineeringMethods for Deformation Fracture and Fatigue PearsonLondon UK 2012
[5] G R Halford Fatigue and Durability of Structural MaterialsASM International London UK 2006
[6] R O Ritchie ldquoMechanisms of fatigue-crack propagation inductile and brittle solidsrdquo International Journal of Fracturevol 100 no 1 pp 55ndash83 1999
[7] S Suresh Fatigue of Materials Cambridge University PressCambridge UK 1998
[8] K N Solanki Two and ree-Dimensional Finite ElementAnalysis of Plasticity-Induced Fatigue Crack Closure AComprehensive Parametric Study Mississippi State Univer-sity Starkville MS USA 2002
[9] A M Alshoaibi and A K Ariffin ldquoFinite element modeling offatigue crack propagation using a self adaptive mesh strategyrdquoInternational Review of Aerospace Engineering (IREASE)vol 8 no 6 pp 209ndash215 2015
[10] A M Alshoaibi ldquoFinite element procedures for the numericalsimulation of fatigue crack propagation under mixed modeloadingrdquo Structural Engineering and Mechanics vol 35 no 3pp 283ndash299 2010
[11] A M Alshoaibi ldquoAdaptive finite element modeling of fatiguecrack propagationrdquo International Journal of Materials Scienceand Applications vol 2 no 3 pp 104ndash108 2013
[12] A M Alshoaibi ldquoAn adaptive finite element framework forfatigue crack propagation under constant amplitude loadingrdquoInternational Journal of Applied Science and Engineeringvol 13 no 3 pp 261ndash270 2015
[13] A M Alshoaibi ldquoComprehensive comparisons of two andthree dimensional numerical estimation of stress intensityfactors and crack propagation in linear elastic analysisrdquo In-ternational Journal of Integrated Engineering vol 11 no 6pp 45ndash52 2019
[14] A Handbook Fatigue and Fracture American Society ofMaterials vol 19 1996
[15] W He J Liu and D Xie ldquoNumerical study on fatigue crackgrowth at a web-stiffener of ship structural details by anobjected-oriented approach in conjunction with ABAQUSrdquoMarine Structures vol 35 pp 45ndash69 2014
[16] K Rege and D G Pavlou ldquoEffect of stop holes on structuralintegrity of offshore structures a numerical modelrdquo in Pro-ceedings of the Institution of Civil Engineers-Maritime Engi-neering ampomas Telford Ltd London UK 2019
[17] H Rhee ldquoFatigue crack growth analyses of offshore structuraltubular jointrdquo Journal of Offshore Mechanics and Arctic En-gineering vol 111 no 1 pp 49ndash55 1989
[18] H Riahi P Bressolette A Chateauneuf C Bouraoui andR Fathallah ldquoReliability analysis and inspection updating bystochastic response surface of fatigue cracks in mixed moderdquoEngineering Structures vol 33 no 12 pp 3392ndash3401 2011
[19] S Qi L X Cai C Bao H Chen K K Shi and H L WuldquoAnalytical theory for fatigue crack propagation rates ofmixed-mode I-II cracks and its applicationrdquo InternationalJournal of Fatigue vol 119 pp 150ndash159 2019
[20] A F Liu Mechanics and Mechanisms of Fracture An In-troduction ASM International London UK 2005
[21] O Demir A O Ayhan and S Iriccedil ldquoA new specimen formixed mode-III fracture tests modeling experiments and
Advances in Materials Science and Engineering 13
criteria developmentrdquo Engineering Fracture Mechanicsvol 178 pp 457ndash476 2017
[22] M Hussain S Pu and J Underwood ldquoStrain energy releaserate for a crack under combined mode I and mode II FractureAnalysisrdquo in Proceedings of the 1973 National Symposium onFracture Mechanics Part II ASTM International WestConshohocken PA USA 1974
[23] H Richard M Sander M Fulland and G Kullmer ldquoDe-velopment of fatigue crack growth in real structuresrdquo Engi-neering Fracture Mechanics vol 75 no 3-4 pp 331ndash3402008
[24] K Tanaka ldquoFatigue crack propagation from a crack inclinedto the cyclic tensile axisrdquo Engineering Fracture Mechanicsvol 6 no 3 pp 493ndash507 1974
[25] P Paris and F Erdogan ldquoA critical analysis of crack prop-agation lawsrdquo Journal of Basic Engineering vol 85 no 4pp 528ndash533 1963
[26] L Coffin ldquoCyclic deformation and fatigue of metalsrdquo eBook Fatigue and Staying Power of Metals Izo IL USSRMoscow Russia 1963
[27] A Wohler ldquoVersuche zur Ermittlung der auf die Eisen-bahnwagenachsen einwirkenden Krafte und die Wider-standsfahigkeit der Wagen-Achsenrdquo Zeitschrift fur Bauwesenvol 10 no 1860 pp 583ndash614 1860
[28] J M Alegre M Preciado and D Ferrentildeo ldquoStudy of thefatigue failure of an anti-return valve of a high pressuremachinerdquo Engineering Failure Analysis vol 14 no 2pp 408ndash416 2007
[29] H Chen Q Wang W Zeng et al ldquoDynamic brittle crackpropagation modeling using singular edge-based smoothedfinite element method with local mesh rezoningrdquo EuropeanJournal of MechanicsmdashASolids vol 76 pp 208ndash223 2019
[30] S Sajith K S R Krishna Murthy and P Robi ldquoPrediction ofAccurate Mixed Mode Fatigue Crack Growth Curves usingthe Parisrsquo Lawrdquo Journal of e Institution of Engineers (India)Series C vol 100 no 1 pp 165ndash174 2019
[31] G R Irwin ldquoAnalysis of stresses and strains near the end of acrack transversing a platerdquo Journal of Applied Mechanicsvol 24 pp 361ndash364 1957
[32] H A Richard F G Buchholz G Kullmer andM Schollmann 2D-and 3D-mixed mode fracture criteria KeyEngineering Materials Trans Tech Publication Stafa-ZurichSwitzerland 2003
[33] H A Richard B Schramm and N-H Schirmeisen ldquoCrackson Mixed Mode loading - ampeories experiments simula-tionsrdquo International Journal of Fatigue vol 62 pp 93ndash1032014
[34] F Erdogan and G C Sih ldquoOn the crack extension in platesunder plane loading and transverse shearrdquo Journal of BasicEngineering vol 85 no 4 pp 519ndash525 1963
[35] S Sajith K S R K Murthy and P S Robi ldquoExperimental andnumerical investigation of mixed mode fatigue crack growthmodels in aluminum 6061-T6rdquo International Journal of Fa-tigue vol 130 p 105285 2020
[36] F Antunes R Branco J Ferreira and L Borrego ldquoStressintensity factor solutions for CTS mixed mode specimenrdquoFrattura ed Integrita Strutturale vol 13 no 48 pp 676ndash6922019
[37] O Demir A O Ayhan S Iric and H Lekesiz ldquoEvaluation ofmixed mode-III criteria for fatigue crack propagation usingexperiments and modelingrdquo Chinese Journal of Aeronauticsvol 31 no 7 pp 1525ndash1534 2018
[38] H Richard Fracture predictions for cracks exposed to super-imposed normal and shear stresses International NuclearInformation System Vienna Austria 1985
[39] X-T Miao Q Yu C-Y Zhou J Li Y-Z Wang andX-H He ldquoExperimental and numerical investigation onfracture behavior of CTS specimen under I-II mixed modeloadingrdquo European Journal of MechanicsmdashASolids vol 72pp 235ndash244 2018
[40] S Sajith S S Shukla K S R K Murthy and P S RobildquoMixed mode fatigue crack growth studies in AISI 316stainless steelrdquo European Journal of MechanicsmdashASolidsvol 80 p 103898 2020
14 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 03) α 90deg
(aw= 05) α 0deg
8 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 30deg
(aw= 05) α 60deg
Advances in Materials Science and Engineering 9
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 90deg
(aw= 07) α 0deg
10 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 30deg
(aw= 07) α 60deg
Advances in Materials Science and Engineering 11
and the cracks all grow with different angles based onequation (2) as the values of KII are higher whereas forcomplete mode I loading (α 90deg) crack paths are domi-nated by the first mode KI and the crack is going to growstraight ampe influence of the first mode (KI) decreases as αincreases and vice versa for mode II
ampe crack direction angles of three geometries withdifferent initial length of cracks (aW 03 05 and 07)are 6deg 10deg and 12deg respectively which are identical to the
experimental angles tested by [39] Furthermore formixed-mode loading in the CTS geometries with loadingangle α 30deg the cracks are propagated to some extentdiffering to the original crack front direction For thesethree geometries the mixed-mode crack propagationangles (aW 03 05 and 07) are about 55deg 85deg and85deg respectively which are almost in agreement withexperimental angles of [39]
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 90deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 11 Comparison of simulated and experimental [35] fatiguelife for loading angle 30deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350 400 450 500000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 12 Comparison of simulated and experimental [35] fatiguelife for loading angle 60deg
12 Advances in Materials Science and Engineering
For the above three loading angles situations (α 90degα 0deg and α 30deg) regardless of the length of the initialcrack for the same load angle (α) the angles of the crackgrowth in each case are close to each other In contrast forcombined mode (III) loading with loading angle α 60deg thedirections of crack are highly dependent on the initial lengthof the crack On the other hand for small crack length (aW 03 and 05) the cracks also grow varying from theinitial crack tip against the clockwise direction with crackpropagation angles of 25deg and 30deg respectively which areidentical to those angles in the experimental test [39]Furthermore if the initial crack length is longer than theprevious cases (aW 07) the direction of the crack growthand the direction of the initial crack happen to be identicaland are presented in Table 3
In order to assess the fatigue life the number of cycles fordifferent loading angles 30 and 60 calculated in this study iscompared with the experimental results observed by [35]under mixed loading mode (III) resulting in clear agree-ments as can be observed in Figures 11 and 12 ampe amountof the applied load is F 16 kN with load ratio R 01whereas the Paris constant as mentioned previously isC 43378times10minus 7 (mmcycle)(MPa
m
radic) and m 26183
[40]
3 Conclusions
For the CTS specimen the finite element analysis of themixed-mode fatigue crack propagation was carried out usingANSYS Mechanical APDL and compared with experimentaldata for various angles of loading ampe predicted values ofΔKeq by the experimental and numerical methods are ob-served to be in good agreement with the present study re-sults ampe mixed-mode fatigue life is predicted andcompared with experimental data for two different loadingangles of 30deg and 60deg Interestingly the theoretically pre-dicted trajectories of the crack growth for all specimens areidentical to the experimental determined paths for differentloading angles Furthermore the predicted values of SIFs (KIand KII) are in good agreement with analytical solutions Itcan be stated that the structure of specimen geometry and itsconfiguration play an important role in obtaining highervalues of mixed-mode SIFs values ampis happens to besignificant in terms of applied loads particularly for highervalues of KII
Data Availability
ampe data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
References
[1] R I Stephens A Fatemi R R Stephens and H O FuchsMetal Fatigue in Engineering John Wiley amp Sons HobokenNJ USA 2000
[2] V V Bolotin Mechanics of Fatigue CRC Press Boca RatonFL USA 1999
[3] C A Brebbia and A V Farahani Fatigue Damage of Ma-terials Experiment and Analysis WIT Press SouthamptonUK 2003
[4] N E DowlingMechanical Behavior of Materials EngineeringMethods for Deformation Fracture and Fatigue PearsonLondon UK 2012
[5] G R Halford Fatigue and Durability of Structural MaterialsASM International London UK 2006
[6] R O Ritchie ldquoMechanisms of fatigue-crack propagation inductile and brittle solidsrdquo International Journal of Fracturevol 100 no 1 pp 55ndash83 1999
[7] S Suresh Fatigue of Materials Cambridge University PressCambridge UK 1998
[8] K N Solanki Two and ree-Dimensional Finite ElementAnalysis of Plasticity-Induced Fatigue Crack Closure AComprehensive Parametric Study Mississippi State Univer-sity Starkville MS USA 2002
[9] A M Alshoaibi and A K Ariffin ldquoFinite element modeling offatigue crack propagation using a self adaptive mesh strategyrdquoInternational Review of Aerospace Engineering (IREASE)vol 8 no 6 pp 209ndash215 2015
[10] A M Alshoaibi ldquoFinite element procedures for the numericalsimulation of fatigue crack propagation under mixed modeloadingrdquo Structural Engineering and Mechanics vol 35 no 3pp 283ndash299 2010
[11] A M Alshoaibi ldquoAdaptive finite element modeling of fatiguecrack propagationrdquo International Journal of Materials Scienceand Applications vol 2 no 3 pp 104ndash108 2013
[12] A M Alshoaibi ldquoAn adaptive finite element framework forfatigue crack propagation under constant amplitude loadingrdquoInternational Journal of Applied Science and Engineeringvol 13 no 3 pp 261ndash270 2015
[13] A M Alshoaibi ldquoComprehensive comparisons of two andthree dimensional numerical estimation of stress intensityfactors and crack propagation in linear elastic analysisrdquo In-ternational Journal of Integrated Engineering vol 11 no 6pp 45ndash52 2019
[14] A Handbook Fatigue and Fracture American Society ofMaterials vol 19 1996
[15] W He J Liu and D Xie ldquoNumerical study on fatigue crackgrowth at a web-stiffener of ship structural details by anobjected-oriented approach in conjunction with ABAQUSrdquoMarine Structures vol 35 pp 45ndash69 2014
[16] K Rege and D G Pavlou ldquoEffect of stop holes on structuralintegrity of offshore structures a numerical modelrdquo in Pro-ceedings of the Institution of Civil Engineers-Maritime Engi-neering ampomas Telford Ltd London UK 2019
[17] H Rhee ldquoFatigue crack growth analyses of offshore structuraltubular jointrdquo Journal of Offshore Mechanics and Arctic En-gineering vol 111 no 1 pp 49ndash55 1989
[18] H Riahi P Bressolette A Chateauneuf C Bouraoui andR Fathallah ldquoReliability analysis and inspection updating bystochastic response surface of fatigue cracks in mixed moderdquoEngineering Structures vol 33 no 12 pp 3392ndash3401 2011
[19] S Qi L X Cai C Bao H Chen K K Shi and H L WuldquoAnalytical theory for fatigue crack propagation rates ofmixed-mode I-II cracks and its applicationrdquo InternationalJournal of Fatigue vol 119 pp 150ndash159 2019
[20] A F Liu Mechanics and Mechanisms of Fracture An In-troduction ASM International London UK 2005
[21] O Demir A O Ayhan and S Iriccedil ldquoA new specimen formixed mode-III fracture tests modeling experiments and
Advances in Materials Science and Engineering 13
criteria developmentrdquo Engineering Fracture Mechanicsvol 178 pp 457ndash476 2017
[22] M Hussain S Pu and J Underwood ldquoStrain energy releaserate for a crack under combined mode I and mode II FractureAnalysisrdquo in Proceedings of the 1973 National Symposium onFracture Mechanics Part II ASTM International WestConshohocken PA USA 1974
[23] H Richard M Sander M Fulland and G Kullmer ldquoDe-velopment of fatigue crack growth in real structuresrdquo Engi-neering Fracture Mechanics vol 75 no 3-4 pp 331ndash3402008
[24] K Tanaka ldquoFatigue crack propagation from a crack inclinedto the cyclic tensile axisrdquo Engineering Fracture Mechanicsvol 6 no 3 pp 493ndash507 1974
[25] P Paris and F Erdogan ldquoA critical analysis of crack prop-agation lawsrdquo Journal of Basic Engineering vol 85 no 4pp 528ndash533 1963
[26] L Coffin ldquoCyclic deformation and fatigue of metalsrdquo eBook Fatigue and Staying Power of Metals Izo IL USSRMoscow Russia 1963
[27] A Wohler ldquoVersuche zur Ermittlung der auf die Eisen-bahnwagenachsen einwirkenden Krafte und die Wider-standsfahigkeit der Wagen-Achsenrdquo Zeitschrift fur Bauwesenvol 10 no 1860 pp 583ndash614 1860
[28] J M Alegre M Preciado and D Ferrentildeo ldquoStudy of thefatigue failure of an anti-return valve of a high pressuremachinerdquo Engineering Failure Analysis vol 14 no 2pp 408ndash416 2007
[29] H Chen Q Wang W Zeng et al ldquoDynamic brittle crackpropagation modeling using singular edge-based smoothedfinite element method with local mesh rezoningrdquo EuropeanJournal of MechanicsmdashASolids vol 76 pp 208ndash223 2019
[30] S Sajith K S R Krishna Murthy and P Robi ldquoPrediction ofAccurate Mixed Mode Fatigue Crack Growth Curves usingthe Parisrsquo Lawrdquo Journal of e Institution of Engineers (India)Series C vol 100 no 1 pp 165ndash174 2019
[31] G R Irwin ldquoAnalysis of stresses and strains near the end of acrack transversing a platerdquo Journal of Applied Mechanicsvol 24 pp 361ndash364 1957
[32] H A Richard F G Buchholz G Kullmer andM Schollmann 2D-and 3D-mixed mode fracture criteria KeyEngineering Materials Trans Tech Publication Stafa-ZurichSwitzerland 2003
[33] H A Richard B Schramm and N-H Schirmeisen ldquoCrackson Mixed Mode loading - ampeories experiments simula-tionsrdquo International Journal of Fatigue vol 62 pp 93ndash1032014
[34] F Erdogan and G C Sih ldquoOn the crack extension in platesunder plane loading and transverse shearrdquo Journal of BasicEngineering vol 85 no 4 pp 519ndash525 1963
[35] S Sajith K S R K Murthy and P S Robi ldquoExperimental andnumerical investigation of mixed mode fatigue crack growthmodels in aluminum 6061-T6rdquo International Journal of Fa-tigue vol 130 p 105285 2020
[36] F Antunes R Branco J Ferreira and L Borrego ldquoStressintensity factor solutions for CTS mixed mode specimenrdquoFrattura ed Integrita Strutturale vol 13 no 48 pp 676ndash6922019
[37] O Demir A O Ayhan S Iric and H Lekesiz ldquoEvaluation ofmixed mode-III criteria for fatigue crack propagation usingexperiments and modelingrdquo Chinese Journal of Aeronauticsvol 31 no 7 pp 1525ndash1534 2018
[38] H Richard Fracture predictions for cracks exposed to super-imposed normal and shear stresses International NuclearInformation System Vienna Austria 1985
[39] X-T Miao Q Yu C-Y Zhou J Li Y-Z Wang andX-H He ldquoExperimental and numerical investigation onfracture behavior of CTS specimen under I-II mixed modeloadingrdquo European Journal of MechanicsmdashASolids vol 72pp 235ndash244 2018
[40] S Sajith S S Shukla K S R K Murthy and P S RobildquoMixed mode fatigue crack growth studies in AISI 316stainless steelrdquo European Journal of MechanicsmdashASolidsvol 80 p 103898 2020
14 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 30deg
(aw= 05) α 60deg
Advances in Materials Science and Engineering 9
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 90deg
(aw= 07) α 0deg
10 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 30deg
(aw= 07) α 60deg
Advances in Materials Science and Engineering 11
and the cracks all grow with different angles based onequation (2) as the values of KII are higher whereas forcomplete mode I loading (α 90deg) crack paths are domi-nated by the first mode KI and the crack is going to growstraight ampe influence of the first mode (KI) decreases as αincreases and vice versa for mode II
ampe crack direction angles of three geometries withdifferent initial length of cracks (aW 03 05 and 07)are 6deg 10deg and 12deg respectively which are identical to the
experimental angles tested by [39] Furthermore formixed-mode loading in the CTS geometries with loadingangle α 30deg the cracks are propagated to some extentdiffering to the original crack front direction For thesethree geometries the mixed-mode crack propagationangles (aW 03 05 and 07) are about 55deg 85deg and85deg respectively which are almost in agreement withexperimental angles of [39]
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 90deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 11 Comparison of simulated and experimental [35] fatiguelife for loading angle 30deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350 400 450 500000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 12 Comparison of simulated and experimental [35] fatiguelife for loading angle 60deg
12 Advances in Materials Science and Engineering
For the above three loading angles situations (α 90degα 0deg and α 30deg) regardless of the length of the initialcrack for the same load angle (α) the angles of the crackgrowth in each case are close to each other In contrast forcombined mode (III) loading with loading angle α 60deg thedirections of crack are highly dependent on the initial lengthof the crack On the other hand for small crack length (aW 03 and 05) the cracks also grow varying from theinitial crack tip against the clockwise direction with crackpropagation angles of 25deg and 30deg respectively which areidentical to those angles in the experimental test [39]Furthermore if the initial crack length is longer than theprevious cases (aW 07) the direction of the crack growthand the direction of the initial crack happen to be identicaland are presented in Table 3
In order to assess the fatigue life the number of cycles fordifferent loading angles 30 and 60 calculated in this study iscompared with the experimental results observed by [35]under mixed loading mode (III) resulting in clear agree-ments as can be observed in Figures 11 and 12 ampe amountof the applied load is F 16 kN with load ratio R 01whereas the Paris constant as mentioned previously isC 43378times10minus 7 (mmcycle)(MPa
m
radic) and m 26183
[40]
3 Conclusions
For the CTS specimen the finite element analysis of themixed-mode fatigue crack propagation was carried out usingANSYS Mechanical APDL and compared with experimentaldata for various angles of loading ampe predicted values ofΔKeq by the experimental and numerical methods are ob-served to be in good agreement with the present study re-sults ampe mixed-mode fatigue life is predicted andcompared with experimental data for two different loadingangles of 30deg and 60deg Interestingly the theoretically pre-dicted trajectories of the crack growth for all specimens areidentical to the experimental determined paths for differentloading angles Furthermore the predicted values of SIFs (KIand KII) are in good agreement with analytical solutions Itcan be stated that the structure of specimen geometry and itsconfiguration play an important role in obtaining highervalues of mixed-mode SIFs values ampis happens to besignificant in terms of applied loads particularly for highervalues of KII
Data Availability
ampe data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
References
[1] R I Stephens A Fatemi R R Stephens and H O FuchsMetal Fatigue in Engineering John Wiley amp Sons HobokenNJ USA 2000
[2] V V Bolotin Mechanics of Fatigue CRC Press Boca RatonFL USA 1999
[3] C A Brebbia and A V Farahani Fatigue Damage of Ma-terials Experiment and Analysis WIT Press SouthamptonUK 2003
[4] N E DowlingMechanical Behavior of Materials EngineeringMethods for Deformation Fracture and Fatigue PearsonLondon UK 2012
[5] G R Halford Fatigue and Durability of Structural MaterialsASM International London UK 2006
[6] R O Ritchie ldquoMechanisms of fatigue-crack propagation inductile and brittle solidsrdquo International Journal of Fracturevol 100 no 1 pp 55ndash83 1999
[7] S Suresh Fatigue of Materials Cambridge University PressCambridge UK 1998
[8] K N Solanki Two and ree-Dimensional Finite ElementAnalysis of Plasticity-Induced Fatigue Crack Closure AComprehensive Parametric Study Mississippi State Univer-sity Starkville MS USA 2002
[9] A M Alshoaibi and A K Ariffin ldquoFinite element modeling offatigue crack propagation using a self adaptive mesh strategyrdquoInternational Review of Aerospace Engineering (IREASE)vol 8 no 6 pp 209ndash215 2015
[10] A M Alshoaibi ldquoFinite element procedures for the numericalsimulation of fatigue crack propagation under mixed modeloadingrdquo Structural Engineering and Mechanics vol 35 no 3pp 283ndash299 2010
[11] A M Alshoaibi ldquoAdaptive finite element modeling of fatiguecrack propagationrdquo International Journal of Materials Scienceand Applications vol 2 no 3 pp 104ndash108 2013
[12] A M Alshoaibi ldquoAn adaptive finite element framework forfatigue crack propagation under constant amplitude loadingrdquoInternational Journal of Applied Science and Engineeringvol 13 no 3 pp 261ndash270 2015
[13] A M Alshoaibi ldquoComprehensive comparisons of two andthree dimensional numerical estimation of stress intensityfactors and crack propagation in linear elastic analysisrdquo In-ternational Journal of Integrated Engineering vol 11 no 6pp 45ndash52 2019
[14] A Handbook Fatigue and Fracture American Society ofMaterials vol 19 1996
[15] W He J Liu and D Xie ldquoNumerical study on fatigue crackgrowth at a web-stiffener of ship structural details by anobjected-oriented approach in conjunction with ABAQUSrdquoMarine Structures vol 35 pp 45ndash69 2014
[16] K Rege and D G Pavlou ldquoEffect of stop holes on structuralintegrity of offshore structures a numerical modelrdquo in Pro-ceedings of the Institution of Civil Engineers-Maritime Engi-neering ampomas Telford Ltd London UK 2019
[17] H Rhee ldquoFatigue crack growth analyses of offshore structuraltubular jointrdquo Journal of Offshore Mechanics and Arctic En-gineering vol 111 no 1 pp 49ndash55 1989
[18] H Riahi P Bressolette A Chateauneuf C Bouraoui andR Fathallah ldquoReliability analysis and inspection updating bystochastic response surface of fatigue cracks in mixed moderdquoEngineering Structures vol 33 no 12 pp 3392ndash3401 2011
[19] S Qi L X Cai C Bao H Chen K K Shi and H L WuldquoAnalytical theory for fatigue crack propagation rates ofmixed-mode I-II cracks and its applicationrdquo InternationalJournal of Fatigue vol 119 pp 150ndash159 2019
[20] A F Liu Mechanics and Mechanisms of Fracture An In-troduction ASM International London UK 2005
[21] O Demir A O Ayhan and S Iriccedil ldquoA new specimen formixed mode-III fracture tests modeling experiments and
Advances in Materials Science and Engineering 13
criteria developmentrdquo Engineering Fracture Mechanicsvol 178 pp 457ndash476 2017
[22] M Hussain S Pu and J Underwood ldquoStrain energy releaserate for a crack under combined mode I and mode II FractureAnalysisrdquo in Proceedings of the 1973 National Symposium onFracture Mechanics Part II ASTM International WestConshohocken PA USA 1974
[23] H Richard M Sander M Fulland and G Kullmer ldquoDe-velopment of fatigue crack growth in real structuresrdquo Engi-neering Fracture Mechanics vol 75 no 3-4 pp 331ndash3402008
[24] K Tanaka ldquoFatigue crack propagation from a crack inclinedto the cyclic tensile axisrdquo Engineering Fracture Mechanicsvol 6 no 3 pp 493ndash507 1974
[25] P Paris and F Erdogan ldquoA critical analysis of crack prop-agation lawsrdquo Journal of Basic Engineering vol 85 no 4pp 528ndash533 1963
[26] L Coffin ldquoCyclic deformation and fatigue of metalsrdquo eBook Fatigue and Staying Power of Metals Izo IL USSRMoscow Russia 1963
[27] A Wohler ldquoVersuche zur Ermittlung der auf die Eisen-bahnwagenachsen einwirkenden Krafte und die Wider-standsfahigkeit der Wagen-Achsenrdquo Zeitschrift fur Bauwesenvol 10 no 1860 pp 583ndash614 1860
[28] J M Alegre M Preciado and D Ferrentildeo ldquoStudy of thefatigue failure of an anti-return valve of a high pressuremachinerdquo Engineering Failure Analysis vol 14 no 2pp 408ndash416 2007
[29] H Chen Q Wang W Zeng et al ldquoDynamic brittle crackpropagation modeling using singular edge-based smoothedfinite element method with local mesh rezoningrdquo EuropeanJournal of MechanicsmdashASolids vol 76 pp 208ndash223 2019
[30] S Sajith K S R Krishna Murthy and P Robi ldquoPrediction ofAccurate Mixed Mode Fatigue Crack Growth Curves usingthe Parisrsquo Lawrdquo Journal of e Institution of Engineers (India)Series C vol 100 no 1 pp 165ndash174 2019
[31] G R Irwin ldquoAnalysis of stresses and strains near the end of acrack transversing a platerdquo Journal of Applied Mechanicsvol 24 pp 361ndash364 1957
[32] H A Richard F G Buchholz G Kullmer andM Schollmann 2D-and 3D-mixed mode fracture criteria KeyEngineering Materials Trans Tech Publication Stafa-ZurichSwitzerland 2003
[33] H A Richard B Schramm and N-H Schirmeisen ldquoCrackson Mixed Mode loading - ampeories experiments simula-tionsrdquo International Journal of Fatigue vol 62 pp 93ndash1032014
[34] F Erdogan and G C Sih ldquoOn the crack extension in platesunder plane loading and transverse shearrdquo Journal of BasicEngineering vol 85 no 4 pp 519ndash525 1963
[35] S Sajith K S R K Murthy and P S Robi ldquoExperimental andnumerical investigation of mixed mode fatigue crack growthmodels in aluminum 6061-T6rdquo International Journal of Fa-tigue vol 130 p 105285 2020
[36] F Antunes R Branco J Ferreira and L Borrego ldquoStressintensity factor solutions for CTS mixed mode specimenrdquoFrattura ed Integrita Strutturale vol 13 no 48 pp 676ndash6922019
[37] O Demir A O Ayhan S Iric and H Lekesiz ldquoEvaluation ofmixed mode-III criteria for fatigue crack propagation usingexperiments and modelingrdquo Chinese Journal of Aeronauticsvol 31 no 7 pp 1525ndash1534 2018
[38] H Richard Fracture predictions for cracks exposed to super-imposed normal and shear stresses International NuclearInformation System Vienna Austria 1985
[39] X-T Miao Q Yu C-Y Zhou J Li Y-Z Wang andX-H He ldquoExperimental and numerical investigation onfracture behavior of CTS specimen under I-II mixed modeloadingrdquo European Journal of MechanicsmdashASolids vol 72pp 235ndash244 2018
[40] S Sajith S S Shukla K S R K Murthy and P S RobildquoMixed mode fatigue crack growth studies in AISI 316stainless steelrdquo European Journal of MechanicsmdashASolidsvol 80 p 103898 2020
14 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 05) α 90deg
(aw= 07) α 0deg
10 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 30deg
(aw= 07) α 60deg
Advances in Materials Science and Engineering 11
and the cracks all grow with different angles based onequation (2) as the values of KII are higher whereas forcomplete mode I loading (α 90deg) crack paths are domi-nated by the first mode KI and the crack is going to growstraight ampe influence of the first mode (KI) decreases as αincreases and vice versa for mode II
ampe crack direction angles of three geometries withdifferent initial length of cracks (aW 03 05 and 07)are 6deg 10deg and 12deg respectively which are identical to the
experimental angles tested by [39] Furthermore formixed-mode loading in the CTS geometries with loadingangle α 30deg the cracks are propagated to some extentdiffering to the original crack front direction For thesethree geometries the mixed-mode crack propagationangles (aW 03 05 and 07) are about 55deg 85deg and85deg respectively which are almost in agreement withexperimental angles of [39]
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 90deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 11 Comparison of simulated and experimental [35] fatiguelife for loading angle 30deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350 400 450 500000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 12 Comparison of simulated and experimental [35] fatiguelife for loading angle 60deg
12 Advances in Materials Science and Engineering
For the above three loading angles situations (α 90degα 0deg and α 30deg) regardless of the length of the initialcrack for the same load angle (α) the angles of the crackgrowth in each case are close to each other In contrast forcombined mode (III) loading with loading angle α 60deg thedirections of crack are highly dependent on the initial lengthof the crack On the other hand for small crack length (aW 03 and 05) the cracks also grow varying from theinitial crack tip against the clockwise direction with crackpropagation angles of 25deg and 30deg respectively which areidentical to those angles in the experimental test [39]Furthermore if the initial crack length is longer than theprevious cases (aW 07) the direction of the crack growthand the direction of the initial crack happen to be identicaland are presented in Table 3
In order to assess the fatigue life the number of cycles fordifferent loading angles 30 and 60 calculated in this study iscompared with the experimental results observed by [35]under mixed loading mode (III) resulting in clear agree-ments as can be observed in Figures 11 and 12 ampe amountof the applied load is F 16 kN with load ratio R 01whereas the Paris constant as mentioned previously isC 43378times10minus 7 (mmcycle)(MPa
m
radic) and m 26183
[40]
3 Conclusions
For the CTS specimen the finite element analysis of themixed-mode fatigue crack propagation was carried out usingANSYS Mechanical APDL and compared with experimentaldata for various angles of loading ampe predicted values ofΔKeq by the experimental and numerical methods are ob-served to be in good agreement with the present study re-sults ampe mixed-mode fatigue life is predicted andcompared with experimental data for two different loadingangles of 30deg and 60deg Interestingly the theoretically pre-dicted trajectories of the crack growth for all specimens areidentical to the experimental determined paths for differentloading angles Furthermore the predicted values of SIFs (KIand KII) are in good agreement with analytical solutions Itcan be stated that the structure of specimen geometry and itsconfiguration play an important role in obtaining highervalues of mixed-mode SIFs values ampis happens to besignificant in terms of applied loads particularly for highervalues of KII
Data Availability
ampe data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
References
[1] R I Stephens A Fatemi R R Stephens and H O FuchsMetal Fatigue in Engineering John Wiley amp Sons HobokenNJ USA 2000
[2] V V Bolotin Mechanics of Fatigue CRC Press Boca RatonFL USA 1999
[3] C A Brebbia and A V Farahani Fatigue Damage of Ma-terials Experiment and Analysis WIT Press SouthamptonUK 2003
[4] N E DowlingMechanical Behavior of Materials EngineeringMethods for Deformation Fracture and Fatigue PearsonLondon UK 2012
[5] G R Halford Fatigue and Durability of Structural MaterialsASM International London UK 2006
[6] R O Ritchie ldquoMechanisms of fatigue-crack propagation inductile and brittle solidsrdquo International Journal of Fracturevol 100 no 1 pp 55ndash83 1999
[7] S Suresh Fatigue of Materials Cambridge University PressCambridge UK 1998
[8] K N Solanki Two and ree-Dimensional Finite ElementAnalysis of Plasticity-Induced Fatigue Crack Closure AComprehensive Parametric Study Mississippi State Univer-sity Starkville MS USA 2002
[9] A M Alshoaibi and A K Ariffin ldquoFinite element modeling offatigue crack propagation using a self adaptive mesh strategyrdquoInternational Review of Aerospace Engineering (IREASE)vol 8 no 6 pp 209ndash215 2015
[10] A M Alshoaibi ldquoFinite element procedures for the numericalsimulation of fatigue crack propagation under mixed modeloadingrdquo Structural Engineering and Mechanics vol 35 no 3pp 283ndash299 2010
[11] A M Alshoaibi ldquoAdaptive finite element modeling of fatiguecrack propagationrdquo International Journal of Materials Scienceand Applications vol 2 no 3 pp 104ndash108 2013
[12] A M Alshoaibi ldquoAn adaptive finite element framework forfatigue crack propagation under constant amplitude loadingrdquoInternational Journal of Applied Science and Engineeringvol 13 no 3 pp 261ndash270 2015
[13] A M Alshoaibi ldquoComprehensive comparisons of two andthree dimensional numerical estimation of stress intensityfactors and crack propagation in linear elastic analysisrdquo In-ternational Journal of Integrated Engineering vol 11 no 6pp 45ndash52 2019
[14] A Handbook Fatigue and Fracture American Society ofMaterials vol 19 1996
[15] W He J Liu and D Xie ldquoNumerical study on fatigue crackgrowth at a web-stiffener of ship structural details by anobjected-oriented approach in conjunction with ABAQUSrdquoMarine Structures vol 35 pp 45ndash69 2014
[16] K Rege and D G Pavlou ldquoEffect of stop holes on structuralintegrity of offshore structures a numerical modelrdquo in Pro-ceedings of the Institution of Civil Engineers-Maritime Engi-neering ampomas Telford Ltd London UK 2019
[17] H Rhee ldquoFatigue crack growth analyses of offshore structuraltubular jointrdquo Journal of Offshore Mechanics and Arctic En-gineering vol 111 no 1 pp 49ndash55 1989
[18] H Riahi P Bressolette A Chateauneuf C Bouraoui andR Fathallah ldquoReliability analysis and inspection updating bystochastic response surface of fatigue cracks in mixed moderdquoEngineering Structures vol 33 no 12 pp 3392ndash3401 2011
[19] S Qi L X Cai C Bao H Chen K K Shi and H L WuldquoAnalytical theory for fatigue crack propagation rates ofmixed-mode I-II cracks and its applicationrdquo InternationalJournal of Fatigue vol 119 pp 150ndash159 2019
[20] A F Liu Mechanics and Mechanisms of Fracture An In-troduction ASM International London UK 2005
[21] O Demir A O Ayhan and S Iriccedil ldquoA new specimen formixed mode-III fracture tests modeling experiments and
Advances in Materials Science and Engineering 13
criteria developmentrdquo Engineering Fracture Mechanicsvol 178 pp 457ndash476 2017
[22] M Hussain S Pu and J Underwood ldquoStrain energy releaserate for a crack under combined mode I and mode II FractureAnalysisrdquo in Proceedings of the 1973 National Symposium onFracture Mechanics Part II ASTM International WestConshohocken PA USA 1974
[23] H Richard M Sander M Fulland and G Kullmer ldquoDe-velopment of fatigue crack growth in real structuresrdquo Engi-neering Fracture Mechanics vol 75 no 3-4 pp 331ndash3402008
[24] K Tanaka ldquoFatigue crack propagation from a crack inclinedto the cyclic tensile axisrdquo Engineering Fracture Mechanicsvol 6 no 3 pp 493ndash507 1974
[25] P Paris and F Erdogan ldquoA critical analysis of crack prop-agation lawsrdquo Journal of Basic Engineering vol 85 no 4pp 528ndash533 1963
[26] L Coffin ldquoCyclic deformation and fatigue of metalsrdquo eBook Fatigue and Staying Power of Metals Izo IL USSRMoscow Russia 1963
[27] A Wohler ldquoVersuche zur Ermittlung der auf die Eisen-bahnwagenachsen einwirkenden Krafte und die Wider-standsfahigkeit der Wagen-Achsenrdquo Zeitschrift fur Bauwesenvol 10 no 1860 pp 583ndash614 1860
[28] J M Alegre M Preciado and D Ferrentildeo ldquoStudy of thefatigue failure of an anti-return valve of a high pressuremachinerdquo Engineering Failure Analysis vol 14 no 2pp 408ndash416 2007
[29] H Chen Q Wang W Zeng et al ldquoDynamic brittle crackpropagation modeling using singular edge-based smoothedfinite element method with local mesh rezoningrdquo EuropeanJournal of MechanicsmdashASolids vol 76 pp 208ndash223 2019
[30] S Sajith K S R Krishna Murthy and P Robi ldquoPrediction ofAccurate Mixed Mode Fatigue Crack Growth Curves usingthe Parisrsquo Lawrdquo Journal of e Institution of Engineers (India)Series C vol 100 no 1 pp 165ndash174 2019
[31] G R Irwin ldquoAnalysis of stresses and strains near the end of acrack transversing a platerdquo Journal of Applied Mechanicsvol 24 pp 361ndash364 1957
[32] H A Richard F G Buchholz G Kullmer andM Schollmann 2D-and 3D-mixed mode fracture criteria KeyEngineering Materials Trans Tech Publication Stafa-ZurichSwitzerland 2003
[33] H A Richard B Schramm and N-H Schirmeisen ldquoCrackson Mixed Mode loading - ampeories experiments simula-tionsrdquo International Journal of Fatigue vol 62 pp 93ndash1032014
[34] F Erdogan and G C Sih ldquoOn the crack extension in platesunder plane loading and transverse shearrdquo Journal of BasicEngineering vol 85 no 4 pp 519ndash525 1963
[35] S Sajith K S R K Murthy and P S Robi ldquoExperimental andnumerical investigation of mixed mode fatigue crack growthmodels in aluminum 6061-T6rdquo International Journal of Fa-tigue vol 130 p 105285 2020
[36] F Antunes R Branco J Ferreira and L Borrego ldquoStressintensity factor solutions for CTS mixed mode specimenrdquoFrattura ed Integrita Strutturale vol 13 no 48 pp 676ndash6922019
[37] O Demir A O Ayhan S Iric and H Lekesiz ldquoEvaluation ofmixed mode-III criteria for fatigue crack propagation usingexperiments and modelingrdquo Chinese Journal of Aeronauticsvol 31 no 7 pp 1525ndash1534 2018
[38] H Richard Fracture predictions for cracks exposed to super-imposed normal and shear stresses International NuclearInformation System Vienna Austria 1985
[39] X-T Miao Q Yu C-Y Zhou J Li Y-Z Wang andX-H He ldquoExperimental and numerical investigation onfracture behavior of CTS specimen under I-II mixed modeloadingrdquo European Journal of MechanicsmdashASolids vol 72pp 235ndash244 2018
[40] S Sajith S S Shukla K S R K Murthy and P S RobildquoMixed mode fatigue crack growth studies in AISI 316stainless steelrdquo European Journal of MechanicsmdashASolidsvol 80 p 103898 2020
14 Advances in Materials Science and Engineering
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 30deg
(aw= 07) α 60deg
Advances in Materials Science and Engineering 11
and the cracks all grow with different angles based onequation (2) as the values of KII are higher whereas forcomplete mode I loading (α 90deg) crack paths are domi-nated by the first mode KI and the crack is going to growstraight ampe influence of the first mode (KI) decreases as αincreases and vice versa for mode II
ampe crack direction angles of three geometries withdifferent initial length of cracks (aW 03 05 and 07)are 6deg 10deg and 12deg respectively which are identical to the
experimental angles tested by [39] Furthermore formixed-mode loading in the CTS geometries with loadingangle α 30deg the cracks are propagated to some extentdiffering to the original crack front direction For thesethree geometries the mixed-mode crack propagationangles (aW 03 05 and 07) are about 55deg 85deg and85deg respectively which are almost in agreement withexperimental angles of [39]
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 90deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 11 Comparison of simulated and experimental [35] fatiguelife for loading angle 30deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350 400 450 500000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 12 Comparison of simulated and experimental [35] fatiguelife for loading angle 60deg
12 Advances in Materials Science and Engineering
For the above three loading angles situations (α 90degα 0deg and α 30deg) regardless of the length of the initialcrack for the same load angle (α) the angles of the crackgrowth in each case are close to each other In contrast forcombined mode (III) loading with loading angle α 60deg thedirections of crack are highly dependent on the initial lengthof the crack On the other hand for small crack length (aW 03 and 05) the cracks also grow varying from theinitial crack tip against the clockwise direction with crackpropagation angles of 25deg and 30deg respectively which areidentical to those angles in the experimental test [39]Furthermore if the initial crack length is longer than theprevious cases (aW 07) the direction of the crack growthand the direction of the initial crack happen to be identicaland are presented in Table 3
In order to assess the fatigue life the number of cycles fordifferent loading angles 30 and 60 calculated in this study iscompared with the experimental results observed by [35]under mixed loading mode (III) resulting in clear agree-ments as can be observed in Figures 11 and 12 ampe amountof the applied load is F 16 kN with load ratio R 01whereas the Paris constant as mentioned previously isC 43378times10minus 7 (mmcycle)(MPa
m
radic) and m 26183
[40]
3 Conclusions
For the CTS specimen the finite element analysis of themixed-mode fatigue crack propagation was carried out usingANSYS Mechanical APDL and compared with experimentaldata for various angles of loading ampe predicted values ofΔKeq by the experimental and numerical methods are ob-served to be in good agreement with the present study re-sults ampe mixed-mode fatigue life is predicted andcompared with experimental data for two different loadingangles of 30deg and 60deg Interestingly the theoretically pre-dicted trajectories of the crack growth for all specimens areidentical to the experimental determined paths for differentloading angles Furthermore the predicted values of SIFs (KIand KII) are in good agreement with analytical solutions Itcan be stated that the structure of specimen geometry and itsconfiguration play an important role in obtaining highervalues of mixed-mode SIFs values ampis happens to besignificant in terms of applied loads particularly for highervalues of KII
Data Availability
ampe data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
References
[1] R I Stephens A Fatemi R R Stephens and H O FuchsMetal Fatigue in Engineering John Wiley amp Sons HobokenNJ USA 2000
[2] V V Bolotin Mechanics of Fatigue CRC Press Boca RatonFL USA 1999
[3] C A Brebbia and A V Farahani Fatigue Damage of Ma-terials Experiment and Analysis WIT Press SouthamptonUK 2003
[4] N E DowlingMechanical Behavior of Materials EngineeringMethods for Deformation Fracture and Fatigue PearsonLondon UK 2012
[5] G R Halford Fatigue and Durability of Structural MaterialsASM International London UK 2006
[6] R O Ritchie ldquoMechanisms of fatigue-crack propagation inductile and brittle solidsrdquo International Journal of Fracturevol 100 no 1 pp 55ndash83 1999
[7] S Suresh Fatigue of Materials Cambridge University PressCambridge UK 1998
[8] K N Solanki Two and ree-Dimensional Finite ElementAnalysis of Plasticity-Induced Fatigue Crack Closure AComprehensive Parametric Study Mississippi State Univer-sity Starkville MS USA 2002
[9] A M Alshoaibi and A K Ariffin ldquoFinite element modeling offatigue crack propagation using a self adaptive mesh strategyrdquoInternational Review of Aerospace Engineering (IREASE)vol 8 no 6 pp 209ndash215 2015
[10] A M Alshoaibi ldquoFinite element procedures for the numericalsimulation of fatigue crack propagation under mixed modeloadingrdquo Structural Engineering and Mechanics vol 35 no 3pp 283ndash299 2010
[11] A M Alshoaibi ldquoAdaptive finite element modeling of fatiguecrack propagationrdquo International Journal of Materials Scienceand Applications vol 2 no 3 pp 104ndash108 2013
[12] A M Alshoaibi ldquoAn adaptive finite element framework forfatigue crack propagation under constant amplitude loadingrdquoInternational Journal of Applied Science and Engineeringvol 13 no 3 pp 261ndash270 2015
[13] A M Alshoaibi ldquoComprehensive comparisons of two andthree dimensional numerical estimation of stress intensityfactors and crack propagation in linear elastic analysisrdquo In-ternational Journal of Integrated Engineering vol 11 no 6pp 45ndash52 2019
[14] A Handbook Fatigue and Fracture American Society ofMaterials vol 19 1996
[15] W He J Liu and D Xie ldquoNumerical study on fatigue crackgrowth at a web-stiffener of ship structural details by anobjected-oriented approach in conjunction with ABAQUSrdquoMarine Structures vol 35 pp 45ndash69 2014
[16] K Rege and D G Pavlou ldquoEffect of stop holes on structuralintegrity of offshore structures a numerical modelrdquo in Pro-ceedings of the Institution of Civil Engineers-Maritime Engi-neering ampomas Telford Ltd London UK 2019
[17] H Rhee ldquoFatigue crack growth analyses of offshore structuraltubular jointrdquo Journal of Offshore Mechanics and Arctic En-gineering vol 111 no 1 pp 49ndash55 1989
[18] H Riahi P Bressolette A Chateauneuf C Bouraoui andR Fathallah ldquoReliability analysis and inspection updating bystochastic response surface of fatigue cracks in mixed moderdquoEngineering Structures vol 33 no 12 pp 3392ndash3401 2011
[19] S Qi L X Cai C Bao H Chen K K Shi and H L WuldquoAnalytical theory for fatigue crack propagation rates ofmixed-mode I-II cracks and its applicationrdquo InternationalJournal of Fatigue vol 119 pp 150ndash159 2019
[20] A F Liu Mechanics and Mechanisms of Fracture An In-troduction ASM International London UK 2005
[21] O Demir A O Ayhan and S Iriccedil ldquoA new specimen formixed mode-III fracture tests modeling experiments and
Advances in Materials Science and Engineering 13
criteria developmentrdquo Engineering Fracture Mechanicsvol 178 pp 457ndash476 2017
[22] M Hussain S Pu and J Underwood ldquoStrain energy releaserate for a crack under combined mode I and mode II FractureAnalysisrdquo in Proceedings of the 1973 National Symposium onFracture Mechanics Part II ASTM International WestConshohocken PA USA 1974
[23] H Richard M Sander M Fulland and G Kullmer ldquoDe-velopment of fatigue crack growth in real structuresrdquo Engi-neering Fracture Mechanics vol 75 no 3-4 pp 331ndash3402008
[24] K Tanaka ldquoFatigue crack propagation from a crack inclinedto the cyclic tensile axisrdquo Engineering Fracture Mechanicsvol 6 no 3 pp 493ndash507 1974
[25] P Paris and F Erdogan ldquoA critical analysis of crack prop-agation lawsrdquo Journal of Basic Engineering vol 85 no 4pp 528ndash533 1963
[26] L Coffin ldquoCyclic deformation and fatigue of metalsrdquo eBook Fatigue and Staying Power of Metals Izo IL USSRMoscow Russia 1963
[27] A Wohler ldquoVersuche zur Ermittlung der auf die Eisen-bahnwagenachsen einwirkenden Krafte und die Wider-standsfahigkeit der Wagen-Achsenrdquo Zeitschrift fur Bauwesenvol 10 no 1860 pp 583ndash614 1860
[28] J M Alegre M Preciado and D Ferrentildeo ldquoStudy of thefatigue failure of an anti-return valve of a high pressuremachinerdquo Engineering Failure Analysis vol 14 no 2pp 408ndash416 2007
[29] H Chen Q Wang W Zeng et al ldquoDynamic brittle crackpropagation modeling using singular edge-based smoothedfinite element method with local mesh rezoningrdquo EuropeanJournal of MechanicsmdashASolids vol 76 pp 208ndash223 2019
[30] S Sajith K S R Krishna Murthy and P Robi ldquoPrediction ofAccurate Mixed Mode Fatigue Crack Growth Curves usingthe Parisrsquo Lawrdquo Journal of e Institution of Engineers (India)Series C vol 100 no 1 pp 165ndash174 2019
[31] G R Irwin ldquoAnalysis of stresses and strains near the end of acrack transversing a platerdquo Journal of Applied Mechanicsvol 24 pp 361ndash364 1957
[32] H A Richard F G Buchholz G Kullmer andM Schollmann 2D-and 3D-mixed mode fracture criteria KeyEngineering Materials Trans Tech Publication Stafa-ZurichSwitzerland 2003
[33] H A Richard B Schramm and N-H Schirmeisen ldquoCrackson Mixed Mode loading - ampeories experiments simula-tionsrdquo International Journal of Fatigue vol 62 pp 93ndash1032014
[34] F Erdogan and G C Sih ldquoOn the crack extension in platesunder plane loading and transverse shearrdquo Journal of BasicEngineering vol 85 no 4 pp 519ndash525 1963
[35] S Sajith K S R K Murthy and P S Robi ldquoExperimental andnumerical investigation of mixed mode fatigue crack growthmodels in aluminum 6061-T6rdquo International Journal of Fa-tigue vol 130 p 105285 2020
[36] F Antunes R Branco J Ferreira and L Borrego ldquoStressintensity factor solutions for CTS mixed mode specimenrdquoFrattura ed Integrita Strutturale vol 13 no 48 pp 676ndash6922019
[37] O Demir A O Ayhan S Iric and H Lekesiz ldquoEvaluation ofmixed mode-III criteria for fatigue crack propagation usingexperiments and modelingrdquo Chinese Journal of Aeronauticsvol 31 no 7 pp 1525ndash1534 2018
[38] H Richard Fracture predictions for cracks exposed to super-imposed normal and shear stresses International NuclearInformation System Vienna Austria 1985
[39] X-T Miao Q Yu C-Y Zhou J Li Y-Z Wang andX-H He ldquoExperimental and numerical investigation onfracture behavior of CTS specimen under I-II mixed modeloadingrdquo European Journal of MechanicsmdashASolids vol 72pp 235ndash244 2018
[40] S Sajith S S Shukla K S R K Murthy and P S RobildquoMixed mode fatigue crack growth studies in AISI 316stainless steelrdquo European Journal of MechanicsmdashASolidsvol 80 p 103898 2020
14 Advances in Materials Science and Engineering
and the cracks all grow with different angles based onequation (2) as the values of KII are higher whereas forcomplete mode I loading (α 90deg) crack paths are domi-nated by the first mode KI and the crack is going to growstraight ampe influence of the first mode (KI) decreases as αincreases and vice versa for mode II
ampe crack direction angles of three geometries withdifferent initial length of cracks (aW 03 05 and 07)are 6deg 10deg and 12deg respectively which are identical to the
experimental angles tested by [39] Furthermore formixed-mode loading in the CTS geometries with loadingangle α 30deg the cracks are propagated to some extentdiffering to the original crack front direction For thesethree geometries the mixed-mode crack propagationangles (aW 03 05 and 07) are about 55deg 85deg and85deg respectively which are almost in agreement withexperimental angles of [39]
Table 3 Continued
ANSYS Experimental test [39]
(aw= 07) α 90deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 11 Comparison of simulated and experimental [35] fatiguelife for loading angle 30deg
Crac
k le
ngth
(mm
)
4500
5000
5500
6000
6500
050 100 150 200 250 300 350 400 450 500000Number of cycles N times 104
ANSYSExperiment (Sajith et al 2020)
Figure 12 Comparison of simulated and experimental [35] fatiguelife for loading angle 60deg
12 Advances in Materials Science and Engineering
For the above three loading angles situations (α 90degα 0deg and α 30deg) regardless of the length of the initialcrack for the same load angle (α) the angles of the crackgrowth in each case are close to each other In contrast forcombined mode (III) loading with loading angle α 60deg thedirections of crack are highly dependent on the initial lengthof the crack On the other hand for small crack length (aW 03 and 05) the cracks also grow varying from theinitial crack tip against the clockwise direction with crackpropagation angles of 25deg and 30deg respectively which areidentical to those angles in the experimental test [39]Furthermore if the initial crack length is longer than theprevious cases (aW 07) the direction of the crack growthand the direction of the initial crack happen to be identicaland are presented in Table 3
In order to assess the fatigue life the number of cycles fordifferent loading angles 30 and 60 calculated in this study iscompared with the experimental results observed by [35]under mixed loading mode (III) resulting in clear agree-ments as can be observed in Figures 11 and 12 ampe amountof the applied load is F 16 kN with load ratio R 01whereas the Paris constant as mentioned previously isC 43378times10minus 7 (mmcycle)(MPa
m
radic) and m 26183
[40]
3 Conclusions
For the CTS specimen the finite element analysis of themixed-mode fatigue crack propagation was carried out usingANSYS Mechanical APDL and compared with experimentaldata for various angles of loading ampe predicted values ofΔKeq by the experimental and numerical methods are ob-served to be in good agreement with the present study re-sults ampe mixed-mode fatigue life is predicted andcompared with experimental data for two different loadingangles of 30deg and 60deg Interestingly the theoretically pre-dicted trajectories of the crack growth for all specimens areidentical to the experimental determined paths for differentloading angles Furthermore the predicted values of SIFs (KIand KII) are in good agreement with analytical solutions Itcan be stated that the structure of specimen geometry and itsconfiguration play an important role in obtaining highervalues of mixed-mode SIFs values ampis happens to besignificant in terms of applied loads particularly for highervalues of KII
Data Availability
ampe data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
References
[1] R I Stephens A Fatemi R R Stephens and H O FuchsMetal Fatigue in Engineering John Wiley amp Sons HobokenNJ USA 2000
[2] V V Bolotin Mechanics of Fatigue CRC Press Boca RatonFL USA 1999
[3] C A Brebbia and A V Farahani Fatigue Damage of Ma-terials Experiment and Analysis WIT Press SouthamptonUK 2003
[4] N E DowlingMechanical Behavior of Materials EngineeringMethods for Deformation Fracture and Fatigue PearsonLondon UK 2012
[5] G R Halford Fatigue and Durability of Structural MaterialsASM International London UK 2006
[6] R O Ritchie ldquoMechanisms of fatigue-crack propagation inductile and brittle solidsrdquo International Journal of Fracturevol 100 no 1 pp 55ndash83 1999
[7] S Suresh Fatigue of Materials Cambridge University PressCambridge UK 1998
[8] K N Solanki Two and ree-Dimensional Finite ElementAnalysis of Plasticity-Induced Fatigue Crack Closure AComprehensive Parametric Study Mississippi State Univer-sity Starkville MS USA 2002
[9] A M Alshoaibi and A K Ariffin ldquoFinite element modeling offatigue crack propagation using a self adaptive mesh strategyrdquoInternational Review of Aerospace Engineering (IREASE)vol 8 no 6 pp 209ndash215 2015
[10] A M Alshoaibi ldquoFinite element procedures for the numericalsimulation of fatigue crack propagation under mixed modeloadingrdquo Structural Engineering and Mechanics vol 35 no 3pp 283ndash299 2010
[11] A M Alshoaibi ldquoAdaptive finite element modeling of fatiguecrack propagationrdquo International Journal of Materials Scienceand Applications vol 2 no 3 pp 104ndash108 2013
[12] A M Alshoaibi ldquoAn adaptive finite element framework forfatigue crack propagation under constant amplitude loadingrdquoInternational Journal of Applied Science and Engineeringvol 13 no 3 pp 261ndash270 2015
[13] A M Alshoaibi ldquoComprehensive comparisons of two andthree dimensional numerical estimation of stress intensityfactors and crack propagation in linear elastic analysisrdquo In-ternational Journal of Integrated Engineering vol 11 no 6pp 45ndash52 2019
[14] A Handbook Fatigue and Fracture American Society ofMaterials vol 19 1996
[15] W He J Liu and D Xie ldquoNumerical study on fatigue crackgrowth at a web-stiffener of ship structural details by anobjected-oriented approach in conjunction with ABAQUSrdquoMarine Structures vol 35 pp 45ndash69 2014
[16] K Rege and D G Pavlou ldquoEffect of stop holes on structuralintegrity of offshore structures a numerical modelrdquo in Pro-ceedings of the Institution of Civil Engineers-Maritime Engi-neering ampomas Telford Ltd London UK 2019
[17] H Rhee ldquoFatigue crack growth analyses of offshore structuraltubular jointrdquo Journal of Offshore Mechanics and Arctic En-gineering vol 111 no 1 pp 49ndash55 1989
[18] H Riahi P Bressolette A Chateauneuf C Bouraoui andR Fathallah ldquoReliability analysis and inspection updating bystochastic response surface of fatigue cracks in mixed moderdquoEngineering Structures vol 33 no 12 pp 3392ndash3401 2011
[19] S Qi L X Cai C Bao H Chen K K Shi and H L WuldquoAnalytical theory for fatigue crack propagation rates ofmixed-mode I-II cracks and its applicationrdquo InternationalJournal of Fatigue vol 119 pp 150ndash159 2019
[20] A F Liu Mechanics and Mechanisms of Fracture An In-troduction ASM International London UK 2005
[21] O Demir A O Ayhan and S Iriccedil ldquoA new specimen formixed mode-III fracture tests modeling experiments and
Advances in Materials Science and Engineering 13
criteria developmentrdquo Engineering Fracture Mechanicsvol 178 pp 457ndash476 2017
[22] M Hussain S Pu and J Underwood ldquoStrain energy releaserate for a crack under combined mode I and mode II FractureAnalysisrdquo in Proceedings of the 1973 National Symposium onFracture Mechanics Part II ASTM International WestConshohocken PA USA 1974
[23] H Richard M Sander M Fulland and G Kullmer ldquoDe-velopment of fatigue crack growth in real structuresrdquo Engi-neering Fracture Mechanics vol 75 no 3-4 pp 331ndash3402008
[24] K Tanaka ldquoFatigue crack propagation from a crack inclinedto the cyclic tensile axisrdquo Engineering Fracture Mechanicsvol 6 no 3 pp 493ndash507 1974
[25] P Paris and F Erdogan ldquoA critical analysis of crack prop-agation lawsrdquo Journal of Basic Engineering vol 85 no 4pp 528ndash533 1963
[26] L Coffin ldquoCyclic deformation and fatigue of metalsrdquo eBook Fatigue and Staying Power of Metals Izo IL USSRMoscow Russia 1963
[27] A Wohler ldquoVersuche zur Ermittlung der auf die Eisen-bahnwagenachsen einwirkenden Krafte und die Wider-standsfahigkeit der Wagen-Achsenrdquo Zeitschrift fur Bauwesenvol 10 no 1860 pp 583ndash614 1860
[28] J M Alegre M Preciado and D Ferrentildeo ldquoStudy of thefatigue failure of an anti-return valve of a high pressuremachinerdquo Engineering Failure Analysis vol 14 no 2pp 408ndash416 2007
[29] H Chen Q Wang W Zeng et al ldquoDynamic brittle crackpropagation modeling using singular edge-based smoothedfinite element method with local mesh rezoningrdquo EuropeanJournal of MechanicsmdashASolids vol 76 pp 208ndash223 2019
[30] S Sajith K S R Krishna Murthy and P Robi ldquoPrediction ofAccurate Mixed Mode Fatigue Crack Growth Curves usingthe Parisrsquo Lawrdquo Journal of e Institution of Engineers (India)Series C vol 100 no 1 pp 165ndash174 2019
[31] G R Irwin ldquoAnalysis of stresses and strains near the end of acrack transversing a platerdquo Journal of Applied Mechanicsvol 24 pp 361ndash364 1957
[32] H A Richard F G Buchholz G Kullmer andM Schollmann 2D-and 3D-mixed mode fracture criteria KeyEngineering Materials Trans Tech Publication Stafa-ZurichSwitzerland 2003
[33] H A Richard B Schramm and N-H Schirmeisen ldquoCrackson Mixed Mode loading - ampeories experiments simula-tionsrdquo International Journal of Fatigue vol 62 pp 93ndash1032014
[34] F Erdogan and G C Sih ldquoOn the crack extension in platesunder plane loading and transverse shearrdquo Journal of BasicEngineering vol 85 no 4 pp 519ndash525 1963
[35] S Sajith K S R K Murthy and P S Robi ldquoExperimental andnumerical investigation of mixed mode fatigue crack growthmodels in aluminum 6061-T6rdquo International Journal of Fa-tigue vol 130 p 105285 2020
[36] F Antunes R Branco J Ferreira and L Borrego ldquoStressintensity factor solutions for CTS mixed mode specimenrdquoFrattura ed Integrita Strutturale vol 13 no 48 pp 676ndash6922019
[37] O Demir A O Ayhan S Iric and H Lekesiz ldquoEvaluation ofmixed mode-III criteria for fatigue crack propagation usingexperiments and modelingrdquo Chinese Journal of Aeronauticsvol 31 no 7 pp 1525ndash1534 2018
[38] H Richard Fracture predictions for cracks exposed to super-imposed normal and shear stresses International NuclearInformation System Vienna Austria 1985
[39] X-T Miao Q Yu C-Y Zhou J Li Y-Z Wang andX-H He ldquoExperimental and numerical investigation onfracture behavior of CTS specimen under I-II mixed modeloadingrdquo European Journal of MechanicsmdashASolids vol 72pp 235ndash244 2018
[40] S Sajith S S Shukla K S R K Murthy and P S RobildquoMixed mode fatigue crack growth studies in AISI 316stainless steelrdquo European Journal of MechanicsmdashASolidsvol 80 p 103898 2020
14 Advances in Materials Science and Engineering
For the above three loading angles situations (α 90degα 0deg and α 30deg) regardless of the length of the initialcrack for the same load angle (α) the angles of the crackgrowth in each case are close to each other In contrast forcombined mode (III) loading with loading angle α 60deg thedirections of crack are highly dependent on the initial lengthof the crack On the other hand for small crack length (aW 03 and 05) the cracks also grow varying from theinitial crack tip against the clockwise direction with crackpropagation angles of 25deg and 30deg respectively which areidentical to those angles in the experimental test [39]Furthermore if the initial crack length is longer than theprevious cases (aW 07) the direction of the crack growthand the direction of the initial crack happen to be identicaland are presented in Table 3
In order to assess the fatigue life the number of cycles fordifferent loading angles 30 and 60 calculated in this study iscompared with the experimental results observed by [35]under mixed loading mode (III) resulting in clear agree-ments as can be observed in Figures 11 and 12 ampe amountof the applied load is F 16 kN with load ratio R 01whereas the Paris constant as mentioned previously isC 43378times10minus 7 (mmcycle)(MPa
m
radic) and m 26183
[40]
3 Conclusions
For the CTS specimen the finite element analysis of themixed-mode fatigue crack propagation was carried out usingANSYS Mechanical APDL and compared with experimentaldata for various angles of loading ampe predicted values ofΔKeq by the experimental and numerical methods are ob-served to be in good agreement with the present study re-sults ampe mixed-mode fatigue life is predicted andcompared with experimental data for two different loadingangles of 30deg and 60deg Interestingly the theoretically pre-dicted trajectories of the crack growth for all specimens areidentical to the experimental determined paths for differentloading angles Furthermore the predicted values of SIFs (KIand KII) are in good agreement with analytical solutions Itcan be stated that the structure of specimen geometry and itsconfiguration play an important role in obtaining highervalues of mixed-mode SIFs values ampis happens to besignificant in terms of applied loads particularly for highervalues of KII
Data Availability
ampe data used to support the findings of this study are in-cluded within the article
Conflicts of Interest
ampe authors declare that they have no conflicts of interest
References
[1] R I Stephens A Fatemi R R Stephens and H O FuchsMetal Fatigue in Engineering John Wiley amp Sons HobokenNJ USA 2000
[2] V V Bolotin Mechanics of Fatigue CRC Press Boca RatonFL USA 1999
[3] C A Brebbia and A V Farahani Fatigue Damage of Ma-terials Experiment and Analysis WIT Press SouthamptonUK 2003
[4] N E DowlingMechanical Behavior of Materials EngineeringMethods for Deformation Fracture and Fatigue PearsonLondon UK 2012
[5] G R Halford Fatigue and Durability of Structural MaterialsASM International London UK 2006
[6] R O Ritchie ldquoMechanisms of fatigue-crack propagation inductile and brittle solidsrdquo International Journal of Fracturevol 100 no 1 pp 55ndash83 1999
[7] S Suresh Fatigue of Materials Cambridge University PressCambridge UK 1998
[8] K N Solanki Two and ree-Dimensional Finite ElementAnalysis of Plasticity-Induced Fatigue Crack Closure AComprehensive Parametric Study Mississippi State Univer-sity Starkville MS USA 2002
[9] A M Alshoaibi and A K Ariffin ldquoFinite element modeling offatigue crack propagation using a self adaptive mesh strategyrdquoInternational Review of Aerospace Engineering (IREASE)vol 8 no 6 pp 209ndash215 2015
[10] A M Alshoaibi ldquoFinite element procedures for the numericalsimulation of fatigue crack propagation under mixed modeloadingrdquo Structural Engineering and Mechanics vol 35 no 3pp 283ndash299 2010
[11] A M Alshoaibi ldquoAdaptive finite element modeling of fatiguecrack propagationrdquo International Journal of Materials Scienceand Applications vol 2 no 3 pp 104ndash108 2013
[12] A M Alshoaibi ldquoAn adaptive finite element framework forfatigue crack propagation under constant amplitude loadingrdquoInternational Journal of Applied Science and Engineeringvol 13 no 3 pp 261ndash270 2015
[13] A M Alshoaibi ldquoComprehensive comparisons of two andthree dimensional numerical estimation of stress intensityfactors and crack propagation in linear elastic analysisrdquo In-ternational Journal of Integrated Engineering vol 11 no 6pp 45ndash52 2019
[14] A Handbook Fatigue and Fracture American Society ofMaterials vol 19 1996
[15] W He J Liu and D Xie ldquoNumerical study on fatigue crackgrowth at a web-stiffener of ship structural details by anobjected-oriented approach in conjunction with ABAQUSrdquoMarine Structures vol 35 pp 45ndash69 2014
[16] K Rege and D G Pavlou ldquoEffect of stop holes on structuralintegrity of offshore structures a numerical modelrdquo in Pro-ceedings of the Institution of Civil Engineers-Maritime Engi-neering ampomas Telford Ltd London UK 2019
[17] H Rhee ldquoFatigue crack growth analyses of offshore structuraltubular jointrdquo Journal of Offshore Mechanics and Arctic En-gineering vol 111 no 1 pp 49ndash55 1989
[18] H Riahi P Bressolette A Chateauneuf C Bouraoui andR Fathallah ldquoReliability analysis and inspection updating bystochastic response surface of fatigue cracks in mixed moderdquoEngineering Structures vol 33 no 12 pp 3392ndash3401 2011
[19] S Qi L X Cai C Bao H Chen K K Shi and H L WuldquoAnalytical theory for fatigue crack propagation rates ofmixed-mode I-II cracks and its applicationrdquo InternationalJournal of Fatigue vol 119 pp 150ndash159 2019
[20] A F Liu Mechanics and Mechanisms of Fracture An In-troduction ASM International London UK 2005
[21] O Demir A O Ayhan and S Iriccedil ldquoA new specimen formixed mode-III fracture tests modeling experiments and
Advances in Materials Science and Engineering 13
criteria developmentrdquo Engineering Fracture Mechanicsvol 178 pp 457ndash476 2017
[22] M Hussain S Pu and J Underwood ldquoStrain energy releaserate for a crack under combined mode I and mode II FractureAnalysisrdquo in Proceedings of the 1973 National Symposium onFracture Mechanics Part II ASTM International WestConshohocken PA USA 1974
[23] H Richard M Sander M Fulland and G Kullmer ldquoDe-velopment of fatigue crack growth in real structuresrdquo Engi-neering Fracture Mechanics vol 75 no 3-4 pp 331ndash3402008
[24] K Tanaka ldquoFatigue crack propagation from a crack inclinedto the cyclic tensile axisrdquo Engineering Fracture Mechanicsvol 6 no 3 pp 493ndash507 1974
[25] P Paris and F Erdogan ldquoA critical analysis of crack prop-agation lawsrdquo Journal of Basic Engineering vol 85 no 4pp 528ndash533 1963
[26] L Coffin ldquoCyclic deformation and fatigue of metalsrdquo eBook Fatigue and Staying Power of Metals Izo IL USSRMoscow Russia 1963
[27] A Wohler ldquoVersuche zur Ermittlung der auf die Eisen-bahnwagenachsen einwirkenden Krafte und die Wider-standsfahigkeit der Wagen-Achsenrdquo Zeitschrift fur Bauwesenvol 10 no 1860 pp 583ndash614 1860
[28] J M Alegre M Preciado and D Ferrentildeo ldquoStudy of thefatigue failure of an anti-return valve of a high pressuremachinerdquo Engineering Failure Analysis vol 14 no 2pp 408ndash416 2007
[29] H Chen Q Wang W Zeng et al ldquoDynamic brittle crackpropagation modeling using singular edge-based smoothedfinite element method with local mesh rezoningrdquo EuropeanJournal of MechanicsmdashASolids vol 76 pp 208ndash223 2019
[30] S Sajith K S R Krishna Murthy and P Robi ldquoPrediction ofAccurate Mixed Mode Fatigue Crack Growth Curves usingthe Parisrsquo Lawrdquo Journal of e Institution of Engineers (India)Series C vol 100 no 1 pp 165ndash174 2019
[31] G R Irwin ldquoAnalysis of stresses and strains near the end of acrack transversing a platerdquo Journal of Applied Mechanicsvol 24 pp 361ndash364 1957
[32] H A Richard F G Buchholz G Kullmer andM Schollmann 2D-and 3D-mixed mode fracture criteria KeyEngineering Materials Trans Tech Publication Stafa-ZurichSwitzerland 2003
[33] H A Richard B Schramm and N-H Schirmeisen ldquoCrackson Mixed Mode loading - ampeories experiments simula-tionsrdquo International Journal of Fatigue vol 62 pp 93ndash1032014
[34] F Erdogan and G C Sih ldquoOn the crack extension in platesunder plane loading and transverse shearrdquo Journal of BasicEngineering vol 85 no 4 pp 519ndash525 1963
[35] S Sajith K S R K Murthy and P S Robi ldquoExperimental andnumerical investigation of mixed mode fatigue crack growthmodels in aluminum 6061-T6rdquo International Journal of Fa-tigue vol 130 p 105285 2020
[36] F Antunes R Branco J Ferreira and L Borrego ldquoStressintensity factor solutions for CTS mixed mode specimenrdquoFrattura ed Integrita Strutturale vol 13 no 48 pp 676ndash6922019
[37] O Demir A O Ayhan S Iric and H Lekesiz ldquoEvaluation ofmixed mode-III criteria for fatigue crack propagation usingexperiments and modelingrdquo Chinese Journal of Aeronauticsvol 31 no 7 pp 1525ndash1534 2018
[38] H Richard Fracture predictions for cracks exposed to super-imposed normal and shear stresses International NuclearInformation System Vienna Austria 1985
[39] X-T Miao Q Yu C-Y Zhou J Li Y-Z Wang andX-H He ldquoExperimental and numerical investigation onfracture behavior of CTS specimen under I-II mixed modeloadingrdquo European Journal of MechanicsmdashASolids vol 72pp 235ndash244 2018
[40] S Sajith S S Shukla K S R K Murthy and P S RobildquoMixed mode fatigue crack growth studies in AISI 316stainless steelrdquo European Journal of MechanicsmdashASolidsvol 80 p 103898 2020
14 Advances in Materials Science and Engineering
criteria developmentrdquo Engineering Fracture Mechanicsvol 178 pp 457ndash476 2017
[22] M Hussain S Pu and J Underwood ldquoStrain energy releaserate for a crack under combined mode I and mode II FractureAnalysisrdquo in Proceedings of the 1973 National Symposium onFracture Mechanics Part II ASTM International WestConshohocken PA USA 1974
[23] H Richard M Sander M Fulland and G Kullmer ldquoDe-velopment of fatigue crack growth in real structuresrdquo Engi-neering Fracture Mechanics vol 75 no 3-4 pp 331ndash3402008
[24] K Tanaka ldquoFatigue crack propagation from a crack inclinedto the cyclic tensile axisrdquo Engineering Fracture Mechanicsvol 6 no 3 pp 493ndash507 1974
[25] P Paris and F Erdogan ldquoA critical analysis of crack prop-agation lawsrdquo Journal of Basic Engineering vol 85 no 4pp 528ndash533 1963
[26] L Coffin ldquoCyclic deformation and fatigue of metalsrdquo eBook Fatigue and Staying Power of Metals Izo IL USSRMoscow Russia 1963
[27] A Wohler ldquoVersuche zur Ermittlung der auf die Eisen-bahnwagenachsen einwirkenden Krafte und die Wider-standsfahigkeit der Wagen-Achsenrdquo Zeitschrift fur Bauwesenvol 10 no 1860 pp 583ndash614 1860
[28] J M Alegre M Preciado and D Ferrentildeo ldquoStudy of thefatigue failure of an anti-return valve of a high pressuremachinerdquo Engineering Failure Analysis vol 14 no 2pp 408ndash416 2007
[29] H Chen Q Wang W Zeng et al ldquoDynamic brittle crackpropagation modeling using singular edge-based smoothedfinite element method with local mesh rezoningrdquo EuropeanJournal of MechanicsmdashASolids vol 76 pp 208ndash223 2019
[30] S Sajith K S R Krishna Murthy and P Robi ldquoPrediction ofAccurate Mixed Mode Fatigue Crack Growth Curves usingthe Parisrsquo Lawrdquo Journal of e Institution of Engineers (India)Series C vol 100 no 1 pp 165ndash174 2019
[31] G R Irwin ldquoAnalysis of stresses and strains near the end of acrack transversing a platerdquo Journal of Applied Mechanicsvol 24 pp 361ndash364 1957
[32] H A Richard F G Buchholz G Kullmer andM Schollmann 2D-and 3D-mixed mode fracture criteria KeyEngineering Materials Trans Tech Publication Stafa-ZurichSwitzerland 2003
[33] H A Richard B Schramm and N-H Schirmeisen ldquoCrackson Mixed Mode loading - ampeories experiments simula-tionsrdquo International Journal of Fatigue vol 62 pp 93ndash1032014
[34] F Erdogan and G C Sih ldquoOn the crack extension in platesunder plane loading and transverse shearrdquo Journal of BasicEngineering vol 85 no 4 pp 519ndash525 1963
[35] S Sajith K S R K Murthy and P S Robi ldquoExperimental andnumerical investigation of mixed mode fatigue crack growthmodels in aluminum 6061-T6rdquo International Journal of Fa-tigue vol 130 p 105285 2020
[36] F Antunes R Branco J Ferreira and L Borrego ldquoStressintensity factor solutions for CTS mixed mode specimenrdquoFrattura ed Integrita Strutturale vol 13 no 48 pp 676ndash6922019
[37] O Demir A O Ayhan S Iric and H Lekesiz ldquoEvaluation ofmixed mode-III criteria for fatigue crack propagation usingexperiments and modelingrdquo Chinese Journal of Aeronauticsvol 31 no 7 pp 1525ndash1534 2018
[38] H Richard Fracture predictions for cracks exposed to super-imposed normal and shear stresses International NuclearInformation System Vienna Austria 1985
[39] X-T Miao Q Yu C-Y Zhou J Li Y-Z Wang andX-H He ldquoExperimental and numerical investigation onfracture behavior of CTS specimen under I-II mixed modeloadingrdquo European Journal of MechanicsmdashASolids vol 72pp 235ndash244 2018
[40] S Sajith S S Shukla K S R K Murthy and P S RobildquoMixed mode fatigue crack growth studies in AISI 316stainless steelrdquo European Journal of MechanicsmdashASolidsvol 80 p 103898 2020
14 Advances in Materials Science and Engineering
