This paper

deals with the effect of crack oblique and its location on the stress intensity

factor mode I (KI) and II (KII) 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 ? leads

to decreasing the value of KI and the maximum value of KII occurs at

?=45o. Furthermore, KII equal to zero at ? = 0o and 90o

while KI equal to zero at ? = 90o. However, there is no sensitive effect to the crack location on the stress intensity

factor while there is a considerable effect of the crack oblique.

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 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 13

Mode I:

……………..(1)

……………..(2)

………………..…..(3)

Mode II:

…….….……..(4)

……………….……..(5)

……..………..(6)

Figure 2: Definition

of the coordinate axis ahead of a crack tip 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

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 combination

of 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 the

principal 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 Solution

KI 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 system

for element PLANE183 17.

2.5

The Studied Cases

To 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 Discussions

KI 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 II

A 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

III

Figures 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. Conclusions

1) 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.