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TJER 2013, Vol. 10, No. 2, 46-51

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*Corresponding author’s e-mail: abdelaziz970@yahoo.fr

A New Scheme for Crack Growth Modeling by Coupling
Modified Quarter Point Crack-Tip Element and the Level

Set Method
Y Abdelaziz

LMS Laboratory, Department of Mechanical Engineering, University of Bechar, Algeria

Received 31 May 2011; accepted 8 December 2012

Abstract: In this paper, an efficient, numerical procedure is presented to track crack growth modeling
without remeshing. The method's key feature is the coupling of a modified quarter-point crack tip ele-
ment (MQPE) with the level set method (LSM) for crack growth problems. The LSM was used to rep-
resent the crack location, including the location of crack tips. The MQPE was used to compute the stress
and displacement fields necessary for determining the rate of crack growth. Numerical test cases includ-
ing various geometrical exceptions (the center-crack plate specimen, the single edge-crack plate speci-
men, and the double-edge crack plate) demonstrate the accuracy, robustness, and efficiency of the
MQPE/LSM coupling. The extrapolation technique was used to estimate numerically the calibration fac-
tor for various specimens. This work confirms the feasibility of the MQPE/LSM to model accurately the
singularity existing in the vicinity of the cracks. It allows an economic and adequate calculation of the
stress intensity factors, which can be introduced into the various criteria of fracture or laws of propaga-
tion of the crack. The new method reduces the need for remeshing, and results agree well with reference
data.

Keywords: Fracture, Crack, Singularity, SIFs, MQPE, Level set method, Calibration factors

MQPELSM )
MQPE

MQPE/LSM

Y Abdelaziz

1. Introduction

Computational mechanics is a flourishing subject in
science and engineering. It is a field in which physical
mechanics problems are solved through mechanics,
computers, and various numerical methods. The finite
element method (Long et al. 2009; Oñate 2009) has
been recognized widely as a well-established numeri-
cal method as it has applications in virtually are areas
of science and engineering. However, for fracture
mechanics (Nestor 2004; Anderson 2005), one of the
major difficulties in simulating crack problems numer-
ically resides in the singular nature of the asymptotic
crack tip fields. In order to capture this singularity
using conventional finite element method, high mesh
density in the crack front region is required. Also,
there is a need for remeshing during crack growth sim-
ulations. However, this task is not always straightfor-
ward, especially for complex geometries.

Over last few decades, many approaches have been
proposed to solve modeling of crack problems. For
example, there have been methods based on the quar-
ter-point finite element (Barsoum 1974), the enriched
finite element method (Gifford and Hilton 1978), the
boundary collocation method (Newman 1971), the
integral equation method (Sneddon 1973), the body
force method (Nisitani 1985), the boundary elements
method (Cruse 1988), and the dislocation method
(Chen and Hasebe 1995), plus mesh-free methods
such as the element-free Galerkin method (Fleming et
al. 1997). To avoid remeshing in modeling crack
problems, diverse techniques were proposed, includ-
ing the incorporation of a discontinuous mode on an
element level (Oliver 1995), a moving mesh technique
(Rashid 1998), and an enrichment technique based on
a partition-of-unity X-FEM (Belytschko and
Black1999).

Barsoum (1974) discovered that by taking the mid-
side nodes of an element that are adjacent to a crack tip
and moving them to the quarter point of the element
side, the singular stress field which occurs at the crack
tip could be produced (Fig. 1). Recently, various stud-
ies (Gray et al. 2003; Alashoaibi and Arifin 2006;
Souiyah et.al. 2009; Abdelaziz 2010) were carried out
to develop this type of element. These efforts are jus-
tified by the effectiveness of the quarter-point finite
element to reproduce a singularity and by its simplici-
ty of integration in major calculation programs without
modifications. Specifically, Gray et al. (2003) pro-
posed a modification of the finite element with a quar-
ter of a point in the vicinity of the cracks and used this
new type of element in the two-dimensional analysis
of the fracture. The standard singular element was
adjusted so that the near-tip crack opening displace-
ment satisfied a known constraint: the coefficient of
the term which is linear in the distance to the tip must
vanish.

The two-dimensional standard quarter point finite
element (QPE) is based upon the three-noded quadrat-
ic element. Fort [0,1], the shape functions for this
element are given by:

(1)

(2)

(3)

Gray et al. (2003) defined new shape functions (Fig.
2) by adding a cubic term:

(4)

(5)

(6)

2. Coupling the Level Set Method with
Modified Quarter-Point Crack Tip
Element (MQPE)

The level set method (LSM) was developed by the
American mathematicians (Osher and Sethian 1988).
It has become popular in many disciplines, including
image processing, computer graphics, computational
geometry, optimization, and computational fluid
dynamics. The aim of this technique is to track the
motion of an interface by embedding the interface as
the zero level set of the signed distance function. The
motion of the interface is matched with the zero level
set of the level set function, and the resulting initial
value partial differential equation for the evolution of
the level set function.

In this work, we propose a numerical scheme cou-
pling the LSM with the Modified Quarter-Point Crack
Tip Element. The LSM was used to represent the crack
location including the location of crack tips. The
MQPE was used to compute the stress and displace-
ment fields necessary for determining the rate of crack
growth.

A crack is described by two level sets (Fig. 3):

1 2 3

t = 0 t = 1

L/4
Crack tip

Figure 1. Standard quarter point finite element (QPE)

A New Scheme for Crack Growth Modeling by Coupling Modified Quarter Point Crack- Tip Elemenet ...

1. a normal level set, (x), which the signed distance
to the crack surface, and

2. a tangent level set (x), which is the signed dis-
tance to the plane including the crack front and
perpendicular to the crack surface.

In a given element, min and max are, respective-
ly, the minimum and maximum nodal values of on
the nodes of that element. Similarly, let min and max
be the minimum and maximum nodal values of on
the nodes of an element, if min max= 0 and min

max = 0, the elements around the crack tip become
degenerated finite elements.

3. Numerical Examples

In order to be able to compare our results with those
in the literature, we determined the calibration factors
of some samples of an isotropic material. Three phys-
ical problems were modeled using the MQPE/LSM
approach. They are a center-crack plate (CCP) speci-
men, a single edge-crack plate (SECP) specimen, and
a double-edge crack plate (DECP) specimen, as shown
in Fig. 5. As these specimens had clean symmetry, we
discounted their studies as a whole. Figure 6 describes
the mesh and boundary conditions applied to the spec-
imens. The analytical calibration is proposed:

(7)

where KI = stress intensity factor; F = charge applied;
W = width of the specimen; B = thickness of the spec-
imen; a = length of the crack.

The general expression of stress intensity factors
(SIFs) by means of the extrapolation technique is
given by:

(8)

where
r = outdistance face of the crack

v = transverse displacement
E = Young's modulus
For plane strain, k = (3-4 )
For plane stress, k = (3- )/ (1+ )

= Poisson's ratio

The analytical calibration factor Y(a/w) can be com-
puted based on the equation in Tada's Handbook
(2000) as follows:

(a) The CCP specimen

(b) The SECP specimen

The results of this study are shown in Figs. 6-8 and
are compared with those of the analytical solution
(Tada and Paris 2000) (Table 1). The caliberation fac-

Figure 3. Construction of initial level set functions

(9)

(10)

(11)

Y Abdelaziz

tor increases with an an increase in crack width. The
results are in good agreement with analytical solution.

As can be seen from figures, the numerical study
highlights the consistent agreement between current
results obtained using MQPE/LSM approach, and the
available analytical results. Besides the consistent
agreement of results, it is necessary to emphasize two
very important points:

* The MQPE was automatically introduced.
* The analysis was carried out without remeshing, and

the initial mesh was maintained despite the change
in the ratio (a/w).

4. Conclusions

Solving crack problems in fracture mechanics is
imperative when quantifying and predicting the behav-
ior of cracked structures under service conditions. To
this end, the accurate evaluation of fracture parameters
such as the stress intensity factors (SIF) is required.
The finite element method has proved to be very well-

suited for the study of fracture mechanics. However,
one of its main drawbacks is that remeshing tech-
niques are required to investigate and follow crack
propagation patterns. Moreover, the crack tip singular-
ity needs to be accurately represented using approxi-
mation methods. Due to the fact that the standard finite
element methods are based on piecewise differentiable
polynomial approximations, they are not well suited to
problems with discontinuous and/or singular solutions.
Typically, finite element methods require significant
mesh refinement to get accurate results. In response to
this deficiency, modified finite elements have been
developed.

In this work, an initial attempt was made in devel-
oping a crack growth prediction tool under the
assumption of linear elastic fracture mechanics. The
tool was based upon the level set method, a MQPE
crack-tip element, and the extrapolation technique for
evaluating a calibration factor. The LSM was used to
represent the crack location, including the location of
crack tips. The MQPE was used to compute the stress
and displacement fields necessary for determining the

Figure 4. Various crack positions

Figure 5. Mesh and boundary conditions

CCP SECP DECP

A New Scheme for Crack Growth Modeling by Coupling Modified Quarter Point Crack- Tip Elemenet ...

rate of crack growth.
In this paper, the performance of this technique is

illustrated and compared with existing solutions for
three examples. The results obtained are compared to
reported solutions and showed an excellent agreement.

This study has demonstrated the utility and poten-
tial of the MQPE elements as a computational fracture
tool to study the singularity near the crack-tip.
Moreover, MQPE coupled with LSM makes possible
the accurate solution of engineering problems in com-
plex domains, which may be difficult to solve using
the standard finite element method.

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Figure 6. Calibration factor Y(a/w) for CCP specimen Figure 7. Calibration factor Y(a/w) for SECP specimen

Crack specimen length/width (a/w) Crack specimen length/width (a/w)

Table 1. Normalized factors

Crack specimen length/width (a/w)

Figure 8. Calibration factor Y(a/w) for DECP specimen

Y Abdelaziz

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