Abstract: This paper deals with theeffect of crack oblique and its location on the stress intensity factor mode I(KI) and II (K11) for a finite plate subjected to uniaxial tension stress. Theproblem is solved numerically using finite element software ANSYS R15 andtheoretically using mathematically equations. A good agreement is observedbetween the theoretical and numerical solutions in all studied cases. We showthat increasing the crack angle f leads to decreasing the value of K1and themaximum value of K11 occurs at f=45o. Furthermore, K11 equal to zeroat f = 0o and 90o while K1equal to zero at f = 90o.However, there is no sensitive effect to the crack location while there is aconsiderable effect of the crack oblique.
Key Words: Crack, angle, location,tension, KI, K11, ANSYS R15.I. INTRODUCTIONFracture can be defined as the process of fragmentationof a solid into two or more parts under the stresses action. Fracture analysisdeals with the computation of parameters that help to design a structure withinthe limits of catastrophic failure. It assumes the presence of a crack in thestructure. The study of crack behavior in a plate is a considerable importancein the design to avoid the failure the Stress intensity factor involved infracture mechanics to describe the elastic stress field surrounding a cracktip.Hasebe and Inohara 1 analyzed the relations betweenthe stress intensity factors and the angle of the oblique edge crack for asemi-infinite plate. Theocaris and Papadopoulos 2 used the experimentalmethod of reflected caustics to study the influence of the geometry of anedge-cracked plate on stress intensity factors K1and Kn.
Kim and Lee3 studied K1and K11 for an oblique crack under normal and shear traction andremote extension loads using ABAQUS software and analytical approach asemi-infinite plane with an oblique edge crack and an internal crack acted onby a pair of concentrated forces at arbitrary position is studied by Qian andHasebe 4. Kimura and Sato 5 calculated K1and K11 of the oblique crackinitiated under fretting fatigue conditions. Fett and Rizzi 6 described thestress intensity factors under various crack surface tractions using an obliquecrack in a semi-infinite body. Choi 7 studied the effect of crack orientationangle for various material and geometric combinations of the coating/substratesystem with the graded interfacial zone. Gokul et al 8 calculated the stressintensity factor of multiple straight and oblique cracks in a rivet hole.Khelil et al 9 evaluated K1numerically using line strain method andtheoretically.
Recentllty, Mohsin 10 and 11 studied theoretically andnumerically the stress intensity factors mode I for center, single edge anddouble edge cracked finite plate subjected to tension stress .Patr ici and Mattheij 12 mentioned that, we candistinguish several manners in which a force may be applied to the plate whichmight enable the crack to propagate. Irwin proposed a classificationcorresponding to the three situations represented in Fig.1. Accordingly, weconsider three distinct modes: mode I, mode II and mode III.
In the mode I, oropening mode, the body is loaded by tensile forces, such that the cracksurfaces are pulled apart in the y direction. The mode II , or sliding mode,the body is loaded by orces parallel to the crack surfaces, which slide overeach other in the x direction. Finally, in the mode III , or tearing mode, thebody is loaded by shear forces parallel to the crack front the crack surfaces,and the crack surfaces slide over each other in the z direction. stress fieldsahead of a crack tip (Fig.2) for mode I and mode II in a linear elastic,isotropic material areas in the follow, Anderson 13 In many situations, a crack is subject to acombination of the three different modes of loading, I, II and III. A simpleexample is a crack located at an angle other than 90° to a tensile load: thetensile load Co, is resolved into two componentperpendicular to the crack, mode I, and parallel to the crack, mode II as shownin Fig.3.
The stress intensity at the tip can then be assessed for each modeusing the appropriate equations, Rae 14, Stress intensity solutions are given in avariety of forms, K can always be related to the through crack through theappropriate correction factor, Anderson 13 where o: characteristic stress, a:characteristic crack dimension and Y: dimensionless constant that depends onthe geometry and the mode of loading.Wecan generalize the angled through-thickness crack of Fig.4 to any planar crackoriented 90° – p from the applied normal stress. For uniaxial loading, thestress intensity factors for mode I and mode II are given by K1= where KI0 is the mode Istress intensity when ? = 0. The crack-tip stress fields (in polar coordinates)for the mode I portion of the loading are given by II. Materials and MethodsBasedon the assumptions of Linear Elastic Fracture Mechanics LEFM and plane strainproblem, K1and K11 to a finite cracked plate for different angles and locationsunder uniaxial tension stresses are studied numerically and theoretically.A. Specimens MaterialThe plate specimenmaterial is Steel (structural) with modulus of elasticity 2.
07E5 Mpa andpoison’s ratio 0.29, Young and Budynas 15. The models of plate specimens withdimensions are shown in Fig.5. B. TheoreticalSolutionValuesof K1and K11 are theoretically calculated based on the following procedure1)Determination of the KIo (K1whenp = 0) based on (7), where (Tada et al 16 ) 2) Calculating K1and K11 toany planer crack oriented (P) from the applied normal stress using (8) and (9).
C. Numerical SolutionK1and K11 are calculated numerically usingfinite element software ANSYS R15 with PLANE183 element as a discretizationelement. ANSYS models at P=0o are shown in Fig.6 with the mesh,elements and boundary conditions.
D. PLANE183 DescriptionPLANE183 is used in this paper as adiscretization element with quadrilateral shape, plane strain behavior and puredisplacement formulation. PLANE183 element type is defined by 8 nodes ( I, J,K, L, M, N, O, P ) or 6 nodes ( I, J, K, L, M, N) for quadrilateral andtriangle element, respectively having two degrees of freedom (Ux , Uy) at each node(translations in the nodal X and Y directions) 17. The geometry, nodelocations, and the coordinate system for this element are shown in Fig.7. E.
The Studied CasesTo explain the effect of crack oblique andits location on the K1and K11, many cases (reported in Table 1) are studiedtheoretically and numerically. III. Results and DiscussionsK1and K11 values are theoretically calculated by (7 – 10) and numerically usingANSYS R15 with three cases asshown in Table 1.A.
Case Study IFigs.8a, b, c, d, e, f, g, h and i explain the numerical and theoretical variationsof K1and K11 with differentvalues of a/b ratio when ? = 0o,15o, 30o, 40o, 45o, 50o, 60o, 70o and 75o, respectively.From these Figs., it is tooeasy to see that the K1> K11 when ? < 45o while K1< K11 when ? > 45o and K1? K11 at ? =45o.
B. Case Study IIA compression between K1and K11 values for differentcrack locations (models b, e and h) at p=30o, 45o and 60owith variations of a/b ratio are shown in Figs. 9a, b, c, d, e, f, g, hand i. From these Figs., it is clear that the crack angle has a considerableeffect on the K1and K11 values but the effect of crack location isinsignificant. Fig.9: Varia tion of K1Num., K1Th.
, K11 Num. and K11 Th. with the variation of a / b for b, e and h model at P = 30, 45 and 60. C. Case Study IIIFigs. 10a, b, c and d explain the variations of K1and K11with the crack angle P = 0o, 15o, 30o, 45o,60o, 75o and 90o for models b, e and h. Fromthese Figs.
, we show that the maximum K1and K11 values appear at P=0o andP=45o, respectively. Furthermore, K11 equal to zero at P = 0oand P = 90o. Generally, the maximum values of the normal and shearstresses occur on surfaces where the P=0o and P=45o,respectively. Fromall Figs.
, it can be seen that there is no significant difference between thetheoretical and numerical solutions. Furthermore, Figs. 11 and 12 are graphicallyillustrated Von.
_Mises stresses countor plots with the variation of locationand angle of the crack, respectively. From these Figs., it is clear that theeffect of crack angle and the effect of crack location are incomparable. Fig.
12: Cou ntor plots of Von._Mises stress with the variation of crack angle at spe cific locat ion. IV. Conclusions1) A good agreement isobserved between the theoretical and numerical solutions in all studied cases.
2) Increasing the crack anglep leads to decrease the value of K1and the maximum value of K11 occurs at p=45.3) K11 vanished at p = 0oand 90o while K1vanished at p = 90o.4) There is no obvious effectto the crack location but there is a considerable effect of the crack oblique. References 1 . N. Hasebe and S. Inohara.
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