Fracture the elastic stress field surrounding a

Fracture can be defined as the process of fragmentation of a solid into two or more parts under the stresses action.

Fracture analysis deals with the computation of parameters that help to design a structure within the limits of catastrophic failure. It assumes the presence of a crack in the structure. The study of crack behavior in a plate is a considerable importance in the design to avoid the failure the Stress intensity factor involved in fracture mechanics to describe the elastic stress field surrounding a crack tip.

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Hasebe and Inohara 1 analyzed the relations between the stress intensity factors and the angle of the oblique edge crack for a semi-infinite plate. Theocaris and Papadopoulos 2 used the experimental method of reflected caustics to study the influence of the geometry of an edge-cracked plate on stress intensity factors KI and KII. Kim and Lee 3 studied KI and KII for an oblique crack under normal and shear traction and remote extension loads using ABAQUS software and analytical approach a semi-infinite plane with an oblique  edge crack and an internal  crack acted  on by a pair  of  concentrated  forces  at  arbitrary  position  is  studied by Qian  and Hasebe 4. Kimura and Sato 5 calculated KI and KII of the oblique crack initiated under fretting fatigue conditions. Fett and Rizzi 6 described the stress intensity factors under various crack surface tractions using an oblique crack in a semi-infinite body. Choi 7 studied the effect of crack orientation angle for various material and geometric combinations of the coating/substrate system with the graded interfacial zone. Gokul et al 8 calculated the stress intensity factor of multiple straight and oblique cracks in a rivet hole. Khelil et al 9 evaluated KI numerically using line strain method and theoretically.

Recentllty, Mohsin 10 and11 studied theoretically and numerically the stress intensity factors mode I for center ,single edge and double edge cracked finite plate subjected to tension stress .Patr ??ci and Mattheij 12 mentioned that, we can distinguish several manners in which a force may be applied to the plate which might enable the crack to propagate. Irwin proposed a classification corresponding to the three situations represented in Figure 1. Accordingly, we consider three distinct modes: mode I, mode II and mode III. In the mode I, or opening mode, the body is loaded by tensile forces, such that the crack surfaces are pulled apart in the y direction.

The mode II, or sliding mode, the body is loaded by shear forces parallel to the crack surfaces, which slide over each other in the x direction. Finally, in the mode III , or tearing mode, the body 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, Figure 1: Three standard loading modes of a crack 12.The stress fields ahead of a crack tip (Figure 2) for mode I and mode II in a linear elastic, isotropic material are as in the follow, Anderson 13Mode I:   ……………..(1)   ……………..

(2)   ………………..…..

(3)Mode II:   …….….……..(4)   …………….

……..(5)   ……..………..(6) Figure 2: Definition of the coordinate axis ahead of a crack tip 13In many situations, a crack is subject to a combination of the three different modes of loading, I, II and III. A simple example is a crack located at an angle other than 90º to a tensile load: the tensile load ?o, is resolved into two component perpendicular to the crack, mode I, and parallel to the crack, mode II as shown in Figure 3.

The stress intensity at the tip can then be assessed for each mode using the appropriate equations, Rae 14. Figure 3: Crack subjected to a combinationof two modes of loading I and II 14.                        Stress intensity solutions are given in a variety of forms, K can always be related to the through crack through the appropriate correction factor, Anderson 13  ,     ……….……….(7)where ?: characteristic stress, a: characteristic crack dimension and Y: dimensionless constant that depends on the geometry and the mode of loading.We can generalize the angled through-thickness crack of Figure 4 to any planar crack oriented 90° ? ? from the applied normal stress. For uniaxial loading, the stress intensity factors for mode I and mode II are given by   ……………….(8)  , ………….

.(9)where KI0 is the mode I stress intensity when ? = 0.  Figure 4:  Through crack in an infinite plate for the general case where theprincipal stress is not perpendicular to the crack plane13.

1.      Materials and Methods Based on the assumptions of Linear Elastic Fracture Mechanics LEFM and plane strain problem, KI and KII to a finite cracked plate for different angles and locations under uniaxial tension stresses are studied numerically and theoretically.2.1  Specimens Material The plate specimen material is Steel (structural) with modulus of elasticity 2.07E5 Mpa and poison’s ratio 0.29, Young and Budynas 15.

The models of plate specimens with dimensions are shown in Figure 5.   Figure 5: Cracked plate specimens.   2.

2   Theoretical Solution Values of KI and KII are theoretically calculated based on the following procedure a)      Determination of the KIo (KI when ? = 0) based on (7), where (Tada et al 16 ) …………………….(10)b)       Calculating KI and KII to any planer crack oriented (?) from the applied normal stress                 using (8) and (9).2.3  Numerical SolutionKI and KII are calculated numerically using finite element software ANSYS R15 with PLANE183 element as a discretization element. ANSYS models at ?=0o are shown in Figure 6 with the mesh, elements and boundary conditions.  Figure 6: ANSYS models with mesh, elements and boundary conditions.  2.4  PLANE183 Description PLANE183 is used in this paper as a discretization element with quadrilateral shape, plane strain behavior and pure displacement 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 and triangle element, respectively having two degrees of freedom (Ux , Uy) at each node (translations in the nodal X and Y directions) 17. The geometry, node locations, and the coordinate system for this element are shown in Figure 7.   Figure 7: The geometry, node locations, and the coordinate systemfor element PLANE183 17. 2.5  The Studied CasesTo explain the effect of crack oblique and its location on the KI and KII, many cases (reported in Table 1) are studied theoretically and numerically.              Table 1: The cases studied with the solution types, models and parameters.

   2.      Results and DiscussionsKI and KII values are theoretically calculated by (7 – 10) and numerically using ANSYS R15 with three cases as shown in Table 1.3.1 Case Study I  Figures 8a, b, c, d, e, f, g, h and i explain the numerical and theoretical variations of KI and KII with different values of a/b ratio when ? = 0o, 15o, 30o, 40o, 45o, 50o, 60o, 70o and 75o, respectively. From these figures, it is too easy to see that the KI > KII when ? < 45o while KI < KII when ? > 45o               and KI ? KII at ? = 45o.2.2  Case Study IIA compression between KI and KII values for different crack locations (models b, e and h) at ?=30o, 45o and 60o with variations of a/b ratio are shown in Figures 9a, b, c, d, e, f, g, h and i.

From these figures, it is clear that the crack angle has a considerable effect on the KI and KII values but the effect of crack location is insignificant. 3.3 Case Study IIIFigures 10a, b, c and d explain the variations of KI and KII with the crack angle ? = 0o, 15o, 30o, 45o, 60o, 75o and 90o for models b, e and h. From these figures, we show that the maximum KI and KII values appear at ?=0o and ?=45o, respectively. Furthermore, KII equal to zero at ? = 0o and ? = 90o. Generally, the maximum values of the normal and shear stresses occur on surfaces where the ?=0o and ?=45o, respectively.From all figures, it can be seen that there is no significant difference between the theoretical and numerical solutions.Furthermore, Figures 11 and 12 are graphically illustrated Von-Mises stresses countor plots with the variation of location and angle of the crack, respectively.

From these figures, it is clear that the effect of crack angle and the effect of crack location are incomparable.   Figure 8: Variation of KI Num., KI Th., KII Num.

and KII Th. with the variation of a / b and ? for model e .  Figure 9: Variation of KI Num., KI Th., KII Num. and KII Th. with the variation of a / b for b, e and h model at ? = 30, 45 and 60.

     Figure 10: Variation of KI and KII with the crack angle: a and b) for model b, e, h and theoretical.                                                                                     c and d) for model d, e, f and theoretical.  Figure 11: Countor plots of Von-Mises stress with the variation of crack location at ? = 45o.   Figure 12: Countor plots of Von-Mises stress with the variation of crack angle at specific location.

  3.      Conclusions1)      A good agreement is observed between the theoretical and numerical solutions in all studied cases.2)      Increasing the crack angle ? leads to decrease the value of KI and the maximum value of KII          occurs at ?=45.3)      KII vanished at ? = 0o and 90o while KI vanished at ? = 90o. 4)      There is no obvious effect to the crack location but there is a considerable effect of the crack oblique.   


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