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On mixed mode crack initiation and direction in shafts: Strain energy density factor and maximum tangential stress criteria

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On mixed mode crack initiation and direction in shafts: Strain energy density factor and maximum tangential stress criteria
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  On mixed mode crack initiation and direction in shafts: Strainenergy density factor and maximum tangential stress criteria Yong Li a,b , Nicholas Fantuzzi b , Francesco Tornabene b, ⇑ a College of Resources and Environmental Science, Chongqing University, Shazheng St. 174, Shapingba, Chongqing 400044, China b DICAM – Department of Civil, Chemical, Environmental and Materials Engineering, University of Bologna, Viale del Risorgimento 2, Bologna 40136, Italy a r t i c l e i n f o  Article history: Received 1 December 2012Received in revised form 1 July 2013Accepted 13 July 2013Available online 25 July 2013 Keywords: Mixed modeStress intensity factorsInitial crack propagation angleFracture criteriaCircular cross-section beam a b s t r a c t A simple method for calculating stress intensity factors of transverse cracked shaft sub- jected to tension, bending and shear is introduced. New formulae of the stress intensityfactors in terms of the crack depth ratio are presented. The critical loads and the normal-izedstrainenergydensityfactorsarecalculatedandgraphicallyshown. Asfarasthemixedmodes are concerned, the initial crack growth angles are determined and mixed load ratiois investigated. The initial crack growth angles and the dimensionless stress intensity fac-tors, obtained by minimumstrain energy density factor criterion and maximumtangentialstress criterion, are compared.   2013 Elsevier Ltd. All rights reserved. 1. Introduction Most applications of fracture mechanics have been concentrated on cracks growing under an opening or mode I mech-anism[1]. However, in many practical applications a cracked component under mixed mode conditions can be encountered frequently[1–4].Theseflawsmaycausemixedmodeloadingduetomixedremoteloading,inclinedcrackundernormal/uni- axial remote loading or mechanical and/or thermal loads combined with arbitrary restraint conditions [2]. For instance, the following cases may occur: a crack initiated in a transverse plane froma tubing shaft, an interface crack under uniaxial loadand/ormixedloads,surfacecrackedtubularjointsappliedintheoffshoreindustry.Mixedmodeaffectsboththedeformationcharacteristics and fracture initiation capacity of the component. Hence, predicting crack growth in mixed mode became animportant subject of fracture mechanics. Various fracture criteria for cracks subjected to mixed mode loading have beenintroduced for the determination of the propagation direction and the critical stress. Griffith [5] introduced a criterion todetermine the conditions to initiate the propagation of a crack. The Maximum Energy Release Rate criterion (MERR) [6,7]followed the Griffith condition and stated that crack growth follows the orientation of maximum energy release rate. Erdo-gan and Sih [8] developed the Maximum Tangential Stress criterion (MTS-criterion) which was one of the first conditionsthat predicted critical stress and crack growth orientation. MTS-criterion stated that the crack growth would occur in thedirection of the maximum tangential stress and would take place when the maximum tangential stress reaches a criticalvalue which only contains the first mode strength toughness. Due to its simply formulation, MTS-criterion became one of the most popular criteria in fracture mechanics. Palaniswamy and Knauss [9] introduced the  G  -criterion which dealt witha criterion of maximum energy release rate to determine both the initial crack propagation direction as well as the condi-tionsofcrackinstabilityintermsoffracturestress,crackorientationangle,andcracklength.Sih[10]proposedtheminimum 0013-7944/$ - see front matter    2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engfracmech.2013.07.008 ⇑ Corresponding author. Tel.: +39 051 2093500; fax: +39 051 2093496. E-mail address:  francesco.tornabene@unibo.it (F. Tornabene).Engineering Fracture Mechanics 109 (2013) 273–289 Contents lists available at ScienceDirect Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech  elasticstrainenergydensityfactorcriterion(SEDF-criterion)whichwasbaseddirectlyonthetotalstrainenergydensity,thatis, the sum of its distortional and dilation components. Hellen and Blackburn [11] presented J-criterion in an attempt to usepath-independent lineintegrals tostudythe problemof crackgrowthunder mixedmode loading. TheocarisandAndrianop-oulos [12] proposedthe named T-criterion whichstatedthat a crackstarts to propagate whenthe dilationstrainenergy at apoint in the vicinity of its tip reaches a critical value and the curve of evaluation of around the crack tip (the elastic–plasticboundary) was used as if it was obtained from the Mises yield condition. Papadopoulos [13–15] invented Det.-criterionwhich was based on the determinant of the stress tensor. It was used to study crack extension angle and the critical stressof fracture under biaxial loading. The local symmetry conditions (LS) proposed by Gol’dstein and Salganik [16] in 1974 re-quired that the crack propagation occurs along a path where  K  II   vanishes.Many improvements on the aforementioned criteria have been done by a lot of researchers. Either critical load value andcrack path or surface growth have been predicted by different criteria in terms of elastic singular stress states and T-stresscomponent.RamuluandKobayashi[17]extendedSEDF-criteriontodynamicmixedmodecrackpropagation.Chenetal.[18] proposedamoresuitablefailurecriterionbasedonaspecificstrainenergycriterionwhichhadbeenusedtomodelthecrackinitiationandpropagationinasinglelapjointwithabrittleadhesiveandaductileadhesive. Nobileet al. [19,20]appliedthestrain energy density theory to determine crack initiation and direction in orthotropic solids, cracked T-beams and circum-ferentially cracked pipes. Bian and Kim [21] and Bian and Taheri [22] investigated two criteria based on the Relative Mini- mumPlastic Zone radius (MPZR) and maximumratio definedfor predictionof fatigue crackinitiationangles whichcouldbeappliedtoinclinedsurfacecrackspecimensandthrough-crackspecimensundermixedmodeloading.Kidaneetal.[23]eval-uated the crack growth direction as a function of temperature, non-homogeneity parameter and crack-tip velocity for func-tionallygradedmaterialsusingbothmaximumtangentialstressandminimumstrainenergydensitycriteria.Ayatollahietal.[24], Maccagnoand Knott [25] investigated the brittle fracturebehavior of composites under mixedmode I/II loadingwhere Nomenclature a  crack depth a ij  auxiliary functions  A ,  A C  area of the uncracked and cracked circular section e  eccentricity of the neural axis E  ,  m  Young’s modulus and Poisson’s coefficient EI  ,  EI  c  bending stiffness of uncracked and cracked section EA  tension stiffness F  N  ( g ),  F  M  ( g ),  F  Q  ( g ) geometric functions under tension, bending moment and shear force G   strain energy release rate D h  the widening of the crack I  ,  I  c  moment of inertia of uncracked and cracked section K  I  ,  K  II  ,  K  III   stress intensity factors for modes I, II, III K   I   ;  K   II   non-dimensional stress intensity factors K  I  N   ;  K  I  M   ;  K  IC   mode I stress intensity factors under tension, bending and toughness M  ,  M  C   bending moment and its critical value M  N  bending moment due to the shifted neutral axis N  ,  N  C   tension and its critical value Q  ,  Q  C   shear force and its critical value R  radius of circular section S  ,  S  min ,  S  C  strain energy density factor and its minimal and critical values S  MQ  ,  S  NQ   strain energy density factors of bending-shear and tension-shear mixed modes S   N  ;  S   M  ;  S   Q   normalized strain energy density factor of tension, bending moment and shear force S   MQ  ;  S   NQ   normalized mixed mode strain energy density factor t   length of crack tip line T  V ,  T  Vcr  dilation strain energy and its critical value U   strain energy of the body a MQ  ,  a NQ   mixed load ratio of bending-shear and tension-shear mixed modes b  slope factor D ,  P 1 ,  P 2  abbreviation symbols g  crack depth ratio h ,  h 0  polar angle and its initial value k ,  l  shearing factor, shear modulus r h ,  r C   circumferential stress and its critical value 274  Y. Li et al./Engineering Fracture Mechanics 109 (2013) 273–289  the experimental results were used to evaluate the minimum SEDF, MTS and maximum energy release rate  ðGÞ  criteria,showing the best agreement with the first criterion. Fatigue crack propagation was investigated under non-proportionalmixed mode loading by Plank and Kuhn [26], and the deviation angle is well predicted via the MTS-criterion. The crack propagation problems in orthotropic and piezoelectric media are studied in [27–32]. A modified maximum tangential stress criterion had been proposed by Viola and Piva [33] and Piva and Viola [34] to predict the crack paths in sheets of brittle materials under biaxial loading. The modes of crack growth in mixed mode conditions were reviewed for theplane and three-dimensional problems in Ref. [35]. Some other aspects related to the present investigation can be found in [36–41]. 2. General framework  Withintheframeworkofbrittlefracture,thewell-knownstrainenergydensitytheoryprovidesamoregeneraltreatmentoffracturemechanicsproblemsbyvirtueofitsabilityindescribingthemulti-scalefeatureofmaterialdamageandindealingwithmixedmodecrackpropagationproblem.Thestrainenergydensityfactortheoryisallowedforpredictingunstablecrackgrowth in mixed mode. The main advantages of this criterion lie in its easy and simplicity, as well as its ability to handlevarious combined loading situations.Thecoreof theSEDF-criterionis theparameterof strainenergydensityfactor S  , whichcanbeexpressedintermsof threestress intensity factors of modes I, II and III  K  i  ( i  = I  ,  II  ,  III  ) for linear elasticity as S   ¼  a 11 K  2 I   þ 2 a 12 K  I  K  II   þ  a 22 K  2 II   þ  a 33 K  2 III   ð 1 Þ where a ij  areauxiliaryfunctionswhichdependontheelasticpropertiesofthematerial,modulusofelasticity, shearmodulusof elasticity  l , Poisson’s coefficient  m , and a polar angle measured from the crack tip  h , as follows [10] a 11  ¼  116 pl ½ð 3  4 m  cos h Þð 1 þ cos h Þ a 12  ¼  116 pl ð 2sin h Þ½ cos h   ð 1  2 m Þ a 22  ¼  116 pl ½ 4 ð 1  m Þð 1  cos h Þ þ ð 1 þ cos h Þð 3cos h   1 Þ a 33  ¼  14 pl ð 2 Þ AbriefapplicationoftheSEDF-criteriontopredictthecrackinitiationanddirectionisbasedontwohypothesizesgivenasfollowing:(1) Crack initiation occurs when the strain energy density factor  S   reaches a critical factor value  S  C  , i.e. S   ¼  S  C   for  h  ¼  h 0  ð 3 Þ The parameter  S  C   depends only on the material, in fact, it is directly related to mode I toughness  K  IC   by the relation S  C   ¼ ð 1 þ m Þð 1  2 m Þ 2 p E  K  2 IC   ð 4 Þ (2) The initial crack growth takes place in the direction of maximumpotential energy density or minimumstrain energydensity  S  min  i.e. @  S  @  h  ¼  0 and  @  2 S  @  h 2  >  0  ð 5 Þ After substituting  a ij  from Eq. (2) into Eq. (1), the derivative of   S   to  h  in plane condition is @  S  @  h  ¼  sin h ð cos h   ð 1 þ 2 m ÞÞ K  2 I   þ ð cos h ð cos h   ð 1 þ 2 m ÞÞ  sin 2 h Þ 2 K  I  K  II   þ sin h ð 1 þ 2 m  3cos h Þ K  2 II   ¼  0  ð 6 Þ As it is well-known, the knowledge of the stress intensity factors plays an important role in fracture control. In fact, thedirection of the crack initiation is only determined by the stress intensity factors (SIFs). In structural applications, the com-bination of standard loading conditions often involves  K  I  ,  K  II   and  K  III   simultaneously. The SIFs can be computed by usingnumerical methods including the singular integral equations, the weight functions, the boundary collocation, the finite ele-ment method and the boundary element approach. Experimental methods have been applied to simple cases in order todetermine the fracture toughness  K  IC   of engineering materials. Stress intensity factors for many configurations are available[42–44]. However, solutions for many complex structural configurations are not yet available in the handbooks. For engi-neering estimations of the strength of cracked bodies, the use of procedures involving a smaller amount of computational Y. Li et al./Engineering Fracture Mechanics 109 (2013) 273–289  275  costcanbesuitable,eventhoughaloweraccuracyisinvolved.RemarkablysimplemethodsforcloseapproximationofSIFsincracked or notched beams have been proposed by Parton and Morozov [45] and by Kienzler and Herrmann [46]. The former method named section method, which is based on the equilibrium condition of the internal forces evaluated in the cross-section passing through the crack tip, takes the stress singularity at the tip of an elastic crack into account. This methodhas beenimprovedand applied by Nobile [47,48]. An extensionof the secondmethod has beenpresentedby Ricci and Viola [49] to estimate the stress intensity factors of cracked T-beams.Inthispaper,thelattermethodbasedonelementarybeamtheoryestimationofstrainenergyreleaseratewhenthecrackis wideninginto a fracture band is applied. In the present work some results in [50] are presented in expanded form. In fact,the details of derivation of stress intensity factors of a transverse crack in a shaft under tension, bending moment and shearforce are proposed and explained. The derived simple formulae of stress intensity factors are combined to SEDF-criterion topredict initial crack growth direction and initiation for single and mixed load modes. The results are compared with thoseobtained by MTS-criterion. 3. Stress intensity factors The approximate method for calculating the stress intensity factors of a crack, proposed by Kienzler and Herrmann [46],based on an elementary beam theory estimation of strain release rate as the crack is widened into a fracture band is devel-opedin the following. Consider a crackedbody, a crackof initial length a  is either extendedto the length a  + D a  as showninFig. 1(a), or widened an amount  D h  as shown Fig. 1(b). As discussed in [46,51,52] the energy release  @  U  / @  a , due to crackextension, is related to the energy release  @  U  / @  h , that is caused by crack widening according to @  U  @  a  ¼  2 b @  U  @  h  ð 7 Þ where  U   is the potential energyavailable for crackgrowthand  b  is usuallya first order factor that in general depends onthegeometry and the crack size.The strainenergyrelease rate G   as the energy dissipatedduring fractureper unit of newlycreatedfracturesurfacearea isclosely related to  @  U  / @  a  as G ¼  1 t  @  U  @  a  ð 8 Þ where  t   is the thickness for an edge cracked beam of rectangular cross-section. Nevertheless, considering a circular cross-section shaft as in Fig. 2(a),  t   can be the length of the tip line for a transverse crack, as shown in Fig. 2(b). t   ¼  4 R  ffiffiffiffiffiffiffiffiffiffiffiffiffiffi g  g 2 p   ð 9 Þ where g  ¼  a 2 R  ð 10 Þ is the dimensionless crack depth (crack depth ratio), and  R  is the radius of circular section.According to the well-known Irwin’s  G   K   relation [53] G ¼  K  2 i E  0  ð 11 Þ Fig. 1.  Stress relief zones of (a) line crack advance; (b) crack band widening.276  Y. Li et al./Engineering Fracture Mechanics 109 (2013) 273–289  where  E  0 = E  /(1   m 2 ) for plane strain,  E  0 = E   for plane stress,  E  0 = l  for mode III fracture. Stress intensity factors  K  i  can be ob-tained from Eqs. (7)–(9) and (11) once  @  U/ @  h and  b  are known K  i  ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   b E  0 2 R  ffiffiffiffiffiffiffiffiffiffiffiffiffiffi g  g 2 p  @  U  @  h s   ð 12 Þ KienzlerandHerrmann[46]proposedaremarkablysimplemethodtoestimate @  U/ @  hviasimplebeamtheory.Thismeth-odis wellillustratedbyconsideringabeamwithasingleedgecracksubjectedtoabendingmoment M   asshowninFig. 2(a).From the elementary beam theory of bending, the widening of the crack by D h  is effective to reduce the bending stiffnessfrom  EI   to  EI  c  along the length D h , and the corresponding change in energy of the beam is D U   ¼  M  2 21 EI     1 EI  c    D h  ð 13 Þ where  EI   is the bending stiffness at the uncracked cross-section and  EI  c  at the cracked cross-section. Here,  I   = p R 4 /4 denotesthe moment of inertia of an uncracked circular section and I  c  ¼  R 4 4  p  cos  1 ð 1  2 g Þ þ 2 ð 1  2 g Þð 1  8 g þ 8 g 2 Þ  ffiffiffiffiffiffiffiffiffiffiffiffiffiffi g  g 2 p h i  ð 14 Þ is the moment of inertia of a cracked circular section. It can also be found in [54] with a different expression.It is worth to note that, when D h ? 0, the incremental strain energy gives @  U  @  h  ¼  M  2 2 E  1 I  c    1 I     ð 15 Þ The argument leading to Eq. (15) can be easily extended to beams under axial loading and shear force as discussed in[50,55]. For beams under tensile load  N  , the incremental strain energy is analogous to Eq. (15) as follows @  U  @  h  ¼  N  2 2 E  1  A c     1  A    ð 16 Þ where  A  = p R 2 is the area of uncracked cross-section of the beam and  A c  ¼  R 2 p  cos  1 ð 1  2 g Þ þ 2 ð 1  2 g Þ  ffiffiffiffiffiffiffiffiffiffiffiffiffiffi g  g 2 p h i  ð 17 Þ is the cross-section area of a cracked portion.Accordingtotheexpressionsin[46,55,56],forbeamssubjectedtoashearforce Q  ,theenergyreleasecanbeobtainedwiththe following equation @  U  @  h  ¼  k Q  2 2 l 1  A c     1  A    ð 18 Þ where  k  is the shearing factor.Thefactor b  isthesecondunknowninEq.(12)andcanbedeterminedfromasymptoticexpansionsforthestressintensityfactorsofverysmallorverylargecracks[56],whichusuallyisoftheorderofone.Suchexpansionshavebeenreportedintheliterature [57] at least for some specimen configurations.SubstitutingEqs. (15), (16)and(18)intoEq. (12), thestressintensityfactorscanbefurtherevaluatedthroughthefollow- ing equations K  I  N   ¼  N  2  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 0 R ð g  g 2 Þ 1 = 2 1  A c    1  A  s   ð 19 Þ K  I  M   ¼  M  2  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 0 R ð g  g 2 Þ 1 = 2 1 I  c    1 I   s   ð 20 Þ K  II   ¼  Q  2  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kb 0 ð 1 þ m Þ R ð g  g 2 Þ 1 = 2 1  A c    1  A  s   ð 21 Þ     a      2      R MM (a) xy,y'x'oo'eaCrack  t (b) Fig. 2.  The cracked beam and cracked cross-section. Y. Li et al./Engineering Fracture Mechanics 109 (2013) 273–289  277
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