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.
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
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.
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.
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
Materials and Methods
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.
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.
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 )
Calculating K1and K11 to
any planer crack oriented (P) from the applied normal stress using (8) and (9).
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.
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.
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.
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.
Case Study I
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 ? =
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
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.
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,
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.
A good agreement is
observed between the theoretical and numerical solutions in all studied cases.
Increasing the crack angle
p leads to decrease the value of K1and the maximum value of K11 occurs at p=45.
K11 vanished at p = 0o
and 90o while K1vanished at p = 90o.
There is no obvious effect
to the crack location but there is a considerable effect of the crack oblique.
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