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 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.
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