crack propagation criteria in three dimensions using the ... · crack propagation criteria in three...
TRANSCRIPT
Crack propagation criteria in three dimensions using the
XFEM and an explicit–implicit crack description
M. Baydoun1∗, T. P. Fries2
1 Aachen Institute of Computational Engineering Sciences, RWTH Aachen, Germany.
email: [email protected]
2 Computational Analysis of Technical Systems, RWTH Aachen, Germany.
February 28, 2012
Abstract
This paper studies propagation criteria in three–dimensional fracture mechanics
within the eXtended Finite Element framework. The crack in this paper is described
by a hybrid explicit–implicit approach as proposed in [16]. In this approach, the crack
update is realized based on an explicit crack surface mesh which allows an investigation
of different propagation criteria. In contrast, for the computation of the displacements,
stresses and strains by means of the XFEM, an implicit description by level set func-
tions is employed. The maximum circumferential stress criterion, the maximum strain
energy release rate criterion, the minimal strain energy density criterion and the ma-
terial forces criterion are realized. The propagation paths from different criteria are
studied and compared for asymmetric bending, torsion, and combined bending and tor-
sion test cases. It is found that the maximum strain energy release rate and maximum
circumferential stress criterion show the most favorable results.
1 INTRODUCTION
1 Introduction
The simulation of three–dimensional crack propagation is a challenging task. The applica-
tion of the standard Finite Element Method (FEM) to fracture is difficult as it requires a
mesh that conforms the crack surface and features a refinement at the crack front. Hence,
during propagation, a frequent remeshing is required which can, especially in three dimen-
sions, hardly be automized. The eXtended Finite Element Method (XFEM) has gained
a lot of attention in the last decade for its advantages in replicating the discontinuity of
the displacement field across the crack surface and the singularity at the crack front with-
out the need for remeshing. Crack propagation using the XFEM was first introduced by
Belytschko et al. [4] and Moes et al. [32] and encompasses at least three major
parts: the crack description, the XFEM formulation and the criteria for the crack update.
The crack in the XFEM can be described explicitly by a surface discretization or implicitly
by means of level set functions, see [7, 37]. An interesting discussion concerning different
topology effects and descriptions of crack surfaces in three dimensions is found in [7]. In
the early papers concerned with two–dimensional fracture mechanics using the XFEM, the
crack was described explicitly in two dimensions [11, 4]. In three dimensions, Sukumar
et al. [43] used the purely explicit description for a plane crack surface. Level set func-
tions were introduced to describe the crack implicitly in [42, 5, 6], and this has developed
a standard in the XFEM for many years, for details see [33, 20]. One possible combination
of the implicit and explicit descriptions, called the vector level set method, was introduced
by Ventura et al. in [45]. In this description, a level set function was defined based on
a geometric crack description in two dimensions. Another mixed description called the hy-
brid explicit–implicit description was introduced by Fries et al. in [16]. This description
applies to two and three–dimensional cracks. In addition to an explicit crack surface mesh,
three level set functions are introduced to facilitate the XFEM implementation. The crack
propagation is realized with respect to the explicit description, i.e. the crack surface mesh.
2
1 INTRODUCTION
An important advantage of the explicit–implicit approach is that it decouples the crack
description from the crack propagation criteria; this is not the case for the purely implicit
crack description that has often been used before. For a systematic study of different prop-
agation criteria, the hybrid approach appears thus highly beneficial.
During propagation, the direction and length of the increments may be obtained by dif-
ferent propagation criteria, see [8]. In this paper, four popular propagation criteria are
applied in the context of the explicit–implicit description: The maximum circumferential
stress criterion (MCSC) introduced in [14], the maximum strain energy release rate criterion
(MSERRC) in [26], the minimal strain energy density criterion (MSEDC) from [39] and the
material forces criterion (MFC) [15]. They cover the whole span of global, local and mixed
propagation criteria. These criteria were introduced in the frame of the standard FEM,
where the crack tips coincide with nodes, the field variables such as the circumferential
stresses, energy density or others are evaluated in the vicinity of the tips. In the XFEM,
the tips do not coincide with nodes, therefore, it is useful to define a set of evaluation points
around each tip. These criteria are rather tricky to implement when using a purely implicit
description of the crack in the XFEM. But when using an explicit description for propa-
gation, great simplifications arise such as the localization of the crack tips, the choice of
evaluation points and the assignment of field variables. The accuracy of the four criteria is
investigated and compared for three test cases in three–dimensions: the bending beam [1],
the Brokenshire test [9] and a mixed mode test case where mode III plays a significant
role [10].
The paper starts with the XFEM formulation for fracture in Section 2. Then the explicit–
implicit description of cracks in the framework of the XFEM is detailed following [16] in
Section 3. Based on the geometric crack description, three level set functions are intro-
duced. The local coordinates associated with the level set functions and the choice of the
enriched nodes are discussed next. The propagation algorithm is detailed in Section 4. The
3
2 XFEM APPROXIMATION FOR CRACK MODELING
maximum circumferential stress criterion is presented in Section 5. The approach using the
maximum strain energy release rate in Section 6, and the minimum strain energy density
is in Section 7. Finally, the material forces criterion is described in Section 8. Numerical
results comparing the four above mentioned propagation criteria are shown in Section 9.
Section 10 ends with the conclusions and perspective work.
2 XFEM approximation for crack modeling
The eXtended Finite Element Method introduced in [32, 4] provides a rather simple treat-
ment of cracks where elements do not conform the crack geometry. A general overview of
the XFEM is found in [17]. The XFEM approximation of the displacement field in (1) is de-
composed into three parts: the continuous classical finite element part, the enrichment that
accounts for the discontinuity in the displacement field and the enrichment that accounts
for the singularities in the stresses and strains at the crack front
u(x) =∑i∈I
Ni(x)ui +∑i∈Icut
N?i (x)
[H(x)−H(xi)
]ai
+∑
i∈IbranchN?i (x)
[ 4∑m=1
(Bm(x)−Bm(xi))bmi
].
(1)
The standard finite element shape functions Ni and the shape functions localizing the
enrichment N?i can have different orders. I is the set of all nodes in the domain, H(x) is
the step enrichment, ai is the unknown of the enrichment at node i and Icut is the set of
nodes whose support is completely cut by the crack. In addition, Bm(x) are four crack-tip
enrichment functions, bmi are the corresponding unknowns of the enrichment and Ibranch is
the set of nodes in the vicinity of the crack front. The tip enrichment functions Bm(x) are
proposed in Belytschko and Black [4] and span the near tip asymptotic fields. They
4
2 XFEM APPROXIMATION FOR CRACK MODELING
are based on polar coordinates (r, θ) in the vicinity of the crack front,
B =
{√r sin
θ
2,√r cos
θ
2,√r sin
θ
2sin θ,
√r cos
θ
2sin θ
}(2)
and have proven their usefulness also in three dimensions. In order to align the enrichment
functions with the crack front, the plane where θ is zero must be tangent to the crack front.
It is seen that a successful implementation of the XFEM encompasses the following steps:
1. Define the enriched nodal sets Icut and Ibranch.
2. Evaluate the enrichment functions. Therefore, level set functions are employed that
imply the polar coordinate system needed in (2).
3. Perform the numerical integration appropriately by considering the discontinuities
and singularities in the enrichment functions, see [42] for details.
It is important to note that, for these three issues, the level set method [37] can be employed;
it thus complements the XFEM very well. The level set method allows to define the enriched
nodes and the step enrichment function directly, see e.g. [42]. It greatly simplifies the
definition of the polar coordinates (r, θ), hence serving a basis for the description of the
branch enrichment functions, see [33]. In addition to that, level set functions help to decide
if an element is cut by the crack. This is a key point in performing the numerical integration
in the XFEM efficiently. In summary, the level set method provides all the information
needed by the XFEM. On the other hand, in order to propagate cracks in purely implicit
descriptions, the level set functions have to be transported by solving a set of Hamilton–
Jacobi equations to steady state, see e.g. [13]. This is not easily achieved especially when
unstructured meshes are involved [20].
An alternative description of the crack, the explicit description, was employed in e.g. [45, 44]
in the context of the XFEM. For example in three dimensions, the crack surface can be
5
3 THE HYBRID EXPLICIT–IMPLICIT DESCRIPTION OF CRACKS
described by a polyhedron with flat triangles. The crack update algorithm is simple and
is achieved by adding surfaces to the existing crack geometry, see e.g. [16]. However, the
explicit description fails to fulfill the above mentioned requirements of the XFEM.
Therefore, the aim of the hybrid explicit–implicit description is to combine the advantages
of the explicit and implicit description of the crack.
3 The hybrid explicit–implicit description of cracks
Each of the descriptions—explicit and implicit—has advantages and disadvantages in crack
propagation within the XFEM. The main goal of the hybrid explicit–implicit formula-
tion [16] is to maintain all the advantages of each description starting with an explicit
description of the crack. From this explicit crack, level set functions are constructed to
account for the needs of the XFEM.
3.1 Explicit description of the crack
In three dimensions, the crack surface is described by a polyhedron with triangles, see Figure
1. The mesh of the crack Γ is a union of triangles Ti with i ∈ {1...Ntri}, Ntri being the
number of triangles,
Γ =
Ntri⋃i=1
Ti. (3)
The crack surface Γ is a closed, orientable 2–manifold in the three–dimensional Euclidean
space. It is defined by a set of nodes in R3 that are related by a connectivity matrix. The
front of the crack is composed by the free edges, see Figure 1.
The crack surface mesh must fulfill the conditions of a manifold. This means the interior,
exterior and the mesh itself have to be well defined, for details see [16, 2].
6
3 THE HYBRID EXPLICIT–IMPLICIT DESCRIPTION OF CRACKS3.1 Explicit description of the crack
Figure 1: A crack surface in three dimensions is explicitly described by a polyhedron with
flat triangles and straight edges. The free edges mark the front, which is shown by the bold
polyline.
3.1.1 Coordinate systems at the crack surface
Since the crack is a manifold in R3, a coordinate system is associated to its interior and
front. Let ni be the normal vector of each triangle Ti. This normal vector is easily defined
by the vertices of the triangle, see [2]. The front is defined by the vertices at the boundary
which are in between the “tips”. The front coordinate system at the tips is defined by
three vectors: a normal vector nk, a tangent tk and a cotangent qk for k = {1...Ntips},
Ntips being the number of vertices at the front. A vertex can actually be shared by more
than two triangles. The normal vectors at a shared vertex are discontinuous since the crack
surface is C0–continuous. Hence, for a node shared by Nstri triangles, a normal vector
may be associated by averaging all the contributing normals nj based on the area of the
corresponding triangles Aj with j = {1...Nstri}
nk =n∗k‖n∗k‖
with n∗k =
∑Nstrij=1 Aj · nj∑Nstrij=1 Aj
∀ k = {1...Ntips}. (4)
7
3.1 Explicit description of the crack3 THE HYBRID EXPLICIT–IMPLICIT DESCRIPTION OF CRACKS
The next step is the computation of a cotangent vector qk at each tip on the front. The
cotangent vector is associated to the edges that compose the crack front. Therefore, for a
tip where multiple edges Nsedg meet, a similar averaging technique based on the lengths lj
of the edges is employed with j = {1...Nsedg}
qk =q∗k‖q∗k‖
with q∗k =
∑Nsedg
j=1 lj · qj∑Nsedg
j=1 lj∀ k = {1...Ntips}. (5)
Finally, the tangent vector tk for each tip k is the cross product of the normal and cotangent
vectors:
tk = qk × nk ∀ k = {1...Ntips}. (6)
The coordinate system at the front of the crack is shown in Figure 2(a). For more details
about the properties of the coordinate systems in the explicit description, see [16].
3.1.2 Extension of the crack surface
For the hybrid approach, an extension of the crack is needed. Using the coordinate system
at the tips, each tip is extended by an increment lk in the direction of the tangent vector
tk with k = {1...Ntips}. Thus, a new set of segments is formed. These segments are then
connected to form new triangles. Figure 2(b) shows the extension of the crack surface by an
increment lk = 5. The crack extension during propagation is done similarly except for one
difference. The extended segments have individual lengths and do not follow the tangent
vectors but rather involve a propagation angle θk with respect to the two–dimensional plane
formed by (tk,nk), see Section 4.
8
3 THE HYBRID EXPLICIT–IMPLICIT DESCRIPTION OF CRACKS3.2 Implicit description of the crack
(a) The coordinate system at the front:
the normal (red), tangent (blue) and
cotangent (green).
(b) The extension of the crack surface by
an increment lk = 5.
Figure 2: The extension of the crack is in the direction of the tangent vector (blue).
3.2 Implicit description of the crack
It was mentioned earlier that the level set method in the XFEM simplifies the implemen-
tation and addresses the requirements of the XFEM, see Section 2. For the XFEM using
the hybrid explicit–implicit description, three level set functions are constructed. This is
in contrast to the standard purely implicit description, as e.g. in [42], where two orthog-
onal signed distance functions are used. And it is also different from the vector level set
method in [45] where one signed level set function is constructed from the crack geometry.
Here, three “natural” level set functions φ1, φ2 and φ3 are constructed based on the explicit
description: two unsigned and one signed distance functions.
• φ1(x) is an unsigned distance function of the crack surface. The value of the level set
function at any point x in the three–dimensional domain is the shortest distance to
the crack surface, see [16, 3]. The isosurface of the level set φ1(x) = 1 is shown in
Figure 3(a).
9
3.2 Implicit description of the crack3 THE HYBRID EXPLICIT–IMPLICIT DESCRIPTION OF CRACKS
• φ2(x) is an unsigned distance function of the crack front . Thus, φ2(x) is the shortest
distance of any point x to the crack front, see [16, 24]. The isosurface of the level set
φ2(x) = 1 is shown in Figure 3(b).
• φ3(x) is a signed distance function of the extension of the crack surface. Thus, φ3(x)
is the shortest distance of x to the extended crack surface of Section 3.1.2. The sign of
φ3(x) is based on the direction of the normal vector on the crack surface that contains
the closest point, see [16]. The isosurfaces of φ3(x) = −1 and φ3(x) = +1 are shown
in Figure 4.
(a) φ1(x) = 1. (b) φ2(x) = 1.
Figure 3: The isosurfaces of the unsigned distances to the crack surface φ1(x) and to the
crack front φ2(x).
10
3 THE HYBRID EXPLICIT–IMPLICIT DESCRIPTION OF CRACKS3.2 Implicit description of the crack
(a) φ3(x) = −1. (b) φ3(x) = +1.
Figure 4: The isosurfaces of the signed distance to the extended crack surface φ3(x).
Each level set φ1(x), φ2(x) and φ3(x) can be evaluated exactly for any point x in R3 for an
arbitrary crack surface mesh. However, the level-set values are typically evaluated only at
the nodes and interpolated into the domain using standard finite element shape functions:
φhj (x) =∑i∈I
Niφj(x) ∀j = {1, 2, 3}. (7)
Then, the crack surface implied by these discretized level set functions is only an approxi-
mation of the crack surface.
3.2.1 Coordinate systems from the implicit description
From the level set functions, two local coordinate systems are defined for any point x in
the domain: a polar coordinate system (r, θ) and a curvilinear system (a, b). The polar
coordinates (r, θ) are evaluated from the level set functions φ2(x) and φ3(x):
r(x) = φ2(x) θ(x) = sin−1 φ3(x)
φ2(x). (8)
11
3.2 Implicit description of the crack3 THE HYBRID EXPLICIT–IMPLICIT DESCRIPTION OF CRACKS
Since the tangent vector in three dimensions is not perfectly tangential to all neighboring
triangles, some further details for the definition of θ(x) are sometimes needed, see [16]. The
polar coordinates in (8) are indeed the same variables required by the branch enrichment
functions in the XFEM formulation of the displacement field, see Equation (2). For the
step enrichment, the Heaviside function may be evaluated by means of the level set function
φ3(x):
H(x) =
0 if φ3(x) ≤ 0,
1 if φ3(x) > 0.(9)
Another local curvilinear coordinate system (a, b) can be evaluated at any point x in R3:
a(x) = r(x) cos(θ(x)) b(x) = r(x) sin(θ(x)). (10)
This coordinate system can be used to define the approximate, implicitly described crack
surface Γ as:
Γ(x) ={
x∣∣∣ b(x) = 0 and max a(x) ≤ 0
}. (11)
One can thus see that the enriched nodal set Icut is easily extracted from a(x) and b(x), see
the nodes of the blue elements in Figure 5. The enriched nodal set Ibranch can be defined
by means of r(x) when specifying the branch enrichment radius Rbranch:
Ibranch ={i ∈ I : r(xi) ≤ Rbranch
}. (12)
The nodal set Ibranch for Rbranch = 0.125 is described by the nodes of the red elements in
Figure 5.
12
4 CRACK PROPAGATION
Figure 5: The element nodes of the blue elements are step enriched. The branch enrichment
is realized at the element nodes of the red elements.
Furthermore, from a(x) and b(x) one may identify all elements of the mesh that are cut
by the crack surface. Let N el be the total number of elements in the mesh and Ielk be the
set of element nodes of element k, then
N cut ={k ∈ {1..N el} : max
i∈Ielka(xi) < 0 and min
i∈Ielkb(xi) ·max
i∈Ielkb(xi) < 0
}. (13)
All nodes of the elements in N cut are step–enriched unless they are already marked for the
branch enrichment. It is thus seen that the polar and curvilinear coordinate systems play
a decisive role in addressing the XFEM requirements from Section 2.
4 Crack propagation
The main disadvantage of the purely implicit description is the crack update. On the other
hand, this is easily realized based on the explicit description, see Section 3.1.2. In order for
the crack to propagate, an update technique similar to the extension algorithm is used with
one difference: the extended segments have individual lengths and do not follow the tangent
13
4 CRACK PROPAGATION
vectors at the tips. Each node is rather extended by a propagation angle θk with respect
to the two–dimensional plane formed by (tk,nk) for k = {1...Ntips}, with Ntips being the
number of nodes at the front, see Figure 6.
(a) The propagation angle θ spans the
two–dimensional plane built by (t,n) at
each node on the front (“tip”).
(b) The new crack surface after propa-
gation is shown in blue.
Figure 6: The crack propagation is realized based on increments at the tips with individual
angles θk and lengths lk .
The geometric update algorithm of the explicit crack description may be summarized
into three main points:
• Starting from an existing crack surface, a basis (tk,nk,qk) is assigned at every tip k
on the front.
• A propagation angle θincrk and a length lk are evaluated at each tip from the propaga-
tion criterion, see Sections 5 – 8.
• The new increments are interconnected to form a new surface mesh. The new and old
surface meshes are combined creating the new crack surface.
14
4 CRACK PROPAGATION 4.1 General comments on propagation criteria
The remaining question is how the propagation angles θincrk and lengths lk are defined, i.e.
“In which direction is the crack propagated and by how much?”. This is answered in the
following sections for different approaches.
4.1 General comments on propagation criteria
In order to answer the above question, one needs to differentiate between two fields: the
global fields evaluated in (x, y, z)–coordinates and local fields evaluated in (r, θ)–coordinates
and (a, b)–coordinates. One can thus differentiate between three principal approaches,
see [8]:
• Local approaches that are based on the local fields at the crack tip such as the maxi-
mum circumferential stress criterion (MCSC) introduced by Erdogan and Sih [14]
and the maximum strain criterion (MSC) introduced by Maiti et al. [29].
• Global approaches that are based on the energy distribution along the crack such as
the maximal strain energy release rate criterion (MSERRC) introduced by Hussain
et al. [26].
• Mixed approaches that are based on the energy distribution along the crack by means
of local fields such as the material forces criterion (MFC) introduced by Gurtin [22]
and the minimal strain energy density criterion (MSEDC) introduced by Sih et
al. [39].
The stress intensity factors are often used for propagation in the XFEM. They are based
on measuring the strength of the singularity at the crack tip and often evaluated using the
interaction integral. The interaction integral is frequently used in two dimensions, yet, it
poses difficulties and uncertainties in three dimensions, see [35, 46]. Hence, the interaction
integral is not employed in this work.
The direction of the crack is directly related to the accuracy of the propagation criterion
15
5 MAXIMUM CIRCUMFERENTIAL STRESS CRITERION – MCSC
used. In the following, the maximum circumferential stress criterion (MCSC), the max-
imum strain energy release rate criterion (MSERRC), the minimal strain energy density
criterion (MSEDC) and the material forces criterion (MFC) are recalled and the influence
of the related fields is emphasized. The efficiency of these criteria was tested using the
standard FEM, e.g. in [8]. In the FEM, the variables of interest for each criterion, i.e.
the circumferential stress for MCSC, energy density for MSEDC, etc. are evaluated in the
vicinity of the front which coincides with nodes. In the XFEM, the tips do not coincide
with nodes but rather lie within elements. Therefore, a set of evaluation points has to be
chosen around selected points on the front which are the “tips”. These criteria are rather
difficult to implement when using a purely implicit description of the crack in the XFEM.
On the other hand, when using an explicit description, the selection of the crack tips, the
placement of evaluation points and the assignment of field variables are greatly simplified.
5 Maximum circumferential stress criterion – MCSC
The maximum circumferential stress criterion was introduced by Erdogan and Sih [14]
for elastic materials. It states that the crack propagates in the direction where the circum-
ferential stress σθθ is maximum. It is a local approach since it is based on local stress fields
around the crack front. The circumferential stress field at the evaluation points around the
tips of the crack front can be formulated in terms of the stress intensity factors:
σθθ =1√2πr
[KI
4
(3 cos
θ
3+ cos
3θ
2
)+KII
4
(−3 sin
θ
2− 3 sin
3θ
2
)]. (14)
The direction of the maximum circumferential stress θincr is expressed by:
∂σθθ∂θ
∣∣∣θ=θincr
= 0, (15)
16
5 MAXIMUM CIRCUMFERENTIAL STRESS CRITERION – MCSC5.1 Point identification
and substituting (14) into (15) leads to:
KI sin(θincr) +KII(3 cos(θincr)− 1) = 0. (16)
Hence, this criterion can well predict the fracture direction in two mixed modes. However,
it is obvious from (14) and (16) that KIII has no effect on σθθ, so it is suggested that the
MCSC is less suitable for situations where mode III is dominant.
5.1 Point identification
Since the MCSC is a mode I/II criterion, the crack is only expected to propagate in the
plane spanned by the normal and tangent vectors at a tip. Therefore, the circumferential
stress is evaluated at points around the crack tips within the plane spanned by (t,n), see
Section 4. First, the points around each tip have to be identified, see Figure 7(a). Each
point within this circle has a unique direction θ with respect to the tangential vector t.
The dissipation in the material is expected to be positive [22] meaning that the crack only
propagates forward, which limits θ to be within −90o and +90o. Furthermore, as suggested
by experiments and theoretical insights, the maximum propagation angle for a pure mode
II is ±70.54o. Therefore, points are placed on the arc between −70.54o to +70.54o from t
in the plane formed by (t,n), see Figure 7(b).
17
5.2 From global to local fields5 MAXIMUM CIRCUMFERENTIAL STRESS CRITERION – MCSC
(a) A set of points are placed on the
circle in the plane (t, n) around each
tip.
(b) The circle is replaced by an arc
within −70.54o and +70.54o from the
tangent vector t.
Figure 7: The point identification algorithm: a set of evaluation points is placed on an arc
around the crack tip.
Evaluation points are placed on this arc for each tip k on the front. The radius r of the
arc is user–defined and investigated in the numerical results. The next step is to define the
circumferential stresses at each point. Each evaluation point on the arc is inside an element
of the mesh. The circumferential stress at the evaluation point is computed based on the
nodal values of the stress fields of the corresponding element nodes. The stress values from
the nodes are projected to the given evaluation point by interpolation.
It is important to note that in C0–continuous approximations, the stress fields are discon-
tinuous over the element boundaries. Therefore, it is recommended to smooth stress fields
at the nodes prior to the interpolation, see [16].
5.2 From global to local fields
The nodal stress values in the domain are given by the XFEM solution. For each evaluation
point p = {1...Nkcp} inside an element—with Nk
cp being the number of points on the arc
around a tip k—, the global smoothed stresses from the nodes are interpolated at the
18
5 MAXIMUM CIRCUMFERENTIAL STRESS CRITERION – MCSC5.3 Parameters and limitations
evaluation point p. In order to project these global stresses to the local system (rp, θp), two
transformations are required. A first transformation from global to local fields is achieved
by the derivatives of (tk,nk,qk):
σtnq = TglσxyzT Tgl =
σtt σtn σtq
σtn σnn σnq
σtq σnq σqq
with Tgl =
∂tk∂x
∂tkdy
∂tk∂z
∂nk∂x
∂nkdy
∂nk∂z
∂qk∂x
∂qkdy
∂qk∂z
, (17)
where σtnq are the local stresses in the (t, n, q)–coordinate system at the point p. The
second transformation Tlp from local to cylindrical coordinates is achieved by a rotation of
the local stresses σtnq:
σrθq = TlpσtnqTlpT =
σrr σrθ σrq
σrθ σθθ σθq
σrq σθq σqq
with Tlp =
cos(θp) sin(θp) 0
− sin(θp) cos(θp) 0
0 0 1
, (18)
where σrθq are the cylindrical local stresses at each point p.
5.3 Parameters and limitations
The circumferential stresses σθθ for all points p = {1...Nkcp} on the arc are evaluated.
The current tip k propagates in the direction θincrk towards the evaluation point with the
maximum circumferential stress. This maximum stress value σk is stored at every tip k. In
addition to the angle, an increment lk is required at every tip. This is achieved by scaling a
user defined increment da by the maximum stress value σk at every tip over the maximum
stress of all the tips:
lk = da · σkmaxi∈1...Ntips{σi}
with σk = max1...Nk
cp
σθθ. (19)
19
6 MAXIMAL STRAIN ENERGY RELEASE RATE CRITERION – MSERRC
With θincrk and lk known for every tip on the crack front, the crack is propagated following
Section 4.
6 Maximal strain energy release rate criterion – MSERRC
The strain energy release rate criterion was introduced by Hussain [26]. The strain energy
rate is understood to be the energy dW required for a crack to increase by an increment
da. The criterion states that among all admissible directions of the crack, the real crack
direction is the one which maximizes the strain energy release rate
G = −dWda
. (20)
The MSERRC has a solid theoretical basis from the energy balance theory [26]. Many
techniques are used to compute G in two dimensions. The most popular ones are the J–
integral introduced by Rice [36], the surface integral introduced by De Lorenzi [28] and
the interaction integral method introduced by Destuynder [12], for details see [8].
The interaction integral is frequently used in two–dimensional fracture application with the
XFEM. Yet, the extension to three dimensions poses difficulties and uncertainties, see [47]
and [46]. The strain energy release rate can be rewritten in terms of the stress intensity
factors.
G =1
E′
(K2
I +K2II
)+
1
2µK2
III, E′ =
E Planestress,
E/(1− v2) Planestrain.(21)
with E being the Young’s modulus and ν the Poisson’s ratio. It can now be seen from (21)
that G does not depend on θ.
20
6 MAXIMAL STRAIN ENERGY RELEASE RATE CRITERION – MSERRC6.1 Approximation of the energy release rate
6.1 Approximation of the energy release rate
It is useful to employ an approximate expression of G introduced in [10] which is again
based on θ. Starting from the effective stress intensity factors:
KI,eff =
[KI
4
(3 cos
θ
3+ cos
3θ
2
)+KII
4
(−3 sin
θ
2− 3 sin
3θ
2
)], (22a)
KII,eff =
[KI
4
(sin
θ
3+ sin
3θ
2
)+KII
4
(cos
θ
2+ 3 cos
3θ
2
)], (22b)
KIII,eff = KIII cosθ
2, (22c)
the stresses in the polar coordinate system can be reformulated such as
σθθ =KI,eff√
2πr, σrθ =
KII,eff√2πr
, σθq =KIII,eff√
2πr. (23)
The energy release rate is approximated by KI,eff , KII,eff and KIII,eff assuming the same
amount of energy is needed under KI, KII and KIII from (21). Thus,
G(θ) ≈ κ+ 1
8µ
[K2
I,eff +K2II,eff
]+
1
2µK2
III,eff . (24)
Substituting (23) in (24), gives the approximation of G(θ) in terms of the stresses in the
cylindrical coordinate system σrθq:
G(θ) =πr
4µ
[(κ+ 1)
(σ2θθ + σ2
rθ
)+ 4σ2
θq
]. (25)
6.2 Parameters and limitations
The value of the energy release rate value G is then defined following (25) at each evaluation
point p = {1...Nkcp} of the arc around every tip at the front. The points are placed following
the algorithm in Section 5.1. A tip k propagates in the direction of the evaluation point with
21
7 MINIMAL STRAIN ENERGY DENSITY CRITERION – MSEDC
the maximum strain energy release rate θincrk , where the propagation angle is the largest
absolute angle of all the maxima of Gk at each tip. The corresponding value of Gk is also
stored at each tip. Then, the increment at every tip is evaluated from
lk = da · Gk
maxi∈1...Ntips
{Gik
} . (26)
In [20], the energy release rate is evaluated over a portion of a cylinder or box around each
tip at the front. In this paper, only points around an arc are placed for two test cases where
mode I and II are dominant. And for a mixed mode I/II/III test case, the above point
identification approach is realized also in a third direction q. The points are then placed
over a portion of a cylinder with width w, radius r and the angle again varies between
−70.54◦ and +70.54◦.
7 Minimal strain energy density criterion – MSEDC
The minimum energy density criterion was first introduced by Sih [38]. It states that the
crack will propagate in the direction of the minimum energy density. This criterion is a
mixed global–local criterion. It is based on the energy distribution around the crack using
local fields (r, θ, q). The strain energy density S is inversely proportional to the distance r
from the crack tip:
S = r · (1
2σijεij) ∀ i, j = r, θ, q; (27)
where the stresses are evaluated following Section 5.2 and the strains εij are defined analo-
gously. The strain energy density is evaluated at points around an arc of radius r.
22
8 MATERIAL FORCES CRITERION – MFC 7.1 Parameters and limitations
7.1 Parameters and limitations
It is noted that the point where the strain energy release rate has a minimum is not unique,
see e.g. [39]. Usually, the smaller absolute angle is considered as the propagation angle θincrk .
The corresponding value of Sk is stored at each tip. The propagation increment at every
tip are defined as follows:
lk = da · Sk
mini∈1...Ntips
{Sik
} . (28)
It is noted that the MSEDC shows stability problems under pure mode II and pure mode
III, for details see e.g. [10, 39].
8 Material forces criterion – MFC
The material forces criterion is introduced by Eshelby in [15]. Material forces are identified
as the driving forces for brittle crack propagation in elastic solids. A duality between
physical and material forces was first discussed by Steinmann [40]. Steinmann’s reasoning
was that any lack of equilibrium in a system denotes an imbalance of the physical forces.
In applications, the most popular material forces application is the J–integral of a crack tip
in fracture mechanics, see [30]. The material force is evaluated by integrating the Eshelby’s
tensor or energy momentum tensor over the solid domain. Following the works of Miehe
and Guerses [23] and [31], a duality exists between the actual stress field σ driving the
displacement field u and a stress like tensor Σ driving the crack increment da. This stress
like tensor is actually the Eshelby tensor and it drives the increment of the crack
Σ = W (∇u) · I− (∇Tu) σ. (29)
It is noted for completeness that the analogy of the material forces and the J–integral is
23
8.1 From FEM to XFEM 8 MATERIAL FORCES CRITERION – MFC
thoroughly discussed in [25, 21]. In two dimensions, the J–integral is computed by an in-
tegral over a contour that encloses the crack. In three dimensions, the contour integral is
replaced by a surface integral.
8.1 From FEM to XFEM
In the finite element approximation, the nodal material force F Ie for an element e is evaluated
by numerical integration over the element volume V e, see [34]
F Ie = −∫V e
(ΣijNI,j) dV
e. (30)
This is in analogy to Steinmann’s duality in [41] between physical forces at the nodes
f Ie = −∫V e
(σijNI,j) dV
e and nodal material forces (30). The resulting total material force
at a node I is defined as the sum of all nodal forces from surrounding elements ne
F I =
ne∑e=1
F Ie . (31)
In the FEM, the largest nodal material force is obtained at the crack tip node. The in-
corporation of the material forces in the XFEM framework was first discussed in [27] and
is fronted by the fact that the crack tip does not coincide with a node. Therefore, this
criterion has to be extended to the vicinity of the crack tip.
8.2 From global to local fields
The nodal global material forces in (x, y, z) are evaluated following equation (31). For this
criterion, a full circle is used around each tip. At each evaluation point p = {1...Nkcp} around
24
8 MATERIAL FORCES CRITERION – MFC 8.3 Parameters and limitations
a tip k, theses global material forces are first interpolated to p. Then they are projected to
the local system (tk,nk,qk) using the same transformation Tgl from Section 5.2:
Ftnq = TglFxyz =
Ft
Fn
Fq
(32)
The resultant material force at each tip F k is equal to the sum of all forces surrounding the
tip.
F k =
Nkcp∑
p=1
F ptnq (33)
8.3 Parameters and limitations
The tip propagates in the direction θincrk as follows
θincrk = tan−1 F
kn
F kt(34)
Furthermore, the length of the increment follows:
lk = da · F k
maxi∈1...Ntips
{F i} . (35)
The MFC proposed here depends on the radius r of the arc surrounding each tip. It
also can be evaluated in (x, y, z), where instead of having an increment of length lk and a
propagation angle θincr, the increment will be directly a three–dimensional vector extending
each individual tip in the global coordinate system. However, in order to be consistent with
the other criteria, we formulate this criterion also rather in terms of θincrk and lk.
25
9 NUMERICAL RESULTS
9 Numerical results
In this section, test cases are studied using mixed modes in the following pairings: mode
I/II, mode II/III and a combination of all three crack modes. The first two test cases are
tested using the above mentioned propagation criteria. The last test case is studied using
the MCSC and MSERRC only. A discussion will conclude every test case.
9.1 Asymmetric bending test
This test case is three–point bending of a beam. It studies a combination of modes I and
II. The experimental setup in [19] is shown in Figure 8(a). Numerical results in three
dimensions in the context of the XFEM are in [1] by means of cohesive cracks. The crack
behavior is tested under two different boundary conditions: when the spring constant k on
the upper left side of the beam is k = 0 and when k = ∞. The initial crack surface is
described by 2 × 5 × 5 triangles, see Figure 8(b). The front of the crack is the bold line
and is composed by 6 frontal nodes or “tips”. The material is assumed to be isotropic and
linear elastic with the following parameters: Young’s modulus is E = 105MPa and Poisson’s
ratio is ν = 0.3. The increment in this test case for all propagation criteria is da = 5. The
number of all elements of the mesh is 5256.
75
150
37.5
75
337.5
337.5 37.5
187.5
(a) Dimensions, supports and loading. (b) The crack surface mesh and front.
Figure 8: Asymmetric bending test case with one edge crack in a beam under three point
bending with a spring on the left upper side.
26
9 NUMERICAL RESULTS 9.1 Asymmetric bending test
The computed crack trajectories for k = ∞ and k = 0 from [19] are shown in Figure
9. The bold line is the typical trajectory of the crack. And the shaded area is an envelope
that contains all trajectories obtained in experiments [19]. In addition, selected tips at the
front do not necessarily have the same propagation angle and length during propagation.
(a) k =∞. (b) k = 0.
Figure 9: Computed crack trajectory and experimental envelope for the asymmetric bending
test case.
Results from all four criteria are shown for three ranges of radii of the arcs where
evaluation points are placed around the tip, see Section 5.1: small r = 0.05−−1.0, medium
r = 1.0−−7.0 and large r = 7.0−−15.0. It is noted that some criteria show a significant
dependence on the radius but the resulting crack surfaces typically change smoothly upon
variations of the radius.
9.1.1 MCSC
Results for the bending test with the MCSC for k = ∞ are shown in Figure 10. The 3D
view shows the final triangular crack surface.
27
9.1 Asymmetric bending test 9 NUMERICAL RESULTS
(a) Small radii (b) Medium radii (c) Large radii
Figure 10: Crack paths using the MCSC for the asymmetric bending test case for k =∞.
It is seen from Figures 10(a) and 10(b) that for small and medium ranges of radii, the
crack trajectories fall within the experimental envelope of Figure 9(a) and are perfectly
two–dimensional in nature as expected for this test case. For the larger range, the side view
clearly shows diverging angles of the tips at the front. On the other hand, the overall path
still fits the admissible trajectory as shown in Figure 9(a). The same study is repeated for
k = 0 and the results are shown in Figure 11.
(a) Small and medium radii (b) Large radii
Figure 11: Crack paths using the MCSC for the asymmetric bending test case for k = 0.
Similar conclusions can be extracted, where the small and medium radii show very
promising results where all the front tips have very similar propagation angles and lengths.
The large range of radii shows varying angles and increments over the depth of the specimen;
the crack is less smooth but still fits the experimental trajectory from Figure 9(b).
28
9 NUMERICAL RESULTS 9.1 Asymmetric bending test
9.1.2 MSERRC
Next, the asymmetric bending test is studied using the maximum strain energy release rate
criterion. Results for different radii and k =∞ are shown in Figure 12 and k = 0 in Figure
13.
(a) Small radii (b) Medium and large radii
Figure 12: Crack paths using the MSERRC for the asymmetric bending test case for k =∞.
For both boundary conditions, it can be seen that the crack surface is rather similar
for all ranges with a slight difference: for small radii, a smooth turning is seen at the
first steps of propagation. The crack first propagates in vertical direction before smoothly
turning towards the correct direction. The crack surfaces for small radii are not as smooth
as expected, see Figures 12(a) and 13(a). No problems are seen for the medium and large
ranges of radii as seen in Figure 12(b) and 13(b). It is noted that in this case, the initial
angles of the increments are oriented immediately towards the final direction, which is in
contrast to small radii.
(a) Small radii (b) Medium and large radii
Figure 13: Crack paths using the MSERRC for the asymmetric bending test case for k = 0.
29
9.1 Asymmetric bending test 9 NUMERICAL RESULTS
9.1.3 MSEDC
The efficiency of the minimal strain energy density criterion is tested next. Results for the
same radii and k =∞ are shown in Figure 14 and results for k = 0 are shown in Figure 15.
(a) Small and medium radii (b) Large radii
Figure 14: Crack paths using the MSEDC for the asymmetric bending test case for k =∞.
For small and medium radii, the crack path is similar to the experimental results but
with small variations in the length of the increments and the crack direction is not always
straight, see the side views in Figure 14(a) and 15(a). On the other hand, for large radii,
the crack tips show diverging angles and increments. From the side view in Figures 14(b)
and 15(b), it can be seen that the crack propagates straight in the first steps, then the tips
diverge from each other and the inner tips fall behind and follow a wrong direction.
(a) Small and medium radii (b) Large radii
Figure 15: Crack paths using the MSEDC for the asymmetric bending test case for k = 0.
For k = 0, it can be seen that optimal results are found for small and medium radii
shown in Figure 15(a). The same conclusion for k = ∞ for large radii can be stated for
k = 0, see Figure 15(b).
30
9 NUMERICAL RESULTS 9.1 Asymmetric bending test
9.1.4 MFC
For the material forces criterion, results for k =∞ are shown in Figure 16 and for k = 0 in
Figure 17.
(a) Small radii (b) Medium radii (c) Large radii
Figure 16: Crack paths using the MFC for the asymmetric bending test case for k =∞.
For small radii, it can be seen that the crack path has a larger propagation angle than
the experimental results in Figure 9 and shows some oscillations. On the other hand, for
larger radii, the crack propagates within the expected range but shows some divergence
after a few propagation steps for k = ∞ and k = 0. The optimal results are found for
medium radii as seen in Figures 16(b) and 17(a).
(a) Small and medium radii (b) Large radii
Figure 17: Crack paths using the MFC for the asymmetric bending test case for k = 0.
9.1.5 Discussion
For the asymmetric bending test case, all criteria had acceptable results at least for some
ranges of radii since the average crack paths fit the curves of the experimental envelope
from [19] as shown in Figure 9. For large radii, it is noticed that the computations end
31
9.2 Torsion test by Brokenshire 9 NUMERICAL RESULTS
earlier than for small and medium ranges. This is caused by the fact that the points around
a crack tip exit the domain, hence the interpolation of stresses and strains is no longer
possible. From the results, it can be easily noticed that the MCSC and the MSERRC
have a superiority over the MSEDC and the MFC for this test case. They both showed
a smoother crack path for small and medium radii. The MSERCC even shows smoother
trajectories for large radii.
9.2 Torsion test by Brokenshire
The following test case is a beam with a skew crack subjected to torsion. The experimental
setup and results are found in [9]. Numerical studies in the frame of the XFEM are in [18].
The experiment is shown in Figure 18(a) with P = 1kN . The initial crack surface is
described by 2 × 15 × 15 triangles and is shown in Figure 18(b). The material is linearly
elastic with Young’s modulus E = 10GPa and Poisson’s ratio ν = 0.3. The two prismatic
supports at the ends of the beam are considered to be rigid bodies. The four criteria are
tested for 4480 elements, an increment da = 5 and under three ranges of the radii for the
point identification: small r = 0.01−−1.0, medium r = 1.0−−5.0 and large r = 5.0−−10.0.
P
50
50
75
75
100
200
200
25
25
45°
45°
(a) Dimensions, supports and loading. (b) The crack mesh and front.
Figure 18: The Brokenshire torsion test case with a skew crack.
32
9 NUMERICAL RESULTS 9.2 Torsion test by Brokenshire
9.2.1 MCSC
Results for the MCSC showing different crack paths for small and medium radii are shown
in Figure 19. A similar behavior to the bending case is seen, where small and medium radii
show smooth and accurate results, see Figure 19(a). The three views of the crack trajectory
show some sort of linearity and symmetry between the propagation angles of tips above
and below the mid of the front. On the other hand, for a larger radius, the crack in the
interior tends to steadily go backwards which defies the principal of maximum dissipation
which states that the crack should always propagate forward, see Figure 19(b).
(a) Small and medium radii
(b) Large radii
Figure 19: Results of the torsion test case using the MCSC with different radii ranges.
9.2.2 MSERRC
The results by the MSERRC in Figure 20 show more stability for varying radii. The crack
propagates forward for all radii. On the other hand, less smoothness and symmetry of the
crack are seen in the interior of the front where the crack front propagation angles do not
vary linearly like the angles from the MCSC as seen from the top view in Figure 20(a).
33
9.2 Torsion test by Brokenshire 9 NUMERICAL RESULTS
(a) Small and medium radii
(b) Large radii
Figure 20: Results of the torsion test case using the MSERRC with different radii ranges.
9.2.3 MSEDC
Crack paths for varying radii using the MSEDC are shown in Figure 21. The same conclu-
sions as for the bending test case concerning radii variations may be drawn. Less symmetry
is achieved with respect to the mid front as seen in Figure 21(a) for small and medium
radii. For large radii, the results are getting less useful where the strain energy density far
from the boundary has a much smaller magnitude when compared to the front near the
boundary.
34
9 NUMERICAL RESULTS 9.2 Torsion test by Brokenshire
(a) Small and medium radii
(b) Large radii
Figure 21: Results of the torsion test case using the MSEDC with different radii ranges.
9.2.4 MFC
The MFC shows results close to MSEDC with less smooth and symmetric crack surfaces
for medium and large radii as shown in Figure 22(b). For small radii, unphysical crack
surfaces are seen where the linearity is lost and oscillations are obtained in Figure 22(a).
In addition, the interior of the front propagates backward and small force magnitudes are
obtained due to the fact that further away from the crack, the magnitude of the material
force is small.
35
9.3 Specimen under modes I, II and III 9 NUMERICAL RESULTS
(a) Small radii
(b) Medium and large radii
Figure 22: Results of the torsion test case using the MFC with different radii ranges.
9.2.5 Discussion
From the crack paths shown above, it is concluded that the MSERRC and MCSC show
the best results. The MCSC shows a better symmetry and linearity of the increments at
each tip on the front. On the other hand, unphysical behavior where the crack propagates
backward is seen for large radii.
9.3 Specimen under modes I, II and III
The following test case is a specimen under mixed modes I, II and III. It follows the
experimental setup in [10] of an aluminum specimen with Young’s modulus E = 75GPa
and Poisson’s ratio ν = 0.35. The variation of the angle α dictates the ratio between the
modes. Angles α = 15◦, 30◦, 45◦, 60◦ without torsion or angles α = 30◦, 45◦, 60◦ with torsion
result in mixed mode I, II and III at the crack, for details see [10]. The initial crack surface
is described by 2 × 4 × 12 triangles, see Figure 23(b). In this case, an angle α = 15◦, zero
torsion and P = 25kN for a mesh with 5660 elements, and an increment da = 0.5 are used.
36
9 NUMERICAL RESULTS 9.3 Specimen under modes I, II and III
yzx
P
P
(a) Dimensions, and loading. (b) The crack mesh and front.
Figure 23: A specimen under a tensile force P = 25KN and an inclination angle α leading
to mixed modes I/II/III at the front.
This test case shows an accentuated mode III compared to the other modes. The MCSC
for varying radii showed unphysical results where the maximum circumferential stress was 0
or negative, i.e. the crack was considered fully under pressure and could not propagate. The
MSERRC showed inconsistent results when evaluation points were placed around an arc.
An extension in the q–direction of the point identification method where points are placed
along a portion of a cylinder showed no change for the MCSC. This behavior is expected
since the MCSC is not able to predict mode III accentuation. Using points around a portion
of a cylinder was then tested for the MSERRC and results for medium radii r ≈ 0.5 and a
width w = 0.5 of the cylinder are shown in Figure 24.
Figure 24: Results using the MSERRC undergoing mixed modes I, II and III.
37
11 ACKNOWLEDGMENTS
This test case made it possible to deduce that the MSERRC is capable in predicting the
crack trajectory in mixed modes I, II and III loading. The results match the experimental
findings in [10].
10 Conclusions
In this paper, different propagation criteria are studied in the context of the XFEM applied
to three–dimensional crack simulations. The hybrid explicit–implicit crack description of
Fries and Baydoun [16] is employed. In this approach, the crack update is realized based
on explicit crack surface meshes which allows for a systematic investigation of different
propagation criteria. This is in contrast to purely implicit crack descriptions where the
crack update is realized based on transport models for the level set functions. The maximum
circumferential stress criterion (MCSC), the maximum strain energy release rate criterion
(MSERRC), the minimal strain energy density criterion (MSEDC) and the material forces
criterion (MFC) are implemented and compared. The four criteria showed acceptable results
for the asymmetric bending. For torsion, the MCSC and the MSERRC perform best and the
MSEDC and the MFC showed considerable problems. The MCSC and the MSERRC were
then tested for mixed mode I/II/III but only the MSERCC was able to provide meaningful
results for this complex situation. Further investigations are still required in particular with
respect of the location of the evaluation points (spheres and cylinders in contrast to circles)
and more systematic investigations of situations where mode III is dominant.
11 Acknowledgments
The authors gratefully acknowledge the computing resources provided by the AICES gradu-
ate school in RWTH Aachen University. The authors also wish to acknowledge the support
of the German Science Foundation in the frame of the Emmy–Noether–research group “Nu-
38
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