Crack propagation criteria in three dimensions using the ... · Crack propagation criteria in three...

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)


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


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)


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


9.1.2 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.


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.


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


9.1.4 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).


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


(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


(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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