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IOP Conference Series: Materials Science and Engineering
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A review of fatigue crack propagation modelling
techniques using FEM and XFEM
To cite this article: K Rege and H G Lemu 2017 IOP Conf. Ser.: Mater. Sci. Eng. 276 012027
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First Conference of Computational Methods in Offshore Technology (COTech2017) IOP Publishing
IOP Conf. Series: Materials Science and Engineering 276 (2017) 012027 doi:10.1088/1757-899X/276/1/012027
A review of fatigue crack propagation modelling techniques
using FEM and XFEM
K Rege and H G Lemu
Department of Mechanical and Structural Engineering and Materials Science,
University of Stavanger, Norway
kristen.rege@uis.no
Abstract. Fatigue is one of the main causes of failures in mechanical and structural systems.
Offshore installations, in particular, are susceptible to fatigue failure due to their exposure to
the combination of wind loads, wave loads and currents. In order to assess the safety of the
components of these installations, the expected lifetime of the component needs to be
estimated. The fatigue life is the sum of the number of loading cycles required for a fatigue
crack to initiate, and the number of cycles required for the crack to propagate before sudden
fracture occurs. Since analytical determination of the fatigue crack propagation life in real
geometries is rarely viable, crack propagation problems are normally solved using some
computational method. In this review the use of the finite element method (FEM) and the
extended finite element method (XFEM) to model fatigue crack propagation is discussed. The
basic techniques are presented, together with some of the recent developments.
1. Introduction
Components which are subjected to fluctuating loads are found virtually everywhere: Vehicles and
other machinery contain rotating axles and gears, pressure vessels and piping may be subjected to
pressure fluctuations (e.g. water hammer) or repeated temperature changes, structural members in
bridges are subjected to traffic loads and wind loads, while those in ships and offshore structures are
subjected to the combination of wind loads, wave loads and currents. If the components are subjected
to a fluctuating load of a certain magnitude for a sufficient amount of time, small cracks will nucleate
in the material. Over time, the cracks will propagate, up to the point where the remaining cross-section
of the component is not able to carry the load, at which the component will be subjected to sudden
fracture [1]. This process is called fatigue, and is one of the main causes of failures in structural and
mechanical components [2]. In order to assess the safety of the component, engineers need to estimate
its expected lifetime. The fatigue life is the sum of the number of loading cycles required for a fatigue
crack to nucleate/initiate, and the number of cycles required for the crack to propagate until its critical
size has been reached [2, 3]. In this paper, computational methods to estimate the lifespan of a
propagating crack whose initial geometry is known will be considered.
Estimations of the fatigue crack propagation rate, da/dN, are normally based on a relation with the
range of the stress intensity factor, ΔK, which is a linear elastic fracture mechanics (LEFM) parameter
for quantifying the load and geometry of the crack. Paris, Gomez and Anderson [4] first proposed the
existence of such a relation in 1961, and its simplest form is the Paris law [5]:
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First Conference of Computational Methods in Offshore Technology (COTech2017) IOP Publishing
IOP Conf. Series: Materials Science and Engineering 276 (2017) 012027 doi:10.1088/1757-899X/276/1/012027
d
d
m
aCK
N= ∆
(1)
where a is the length of an edge crack (or half the length of an internal crack), N is the number of
loading cycles, and C and m are scaling constants. Due to its simplicity, the Paris law is only
applicable for intermediate values of ΔK (region II in Figure 1), under constant amplitude cyclic
loading [6]. Furthermore, the constants C and m are influenced by the applied stress ratio R =
Kmin/Kmax, and crack closure effects are not taken into account. Therefore, a number of different crack
propagation laws have been proposed, each one taking different factors into account [7]. One of the
most detailed and commonly used is the so-called NASGRO equation [8]:
( )
( )
th
max
c
1
1
d
d1 1
p
m
q
K
f
aK
CK
NR K
K
∆
−
−∆
= ⋅∆ ⋅
−
−
(2)
where ΔKth is the threshold value for fatigue crack propagation, Kc is the critical stress intensity factor
at which fracture occurs, ΔK = Kmax – Kmin, R = Kmin/Kmax, f is a crack opening function and C, m, p and
q are empirical constants. Note that for a given material, C and m do not have the same numerical
values in different crack propagation laws.
Figure 1. Typical fatigue crack growth curve.
The loading and displacement of a crack can be described by the three modes of fracture, each with
its own stress intensity factor; mode I (tensile opening, KI), mode II (in-plane sliding, KII) and mode III
(tearing/out-of-plane shear, KIII) [6]. The different modes require different values for the constants in
the crack propagation law. In the case of mixed-mode fatigue, it may be necessary to use an effective
mixed-mode stress intensity factor [9], for instance as given in [10], or a modified crack propagation
law [11].
The crack propagation life can be estimated by integrating equation (1) or (2). However, the stress
intensity factors Kmax and Kmin are normally functions of the crack length a, and depend on the
geometry of the structure. Analytical integration of equations (1) and (2) is therefore rarely viable for
complicated geometries. Instead, crack propagation problems are normally solved using some
computational method, e.g. the Finite Element Method (FEM) [12]. The crack propagation process is
then solved in a step-wise manner. For each step, the crack is advanced a small length, and the number
of cycles required for the next crack increment is estimated using one of the crack propagation laws. In
Fracture
Region I
Region II
Region III
m
ΔKth
d
lg da
N
lg ΔK
3
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First Conference of Computational Methods in Offshore Technology (COTech2017) IOP Publishing
IOP Conf. Series: Materials Science and Engineering 276 (2017) 012027 doi:10.1088/1757-899X/276/1/012027
order to accomplish this, the computational method needs to perform the following tasks within each
step [12, 13]:
1. Computation of the minimum and maximum stress and displacement fields within the cracked
component.
2. Evaluation of the minimum and maximum stress intensity factors for the crack.
3. Evaluation of the direction for further crack propagation.
4. Generation of a representation of the crack advancement.
This process is repeated until the critical stress intensity factor is reached; Kmax = Kc, and the number
of experienced cycles are summed to obtain the crack propagation life.
The main challenge when using the finite element method to estimate the fatigue crack
propagation, lies in the fourth task in the list above [12]. In order to evaluate stress intensity factors for
the advanced crack, the finite element mesh needs to be updated, a process which has been shown to
be challenging [6]. This has led to the development of alternative computational methods to handle
propagating cracks, among them the eXtended Finite Element Method (XFEM) [14, 15], the Boundary
Element Method (BEM) [16], hybrids between finite and boundary element methods, e.g. the
Symmetric Galerkin Boundary Element Method – Finite Element Method (SGBEM-FEM) [17] and
the Scaled Boundary Finite Element Method (SBFEM) [18], and meshless methods [19].
In this review, the use of the finite element method (FEM) and the extended finite element method
(XFEM) to model fatigue crack propagation will be discussed. The basic techniques will be presented,
together with some of the recent developments. The review will focus on the modelling techniques,
and only to a less extent on their applications.
2. Crack propagation be the finite element method
2.1. Computation of stress and displacement fields, and the stress intensity factor
The first issue when estimating fatigue crack propagation rates using the finite element method is the
computation of sufficiently accurate values for the stress intensity factor for the crack at the maximum
and minimum applied loads during each cycle. In order to compute the stress intensity factor, the
stress and displacement fields of the whole component are also needed. Three simple methods for
computing the stress intensity factors of mode I cracks from a finite element stress field were
presented by Chan, et al. [20] already in 1970. These methods were called the stress method, the
displacement method and the line integral method, and were all computed using ordinary linear
(constant-strain) triangle elements, with a high degree of refinement at the crack tip.
2.1.1. The stress method. In the stress method, nodal stress values are extrapolated to the crack tip. It
is often easiest to do this extrapolation along the crack plane, in which case the stress intensity factor
is related to the stress normal to the crack plane, σyy, by [6]:
I0
lim 2π
yy
r
Kr
σ
→
=
(3)
where r is the distance from the crack tip. The nodal values of KI are plotted as a function of r, and
extrapolated to r = 0. As ordinary finite elements are not able to represent the stress singularity at the
crack tip, the nodal values of KI closest to the tip should be omitted when performing the extrapolation
[20]. The singularity of the stress field also causes this method to be one of the least accurate methods
available [6]. A similar approach is also possible for mode II and mode III cracks.
2.1.2. The displacement method. The second method is called the displacement method, and involves
a relation between the stress intensity factor and the crack-opening displacement uy [20]. For mode I
loading, this relation is given as [6, 20]:
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IOP Conf. Series: Materials Science and Engineering 276 (2017) 012027 doi:10.1088/1757-899X/276/1/012027
I0
2
lim 4
y
r
Eu
Kr
∗
→
π
=
(4)
where:
2
, plane stress
, plane strain
1
E
EE
ν
∗
=
−
(5)
E is the elastic modulus, while ν is Poisson's ratio. As the opening displacement uy is equal to zero at
the crack tip, the nodal values of uy near the crack tip must be disregarded when performing the
extrapolation of KI to r = 0. The displacement method does generally give more accurate estimations
than the stress method [6].
2.1.3. Finite elements at the crack tip. In order to reduce the mesh quality required for the
displacement method, special crack tip elements were developed in the 1970’s, which were able to
describe the singularity which exists in the near-crack stress field. One of them was the isoparametric
bilinear rectangle (quadrilateral) 4-node element with special shape functions, proposed by Tracey
[21] in 1971, which should be used at the crack tip. Four years later, Henshell and Shaw [22] declared
in the title of their paper that “Crack tip finite elements are unnecessary”. They could show that the
crack tip singularity could be represented by 8-node isoparametric quadratic rectangle elements, if the
mid-side nodes closest to the crack tip were moved to the “quarter points,” i.e. ¼ of the element length
away from the crack tip. Similar 2D elements, as well as three-dimensional elements, were
independently developed by Barsoum, and presented in the same journal, two issues later [23].
When used at the crack tip, these so-called "quarter-point elements" [24] have a reasonably
accurate stiffness, but the local values of stress and displacement within the quarter-point elements are
poor [22]. The nodal displacements for these elements should therefore be omitted when calculating K.
Recommendations for generating a suitable mesh with quarter-point elements for evaluating KI by the
displacement method have been given by Menandro, et al. [25] and Guinea, et al. [26], and a typical
example of the crack tip mesh is shown in Figure 2. Note that the quarter-point (8-node quadratic
rectangle) elements are collapsed down to triangles, where each element has three nodes located at the
crack tip.
Figure 2. (a) Typical rosette pattern for the
FEM mesh at a LEFM crack tip, and (b) detail
of the two inner rings, with nodes shown. Note
the use of quarter-
tip elements only in the
innermost ring.
2.1.4. Energy release rate methods. The potential energy decrease per unit crack advance is called the
energy release rate, G, and may be used to characterise crack growth in a linear or nonlinear elastic
body [27-29]. Rice [27] showed that the energy release rate could be computed by a path independent
line integral, which for a two-dimensional problem is defined by:
ddJ Wy s
x
Γ
∂
= −⋅
∂
∫u
T
(6)
(a)
(b)
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where Γ is a curve surrounding the crack tip, W(x, y) is the strain-energy density field, and x, y are the
Cartesian coordinates being parallel and normal to the crack tip, respectively. T is the traction vector
associated to the outward normal to Γ, u is the displacement vector, and ds is an incremental arc length
along Γ.
The third method to compute the stress intensity factors mentioned by Chan, et al. [20] is the line
integral method. This method consists of numerically evaluating the J integral, equation (6), for the
finite element solution over an arbitrary path surrounding the crack tip. In the case of small-scale
yielding, the energy release rate may be related to the mode I stress intensity factor by the following
equation [20, 27]:
I
K GE JE
∗∗
= =
(7)
The J integral may be evaluated at a remote contour, like the outer boundary of the geometry [20],
which improves its numerical accuracy, compared with the stress method and the displacement
method. The application of this method to three-dimensional problems is more difficult, however,
because the integral becomes a surface integral. It is difficult both to define the surface and to perform
the numerical integration [29]. This led to the development of alternative methods to evaluate the
energy release rate, e.g. the virtual crack extension methods developed by Parks [28], Hellen [30] and
deLorenzi [29]. The most accurate and efficient [6] method for the numerical evaluation of the energy
release rate seems to be the domain integral method, formulated by Shih, et al. in 1986 [31]. They
compute the J integral by using the divergence theorem, in which the three-dimensional surface
integral is transformed into a volume domain integral, which is evaluated using Gaussian quadrature.
For mixed mode I+II loading with small-scale yielding, the following relationship exists between
the energy release rate and the stress intensity factors [14, 27]:
( )
22
I II
1
JG K K
E
∗
= = +
(8)
In order to extract the stress intensity factors from the J integral, Yau, et al. [32] developed a technique
using an interaction integral. The combination of the interaction integral with the domain integral
method is shown in [14, 33].
The main advantage of the energy release rate methods is that accurate estimations for the stress
intensity factor may be obtained even with a relatively coarse mesh [26]. A finer mesh, with quarter-
point elements at the crack tip, is required if determination of the stress field is part of the objective.
The domain integral is often recommended for practical use. Some find it easier to implement in
certain FE codes (because quarter-point elements are not needed) [34], and others have found it to be
more stable than the displacement method [24]. On the other hand, if quarter-point elements may
easily be used in a FE code, the displacement method does not need any specialised post-processing
routine to obtain the stress intensity factor. Both the displacement method and the domain integral
method are able to accurately predict the stress intensity factor [24]. The displacement method is still
widely used [9, 26, 35], and Guinea, et al. [26] question whether the domain integral actually is the
most efficient.
2.2. Evaluation of the direction for further crack growth
If a crack is subjected to a mixed-mode loading, the propagating crack seeks the path of least
resistance. Several theories have been proposed to choose this path. The three most used [35] are the
criteria of maximum tangential (circumferential) stress [36], maximum energy release rate [37] and
minimum strain energy density [38]. Other criteria include the criterion of maximum dilatational strain
energy density [39] and the criterion of minimum accumulated strain energy [11]. There is no general
agreement about which criterion should be used for a given material, but Bittencourt, et al. [24] have
shown that the three former criteria predict basically the same crack growth trajectory for poly(methyl
methacrylate) (PMMA), which is a brittle material. The criterion of maximum tangential stress is often
applied in FEM simulations of fatigue crack growth, because it is simple to implement, as it has an
approximate explicit solution for the crack growth direction θ as a function of KI and KII [9, 14, 35].
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To implement the evaluation of the direction for further crack propagation in FE codes is generally not
a problem [12].
2.3. Representation of the crack advancement
When the stress intensity range and the crack growth direction have been found, the number of cycles
required for the crack to propagate a distance Δa may be estimated by the crack propagation law. As
the crack propagation law is not linear with respect to Δa, the stress intensity factor range ΔK needs to
be re-evaluated after the increment Δa. Reducing this increment will increase the accuracy of the
solution, at the cost of increased computational effort.
In order to re-evaluate ΔK, the crack increment needs to be represented by the finite element mesh.
As noted in section 2.1, some mesh refinement at the crack tip is normally needed. When the crack tip
moves due to the crack increment, the focused region of the mesh should follow. The most common
technique when modelling propagating fatigue cracks under LEFM conditions is to perform global or
local re-meshing [9, 24, 35, 40]. Local re-meshing is generally preferred, due to the lower
computational effort, compared to global re-meshing.
Local re-meshing was employed already in 1986 by Højfeldt and Østervig [40] to predict the shape
and crack propagation life of fatigue cracks in shafts with a diameter transition. They employed the
displacement method and quarter-point elements to estimate the stress intensity factors of the three-
dimensional crack, and the criterion of minimum strain energy density to estimate the crack growth
direction. Bittencourt, et al. [24] and Miranda, et al. [35] used the local re-meshing technique, together
with quarter-point elements and the displacement technique, in their methodology for assessment of
fatigue crack propagation in two-dimensional components. The re-meshing technique was also used
by Alegre and Cuesta [9] to model mode I+II crack propagation in a valve.
The local re-meshing technique generally consists of four steps [24], as illustrated in Figure 3; (a)
removing the existing mesh around the crack, (b) advancing the length of the crack, (c) applying
quarter-point elements in a uniform rosette pattern at the crack tip, and (d) generating a new mesh in
the open area.
(a) (b)
(c) (d)
Figure 3. Local re-
meshing technique. (a) Existing mesh. (b) The mesh around the crack tip is
removed, and the crack is advanced. (c) Rosette of elements are applied at the crack tip. (d) The
remaining area is meshed.
Recent research in crack propagation modelling by FEM often deals with plasticity induced crack
closure, e.g. [41-43], in which the material plasticity is explicitly taken into account in the finite
element analysis. Re-meshing is then highly cumbersome, with respect to both computational and
programming effort, because the plastic strain history needs to be mapped from the old mesh to the
new one [6]. Furthermore, accuracy is lost during this mapping process. As an alternative, it is
possible to create a single mesh which accommodates crack growth, by being refined along the
assumed crack propagation path. The disadvantage of this approach is that the crack path and shape
are predetermined by the mesh [6, 12]. In addition, the crack increment Δa has to correspond to the
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element size, making the crack growth response mesh dependent. By employing micrometre-size
elements, this approach is nevertheless widely used [42-43].
In order to advance the crack through the refined mesh, the following techniques may be used [6,
44]: (a) Removing elements along the crack front once a failure criterion is reached, (b) releasing the
nodes at the crack tip at specific load steps or according to a failure criterion, so that these elements
are no longer connected, or (c) using cohesive elements. The cohesive elements are zero-thickness
elements which are placed in between the ordinary elements, and for which a certain force-
displacement law is specified. When computing fatigue crack propagation with plasticity induced
crack closure, the node release technique is the most common to use [45], but cohesive elements are
also used by some researchers [42, 43, 46]. It should be noted that the plastic crack tip field does not
contain the
1r
singularity [6], making quarter point elements unnecessary when studying plasticity
induced crack closure.
Colombo and Giglio [12] recently developed a crack advancement technique for LEFM conditions
which is a combination of the local re-meshing and node release techniques. Their technique may be
described by four steps: (a) the crack increment is projected on the FE mesh, (b) the elements touched
by the crack increment are deleted, (c) the open area is re-meshed by debondable triangular elements
which conform to the crack increment, and (d) the nodes along the crack increment are released. This
technique limits the re-meshing to the absolute minimum, while still keeping the crack growth mesh-
independent. The resulting mesh is inadequate for evaluating the stress intensity factors using the
displacement technique, however. Therefore, the submodelling technique [47, 48] is used, in which the
displacements from this mesh are imposed on an independent focused mesh which represents the crack
tip using quarter-point elements (as shown in Figure 2(a)). The displacement field of this focused
mesh is then used to estimate the stress intensity factor. The submodelling technique has also been
used by Schöllmann, et al. [49] in their fatigue crack growth simulation software.
2.4. Current trends
In recent years the research focus seems to have shifted from simple geometries modelled under
LEFM conditions towards three-dimensional cracks in complex geometries or with plasticity induced
crack closure. A review of the literature on finite element analysis of three-dimensional fatigue cracks
is found in [50], while an overview of the finite element analysis of plasticity induced crack closure is
found in [45]. We will here limit our discussion to three of the latest works.
Aguilar Espinosa, et al. [41] have used four-node quadrilateral elements with the node release
technique in Abaqus, in order to estimate the crack opening and closing stress intensity factors for a
crack in a four-point bending specimen subjected to fatigue loading. The contact of the crack flanks
was simulated by using Abaqus' surface contact boundary condition. However, even with a mesh
refined to 4 µm elements at the crack tip, the estimated crack opening and closing loads are
significantly lower than the experimentally determined values.
Solanki, et al. [45] stated that the physics of fatigue crack growth is not taken into account when
the node release technique is used. García-Collado, et al. [42] illustrate some of the differences
between the results obtained by using a physics-based cohesive element technique, and those obtained
by using the node release technique, like differences in the plastic strain field around the crack tip.
They used an element size of 15 µm at the crack tip, with 13 213 nodes in total to model a compact-
tension specimen. In order to reduce the mesh refinement required at the crack tip, Hu, et al. [43]
propose a special singular element to be used at the crack tip when considering plasticity induced
crack closure in two-dimensional geometries. This single element covers the whole crack tip, and its
radius should be equal to the plastic zone length. The element is based on a cohesive zone model, and
is used to simulate variable amplitude propagation of a mode I fatigue crack.
3. Crack propagation by the extended finite element method
As indicated in section 2.3, the methods for evaluating fatigue crack growth by the ordinary finite
element method do normally contain one of two shortcomings: The methods are either mesh-
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First Conference of Computational Methods in Offshore Technology (COTech2017) IOP Publishing
IOP Conf. Series: Materials Science and Engineering 276 (2017) 012027 doi:10.1088/1757-899X/276/1/012027
dependent, or requires re-meshing. In order to avoid both these shortcomings, Belytschko and Black
published a mesh-independent method with minimal re-meshing in 1999 [14]. This method was
further developed by Moës, Dolbow and Belytschko [51] into a mesh-independent method without any
re-meshing. The method has later become known as the eXtended Finite Element Method (XFEM)
[15, 44], and has become widely popular [52] for solving continuum mechanics problems containing
discontinuities like cracks and material interfaces. In addition to crack propagation problems, XFEM
has been employed to solve two-phase flows and fluid-structure interaction problems [15], and has
even been proposed as an applicable tool to predict the deformation of a brain subjected to a surgeon's
cut [53].
As illustrated in Figure 4, XFEM uses a non-conforming mesh to model the crack (or other
discontinuities), i.e. the cracks are modelled independently of the mesh [44]. This is made possible by
"enriching" the elements cut by the crack, by adding special shape functions to take care of the local
discontinuities and singularities around the crack [15]. The mesh-independency makes it possible to
use the same mesh for all stages of a growing fatigue crack. Even though the modelling of the crack is
said to be independent of the mesh, Ren and Guan [54] clearly illustrate that the level of mesh
refinement at the crack influences the accuracy of the representation of a three-dimensional crack. The
mesh-independency does rather imply that the crack growth increment and orientation may be chosen
independently of the mesh.
Figure 4.
Illustrative sketches of (a) a
conforming FEM mesh and (b) a non-
conforming XFEM mesh. ▲ indicates tip
enriched nodes, while ● indicates step
enriched nodes. Element sizes not to scale.
We will here briefly review XFEM for the purpose of modelling fatigue cracks. An extensive
review of XFEM in general is found in [15], and some of its numerous applications are reviewed in
[55].
3.1. Representation of the crack
As the extended finite element method mesh does not normally conform to the crack, it is necessary to
formulate the position and shape of the crack face and the position of the crack tips mathematically.
This is normally (but not necessarily) done using the level set method [15, 44, 56, 57]. An arbitrary
crack may be described by two level set curves. The first level set curve, ϕ(x,t), is the signed-distance
function, which is equal to zero along the whole crack surface, as well as at its tangential extensions at
its ends, as shown in Figure 5(a) [15, 56]:
( )
( )
, min ,
t
t
Γ
Γ
Γ
φΩ
∈
=± − ∀∈
x
x xx x
(9)
where the sign is positive and negative on opposite sides of the crack. xΓ is the closest point to x on the
crack face Γ(t). Ω represents the domain of the solid.
While ϕ(x,t) represents the position and geometry of the crack face, it does not specify where the
crack ends. This is accomplished by a second level set curve, γ(x,t), which is orthogonal to Γ(t) and
equal to zero at the crack front [15], as shown in Figure 5(b). The crack front corresponds to the crack
tip for a two-dimensional crack, or to the perimeter of a three-dimensional crack [57].
(a) (b)
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Figure 5. Level set curves (a) ϕ(x,t) and (b) γ(x,t) for a 2D crack.
Given these two level sets, the geometry and position of the crack is given by [15, 56]:
( ) ( ) ( )
{ }
:,0 ,0t tt
Γφ γ
= =∧≤xx x
(10)
As the crack grows, Γ(t) is updated, based on the crack increment length and direction. One of the
possible ways to update the crack representation is the fast marching method used by Sukumar, et al.
[57]. Alternative methods are mentioned in [15].
The values of the level sets are normally saved at the element nodes, and interpolated within the
finite elements using standard shape functions [15]. It should be noted that this interpolation makes the
accuracy of the crack representation dependable on the element size and the number of nodes per
element, as illustrated by Ren and Guan [54].
3.2. The stress and displacement fields
The fundamental requirement of any finite element method is its ability to represent the stress and
displacement fields of the loaded solid. With a crack being defined by the level set method, the
extended finite element method needs to be able to compute these fields with sufficient accuracy. In
order to take the discontinuity and the singularity around the crack into account, the XFEM introduces
an enrichment to the finite elements which are cut by the crack [15, 44]. More precisely, "a node is
enriched if its support is cut by the crack" [51]. The nodes are enriched by the introduction of an
additional set of degrees of freedom, q. The nodes do still contain their traditional degrees of freedom,
u, meaning that the enriched nodes have a higher number of degrees of freedom than the ordinary
nodes. The displacement field within the elements is then approximated by the following expression:
() () ( ) ( ) ( )
standard FE approx. enrichment
hi i i ii
iI iI
NN
ψψ
∗
∗
∈∈
= + ⋅ −
∑∑
u x xu x x x q
(11)
which is the standard XFEM approximation [15]. The enriched nodes I* are a subset of all the nodes I,
Ni(x) are the standard FEM shape functions and ψ(x) is called the enrichment function.
( )
i
N
∗
x
are
some functions which have the partition of unity property [15]:
( )
1
i
iI
N
∗
∗
∈
=
∑
x
(12)
ϕ = +0.2
ϕ = 0.0
ϕ = +0.1
ϕ = −0.1
ϕ = −0.2
ϕ = −0.3
ϕ = −0.4
ϕ = +0.3
Γ
γ = +0.3
γ = +0.2
γ = +0.1
γ = 0.0
γ = −0.1
γ = −0.2
γ = −0.3
Γ
(a)
(b)
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As the standard FEM shape functions Ni(x) do have the partition of unity property [44],
( )
i
N
∗
x
are
usually chosen to be equal to the Ni(x). The term −ψ(xi) is called shifting, and is used to remove the
effects of qi on the nodes and ensure compatibility across elements, i.e. to ensure that uh(xi) = ui.
The enrichment function is responsible for introducing discontinuities and singularities to the
displacement field. Belytschko and Black [14] proposed the enrichment functions that describe the
singular displacement field around the crack tip under LEFM conditions in 1999:
( )
tip
cos , sin , sin sin , cos sin
222 2
r
θθθ θ
ψ θθ
=
x
(13)
Here, r and θ are the polar coordinates with the origin at the crack tip, and θ = 0 along the parallel
extension of the crack into the material.
The above enrichment is based on linear elastic fracture mechanics (LEFM), and is therefore only
representative for cases where the crack tip plastic zone is considered to be small [44]. For the case of
significant plasticity, an alternative set of crack tip enrichment functions was developed by Elguedj, et
al. [58] in 2006. They still assume that the plasticity is confined to a region near the crack tip, and
apply the Hutchinson-Rice-Rosengren (HRR) solution for a power-law hardening material, proposing
the following crack tip enrichment functions:
( )
11
tip
sin , cos , sin sin , cos sin , sin sin3 , cos sin3
222 2 2 2
n
r
θθθ θ θ θ
ψ θθ θ θ
+
=
x
(14)
n is here the hardening exponent of the material.
Even though the enrichment functions, equations (13) & (14), are discontinuous along θ = ±π [14],
i.e. throughout the crack, they are not readily applicable to describe long, severely curved or three-
dimensional cracks [14, 51]. Hence, Belytschko and Black stated their method to require “minimal re-
meshing.” Moës, et al. [51] removed the need of re-meshing altogether, by introducing an additional
enrichment, to be used for the enriched nodes along the crack, away from the crack tip. This
enrichment models the discontinuous displacement field over the crack by using the sign function [15,
51]:
()
()
()
( )
()
step
1 if , 0,
sign , 0 if , 0,
1 if , 0.
t
tt
t
φ
ψφ φ
φ
− <
= = =
>
x
xx
x
(15)
This enrichment function is equal to −1 and +1 on opposite sides of the crack, which is normally
represented by the level set ϕ(x,t). Moës, et al. [51] used the symbol H(x) for this enrichment function,
and subsequent authors have often referred to it as the Heaviside function, e.g. [53, 55, 57, 59], even
though the Heaviside step function formally has the property H(ϕ(x)) = 0 if ϕ(x) ≤ 0 [15]. However,
the sign and the Heaviside step functions do actually lead to identical results because they span the
same approximation space [15].
By using the crack tip and step enrichments, an appropriate approximation for the stress and
displacement fields may be obtained, both for two-dimensional [51, 60] and three-dimensional [57,
59] cracks. If we divide the additional degrees of freedom, q, into those around the crack tip, b
(indicated by ▲ in Figure 4(b)), and those remaining, a (indicated by ● in Figure 4(b)), we can rewrite
equation (11) [15]:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
, tip
step step tip tip
h j jj
i i i ii i ii
iI j
iI iI
NN N
φγ
ψ ψ ψψ
∗∗
∗∗
∈∈∈
= + ⋅ − + ⋅−
∑ ∑ ∑∑
ux xu x x xa x x xb
(16)
It should be noted that integration of the weak form of the finite element formulation is not straight
forward when the elements contain singularities and discontinuities, because the standard Gauss
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quadrature requires a smooth integrand and a finite order polynomial [44]. Special integration
techniques have therefore been developed for this purpose [15, 44, 51, 58].
3.3. Evaluation of the stress intensity factor and the direction for further crack growth
The stress intensity factor is commonly extracted from the extended finite element solution by
employing the domain form of the interaction integral [14, 51, 54, 57, 58], as explained in section
2.1.4. It is likely that this method is preferred over the displacement method, because the crack tip
displacements are less readily available in the XFEM results than in corresponding FEM results.
Moës, et al. [51] used a proper mesh and demonstrated computation of the stress intensity factors for
two-dimensional mode I+II cracks within 1–2 % deviation from the analytical values. The XFEM
enrichment takes care of the near-crack singularity, and linear elements can therefore be used even for
the LEFM analyses.
The direction for further crack growth is often evaluated using the criterion of maximum tangential
stress [36], e.g. in [14, 51, 54, 60], just like in fatigue analyses using the ordinary FEM. The angle of
propagation is estimated using a function of KI and KII, and the level sets describing the crack are
updated accordingly. The relation between the angle of propagation and the level set updates are
illustrated in [44].
3.4. Fatigue crack propagation and current trends
The first application of the extended finite element method to fatigue crack growth propagation was
illustrated already by Moës, et al. [51] in 1999. They modelled the propagation of mode I+II fatigue
cracks propagating from two holes in a two-dimensional plate with 2650 nodes. Two different values
for the crack increment size are used, but the numerical results are not compared to any experimental
ones.
Sukumar, et al. [57] presented the first XFEM simulation of planar fatigue cracks propagating
through a three-dimensional solid (i.e. pure mode I cracks) in 2003. They use the level set method to
track the position of the crack front, and the fast marching method to advance the crack front. Using
three-dimensional meshes of 24x24x24 eight-node (linear) hexahedral elements, they compute the
stress intensity factors for a planar penny-shaped crack with errors between 0.4 % and 2.9 %, while for
a planar elliptical crack the errors are between 0.6 % and 3.7 %. The fatigue simulations correctly
predicted an initially elliptical crack to develop into a penny-shaped crack, but the crack propagation
life is not compared to any experimental results.
Comparisons of XFEM fatigue simulations to experimental results have recently been published by
Bergara, et al. [59]. In this work, they have used the XFEM-based LEFM approach which is
implemented in Abaqus to simulate the growth of a semi-elliptical crack located at the side of a beam
specimen subjected to four-point bending. The specimen is completely modelled in three dimensions
by 8-node hexahedral elements, with refinements around the crack and the boundary condition sites.
With 173 892 nodes in total, approximately one week was required to run one fatigue crack
propagation simulation using three 3.40 GHz processors. Bergara, et al. report excellent
correspondence between the numerical and experimental crack propagation histories, as well as good
agreement for the evolution of the crack geometry. However, there are large differences between the
experimental and numerical stress intensity factor ranges; up to approximately 40 %. This seems
remarkable given the fine mesh and good correspondence in the crack propagation histories.
Zhan, et al. [3] have proposed an entire framework for the fatigue life prediction of metallic
components, where the crack initiation life is evaluated using FEM with continuum damage
mechanics, while the crack propagation life is evaluated using XFEM. The initiated crack is set to be
0.1 mm long. Computational and experimental results have been compared for a mode I crack in a
fuselage structure, modelled by 8-node hexahedral elements, using 20 106 nodes for half the model.
The predicted crack propagation life was 28 % shorter than the experimental one, indicating a
potential for improving the technique.
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A new method for reducing the required structural mesh density near the crack has recently been
proposed by Ren and Guan [54]. Their method is intended for the crack growth analysis of three-
dimensional and arbitrarily shaped cracks subjected to mixed mode loading. They note that the level
set method is not able to fully represent the crack front surface if the crack has a complicated geometry
or if the mesh is relatively coarse. It is therefore proposed to use an individual mesh for the crack, and
let the crack geometry in the XFEM model of the structure be described by this mesh, instead of the
level set method. It is shown that this method makes the values for the stress intensity factor range
more uniform along the crack front, and they are also less dependent on the structural mesh. However,
the crack mesh is shown to be very fine in the illustrations, and necessarily needs to be updated (re-
meshed) as the crack propagates. It has not been shown whether the computation time saved by using
a coarser overall mesh makes up for the time required to re-mesh the crack for each propagation
increment. Still, it is possible that this technique is more efficient for analysing fatigue crack
propagation in large and complex components.
4. Discussion
The studies reviewed in section 3.4 indicate that even though XFEM is capable of accurately
predicting stress intensity factors for simple “handbook” geometries, further development is required
in order to make accurate predictions of stress intensity factors and fatigue crack propagation lives for
more complex geometries. It should be noted that the accuracy of the predicted crack propagation life
will always be limited by the fracture mechanics assumptions and the crack propagation law which is
used, however. On the other hand, even though the XFEM was developed to reduce the computational
effort required to simulate fatigue crack propagation, none of the studies reviewed do actually
compare computation times between FEM and XFEM. This is remarkable.
In 1986 Højfeldt and Østervig [40] were able to predict the fatigue crack propagation life of shafts
with shoulder fillets, with errors below 20 % in just 20 400 s (CPU-time). These results were obtained
using the ordinary FEM with a three-dimensional mesh and the local re-meshing technique. By
exploiting symmetry, only half the shaft was modelled, using only 1160 nodes. One would expect that
after 30 years of research, either the accuracy of the computational results would increase, or the
computation time would decrease, especially with the introduction of XFEM. The works considered in
this review are inconclusive in this regard, mainly because computation times (with processor
frequencies) are rarely stated. In order to choose between the various computational techniques for
modelling fatigue crack propagation, and their variants, such information is essential.
Yazid, et al. state that “about the only drawback” of the present XFEM “is the need for a variable
number of degrees of freedom per node” [55]. This is mainly a challenge when XFEM is incorporated
into existing FEM codes [15]. A more important drawback, as indicated above, for both FEM and
XFEM, seems to be the large number of nodes and long computation time required. FEM and XFEM
are challenged by methods requiring significantly fewer nodes, like the boundary element method
(BEM) [16] and the SGBEM-FEM by Nikishkov, Park and Atluri [17]. Recent comparisons by Dong
and Atluri [52, 61] indicate that the SGBEM-FEM is both more accurate and more efficient than
XFEM to analyse propagating fatigue cracks under LEFM conditions.
5. Conclusions
Both the ordinary Finite Element Method (FEM) and the eXtended Finite Element Method (XFEM)
are able to provide good predictions for the fatigue crack growth propagation in simple geometries.
While the FEM mesh conforms to the fatigue crack, the crack is only implicitly modelled in XFEM,
independently of the mesh.
In FEM, the domain integral method is often recommended for evaluating the stress intensity
factors, while the displacement method is still often used in practice. The domain integral method is
commonly used also in XFEM. The criterion of maximum tangential stress is chosen for determining
the crack propagation path in the majority of the analyses. Under linear elastic fracture mechanics
(LEFM) conditions, local re-meshing is used to advance the crack in FEM, whereas either the node
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release technique or cohesive elements are used under elastic-plastic fracture mechanics conditions. In
XFEM, it is common to use the level set method to describe the crack geometry.
As the complexity of the problem increases, the accuracy of both methods decreases. The latest
research therefore focuses on applying and improving these techniques to complex problems, like
three-dimensional crack propagation under mixed-mode loading. In the current literature, there is also
an increasing focus on modelling plasticity induced closure and fatigue propagation under variable
amplitude loading using FEM.
For both methods, there is also a focus on improving their efficiency, by introducing new
techniques. Even though the XFEM was developed to reduce the computational effort required for
crack propagation problems, none of the studies reviewed do actually compare computation times
between FEM and XFEM. This is remarkable, and should be considered for later studies.
Acknowledgements
This work is part of a PhD study supported by the Norwegian Ministry of Education and Research.
The financial support is acknowledged.
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