INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Int. J. Numer. Meth. Engng 2003; 58:103–126 (DOI: 10.1002/nme.761)
Extended nite element method for quasi-brittle fracture
Stefano Mariani and Umberto Perego∗;†
Dipartimento di Ingegneria Strutturale; Politecnico di Milano; Piazza L. da Vinci 32; Milan 20133; Italy
SUMMARY
A methodology for the simulation of quasi-static cohesive crack propagation in quasi-brittle materials
is presented. In the framework of the recently proposed extended nite element method, the partition
of unity property of nodal shape functions has been exploited to introduce a higher-order displacement
discontinuity in a standard nite element model. In this way, a cubic displacement discontinuity, able
to reproduce the typical cusp-like shape of the process zone at the tip of a cohesive crack, is allowed
to propagate without any need to modify the background nite element mesh. The eectiveness of the
proposed method has been assessed by simulating mode-I and mixed-mode experimental tests. Copyright
? 2003 John Wiley & Sons, Ltd.
KEY WORDS:
quasi-brittle fracture; extended nite element method
1. INTRODUCTION
Fracture phenomena in quasi-brittle materials are characterized by a transition from the
initial undamaged state to a state of diused damage with developing micro-cracks, up to
a nal state of complete localized damage with propagating macroscopic cracks. Nowadays,
nite element (FE) simulations can be successfully carried out in the early stage of damage
evolution by means of regularized damage models [1–5], where the regularization is necessary to prevent the mesh dependence occurring as a consequence of the loss of ellipticity of
the boundary value problem in the softening regime. However, as damage grows, the dissipative mechanisms tend to localize along always narrower bands, while the material in the
rest of the body relaxes and tends to behave elastically. The bandwidth of the process zone
(PZ), i.e. the region where the dissipative phenomena localize, needs to be resolved by an
adequate number of nite elements to obtain meaningful results. In this stage, continuum
models would require a progressive mesh renement, unconvenient from the computational
∗ Correspondence
to: Umberto Perego, Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Piazza L.
da Vinci 32, Milan 20133, Italy.
[email protected]
† E-mail:
Contract=grant sponsor: MIUR-Conancing Program 2000; contract=grant number: MM08161945-001
Contract=grant sponsor: LSC-Politecnico di Milano
Copyright ? 2003 John Wiley & Sons, Ltd.
Received 22 March 2002
Revised 10 December 2002
Accepted 23 December 2002
104
S. MARIANI AND U. PEREGO
viewpoint and dicult to justify on physical grounds since, in reality, material separation
is occurring. Analysis procedures envisaging a transition from continuum models to models incorporating a discrete displacement discontinuity appear to be a viable and promising
alternative [6–8]. Other simulation methods, alternative to the nite element approach, like
e.g. boundary elements and meshless methods, are worth mentioning but are not considered
here [9–11].
The most direct way to allow for displacement jumps, seems to be to conne the crack
development to inter-element boundaries [12–14]. However, since the discontinuity pattern is
conditioned by the mesh layout, in most cases a realistic crack path is obtained with this
approach only with a very ne mesh or, again, with a mesh adaptation strategy. Alternative approaches, based on FE displacement models with embedded discontinuities (see e.g.
References [15–18] and Reference [19] for a recent review and references therein) appear to
be more exible and eective. Within this latter category, an approach, recently proposed in
the literature and based on the partition of unity property of standard FE shape functions, is
followed herein.
Partition of unity FE methods (also termed in the literature ‘generalized FE methods’),
have been presented by Babuska and co-workers [20–22]. By exploiting the property that the
nodal shape functions sum up to unity everywhere in the domain to be modelled, displacement
FE approximate solutions can be enhanced by augmenting the interpolation basis through
ad hoc assumed local functions. In this way, local known features of the exact solution
can be explicitly added to the standard FE approximation elds [22]. The same concept
has been extensively exploited by Belytschko and co-workers [23, 24] for the simulation of
brittle fracture propagation in elastic structures (they use the term ‘extended FE method’ to
denote their technique). Applications of the extended FE method to quasi-brittle fracture with
a cohesive PZ have been recently presented by Wells and Sluys [25] and Moes and Belytschko
[26] (the latter when the present paper was already under review). One of the main advantages
of the extended FE approaches is that the modelling of the displacement discontinuity is
almost completely independent of the pre-existing mesh. Therefore, the desired accuracy in
the modelling of the displacement discontinuity can be achieved at a very reasonable cost,
without modifying the background mesh.
One of the peculiar aspects of a cohesive PZ developing within a linear-elastic domain is
that the tractions between the two sides of the discontinuity lead to a reduction of the stress
concentration ahead of the PZ tip. Therefore, the asymptotic solutions for displacements of
linear elastic fracture mechanics, which were used in Reference [23] to enhance the displacement eld in the crack tip region, are of no use in the present context. Rather, an accurate
modelling of the deformation and stress state in the bulk of the solid ahead of the tip has to
be obtained by enhancing the approximation elds via, e.g. high-order polynomial bases. Local enhancement of the displacement eld by means of quadratic polynomials, discontinuous
across the fracture surface, is here adopted for three-node constant strain triangles. Extended
FE solutions are obtained with a cubic displacement model of the crack opening and sliding
which correctly reproduces the typical cusp-like opening prole in the PZ region, at the cost
of additional degrees of freedom only at the vertex nodes of the elements whose support is
crossed by the discontinuity.
The eectiveness of the proposed methodology has been tested simulating two standard
experimental tests: a mode I crack growth in a wedge-splitting test and a mixed I–II mode
crack growth in an asymmetric three-point bending test. Accurate crack patterns, in accordance
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
EXTENDED FINITE ELEMENT METHOD FOR QUASI-BRITTLE FRACTURE
105
with the experimental evidences, have been obtained in both cases with relatively large crack
advancement steps.
2. QUASI-STATIC ANALYSIS OF FRACTURING MATERIALS:
GOVERNING EQUATIONS
Let us consider a two-dimensional domain with boundary = t ∪ u with t ∩ u = ∅, such
that tractions are imposed on t and displacements on u . Let d denote a propagating internal
discontinuity (Figure 1) consisting of two parts: a rst part where frictionless interaction
between the opposite sides is allowed only upon closure under compressive stress states and
a second part corresponding to the actual process zone where cohesive interaction between
the opening (and possibly sliding) sides is taking place. Plane strain conditions are assumed,
so that the discontinuity line will be more appropriately referred to as discontinuity surface.
Even though, in general, more than one crack can propagate at the same time in a body
under suitable loading conditions, in this paper it is assumed that only one crack, starting from
the boundary of , can generate and propagate. Similarly, during the analysis, possible crack
branching phenomena are not accounted for. This is not a limitation of the proposed approach
but, ruling out the possibility of bifurcation phenomena allows to simplify the presentation.
The equilibrium conditions for the above-dened solid are
CT + b = 0
n = t
= −t+
m
on d+ ;
in \d
(1)
on t
(2)
= t−
m
on d−
(3)
t
n
Γu
Γt
Γd
Ω
Γd+
–
Γd
P–
P+
P m
s
wn
ws
Figure 1. Geometry of modelled domain and notation.
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
106
S. MARIANI AND U. PEREGO
Here is the column matrix gathering the components of the stress tensor (termed hereafter as
stress vector); b and t are the assigned external loads per unit volume and surface, respectively;
C is the dierential compatibility operator; n and m are matrices containing the components
of the unit outward normal n to and of the unit normal m to d , dened by the direction
of propagation as shown in Figure 1. Accordingly, d+ and d− dene the two sides of d
acted upon by tractions t+ and t− , which express the cohesive interaction in the PZ. The
equilibrium condition across surface d reads
t ≡ t− = −t+
(4)
Under the assumption of linearized kinematics, the compatibility conditions in \d and
along u are given by
U = Cu in \d
(5)
u = u
(6)
on u
U and u being the strain and the displacement vectors, respectively, and u the assigned values
of displacements along u .
The displacement discontinuity w across d can be expressed in terms of the vector u
computed on the two sides of the discontinuity, i.e.:
w = u|d+ − u|−
(7)
d
3. CONSTITUTIVE MODEL
According to the denition of the problem given in the previous section, the considered body
is assumed to be composed of two materials: the bulk material and the material in the fracture
PZ. For the present purposes, the bulk material is assumed to be linear elastic:
= D U
in \d
(8)
D being the matrix of bulk stiness moduli.
A simple mixed-mode cohesive model with damage is here assumed for the material in the
PZ under predominantly tensile stress states (normal traction component tn ¿0, see below)
[6, 27]. Let wn and ws be the opening and sliding components, respectively, of the displacement
discontinuity vector w = {wn ws }T in the local reference frame of d . Furthermore, let tn and
ts be the corresponding normal and tangential traction components.
Neglecting possible dilatancy eects due to coupling between normal and shear opening
displacements, pure mode I cohesive fracture propagation is rst considered. Let tnM denote
the activation threshold of the undamaged material for normal damage initiation (Figure 2(a)).
A damage mechanism with linear softening branches is assumed.
If G is dened as the cohesive fracture energy which can be dissipated in pure mode I,
a damage variable D is introduced as
G
(9)
D=1 −
G
G denoting the cohesive energy left to be dissipated for given opening displacement under
the assumption that complete closure of the crack is obtained upon unloading (Figure 2(a)),
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
EXTENDED FINITE ELEMENT METHOD FOR QUASI-BRITTLE FRACTURE
107
tn
tnM
Qn
– 1 – D Qn
D
(a)
1–D= G
G
wn
wnU
t
t M= tnM
(b)
Q
– 1– D Q
D
1–D=G
G
w U= wnU
w
Figure 2. Assumed rigid–damage interface law and notation: (a) pure mode I; and
(b) eective law for mixed mode.
without residual permanent deformations. Let wnU denote the limit displacement beyond which
the process zone is completely damaged and no tractions can be transmitted. One has
G = 12 tnM wnU
(10)
Let Qn = −tnM =wnU be the negative slope of the softening branch in Figure 2(a). Upon unloading, the cohesive crack is assumed to close following the linear path (Figure 2(a)):
1−D
Qn wn
(11)
tn = −
D
In the more general case of mixed mode cohesive crack propagation, an eective cohesive
damage law is dened (see Figure 2(b)). In this case, propagation is assumed to take place
initially in mode I, in a direction orthogonal to the maximum tensile principal stress. Therefore,
it seems reasonable to assume for the eective cohesive law a limit eective traction t M = tnM .
Since the cohesive fracture energy G is assumed to be invariant with respect to the propagation
mode, it follows also that wU = wnU . The eective opening displacement is dened, according
to References [6, 27], as
(12)
w = wn2 + 2 ws2
The eective mixed mode cohesive law is then dened as in Figure 2(b) with a linear softening
branch of negative slope Q = −t M =wU and with an eective damage D = 1 − G= G.
The normal and tangential traction components tn and ts can be derived from the eective
quantities enforcing a total incremental work equivalence (a superposed dot denotes rates):
t ẇ = tn ẇn + ts ẇs
Copyright ? 2003 John Wiley & Sons, Ltd.
∀ẇn ; ẇs
(13)
Int. J. Numer. Meth. Engng 2003; 58:103–126
108
S. MARIANI AND U. PEREGO
with
ẇ =
ws
wn
ẇn + 2 ẇs
w
w
(14)
It follows that the equivalence is satised i
tn =
t
wn ;
w
ts = 2
t
ws
w
(15)
From (12) and (15) it also follows that
t=
tn2 +
ts2
2
(16)
From the denitions of w (12) and t (16) it appears that 2 can be interpreted as a coupling
coecient allowing to assign dierent weights to sliding and opening mechanisms [27].
Damage activation in the PZ is modelled by introducing an activation function f and the
relevant damage loading/unloading conditions:
f = t − t M (1 − D);
f60;
Ḋ¿0;
fḊ = 0
(17)
Under the assumption of continuous damage loading (Ḋ¿0), the rate equations for the
eective law can be easily integrated leading to
t = t M + Qw;
D=
w
wU
(18)
To dene the tangent stiness matrix for the rate problem, it is necessary to distinguish
between crack initiation (w = 0) and crack propagation from an already initiated state (w¿0).
In the rst case, one has tn = tnM and ts = 0 and the initiation is in pure mode I:
t˙n = Qẇn
while t˙s is governed by the local elastic response
(w¿0), the rate behaviour ṫ = D ẇ of the PZ is
matrix D which is given by

1 0

1−D



−
Q


D

0 2





D =
wn2


1 0

1  w2
1−D


Q
−
+ Q


2

D
D  wn ws
0 


2 2
w
(19)
of the bulk material. In the latter case
governed by the local tangent stiness
if f¡0 or ḟ¡0

wn ws
w2 
 if f = ḟ = 0
2 
4 ws
2
w
2
(20)
It is worth noting that the present model has been conceived for being used under
predominantly tensile stress states (tn ¿0) since it is not able to reproduce frictional
eects and non-associative behaviour under compressive stress states. Cohesive coupled models
of similar type have been recently used to simulate crack growth in concrete-like
materials and aluminium under quasi-static as well as dynamic loading conditions, see
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
EXTENDED FINITE ELEMENT METHOD FOR QUASI-BRITTLE FRACTURE
109
e.g. References [14, 28, 29]. More accurate descriptions of the PZ behaviour for quasi-brittle
materials have been proposed e.g. by Carol et al. [30] and Cocchetti et al. [31], and will be
considered for future implementations of the proposed methodology.
4. FINITE-ELEMENT IMPLEMENTATION
4.1. Principle of virtual power
Let U be the space of admissible displacements u in ; i.e. such that u = u on u , u possibly
discontinuous on d and u ∈ C 0 in \d . By introducing the test functions v ∈ U0 (with zero
prescribed displacements on u ), the weak form of the incremental equilibrium equations
reads
−
T
T ˙
T ˙
T +
U (v)˙ (u̇) d =
v b d +
v t dt +
v ṫ (u̇) dd +
vT ṫ (u̇) dd
\d
\d
T ˙
v b d +
=
\d
T ˙
(21)
v t dt −
t
d−
d+
t
d
wvT ṫ(u̇) dd
∀v ∈ U0
where wv is dened analogously to w (see Equation (7)) by simply replacing u with v.
In Equation (21) use has been made of the fact that, in view of the assumed linearized
kinematics, d ≡ d+ ≡ d− .
Taking into account the constitutive laws for the bulk material (8) and for the cohesive
part of d (Section 3), the structural problem can be expressed in variational form as
nd u ∈ U:
U (v)D U̇(u̇) d +
T
\d
d
wvT RT D Rẇ dd
T ˙
v b d +
=
\d
vT t˙ dt
∀v ∈ U0
t
(22)
where R is the orthogonal transformation matrix relevant to the x − y → s − m mapping (see
Figure 1).
4.2. Enhanced discontinuous nite element interpolation
The numerical simulation of the cohesive crack growth is here formulated within the frame
of the extended FE method, where the partition of unity property of the nodal shape functions
is exploited to construct enriched interpolation elds.
Let us consider a two-dimensional domain : an unstructured triangulation is introduced,
with nodes (belonging to set I ) placed only at element vertices. Let i (x), i ∈ I , x being the
position vector, be the usual piecewise linear nodal shape functions of three-node triangular
elements. Each i (x) has two important features that can be exploited:
• it has a local, compact support !i consisting of the elements that share the vertex node i;
• it is a partition of unity subordinate to the (open) covering {!i } of , i.e.
i (x) = 1 ∀x ∈ (23)
i∈I
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
110
S. MARIANI AND U. PEREGO
The standard FE discretized displacement eld reads
uh (x) =
i∈I
i (x)ui0
(24)
ui0 being the degrees of freedom (DOFs) at node i. According to the so-called generalized FE method [21, 22], an enhanced approximation eld is constructed multiplying the
standard shape functions i (x) by ad hoc assumed space functions with better representation properties (e.g. incorporating known parts of the exact solution of the problem as in
References [22, 23]).
As recently proposed by Belytschko and co-workers in their extended nite element method
[23], discontinuous displacement elds across d , can be handled by enriching the generalized
interpolation eld as follows:
uh (x) =
i∈I
i (x)ui0 +
3
j∈ J k=0
H(x)j (x)
E
kj (x)ukj
(25)
where the set J gathers those nodes, except for the current d tip node (for reasons which
will be explained at the end of this section), whose support !j is cut by d ; kj (x) are the
E
the relevant additional
enhancement functions that have been introduced at node j and ukj
DOFs. H(x) is the generalized Heaviside step-function, dened as
H(x) =
+1
if (x − x∗ )T m¿0
−1
if (x − x∗ )T m¡0
(26)
x∗ being the closest point projection of x onto d and m being the normal to d (see Figure 1).
Following [21], a cubic displacement interpolation eld is obtained adopting:
0j (x)
=1
x − xj 2
1j (x) =
hj
y − yj 2
(x)
=
2j
hj
x − xj
y − yj
3j (x) =
hj
hj
(27)
where xj ; yj are the j-th node co-ordinates and hj is a typical characteristic dimension of the
elements that share vertex node j. The scaling factor hj is introduced to reduce pollution errors
in the numerical solution; its purpose is to maintain the product j (x) kj (x) of the same order
of magnitude of j (x) in the whole support !j of node j. The above-dened enhancement
functions 1j ; 2j ; 3j vanish in correspondence of their reference node j. Furthermore, since
each kj is multiplied by the standard shape function j (x), the enhanced interpolation eld
vanishes along the element edge opposite to the reference node j.
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
EXTENDED FINITE ELEMENT METHOD FOR QUASI-BRITTLE FRACTURE
111
y
1
H = –1
Γd
d
x
1
H = +1
Figure 3. Unit triangular three-node element used to describe zero energy modes associated to opening
and/or sliding displacement discontinuities along d .
(a)
(b)
(c)
Figure 4. Fundamental zero energy deformation modes associated to rd2 for the three-node
discontinuously enhanced element.
Nodes belonging to set J are enriched and, in the present two-dimensional setting, possess
10 DOFs each. However, only nodes whose supports are cut by d , belong to J . The remaining
nodes in the mesh remain endowed only with the standard FE DOFs u0 and, therefore, the
computational cost is not signicantly increased.
Let us consider an enhanced triangular three-node element: according to Equation (25) this
element possesses 30 DOFs. The kinematic role of the enhanced DOFs can be understood
considering vanishing cohesive properties, as it happens when the eective crack opening w
exceeds its limit value wU (non-interacting crack sides, see Section 3), and then carrying out
an eigen-analysis of the reference unit triangle, cut by d at y = d¡1, shown in Figure 3.
The eigen-analysis shows that the element stiness matrix has a rank deciency rd = 10,
which can be decomposed into three contributions rd = rd1 + rd2 + rd3 . Here, rd1 = 3 is due to
rigid-body motions of the element as a whole without crack opening and sliding; rd2 = 3 is
due to rigid-body motions of the element part where H = +1 with respect to the element
part where H = −1. These modes have to be present in any displacement model aimed
at a locking-free description of crack opening [32]. When cohesive forces are transmitted
between the two sides of the crack (w¡wU ), no singularity in the element stiness matrix
is associated to these modes. When w¿wU , the three rd2 zero-energy modes appear in the
element stiness matrix, which do not lead to energy-free modes in the global stiness matrix
until a collapse mechanism develops. Figure 4 shows the three fundamental zero-energy
K
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
112
S. MARIANI AND U. PEREGO
modes associated to: (a) uniform crack opening (mode I); (b) uniform crack sliding (mode
II); (c) relative rigid-body rotation of the two parts of the element. The remaining part of
the rank deciency, rd3 = 4, is due to four independent non-zero combinations of DOFs in
the enhanced part of the interpolation eld (25), see Reference [21], which give rise to zero
displacement throughout the element. This means that the element stiness matrix maintains
a rank deciency rd = rd3 = 4 even if rigid-body motions are constrained and the cohesive
properties are non-zero. Terms which cause rd3 give rise to a singular generalized stiness
does not aect the
of the nite element aggregate. However, the rd3 singularity of K
matrix K
approximate generalized FE solution, i.e. no global zero-energy modes can appear. Despite the
a unique solution in terms of displacement and
singularity of the generalized stiness matrix K,
stress elds can be obtained by means of specically devised algorithms (see Reference [21]).
Let e be the domain of the triangular element e and let e1 ; e2 ; e3 denote its nodes. To show
details of the extended FE implementation, let us suppose that all nodes of element e belong
to set J and let de denote the portion of d belonging to e . The approximated displacement
eld uh (x) for in-plane displacements reads
 0 
ue 1 










0 


u


e
2 








 u0 




e3 






 uxh (x) 
.
0
0
0 .
E
E
E
h
u (x) =
=[e1 e2 e3 . e1 e2 e3 ] · · ·


 uh (x) 




y

E 

ue1 


(28)










E 


 ue2 






 E 

ue 3
 0
 ue 
.
= [e0 .. eE ] · · · = e (x)ue ;
 E
ue
x ∈ e
where
0
=
0
0
E
=H
0
0
;
= e1 ; e2 ; e3
1
0
0
2
0
3
0
0
0
0
(29)
0
1
2
3
are the matrices of standard linear and extended shape functions, respectively; u0 is
a 2-component array, while uE is an 8-component array. Here (x) is the shape function of
node . For compactness reasons, in Equations (28), (29) and in the remainder of the section
the dependency of functions , and H on the position vector x is suppressed.
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
113
EXTENDED FINITE ELEMENT METHOD FOR QUASI-BRITTLE FRACTURE
The strain eld is given by
Uh (x) =
 h
x (x)
















yh (x)
 0


 ue 
E
Be 3 ] · · ·

 E

.
=[Be01 Be02 Be03 .. BeE1 BeE2
ue

h
(x)
2xy
 0
u 
. E  e
0 .
= [Be . Be ] · · · = Be (x)ue ;

 E

(30)
x ∈ e \de
ue
where

; x

B0 = 
 0
; y


BE = H 
0


; y 

; x
(
0 ); x
(
0
(
1 ); x
(
2 ); x
0
0 ); y
(
(
0
1 ); y
(
2 ); y
3 ); x
0
0
(
0
0

0
(
0 ); y
(
1 ); y
(
2 ); y
(
3 ); y
(
0 ); x
(
1 ); x
(
2 ); x
(
3 ); x
3 ); y


(31)
The discretized displacement discontinuity eld along de can be expressed as
wh (x) = uh |de+ − uh |− =
de
3
j∈J k=0
.
= [0 0 0 .. WeE1 WeE2
2j (x)
E
kj (x)ukj
 0

 ue 

E
We3 ] · · ·

 E

(32)
ue
 0

 ue 

..
E
= [0 . We ] · · · = We (x)ue ;

 E

x ∈ de
ue
where
WE
=2
0
0
0
1
2
0
Copyright ? 2003 John Wiley & Sons, Ltd.
0
3
0
0
0
0
1
0
2
(33)
3
de
Int. J. Numer. Meth. Engng 2003; 58:103–126
114
S. MARIANI AND U. PEREGO
The semi-discretized spatial version of the variational statement (22) thus reads
Ae
e \de
(Be ) D Be d+
(We ) R D RWe dd
T
T
T
u̇e −
T ˙
(e ) b d− (e ) t dt = 0
e \de
de
T ˙
te
(34)
where te denotes the (possible) portion of the external boundary t belonging to the element
e and A represents the assemblage operator.
e is partitioned according to the adopted arrangement
The extended element stiness matrix K
of vector ue (see Equation (28)), that is

e =
K
Ke00
Ke0E
KeE0
KeEE


(35)
Terms in Ke00 represent the standard stiness matrix of the underlying three-node constant
strain triangle, while the other terms account for the adopted enriched formulation. The term
KeEE contains, through matrix D , the non-linear contributions of the dissipative constitutive
model of the fracture PZ. The expression of the algorithmic tangent stiness matrix implemented in the extended FE code is shown in Table I.
In linear elastic fracture mechanics, the asymptotic solution in the crack tip region is available and in References [22, 23] it has been incorporated in the enhanced displacement model
of the element containing the current tip. For the considered mixed-mode cohesive crack propagation problem, the standard enhancement (25) of the displacement model has been adopted
also in the elements around the tip of the process zone. It should be noted that the typical
cusp-like shape of the crack opening in the tip region, where each side has an s-shaped prole,
can be well reproduced by the adopted cubic discontinuous displacement model.
For computational convenience, the element that contains the current d -tip is sub-triangulated (see Figure 5) introducing an additional node at the tip (node marked with a square in the
gure). Even though this is not strictly necessary, adding that node made our
implementation simpler at a very limited cost (the degrees of freedom need already be
augmented when the enhanced model is activated), allowing to identify in an unambiguous way the regions where H(x) = ±1. Furthermore, also without the additional node, a
sub-triangularization of the element containing the tip would be required for the computation
of the integrals. Only the two nodes whose supports are crossed by d are enhanced in the
element containing the tip (nodes marked with circles in the right closed-up view in the gure). In this way the resulting displacement eld is discontinuous only in the sub-region cut
by d , while in the two forward sub-regions shaded in the gure the displacement eld is still
continuous.
In all elements with at least one enhanced node, six Gauss points are used for the integration
in triangular regions. For those elements which are cut by d in a triangular and a quadrilateral
subregions, the quadrilateral part is further subdivided (only for integration purposes) into
two triangles. In each of them, six Gauss points are used. Since the stiness matrix contains
fourth-order polynomial terms, the adopted rule provides an exact integration. The integral
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
EXTENDED FINITE ELEMENT METHOD FOR QUASI-BRITTLE FRACTURE
115
Table I. Finite-step algorithmic treatment of the cohesive law.
1. Compute trial elastic update:
t = (‘) t −
(‘+1) tr
1 − (‘) D (‘+1)
w − (‘) w
Q
(‘) D
2. Check damage activation criterion:
(‘+1)
(a) If
(‘+1)
tr
f = (‘+1) t tr − t M (1 − (‘) D)
tr
f ¡0 then:
• update eective traction t and damage D:
(‘+1)
(‘+1)
t =
D =
(‘+1) tr
t
(‘)
D
• compute cohesive tangent stiness matrix:
(‘+1)
(b) Else if
(‘+1)
1 − (‘) D
D = − (‘)
Q
D
1
0
0
2
tr
f ¿0 then:
• update eective traction t and damage D:
(‘+1)
(‘+1)
t = t M + Q(‘+1) w
(‘+1)
D =
w
wU
• compute cohesive tangent stiness matrix:
1 0
1 − (‘+1) D
(‘+1)
D = − (‘+1)
Q
D
0 2

(‘+1) 2
wn

(‘+1) w 2

1
+ (‘+1) Q 
D 
 2 (‘+1) wn (‘+1) ws
(‘+1) w 2
2

wn (‘+1) ws

(‘+1) w 2



(‘+1) 2

ws
4
2
(‘+1) w
(‘+1)
along de (the second integral in Equation (34)) contains sixth-order polynomial terms, which
are integrated exactly with four Gauss points.
4.3. Criteria for crack propagation
According to a standard methodology, the advancement of the tip of the discontinuity surface
d is assumed to be governed by the following criteria: (a) d growth takes place as soon as
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
116
S. MARIANI AND U. PEREGO
Γd
Γd
Figure 5. Sketch of d tip region: displacement eld discontinuous across d . Circles
mark enhanced nodes.
the limit tensile carrying capacity t M of the material (see Figure 2(b)) is attained at the node at
the current d tip; (b) d grows in a direction perpendicular to the maximum principal stress.
The evolution of the cohesive PZ behind the tip of d depends on the structural response
to the load increment and its length is an outcome of the numerical computations. The actual
crack (intended as material separation) is conceived as the portion of d where damage D = 1,
and starts at the beginning of the PZ, in correspondence of the rst Gauss point where no
more normal tensile tractions are transmitted between the crack sides.
Possible crack closures (wn ¡0) are dealt with adding a ctitious very high compressive
stiness in the t − w relation. It is worth noting that in the numerical examples, due to the
monotonically increasing load, closures never occurred.
5. SOLUTION OF THE FINITE-STEP DISCRETIZED PROBLEM
N
Let the time interval of interest T = [(0) ; (end) ] be subdivided in time steps so that T = ‘=0
[(‘) ; (‘+1) ]. Under quasi-static loading conditions, at the beginning of the nite step [(‘) ;
(‘+1)
] it is assumed that the solution is known in terms of: displacements (‘) u, strains and
stresses (‘) U; (‘) in \d ; displacement discontinuities (‘) w and tractions (‘) t on d .
A standard Newton–Raphson iterative scheme is adopted here. At each iteration an estimate
of the displacement increment (‘+1) u over the time step is obtained from the current tangent
matrix. The corresponding estimates of strain and stress increments are obtained enforcing
compatibility and Hooke’s law in the bulk material:
(‘+1)
u = (‘) u + (‘+1) u
(‘+1)
U = (‘) U + C((‘+1) u)
(‘+1)
= (‘) + D C((‘+1) u)
in \d
(36)
The estimate at the current iteration of the displacement discontinuity across d is given by
(‘+1)
w = (‘) w + (‘+1) u|d+ − (‘+1) u|−
d
on d
(37)
The non-linear response of the cohesive PZ is obtained through analytical integration, as in
(18) at each iteration for assigned increment of the displacement discontinuity (‘+1) w − (‘) w.
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
EXTENDED FINITE ELEMENT METHOD FOR QUASI-BRITTLE FRACTURE
117
Equations (18) are exact only under the assumption of continuous loading within the step.
Since local unloading may occur also in the case of monotonically increasing global loading
conditions, this assumption cannot be veried a priori. Therefore, in the spirit of the widely
used backward-dierence integration, the consistency condition f = 0 is here enforced only
at the end of the step, which may imply an approximation in the case of unloading occurring within the step. This kind of approximation is customary whenever dissipative material
behaviour is involved.
First, a trial elastic update is computed (step 1 in Table I). Hence, the damage activation
criterion (17) is checked (step 2 in Table I). In case of elastic unloading (case (a)), damage
does not increase and the cohesive tangent stiness matrix (‘+1) D in the local s − m reference
frame reduces to the elastic-damaged one. Otherwise, if the damage activation criterion is
violated by the trial elastic increment (case (b)), the tractions (‘+1) t, the damage variable
(‘+1)
D and the tangent stiness matrix (‘+1) D are updated as shown at step 2, point (b) in
Table I. On the basis of the new estimate of the tractions along d , the equilibrium residual
is computed and convergence is checked. If the convergence criterion is not satised, a new
iteration is carried out.
For each time step, the incremental structural response is sought assuming xed position
of the crack. Crack tip advancement is allowed only at the end of the step, after equilibrium
convergence has been reached. At the end of each time step, the criterion for d growth
discussed in Section 4.3 is checked. If the criterion is satised, the d -tip is allowed to
advance in the prescribed direction, under xed boundary and loading conditions, of a length
which is computed from an assigned value a in such a way that the tip is as close as
possible to the element centroid. This provision avoids excessive distortion when the element
containing the tip is sub-triangulated (see Section 4.2).
The stress state in the node at the d -tip, to be used in the d growth criterion of Section 4.3,
is evaluated as follows. First, the Nq Gauss points belonging to a semi-circle G ahead of
the PZ tip, with a radius up to eight times the element characteristic dimension (see the
shaded areas in Figure 6), are identied. Then, for each in-plane stress component, a complete
(a)
(b)
Figure 6. Computation of stress state at d tip: patch to be used for stress recovery
procedure is constituted by shaded elements ahead of crack tip: (a) patch not interacting
with boundary ; and (b) patch interacting with boundary .
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
118
S. MARIANI AND U. PEREGO
fourth-order polynomial eld ∗ (x; ar ), depending on coecients ar (r = 1; : : : ; 15), is tted
to the stress values h (xq ), at the Gauss points xq ∈ G , using the least-squares optimization:
min
ar
Nq
∗
( (xq ; ar ) − (xq ))
h
2
;
x q ∈ G
(38)
q=1
The computed stress eld ∗ is nally extrapolated to the d -tip node.
6. NUMERICAL APPLICATIONS
6.1. Wedge splitting test
A wedge splitting (WS) test, with geometry and applied loading conditions shown in Figure 7
∼ 1 , has been recently used by Saouma and co-workers
and with a thickness to width ratio =
2
[33] to determine the fracture toughness of concrete-like materials and to accurately measure
PZ features by means of embedded optical bres.
This test has been used as a rst benchmark for the proposed numerical methodology. The
specimen has been discretized using the background FE meshes depicted in Figure 8: the rst
one (Figure 8(a)) consists of 781 nodes and 1494 plane-stress three-node triangular elements
and is structured (though non-symmetric) in the ligament region, where crack propagation
takes place; the second one (Figure 8(b)) consists of 827 nodes and 1580 plane-stress threenode triangular elements and is unstructured in the ligament region. Owing to the symmetry
in the geometry and boundary conditions, a mode I rectilinear crack growth is expected. The
structured mesh has been distorted slightly pushing the elements upwards in the ligament
region, so preventing interelement lines to be exactly coincident with the expected crack path
(see the enlarged view in Figure 10). The thick lines in Figure 8 represent the pre-formed
notch.
For mode I fracture tests, a stress recovery procedure similar to (38) but based on thirdorder polynomial elds was adopted in Reference [34], with G consisting only of the two
200
70
P
30
200
B=160
P
30
85
th=97
Figure 7. Wedge splitting test: loading conditions and geometry. Dimensions are
in mm; th means thickness.
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
EXTENDED FINITE ELEMENT METHOD FOR QUASI-BRITTLE FRACTURE
119
(a)
(b)
Figure 8. Wedge splitting test: adopted: (a) structured; and (b) unstructured FE meshes.
elements ahead of the crack which share the tip node and whose displacement eld is continuous (shaded elements in Figure 5). Using this methodology, a perfectly rectilinear crack path
could be obtained only in the case of a symmetric structured background mesh. In contrast,
non-symmetric meshes gave rise to spurious oscillation of the crack path. The poor results obtained in that case with non-symmetric meshes motivate the enhanced stress recovery methodology adopted in this paper.
The following parameter values have been adopted for the bulk material and the cohesive
law: E = 25 200 MPa, = 0:2; Q = −54:45 MPa=mm, t M = 3:3 MPa. The parameters used for
the cohesive model lead to an eective fracture energy G = 100 J=m2 .
The analyses have been carried out controlling the crack mouth opening displacement
CMOD, for assigned reference crack advancement length a. According to the experiments
in Reference [33], the CMOD is intended as the variation in length of the segment B (see
Figure 7) originally of 160 mm.
The eects of the assumed reference crack advancement a on the global P vs CMOD
response and on the crack propagation path are shown in Figures 9 and 10, respectively, for
2 = 1:0. The global specimen response is only slightly aected by the size of a, while
almost perfectly straight crack paths are obtained for any adopted value of a. No analyses
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
120
S. MARIANI AND U. PEREGO
4000
experiment
∆a=7.5 (mm)
∆a=10 (mm)
∆a=15 (mm)
40
20
∆a == 7.5
initial crack tip
2000
y
P (N)
3000
0
∆a = 10
1000
∆a = 15
-20
0
0
0.1
0.2
0.3
0.4
0.5
CMOD (mm)
Figure 9. Wedge splitting test, structured mesh:
eect of a on P vs CMOD response (2 = 1:0).
-40
100
120
140
160
180
200
x
Figure 10. Wedge splitting test, structured mesh:
eect of a on crack path (2 = 1:0). Close-up
of the ligament region with underlying mesh.
have been run using values of a¿15 mm in order to prevent spurious numerical closures
in the PZ; as a reference, a value of a of the order of a few (2–4) times the characteristic
dimension of the elements in the ligament region can be typically considered as an upper
bound.
Owing to the almost perfectly straight mode I crack growth, results are not inuenced by
2 . A reasonable agreement between experimental measurements and numerical results can be
observed in Figure 9; a better agreement could be obtained at peak load and in the initial stage
of the softening regime by a more rened calibration of the PZ model parameters, which,
however, is not within the scope of this work.
In Figure 10 just minor oscillations of the nal d congurations with respect to the symmetry axis (y = 0) show up, possibly due to the proximity of the boundary which reduces
the extension of the element patch to be used for the stress recovery. Notice that results
are here represented in an enlarged view of the ligament region, the whole specimen width
being 200 mm; only by zooming on this region, the above-mentioned oscillations can be
appreciated.
For 2 = 1:0 and a = 7:5 mm, the evolution of the opening discontinuity along d (amplied by a factor 200) and the distribution of the cohesive tensile tractions in the PZ up
to a CMOD = 0:4 mm (corresponding to an almost complete propagation through the initial
ligament) are illustrated in Figure 11. Even though the reference crack advancement
a = 7:5 mm is larger than the typical element size of 5 mm, the adopted cubic modelling
of the displacement discontinuity allows an accurate description of the cusp-like opening prole in the PZ. Good agreement with the experimental results reported in Reference [33] is
obtained in terms of PZ length 30 mm in the central and nal stages of crack growth
(see, e.g. the snapshots corresponding to CMOD = 0:4 mm). As a consequence of the stepwise almost xed crack advancement length, for larger values of a it may happen (even
though this never occurs in our numerical tests) that the PZ is only partially active. In this
case, spurious compressive states might appear ahead of the PZ tip. Shorter values of a
should be assigned in this case and the step re-run. A more rigorous implicit formulation
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
EXTENDED FINITE ELEMENT METHOD FOR QUASI-BRITTLE FRACTURE
CMOD = 0.1 (mm)
30
30
20
20
10
10
y
y
CMOD = 0.1 (mm)
0
0
-10
-10
-20
-20
-30
110 120 130 140 150 160 170 180 190 200 210
-30
110 120 130 140 150 160 170 180 190 200 210
x
x
CMOD = 0.2 (mm)
30
20
20
10
10
y
y
CMOD = 0.2 (mm)
30
0
0
-10
-10
-20
-20
-30
110 120 130 140 150 160 170 180 190 200 210
-30
110 120 130 140 150 160 170 180 190 200 210
x
x
CMOD = 0.4 (mm)
30
20
20
10
10
y
y
CMOD = 0.4 (mm)
30
0
0
-10
-10
-20
-20
-30
110 120 130 140 150 160 170 180 190 200 210
-30
110 120 130 140 150 160 170 180 190 200 210
x
121
x
Figure 11. Wedge splitting test, structured mesh: evolution of crack pattern (
a = 7:5 mm; 2 = 1:0).
Details of crack opening and sliding (left column) and cohesive interaction in the PZ (right column).
where the exact crack advancement length is part of the step solution is currently under
study.
In Figure 12 the P vs CMOD plots obtained with the structured and unstructured background
meshes of Figure 8 are compared, in the case of assigned a = 7:5 mm and 2 = 1:0. It can
be seen that the underlying mesh has limited eect on the test response, provided that the two
meshes have the same capability to resolve the stress state ahead of the d tip, i.e. the same
element characteristic dimension. This is further elucidated in Figure 13, where the almost
straight d path obtained with the unstructured background mesh is shown.
6.2. Mixed-mode three-point bending test
The capabilities of the proposed methodology in mixed-mode I–II crack growth have been
assessed simulating a three-point bending (TPB) test on an asymmetric single edge notched
specimen (Figure 14). The notch position is determined by the oset ratio [11, 35], dened
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
122
S. MARIANI AND U. PEREGO
4000
experiment
structured mesh
unstructured mesh
3000
40
initial crack tip
2000
y
P (N)
20
0
1000
-20
0
0
0.1
0.2
0.3
0.4
0.5
CMOD (mm)
-40
100
120
140
160
180
200
x
Figure 12. Wedge splitting test: eect of the underlying mesh (structured vs unstructured) on P
vs CMOD response (
a = 7:5 mm; 2 = 1:0).
Figure 13. Wedge splitting test, unstructured
mesh: crack path (
a = 7:5mm; 2 = 1:0). Closeup of the ligament region with underlying mesh.
230
P
P
75
19
χS
th=25
S=102.5
Figure 14. Mixed-mode three-point bending test:
loading conditions and geometry. Dimensions are
in mm; th means thickness.
Figure 15. Mixed-mode TPB test:
adopted FE mesh.
as the ratio between the distance of the notch from the mid-span cross-section and half of
the span.
If 60:7, John and Shah [35] experimentally showed that, under quasi-static loadings,
a single crack initiate from the notch and propagates up to the specimen failure. For ¿0:7,
cracking initiates at the notch while the nal failure involves cracking at the mid-span
cross-section. Bifurcation phenomena, leading to multiple crack growth, are not included in
the present version of the proposed formulation. Hence, attention is restricted to cases with
60:5.
The underlying uniform mesh of 5520 plane stress three-node triangular elements and
2883 nodes is represented in Figure 15, along with the loading condition. The following
values of the constitutive parameters, equal to those proposed in Reference [11] for an
analysis of the same test, were assumed: bulk material E = 31 370 MPa, = 0:2; cohesive
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
EXTENDED FINITE ELEMENT METHOD FOR QUASI-BRITTLE FRACTURE
123
5000
4000
P (N)
3000
2000
χ=0.00
χ=0.25
χ=0.50
1000
0
0
0.025
0.05
0.075
0.1
u (mm)
Figure 16. Mixed-mode TPB test: eect of oset ratio on P vs u response (
a = 3:5 mm; 2 = 1:0).
model Q = −57 MPa=mm, t M = 4:4 MPa, which correspond to an eective fracture energy
G = 170 J=m2 . According to what was proposed in Reference [35] and, recently, in Reference [11], 2 = 1:0 is adopted for this test, so that the eective discontinuity is given by
w = wn2 + ws2 .
Since the adopted cohesive law does not incorporate frictional and interlocking eects (as
anticipated in Section 3), which play an important role under mixed-mode loading conditions,
the simulation is not expected to provide accurate results in terms of P vs u plots. However,
the following two important features of the specimen response evidenced by the experiments
(see Reference [35]) are correctly reproduced: for increasing oset ratio of the notch, the
initial elastic response in the load P vs displacement u plane turns out to be stier and the
peak load increases (Figure 16); in the initial growth stage, crack propagates inclined of an
almost constant angle with respect to the notched cross-section (Figure 17).
For = 0:00, 0.25, 0.50 and a = 3:5 mm, the numerical crack paths at a vertical displacement u = 0:09 mm, are compared in Figure 17 to the corresponding available experimental crack paths. The rst case ( = 0:00) is here used as a further test for the numerical
procedure under mode I loading conditions: as for the wedge splitting test, symmetry is globally preserved up to the region immediately below the external load P, where bifurcation and
branching can occur (see Reference [36]) due to the isotropy of the stress state. In the case of
= 0:25, the crack grows with an almost constant slope and it is shown to be not inuenced
by the mesh layout. This result is further evidenced in the case of = 0:50, where numerical
results compare well with the experimental path.
7. CONCLUSIONS
A numerical methodology has been proposed to simulate quasi-static bidimensional crack
propagation in quasi-brittle materials. The method is based on the partition of unity concept
[20, 22], which allows to introduce in a simple and rational way discontinuous as well as
higher-order local displacement interpolation elds [21, 23, 34] into the discretized model.
The proposed technique applies to quasi-brittle cohesive fracture the so-called extended niteelement method originally developed by Belytschko and co-workers [23, 24].
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
124
S. MARIANI AND U. PEREGO
χ=0.00
initial crack tip
(a)
initial crack tip
χ=0.25
(b)
χ= 0.50
initial crack tip
(c)
Figure 17. Mixed-mode TPB test: comparison between available experimental (dashed lines) [35] and
numerical (continuous lines) crack paths: (a) = 0:00; (b) = 0:25; and (c) = 0:50.
The main features of the present methodology can be summarized as follows:
• In the considered two-dimensional setting, the crack path is described by a piecewise-
straight line. The orientation of each propagation segment is a result of the analysis
and is completely independent of the underlying nite element mesh. No remeshing
is required in the region near the tip of the propagating crack. Only a local subtriangularization is introduced for computational convenience so that the tip of the PZ
always coincide with a vertex of a nite element. Ad hoc quadrature rules have been devised both in the nite elements crossed by the discontinuity and along the discontinuity
itself.
• The nite element discretization follows in a natural and consistent way from a variational
formulation of the rate problem which incorporates the contribution of the displacement
discontinuity.
• The discontinuity has been embedded into the nite element model by locally enriching
the displacement eld with cubic discontinuous polynomial terms. This allows us to
correctly reproduce the typical cusp-like shape of the discontinuity in the PZ and a ne
resolution of the displacement eld in the surrounding region.
• The simple constitutive model originally proposed by Camacho and Ortiz in Reference [27] has been adopted for the cohesive process zone. This model is widely used
to simulate fragmentation, quasi-brittle as well as ductile fracture under both quasi-static
and dynamic loading conditions. In the present rst implementation, bifurcation and
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
EXTENDED FINITE ELEMENT METHOD FOR QUASI-BRITTLE FRACTURE
125
branching phenomena have been ruled out so that only cases with one propagating discontinuity can be handled.
• A consistent Newton–Raphson iterative scheme has been formulated. The implemented
numerical procedure assigns piecewise-straight crack advancements of variable length.
The adopted cubic interpolation of the displacement discontinuity along the crack path
allows for an accurate description of the traction eld in the process zone. The algorithm
has proven to be robust and computationally eective. No special convergence problems
have been encountered during the numerical tests.
• Since the advancement criterion is based on the maximum tensile principal stress, a high
resolution of the stress eld is required. This has been obtained by best tting a smooth
high-order polynomial solution to the computed stresses at Gauss points in the region
ahead of the current process zone tip. The procedure has provided accurate results both
under mode I and mixed-mode loadings.
The considered numerical simulations have conrmed the exibility and eectiveness of
the method for the modelling of crack growth under general mode I and mixed-mode loading
conditions. Accurate results, reproducing the main qualitative features of the experimental tests
have been obtained both for a wedge splitting test and for a three-point bending test on an
asymmetric single-notched specimen.
Future developments, currently in progress, include extension to multiple cracks and more
accurate description of the process zone behaviour incorporating friction and dilatancy.
ACKNOWLEDGEMENTS
The present work has been carried out within the context of the MIUR co-nancing research project
MM08161945-001 2000, with the additional support of LSC Project-Politecnico di Milano.
REFERENCES
1. Pijaudier-Cabot G. Nonlocal damage. In Continuum Models for Materials with Microstructure, Muhlhaus HB
(ed.). Wiley: New York, 1995; 105 –143.
2. Peerlings RHJ, de Borst R, Brekelmans WAM, Geers MGD. Gradient-enhanced damage modelling of concrete
fracture. Mechanics of Cohesive-Frictional Materials 1998; 3:323–342.
3. Ferrara L, di Prisco M. Mode I fracture behaviour in concrete: nonlocal damage modelling. Journal of
Engineering Mechanics (ASCE) 2001; 127:678– 692.
4. Comi C. A non-local model with tension and compression damage mechanisms. European Journal of Mechanics
– A/Solids 2001; 20:1–22.
5. Comi C, Perego U. Fracture energy based bi-dissipative damage model for concrete. International Journal of
Solids and Structures 2001; 38:6427– 6454.
6. Jirasek M, Zimmerman T. Embedded crack model: I. Basic formulation. International Journal for Numerical
Methods in Engineering 2001; 50:1269 –1290.
7. Jirasek M, Zimmerman T. Embedded crack model: II. Combination with smeared cracks. International Journal
for Numerical Methods in Engineering 2001; 50:1291–1305.
8. Oliver J, Huespe AE, Pulido MDG, Chaves E. From continuum mechanics to fracture mechanics: the strong
discontinuity approach. Engineering Fracture Mechanics 2002; 69:113–136.
9. Cen Z, Maier G. Bifurcations and instabilities in fracture of cohesive-softening structures: a boundary element
analysis. Fatigue and Fracture of Engineering Materials and Structures 1992; 15:911–928.
10. Maier G, Frangi A. Symmetric boundary element method for ‘discrete’ crack modelling of fracture processes.
Computer Assisted Mechanics and Engineering Science 1998; 5:201–226.
11. Belytschko T, Organ D, Gerlach C. Element-free Galerkin methods for dynamic fracture in concrete. Computer
Methods in Applied Mechanics and Engineering 2000; 187:385 –399.
12. Bocca P, Carpinteri A, Valente S. Mixed mode fracture of concrete. International Journal of Solids and
Structures 1991; 27:1139 –1153.
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126
126
S. MARIANI AND U. PEREGO
13. Xu X-P, Needleman A. Numerical simulation of fast crack growth in brittle solids. Journal of the Mechanics
and Physics of Solids 1994; 42:1397–1434.
14. Ortiz M, Pandol A. Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation
analysis. International Journal for Numerical Methods in Engineering 1999; 44:1267–1282.
15. Klisinski M, Runesson K, Sture S. Finite element with inner softening band. Journal of Engineering Mechanics
(ASCE) 1991; 117:575 –587.
16. Simo JC, Oliver J, Armero F. An analysis of strong discontinuities induced by strain softening in rate-independent
inelastic solids. Computational Mechanics 1993; 12:277–296.
17. Lot HR, Shing PB. Embedded representation of fracture in concrete with mixed nite elements. International
Journal for Numerical Methods in Engineering 1995; 38:1307–1325.
18. Bolzon G, Corigliano A. Finite elements with embedded discontinuity: a generalised variable formulation.
International Journal for Numerical Methods in Engineering 2000; 49:1227–1266.
19. Jirasek M. Comparative study on nite elements with embedded cracks. Computer Methods in Applied
Mechanics and Engineering 2000; 188:307–330.
20. Melenk JM, Babuska I. The partition of unity nite element method: basic theory and applications. Computer
Methods in Applied Mechanics and Engineering 1996; 139:289 –314.
21. Duarte CA, Babuska I, Oden JT. Generalized nite element methods for three-dimensional structural mechanics
problems. Computers and Structures 2000; 77:215 –232.
22. Strouboulis T, Copps K, Babuska I. The generalized nite element method. Computer Methods in Applied
Mechanics and Engineering 2001; 190:4081– 4193.
23. Moes N, Dolbow J, Belytschko T. A nite element method for crack growth without remeshing. International
Journal for Numerical Methods in Engineering 1999; 46:131–150.
24. Sukumar N, Moes N, Moran B, Belytschko T. Extended nite element method for three-dimensional crack
modelling. International Journal for Numerical Methods in Engineering 2000; 48:1549 –1570.
25. Wells GN, Sluys LJ. A new method for modelling cohesive cracks using nite elements. International Journal
for Numerical Methods in Engineering 2001; 50:2667–2682.
26. Moes N, Belytschko T. Extended nite element method for cohesive crack growth. Engineering Fracture
Mechanics 2002; 69:813–833.
27. Camacho GT, Ortiz M. Computational modelling of impact damage in brittle materials. International Journal
of Solids and Structures 1996; 33:2899 –2938.
28. Ruiz G, Ortiz M, Pandol A. Three-dimensional nite element simulation of the dynamic Brazilian tests on
concrete cylinders. International Journal for Numerical Methods in Engineering 2000; 48:963–994.
29. Arun Roy Y, Dodds RH. Simulation of ductile crack growth in thin aluminum panels using 3-D surface cohesive
elements. International Journal of Fracture 2001; 110:21– 45.
30. Carol I, Prat PC, Lopez CM. Normal=shear cracking model: application to discrete crack analysis. Journal of
Engineering Mechanics (ASCE) 1997; 123:765 –773.
31. Cocchetti G, Maier G, Shen XP. Piecewise linear models for interfaces and mixed mode cohesive cracks.
Computer Modeling in Engineering and Sciences 2002; 3:279 –298.
32. Jirasek M, Belytschko T. Computational resolution of strong discontinuities. In Proceedings of the Fifth World
Congress on Computational Mechanics (WCCM V), Mang HA, Rammerstorfer FG, Eberhardsteiner J (eds).
ISBN 3-9501554-0-6, Vienna University of Technology, 2002.
33. Denarie E, Saouma VE, Iocco A, Varelas D. Concrete fracture process zone characterization with ber optics.
Journal of Engineering Mechanics (ASCE) 2001; 127:494 –502.
34. Mariani S, Perego U. A PU-FE approach to quasi-brittle fracture. In Proceedings of the 15th AIMETA Congress
of Theoretical and Applied Mechanics, Augusti G, Mariano PM, Sepe V, Lacagnina M (eds). Taormina (Italy),
2001.
35. John R, Shah SP. Mixed-mode fracture of concrete subjected to impact loading. Journal of Structural
Engineering (ASCE) 1990; 116:585 – 602.
36. Bolzon G, Cocchetti G. On a case of crack path bifurcation in cohesive materials. Archive of Applied Mechanics
1998; 68:513–523.
Copyright ? 2003 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2003; 58:103–126