Determination of stress intensity factor for a crack emanating from a hole in

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Determination of stress intensity factor for a crack emanating from a hole in

  1. 1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME373DETERMINATION OF STRESS INTENSITY FACTOR FOR A CRACKEMANATING FROM A HOLE IN A PRESSURIZED CYLINDER USINGDISPLACEMENT EXTRAPOLATION METHODAKASH.D.A (1), Anand.A (2), G.V.GNANENDRA REDDY(3), SUDEV.L.J(4)(1)Department of Mechanical Engineering, SJCIT Chickaballapur, 562101(2)Department of Mechanical Engineering, Vidyavardhaka College of engineering, Mysore, 570002(3)Department of Mechanical Engineering, SJCIT Chickaballapur, 562101(4)Department of Mechanical Engineering, Vidyavardhaka College of engineering, Mysore, 570002ABSTRACTThe machine elements with cylindrical profile such as pressure vessel, cylindricalshells, which have been used extensively as the structural configuration in aerospace andshipping industries needs to be leak proof. But however it’s not possible to fabricate 100%leak proof pressure vessel / cylindrical shell as the industrial materials do not have uniformcomposition. Thus defects or cracks are inevitable in their substructure, also during theirservice life a crack may initiate on an internal/external boundary of circular cylinder whichhas important influence on stress distribution in the structure. Hence the structuralassessments of hallow cylinders ranging from thick walled pressure vessel to thin walledpipes has to be carried out, that in-turn relay’s on availability of Stress Intensity Factor (S.I.F) for fracture analysis. The magnitude of the S.I.F determines the propagation of crack.In this paper, Considerable effort has been devoted for computation of the S.I.F of crackemanating from a hole in pressurized cylinder. The objective of this work is to determine SIF(Plane Strain) for a crack emanating from a hole in a Pressurised cylinder using FiniteElement Method (FEM). From this study it was observed that the value of SIF rises suddenlywhen the crack tip is near to the hole and it stabilises as the crack tip move far from the hole.The SIF values evaluated for different crack length using FEM is normalised with theanalytical values obtained from theoretical equation with respect to (a/D) ratio whichprovides important information for subsequent studies such as the crack growth ratedetermination and prediction of residual strength with plane strain and plane stressconditions.INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERINGAND TECHNOLOGY (IJMET)ISSN 0976 – 6340 (Print)ISSN 0976 – 6359 (Online)Volume 4, Issue 2, March - April (2013), pp. 373-382© IAEME: www.iaeme.com/ijmet.aspJournal Impact Factor (2013): 5.7731 (Calculated by GISI)www.jifactor.comIJMET© I A E M E
  2. 2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME374Keywords: Fracture Mechanics, S.I.F and crack emanating from Hole1. INTRODUCTIONPressure vessel and cylindrical shells have been used extensively as the structuralconfiguration in aerospace and shipping industries needs to be leak proof. However newadvancement in computers has made Finite Element Analysis (FEA) a practical tool in thestudy of pressure vessels [1], especially in determining stresses in local areas such aspenetrations and service holes. The most likely places for crack initiating and development ofcracks are the service holes. Due to the high stress concentration in this area cracks may grow intime, leading to a loss of strength and the reduction of the lifetime of the product as shown inFigure1. If the structure is concerned with different loading, the crack behaviour must be assessedin order to avoid catastrophic failures. For this, the knowledge of the crack size, service stress,material properties and Stress Intensity Factor (SIF) is required [2]. Hence the structuralassessments of hallow cylinders /shell ranging from thick walled pressure vessel to thinwalled pipes has to be done, that in-turn relay’s on availability of stress intensity factor forfracture and fatigue analysis. Thus it has been recognized that the stress intensity factor is animportant parameter to determine the safety of a cracked component, but the basic practicalproblem a designer faces, is to make a decision to opt the method for determining stressintensity factors. It is not easy to strike a balance between the accuracy of the method, timerequired to get a solution, and cost. Numerous equations for stress intensity factors areavailable in the literature [1 – 6]. These factors represent various geometries and loadingconditions of fundamental importance in the prediction of structural failure of crackedcylindrical bodies. In all there are probably more than 600 formulas for calculating K valuesfor different crack configurations, body geometries, and loading situations.Here the objective of the work is to determine S.I.F for a crack emanating from a hole ina pressurized cylinder.Fig.1: Larger crack formed by the link-up of fatigue cracks at adjacent rivets.2. FRACTURE MECHANICSFracture mechanics involves a study of the presence of the cracks on overall propertiesand behaviour of the engineering component. The process of fracture may be initiated at defectlocations like micro-cracks, voids, and the cavities at the grain boundaries. These defects canlead to the formation of a crack due to the rupture and disentanglement of molecules, rupture ofatomic bonds or dislocation slip [3].Cracked body can be subjected to three modes of loads as shown in Figure 2. In somecases, body may experience combination of the three modes:
  3. 3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME3751. Opening mode: The principal load is applied normal to the crack surfaces, which tendsto open the crack. This is also referred as Mode I loading (Figure 2a).2. In-plane shear mode: This mode corresponds to in-plane shear loading which tends toslideOne crack surface with respect to the other. This is also referred as Mode II loading(Figure2b).3. Out-of-plane shear mode: This is the tearing and anti-plane shear mode where the cracksurfaces move relative to one another and parallel to the leading edge of the crack (Figure 2c).(a) (b) (c)Fig. 2: Three modes of loading that can be applied to a crackThe Stress Intensity Factor (SIF) is one the most important parameters in fracturemechanics analysis. It defines the stress field close to the crack tip and provides fundamentalinformation of how the crack is going to propagate. In this study, A typical and practical pointmatching technique, called Displacement Extrapolation Method (DEM) is chosen for thenumerical analysis method. Plane strain assumption is valid for very thin-walled structures; theevaluation of S.I.F (KI) by Displacement Extrapolation Method (DEM) is as discussed bellow forplane strain condition.The stress intensity factors at a crack for a linear elastic fracture mechanics analysis maybe computed using the KCALC command. The analysis uses a fit of the nodal displacements inthe vicinity of the crack. The actual displacements at and near a crack for linear elastic materialsare)1(2211krGKu ++=π(1))k1(2rG2Kv 1+π+= (2)π+=2rGK2w 111(3)
  4. 4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME376Where:u, v, w = displacements in a local Cartesian coordinate system as shown in figure 3r, θ = coordinates in a local cylindrical coordinate system as shown in figure 3.G = shear modulusvK+=1νIn Plane Stress (4)K= 3 - 4 In Plane strain…………………………..(5)‫ݒ‬ = Poissons ratioFor Mode-1, SIF at crack tip is expressed asrVk1G2K1∆+π= (6)Where ∆v, are the motions of one crack face with respect to the other.Then A and B are determined so thatBrArV+= (7)At points J and K.Next, let r approach 0ArVlim 0r =→ (8)Fig. 3: Nodes Used for the Approximate Crack-Tip Displacements for Full crack ModelThus, Equation 5 becomes:mmmmNkGAK 21122+= π (9)3. OBJECTIVE OF WORK AND METHODOLOGYThe objective of this work is to determine S.I.F for a longitudinal crack emanating from ahole in a pressurized cylinder as shown in figure 4. The objective is achieved by developing amodel of a cylinder with hole and a through crack using CATIA V5 software .The CATIA modelis imported to ANSYS.The FE model is meshed using 8-node quadrilateral doubly curvedSHELL 93 elements in the pre-processor of the ANSYS software . Further as a part of the finiteelement work, a mesh sensitivity Study was conducted.
  5. 5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME377A shell with a longitudinal crack was meshed using three different mesh densities.Mainly, the area around the imperfection was modelled with a finer mesh. Further the cracktip singular elements were created using KSCON command. For this model there are 36singular elements around the crack tip and the radius of the first row elements is ∆a (Where ∆a =a/100).The model is then solved (Static Analysis) by subjecting it to an internal pressure of 1MPaload with appropriate boundary conditions. Then the S.I.F is evaluated in general postprocessorby using KCALC command.The geometry of the meshed test model with crack tip singular elements in ANSYS 12 isas shown in the Figure 5.The material considered is 304 steel (ASME). The material is assumedto be linear elastic with young’s modulus of 2.5GPa and poisons ratio 0.3Where,D= Diameter of the hole (20mm),a= Half Crack lengthσ=applied hoop stress (Pr/t)P= Internal pressure 1MPat=Thickness of the cylindrical shell10mmFig.4: Geometry of model(a) (b) (c)Fig. 5: (a) Finite Element Model meshed with Boundary Condition (b) Zoomed View ofelements near crack tip (c) Zoomed View of Crack Tip Singular Elements
  6. 6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME3784. VALIDATION OF PROPOSED METHODOLOGYThe methodology proposed to determine Mode-1 S.I.F in previous section is validatedusing the practical problem “Determination of S.I.F of longitudinal cracks in a pressurizedcylindrical shell” from reference 3.A cylindircal shell with varying longitudinal crack length givenin reference 3 is as shown in Figure 6.Fig.6: Longitudinal crack in internally pressurized cylinderMode I S.I.F (KI) is given byKI(Theo)= )(f. 1 απσ a ………………………………………………(10)Where)()07.029.152.01()(f 321Rtaxxxx=−++=αThe half crack length was varied from 20mm to maximum half crack length of 439.53mm .Themaximum crack length in a given dimensions of cylindrical shell was determined using curvatureparameter β [9]4 2)1(12Rtaν−=β ……………………….(11)if β =8 for longitudinal cracks , thickness of the cylindrical shell is 10mm and radius of thecylindrical shell is 1000mm then the maximum crack length for given set of cylindrical shelldimension is 439.53mm.The values of S.I.F obtained by the theoretical and FEA is given in table 1.
  7. 7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME379Table 1: Mode-1 S.I.F (KI) Using FEA and theoretical KI(Theo) for different Half Cracklengths (a)Fig.7: variation of theoretical and FEA values of S.I.F Vs crack length for longitudinalcrack in Pressurized cylinderFrom the Fig 7 it is indicated that the results which were obtained by using the finiteelement method are in good agreement with theoretical equation for a longitudinal through crackemanating in internally pressurized cylindrical shell with an average percentage of error 1.53%which is negligible. Thus the proposed methodology to determine the Mode-1 S.I.F forlongitudinal cracks in pressurised cylindrical shell is validated against a standard Procedure.Half CrackLength (a)mmMode -I SIF byFEAmmMPaMode -I SIFby AnalyticalmmMPa% error20 821.49 851.322 3.5040 1282.6 1330.62 3.6160 1749.6 1828.751 4.3380 2265.4 2352.63 3.71100 2831.4 2931.63 3.42120 3448.9 3555.117 2.99140 4114.8 4221.655 3.42180 5570.8 5669.381 1.74220 7168.4 7247.292 1.09260 8886.5 8933.523 0.53300 10711 10710.546 0.00340 12634 12561.166 -0.58380 14608 14471.544 -0.94400 15685 15446.58 -1.54439.532 17798 17398.911 -2.29
  8. 8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME3805. RESULTS AND DISCUSSIONSFor the problem “Determination of SIF for a Crack Emanating from a Hole in aPressurized cylinder” the test models containing a through crack emanating from a hole aremeshed and with plane strain condition it was internally pressurized and respective SIF’s arecalculated.Theoritical Mode-1 is calculated using the relationmmmmNa)Theo(K 2eff1 πσ= …………………… (12)The variation of normalized Stress Intensity Factor (KI/KO) (By Plane Strain Method)with respect to a/D ratio [actual crack length (a) to the Diameter of the hole (D)] is as shown inFigure 8. The normalized SIF (KI/KO) is used to obtain the characteristic curve of SIF whichdepends only on the geometrical factor and its variation within the given domain (a/D).It’s observed that as the crack is near to the hole the stress concentration around holes hasa strong influence on the SIF value. For a/D ratio 0.5 there is a steep rise in SIF KI, this is due tocrack is small and the crack tip is near to stress concentration at the hole from which crack inemanating. As the crack grows further (for a/D ranging from 0.1 to 25) the crack tip moves farfrom the stressed areas hence the value of SIF increases the system will fail with increase in cracklength.Table 2: The normalized Stress Intensity Factor (KI/KO) with respect to a/D ratioHalfCrackLength (a)mmHoledia(D)mma/D Mode -I SIF byFEA(KI)mmMPaaeffmmMode -I SIF byTheoritical(Ko)mmMPaKI / Ko10 20 0.50 743.96 15 686.51 1.0836820 20 1.00 853.01 20 792.72 1.07605940 20 2.00 1085.8 30 970.88 1.11837260 20 3.00 1310.4 40 1121.07 1.16888280 20 4.00 1540.1 50 1253.40 1.228742140 20 7.00 2302.5 80 1585.43 1.452284180 20 9.00 2881.6 100 1772.57 1.625663200 20 10.00 3191.3 110 1859.09 1.716596220 20 11.00 3514.8 120 1941.75 1.810118260 20 13.00 4200.3 140 2097.33 2.002688300 20 15.00 4932.1 160 2242.14 2.199727320 20 16.00 5313.5 170 2311.15 2.299075380 20 19.00 6519.8 200 2506.79 2.600855400 20 20.00 6938.3 210 2568.70 2.701098
  9. 9. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME381Fig.8: Variation of normalized S.I.F for a crack emanating from circular hole inpressurized cylinder Vs a/DThe Deformed Geometries for crack length (a) of 20 mm is as shown in Figures 9, Themaximum Von-Misses stress is found to be at crack tip.Fig 9: (a) VonMises Stress Distribution for Pressurised cylinder containing hole diaD=20mm and crack length a=20mm6. CONCLUSIONThe problem of determining stress intensity factors for a crack emanating from a hole in apressurized cylinder is of prime importance in damage tolerance analysis. In the present studyANSYS12, unified FEA software is chosen. It has the required pre-processing capabilitiesfor finite element modeling and analysis of cracked shell structures as demonstrated in thispaper.
  10. 10. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME382The variation of normalized S.I.F (Ki/Ko) with respect to a/D ratio is used to obtain thecharacteristic curve of SIF which depends only on the geometrical factor and its variation withinthe given domain (a/D).The fracture mechanics analysis on “effects of pressure bulging near the crack on thestress intensity factor” is described in this paper that provides an explanation for the lowerstrength of cracked cylinders.The presented stress intensity factors in this paper are essential to predict(1) Mixed mode fracture under static, dynamic and sustained loads(2) Residual strength(3) Crack growth life under cyclic loading conditions.However there is a clear need to verify the predictions using experimentalinvestigations, but the method used in this paper can be utilized for calculating the stressintensity factor for many other loading cases and many values of the crack length. This providesimportant information for subsequent studies, especially for fatigue loads, where stress intensityfactor is necessary for the crack growth rate determination.REFERENCES[1] Heckman david: Finite element analysis of pressure vessels, MBARI 1998[2] Chuin-Shan Chen, Paul A. Wawrzynek, and Anthony R. Ingraffea, 1999, “Crack GrowthSimulation and Residual Strength Prediction in Airplane Fuselages”, NASA/CR-1999-209115.[3] “Fracture mechanics-Fundamentals and Applications” by T.L.Anderson published by Taylorand Francis group 2005[4] Gustavo V. Guinea, Jaime Planas and Manuel Elices, 2000, “KI Evaluation by theDisplacement Extrapolation Technique”, Engineering Fracture Mechanics 66 (2000), pp. 243-255.[5] Miloud Souiyah, A. Muchtar, Abdulnaser Alshoaibi and A.K. Ariffin, 2009, “Finite ElementAnalysis of the Crack Propagation for Solid Materials” American Journal of Applied Sciences 6(7), pp. 1396-1402.[6] Craig A. Barwell, Lorenz.,A Study of Failure in Small Pressurized Cylindrical ShellsContaining a Crack., NASA/CR-1998-208454[7] Yao-Chen Li, 1984, “The Finite Element Method By Employing The Singular Element WithConcordant Displacement At The Crack Tip” Engineering Fracture Mechanics, Vol. 19, No. 5,pp. 959-972.[8] Agne Karlsson and Jan Backlund, 1978, “Summary of SIF Design Graphs for CracksEmanating From Circular Holes” International Journal of Fracture, Vol. 14, No. 6[9] Richard D. Young“Non-linear local bending response and bulging factors for longitudinaland circumferential cracks in pressurised cylindrical shells”.[10] Kannan.P, K.Balamurugan and K. Thirunavukkarasu, “Experimental Investigation on theInfluence of Silver Interlayer in Particle Fracture of Dissimilar Friction Welds”, InternationalJournal of Mechanical Engineering & Technology (IJMET), Volume 3, Issue 2, 2012,pp. 32 - 37, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.[11] I.M.Jamadar, S.M.Patil, S.S.Chavan, G.B.Pawar and G.N.Rakate, “Thickness Optimization ofInclined Pressure Vessel using Non Linear Finite Element Analysis using Design by AnalysisApproach”, International Journal of Mechanical Engineering & Technology (IJMET), Volume3, Issue 3, 2012, pp. 682 - 689, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.

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