ABSTRACT

This paper

deals with the influenceof 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 theor.ly using mathematically equations. A good agreement

is observed between the theor. and Num.solutions in all studied cases. We show that increasing the crack oblique ? leads

to decreasing the value of KI and the max.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 influenceto the crack location on the stress intensity

factor while there is a considerable influenceof the crack oblique.

Key Words: Crack, oblique, location, tension,

KI, KII, ANSYS R15.

1. 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 oblique 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 influenceof crack orientation oblique

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 theor.ly. Recentllty, Mohsin 10

and11 studied theor.ly 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 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 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,

Fig. 1: Three standard loading modes of a crack 12.

The 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

Mode I:

……………..(1)

……………..(2)

………………..…..(3)

Mode II:

…….….……..(4)

……………….……..(5)

……..………..(6)

Fig. 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 oblique 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 Fig. 3. The stress intensity at the tip can then be assessed for

each mode using the appropriate equations, Rae 14.

Fig. 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 obliqued through-thickness crack of Fig. 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.

Fig.

4: Through crack in an infinite plate for the

general case where the

principal stress is not perpendicular

to the crack plane13.

2.

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 obliques and

locations under uniaxial tension stresses are studied numerically and theor.ly.

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 Fig. 5.

Fig. 5: Cracked plate specimens.

2.2

Theor. Solution

Values

of KI and KII are theor.ly 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 Num.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 Fig. 6 with the mesh, elements and boundary conditions.

Fig. 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 trioblique

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.

Fig. 7: The geometry, node locations, and the coordinate system

for element PLANE183 17.

2.5

The Studied Cases

To explain the influenceof

crack oblique and its location on the KI and KII, many cases (reported in Table

1) are studied theor.ly and numerically.

Table 1: The

cases studied with the solution types, models and parameters.

3.

Results and Discussions

KI and

KII values are theor.ly calculated by (7 – 10) and numerically using ANSYS R15

with three cases as shown in Table 1.

3.1 Case Study I

Fig.s

8a, b, c, d, e, f, g, h and i explain the Num.and theor. Differences 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 Fig.s, it is too

easy to see that the KI > KII when ? < 45o while KI < KII
when ? > 45o and

KI ? KII at ? = 45o.

3.2

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 Differences of a/b ratio are shown in Fig.s 9a, b, c, d, e, f, g, h and i.

From these Fig.s, it is clear that the crack oblique has a considerable influenceon

the KI and KII values but the influenceof crack location is insignificant.

3.3 Case Study

III

Fig.s 10a, b, c and d explain the Differences of KI and KII with

the crack oblique ? = 0o, 15o, 30o, 45o,

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

these Fig.s, we show that the max.KI and KII values appear at ?=0o

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

and ? = 90o. Generally,

the max.values of the normal and shear stresses occur on surfaces where the ?=0o

and ?=45o, respectively.

From all Fig.s, it can be seen that there is no significant difference

between the theor. and Num.solutions.

Furthermore, Fig.s

11 and 12 are graphically illustrated Von._Mises stresses countor plots with

the Difference of location and oblique of the crack, respectively. From these Fig.s,

it is clear that the influenceof crack oblique and the influenceof crack

location are incomparable.

Fig. 8: Difference of KI Num., KI Th., KII Num. and KII Th. with the Difference

of a / b and ? for model e .

Fig. 9: Difference of KI Num., KI Th., KII Num. and KII Th. with the Difference

of a / b for b, e and h model at ? = 30, 45 and 60.

Fig. 10: Difference of KI and KII with the crack oblique: a and b) for

model b, e, h and theor..

c and d) for model d, e, f and theor..

Fig.

11: Countor plots of Von._Mises stress with the Difference of crack location at

? = 45o.

Fig.

12: Countor plots of Von._Mises stress with the Difference of crack oblique at

specific location.

4. Conclusions

1) A good agreement is observed between the theor.

and Num.solutions in all studied cases.

2)

Increasing the crack oblique ? leads to decrease

the value of KI and the max.value of KII occurs at ?=45.

3)

KII vanished at ? = 0o and 90o

while KI vanished at ? = 90o.

4)

There

is no Clear influence.to the crack location but there is a considerable influenceo f the crack oblique.

5. References

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