Abstract: This paper deals with the

effect 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. The

problem is solved numerically using finite element software ANSYS R15 and

theoretically using mathematically equations. A good agreement is observed

between the theoretical and numerical solutions in all studied cases. We show

that increasing the crack angle f leads to decreasing the value of K1and the

maximum value of K11 occurs at f=45o. Furthermore, K11 equal to zero

at f = 0o and 90o while K1equal to zero at f = 90o.

However, there is no sensitive effect to the crack location while there is a

considerable effect of the crack oblique.

Key Words: Crack, angle, location,

tension, KI, K11, ANSYS R15.

I.

INTRODUCTION

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.

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 K1and Kn. Kim and Lee

3 studied K1and K11 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 K1and K11 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 K1numerically using line strain method and

theoretically. Recentllty, Mohsin 10 and 11 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 ici 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 Fig.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 orces 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.

stress fields

ahead of a crack tip (Fig.2) for mode I and mode II in a linear elastic,

isotropic material are

as in the follow, Anderson 13

In 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 90° to a tensile load: the

tensile load Co, is resolved into two component

perpendicular to the crack, mode I, and parallel to the crack, mode II as shown

in Fig.3. The stress intensity at the tip can then be assessed for each mode

using the appropriate equations, Rae 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

where o: 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 Fig.4 to any planar crack

oriented 90° – p from the applied normal stress. For uniaxial loading, the

stress intensity factors for mode I and mode II are given by K1=

where KI0 is the mode I

stress 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 Methods

Based

on the assumptions of Linear Elastic Fracture Mechanics LEFM and plane strain

problem, K1and K11 to a finite cracked plate for different angles and locations

under uniaxial tension stresses are studied numerically and theoretically.

A.

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 Fig.5.

B. Theoretical

Solution

Values

of K1and K11 are theoretically calculated based on the following procedure

1)Determination of the KIo (K1when

p = 0) based on (7), where (Tada et al 16 )

2)

Calculating K1and K11 to

any planer crack oriented (P) from the applied normal stress using (8) and (9).

C.

Numerical Solution

K1and K11 are calculated numerically using

finite element software ANSYS R15 with PLANE183 element as a discretization

element. ANSYS models at P=0o are shown in Fig.6 with the mesh,

elements and boundary conditions.

D.

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 Fig.7.

E.

The Studied Cases

To explain the effect of crack oblique and

its location on the K1and K11, many cases (reported in Table 1) are studied

theoretically and numerically.

III.

Results and Discussions

K1and K11 values are theoretically calculated by (7 – 10) and numerically using

ANSYS R15 with three cases as

shown in Table 1.

A.

Case Study I

Figs.

8a, b, c, d, e, f, g, h and i explain the numerical and theoretical variations

of K1and K11 with different

values of a/b ratio when ? = 0o,

15o, 30o, 40o, 45o, 50o, 60o, 70o and 75o, respectively.

From these Figs., it is too

easy to see that the K1> K11 when ? < 45o while K1< K11 when ? > 45o and K1? K11 at ? =

45o.

B.

Case Study II

A compression between K1and K11 values for different

crack locations (models b, e and h) at p=30o, 45o and 60o

with variations of a/b ratio are shown in Figs. 9a, b, c, d, e, f, g, h

and i. From these Figs., it is clear that the crack angle has a considerable

effect on the K1and K11 values but the effect of crack location is

insignificant.

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 III

Figs. 10a, b, c and d explain the variations of K1and K11

with the crack angle P = 0o, 15o, 30o, 45o,

60o, 75o and 90o for models b, e and h. From

these Figs., we show that the maximum K1and K11 values appear at P=0o and

P=45o, respectively. Furthermore, K11 equal to zero at P = 0o

and P = 90o. Generally, the maximum values of the normal and shear

stresses occur on surfaces where the P=0o and P=45o,

respectively.

From

all Figs., it can be seen that there is no significant difference between the

theoretical and numerical solutions.

Furthermore, Figs. 11 and 12 are graphically

illustrated Von._Mises stresses countor plots with the variation of location

and angle of the crack, respectively. From these Figs., it is clear that the

effect 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.

Conclusions

1)

A good agreement is

observed between the theoretical and numerical solutions in all studied cases.

2)

Increasing the crack angle

p leads to decrease the value of K1and the maximum value of K11 occurs at p=45.

3)

K11 vanished at p = 0o

and 90o while K1vanished at p = 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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