MECHANICS
OF COHESIVEFRICTIONAL MATERIALS,
VOL.
1,
32 1347
(1
996)
‘Crushcrack’
:
a nonlocal damage model for concrete
M.
di Prisco’ and J. Mazars’
‘Department
of
Structural Engineering, Politecnico di Milano, 20133 Milan. Italy
‘Lab.
de Mecanique et Technologie, Ecole Normale Superieure de Cachan
94235
Cachan Cedex, France
SUMMARY
A
model
is
presented based on the nonlocal damage theory.
It
sets out to describe the behavior of concrete under
freevariable loads, which are constant in sign. Its purpose is to analyze shear behavior and high straingradient
localized problems, and it takes Mazar’s model as a starting point with reference to the basic idea of a scalar
isotropic nonlocal damage controlled by principal tensile strains. In addition, the other two main features are an
internal variable denoted to the control or reversible volumetric expansion in compression, and irreversible
strains aimed at modelling crushing in compression and cracks both in tension and compression.
As
a
consequence, inducedanisotropy, dilatancy and pathdependency can be reproduced. In particular, the modelling
of micro and macrocracks makes it possible
to
capture mixedmode cracking as well as aggregate interlock,
which requires a residual stiffness to guarantee the transmission of transversal and normal stresses for assigned
slips.The model requires the knowledge of the material response in uniaxial tension and compression, and biaxial
compression tests which can be introduced directly by adopting experimental curves, or by means of a reduced
number of parameters. The effectiveness of the model is shown through comparisons with several sets of
experimental tests on both small specimens, assumed
to
be homogeneous, and boundary value problems.
KEY
WORDS
damage; dilatancy; failuremodes; fractures; localization; shear
1.
INTRODUCTION
Fracture in concrete is a difficult problem because it induces localization and discontinuity in the
displacement field, not only at the microlevel, but also at the meso and macrolevel. Although for a
metal it is possible to identify the constitutive behaviour on a mesoscale structure, assuming a
sufficient homogeneity in the test, concrete is strongly affected by its considerable degree of
heterogeneity (14 magnitude orders characterize its constituent aggregates) and only the initial elastic
behaviour can be identified on a mesoscale structure, without the occurrence of localization on a
meso and macrolevel.
Research has been examining localization and softening problems carefully over recent years, in
order to establish which models are meshindependent in the descending branch of the constitutive
relations, and evaluate their efficiency in analysing concrete structures.
De Borst, Sluys and Pamin have provided enlightening contributiom4 to current thinking,
clarifying the advantages and disadvantages of different approach and different failure modes. All the
approaches introduce a characteristic length
so
as to avoid confining localization to zerovolume
zones with the progressive reduction to zero of dissipative energy.
now well supported by extensive
computational experience for both the microplane and Cauchy continuum formulations.’’
’
The proposed model follows the nonlocal damage
CCC 1082501 0/96/04032127
0
1996 by John Wiley
&
Sons, Ltd.
Received 17 October 1995
Revised
23
February 1996
322
M. DI
PRISCO
AND
J.
MAZARS
Assuming that the physical explanation has been mainly phenomenologic and empirical, BazantI2
sought the main source of nonlocality in the interactions among adjacent microcracks. According to
fracture mechanics, he carefully analysed the micromechanics of cracking and damageI3,l4 and tried
to deduce the proper
form
of the nonlocal spatial integral.
This model may be regarded as a further development of a previous one,l5 and is aimed at giving a
more accurate response to multiaxial stresses,16 shear, and reinforcing steelconcrete interaction
problem^.'^^''
The limits of the previous model, implemented in its nonlocal updated version in the
Cesar FE Code, become apparent especially for high straingradient problems like the punch test,
shear box test and dowel action." The weak points appear to be the lack
of
irreversible strains in
tension and compression, so that damage is overestimated and dilatancy effects are neglected.
Consequently, inelastic strains have now been introduced in the model, and are represented by a
local tensor if associated with crushing, and by a nonlocal tensor if associated with cracking. The
latter represents a nonlocal measure of the discontinuities in the displacement field at the interface of
growing micro and macrocracks; an average mesoscale compliance is assumed in order to describe
representative concrete volumes affected by interlocked cracks. At the same time, an internal variable
is added to control the transverse reversible strains, described in elasticity by Poisson's coefficient
and related to damage accumulated in compression.
The evolution law of damage is strictly related to experimental tests and their relative
interpretation as presented by Van Mier.20. Owing to the nonlocality of the irreversible strains and to
the unusual potential functions expressed in terms of total strains, the terms which appear in the
dissipative power are not all positive: for a general path some thermodynamical constraints are
imposed to satisfy the positiveness of the dissipative power.2' These aspects will be discussed in
detail in a forthcoming paper.
Although several authors have proposed well known approaches for modelling the permanent
strains related
to
damage and plasticity, simplifying the observation of the thermodynamic constraints
from a mathematical point of view,2229 these approaches meet serious numerical convergence
problems when implemented in a nonlocal version.
2.
PHYSICAL ASSUMPTIONS
As shown by Van Mie?' with reference to multiaxial stresspaths, three kinds of failure can occur in
concrete cube specimens and each one is associated to a different, well established fracture mode
(Fig.
1):
tensile fracture, short inclined shear planes in two directions or pronounced shear band. The
last two modes characterize uniaxial and biaxial compression and a separating line can be traced
(U*
=
0), which indicates the passage from expansion in two principal directions to only one. The
same failure modes were observed on cube specimens by Nelissen3' with reference to biaxial stress
space
(03
=
0): the lack of the third stress component makes the crack pattern more regular.
Fracture in mode I is
s
tricky problem because
it
always shows a localization plane with the normal
vector parallel to the tensile load: this peculiarity prevents a mechanical description in terms of
homogeneous stress and
train.^'
By contrast, according to Nelissen,
in
uniaxial and biaxial
compression the measure of the strain component parallel to the loading directions could be
performed in that the normal vector to the localization planes is always at right angles to the loading
directions.
To
limit the instability concerning the localization in tensile fracture, Bazant and
PijaudierCabot3* suggested a refined test, called the
P.I.E.D.
test, which adopts steel ties coupled in
parallel with concrete specimens (in Fig.
2
please see the experimental device proposed by
Ramtani33). It introduces a lot of unsolved problems, but tries to homogeneize the tensile fracture
mode making the measure of the tensile strain performable also in the softening branch.
CRUSHCRACK
323
/
Failure
Modes of BiaxiaUy Load,:d
Concrete
(Nelissen
1972)
Figure
1,
Failure modes in concrete according to Reference 30
( U,
= 0)
and Referenci:
20
(oZ
=
Pu3;
0
<
/3
<
0.
I).
FORCE
Figure
2.
Uniaxial tension according to
PIED
test:33 test setJp.
324
M.
DI
PRISCO
AND
J.
MAZARS
Van Mier”, van Vliet and van Mier35 have pointed out that both
in
concrete prismatic and in
cylindrical specimens subjected to uniaxial compression localization occurs, despite all the
precautions adopted when performing the tests. The angle of the shear band was also i nve~t i gat e.~~
Their experimental evidence seems to highlight that the ‘stressstrain curve of concrete is not a
material property but a mix of structure and mechanical behaviour’
,34
where boundary conditions
play a significant role. Other authors, testing cylindrical specimens in uniaxial compression, observed
crack patterns which show cracks oriented only in the same direction as the principal compressive
stresses36337 when interface friction was eliminated.
Although the formation of shear band in compression might therefore be considered an unsolved
question, two main procedures may be adopted to describe the concrete mechanical behaviour. The
first is based on the description of a failure surface, assuming it is not affected by localization.
Furthermore, a potential function must be introduced related to a finite number of parameters which
allow for the description of the softening behaviour. The choice of these parameters can be made only
by an indirect identification procedure which requires the investigation of a BVP (boundary value
problem). It must be noted that, according to this procedure, the defect distribution, as well as the
characteristic length introduced to avoid mesh sensitivity, also affect the mechanical test response.
The second identification procedure starts from the hypothesis that the three different failure modes
highlighted by Nelissen3’ could be associated to three different types of mechanical behaviour, each
also identifiable in the softening branch by means of suitable experimental tests (cube specimens or
cylindrical specimens where any shear band has been observed and P.I.E.D. test). In this case BVPs
are helpful in checking the defect influence and the parameter role, like the characteristic length, in
fracture propagation.
In principle both identification procedures can be adopted, but the proposed model is conceived to
describe concrete behaviour according to the second identification procedure. It is assumed that any
concrete fracture is a combination of these three basic failure modes (uniaxial tension, uniaxial and
biaxial compression), Each of them involves a different damage evolution law. Damage is controlled
only by a function of the principal tensile strains and, in particular, this function is defined according
to a nonlocal kinematic description of the displacement field. According to Pant a~opoul ou,~~ the cc
model follows the strainbased criterion. For this reason
it
is mathematically simple. As such
it
allows
for a gradual reduction of strength with increasing transverse tensile strain, which is consistent with
the biaxial stress experiments, and enables the stiffness of the confining device to be taken into
account. Besides fracture modes, crushing is also taken into account due to the high value of
compressive principal strains: assuming that the contact area which transmits the stress does not
change, it involves only permanent strains and any influence on material stiffness is not taken into
consideration. It is controlled by an internal variable
(s,
see equation
(19)),
which can be regarded as
a kind
of
damage associated
to
a volume contraction, owing to a pore collapse. This choice agrees
with the experimental evidence:38 thus ‘failure may be closer to plastic flow and damage is no longer
a result of unstable crack propagation’. As a consequence, in the cc model consistency is imposed ‘a
priori’
in an integrated form only along those radial stress paths which concur with the four failure
modes mentioned above.
As is well known, a uniaxial compression test shows significant transverse strains, of which only a
small part is reversible. This reversibility is assumed to be related to an increasing Poisson’s
coefficient
v.
A clear description of its evolution is made possible by the introduction of another
internal variable
6,
related only to damage in compression: this assumption neglects any variation of
v
for the tensile test because of the lack of test results. It is interesting to notice that the model presented
here remains elastic and isotropic, when the yield functions which control the evolution
of
the three
internal variables
(D,
6,
s)
are not activated. It considers the elastic constants
E,
v
associated with
2
interval variables
( D,
6)
which control their evolution for damaged states.
By contrast, the
CRUSHCRACK
325
introduction of irreversible strains allow
us
to describe the relative induced anisotropy. The modulus
of these permanent strains is expressed as a function of damage: this choice also simplifies the
identification of each damage evolution law associated to the three different failure modes.
Concrete shows rough cracks capable
of
transmitting substantial interface stresses even when their
openings are large,39 owing to aggregate interlock action. Consequently, a model capable of fitting
this behaviour has to follow the mechanical response up to the macrocracking phase. Obviously, this
implies the modelling of the mechanical behaviour of a macroscopically cracked material region and
so
characterized by a discontinuous displacement field with a sort of ‘generalized’ continuum. In
fracture mechanics, damage is regarded as a suitable means of describing mechanical behaviour and
stresspatterns until the peak load, which coincides in general with microcracking, leaving the
description
of
macrocracking and its propagation to
LEFM
(linear elastic fracture mechanics), as
pointed out by Mazars,’’ Mazars and Bazant?’ and
Mazars
and PijaudierCabot!’ On the other hand,
the need to model the shear behaviour of R/C structures where aggregate interlock plays a significant
role, has induced several authors to take into account reduced shear stiffness, smearing the localized
interface displacements on the crackdistance!* At the same time Ha~sanzadeh~~ and Nooru
M~ h a me d ~ ~ have experimentally investigated the complex onset and propagation of cracks when
mode
I
and mode I1 interact, which involves pathdependency. The tensor
E$+(x),
introduced in the c
c model (equation
(12)),
provides an average measure of the discontinuities located along the cracks:
a
sort
of a macroscale kinematic compliance is assumed to maintain a traditional approach by means
of
F.E.
with the displacement field described through polynomial functions
E
C,,.
The irreversible
strain tensor is assumed to be nonlocal and related to the gradient of a potential function of the non
local strain invariant
E+.
The result is a model with ‘rotating’ cracks, but the crack orientation is
related to the strain path according to Ha~sanzadeh.~~
3. ANALYTICAL DESCRIPTION
The model considers damage
(D)
to be an isotropic, internal, scalar variable, as in Mazar’s previous
m~d e l:’ ~ this choice is a simplifying assumption rather than an accurate description
of
the behavior
of
concrete. Damage represents the reduction in the area capable of transferring stress which decreases
with the propagation of microcracking, and decreases even more with macrocracks.
It
is associated to
the stiffness controlled by Young’s modulus
E.
The volume also changes as the cracking process
develops. The irreversible increase in volume is described by means of irreversible strains strictly
related to damage, while the reversible part is associated to the evolution of
v,
Poisson’s coefficient.
Its evolution, which is assumed to be monotonic in order to respect thermodynamic principles, is
described by another internal variable 6, and can reach a maximum value fixed lower than 0.5 to keep
the elastic stiffness matrix isotropic. The small strain and displacement assumption and isothermal
processes are considered. Having introduced the inelastic strain tensor rate
&$,
which may be split into
a nonlocal tensor related to cracking
k;+(x)
and
a
local tensor related to crushing
&:(x),
and
assuming the additive cumulation with elastic strains k;,
E..
r/
=
E!!
r/
+
&i!
r/
=
&?!
r/
+
&I!
r/+ r/
+
&I?
=
&;
+
&;
(2)
the stressstrain relationship becomes
According to this scalar description of damage, it may be assumed that the concrete is isotropic up to
failure in its mechanical constitutive behaviour, but it
also
shows induced anisotropy owing to
inelastic strains
E;.
326
M. DI
PRISCO
AND
J.
MAZARS
The internal variables
D
and
6
range from
0
for the virgin material, to 1 and
(vlim/vO

1)
respectively at asymptotic failure
( E ~
+.
CO).
A limit is necessary for positive inelastic nonlocal
strain components: this limit can be established for rough cracks in terms of an average quantity
which guarantees the interlocking of the discontinuities and consequently the transmission of stresses
with consequent elastic strains. It can be roughly estimated as
max
E$
<
d,/21,
where
d,
is the maximum aggregate size and
I,
is the characteristic length of the material.' Beyond
this limit,
D
is imposed as equal to 1, which means stiffness vanishes, showing that the model in its
present state cannot describe the unilateral character of concrete.
In
the case of strainsoftening, spurious localization and lack of mesh objectivity may occur. To
avoid these unrealistic features, the approach of treating the elastic part of the strain as local,
suggested by PijaudierCabot and Bazant,' is adopted here.
The nonlocal equivalent strain
E+,
which is mainly related to opening mode 1 in microcracking
and to mixed modes
1
and 2 when macrocracks appear, controls the growth of all the damage laws
associated to the three kinds of failure indicated by Van Mier.20 This nonlocal invariant remains the
key to the model, as in the previous one," and is defined asI6
E+(x)
=
__
E+( s) ( P( s

x)
dV
V,(x>
'J
v
with
E:
are the principal strains defined as in (2);
cp(s

x)
is the weighting function
E
C,;
and
V,(x)
is the
representative volume at point
X:
As specified by Bazant and PijaudierCabot,6
K
can be taken as being equal to
fi,
2
or
( 6f i ) 1'3
respectively for
lD,
2D
or 3D problems; 1, is the characteristic length of the nonlocal continuum and
is proportional to the smallest size of the damage localization zone.
An initial elastic domain characterizes the model and is represented by all the points in the
principal strain space inside the surface:
(9
1
F(,E+)
=
E+

Et()
GO.
According to a formalism, which is similar to the associate plasticity one, when the surface is reached
the first time
( F
=
0),
damage appears and the yield surface is described by
F
=
F,
U
F2:
FI[E~;., D,
8,
~,(ci/),
~ t ( 0 ~ ) l
=
@$,(E+)
+
utDt(E+)

DdO,
(10)
where
U,,
a,,
D,,
D,,
6,
will be defined in the following.
CRUSHCRACK
The evolution is described by
327
where
c
takes into account the confinement for multiaxial compression stress states,f, and& are two
parameters which have to be identified and the
( )
brackets indicate the negative part of the
expression placed within them [i.e.
(x)
=
0
if
x>
0;
(x)
=
x
if
x
<
01.
Damage is split into three parts according to the three failure modes mentioned above:20
D
=
acDc
+
aPt
=
a,(a,)(P

a(&;)lD:(E+)
+
[a($)

W 3 E + ) J
+
at ( ~*,>W+)
(14)
where
DL(E+)
is the damage evolution law associated to a perfect biaxial compression (only one point
in the biaxial domain

<
0)
where only one principal strain is not negative;
DL'(E+)
is the
damage evolution law corresponding to a prevalently uniaxial compression behaviour where two
principal strains are not negative;
D@+)
is the damage evolution law associated to uniaxial tension.
The
a,,
at
and
q
factors are expressed as nondimensional functions of the principal strains.
=
Adopting the notation:
(E:)+
=
E::
+E:,
(1
5 )
in which
E:
are the positive elastic principal strains owing to positive stresses and
E:!
are the positive
elastic principal strains owing to negative stresses (Poisson's effect),
a,
and
a,
are defined as:''
with
Function
&( E+),
function:
is related only to damage in compression
D,(E+)
according to a trigonometric
M

vo
6,
=
~
[sin2($n)@(E+)],
vO<M
<
0.5,
VO
where
M
and
n
are positive constants.
As mentioned above, crushing in compression is taken into account too, but following the
assumptions of the model, concrete behaviour is modelled by means of irreversible strains without
any influence on damage. The uncoupling between crushing and cracking permits us to take into
account the former as a local process because it does not cause softening; thus, to simplify the
FE
implementation, a local description was preferred in spite of the previous choice.4s
To
'symmetrize'
the model
a
function,
G( L),
is
introduced:
328
M. DI
PRISCO AND J. MAZARS
0.0005
I
I

0.001 0
0.0010 0.0005 0.0000
0.
a05
Figure
3.
Evolution of the initial elastic domain for radial paths in the principal strain space.
where
and
E,,,
is the threshold below which the behaviour is perfectly elastic.
The evolution is controlled by the interval variable
s
defining
s =
E!!

E,~:
g,
,
g2
are parameters which have to be identified. The identification of
g(E_)
is related to the fourth
kind of failure, typical of hydrostatic processes.
It is interesting to remark that initial elastic domain may be performed in the principal strain space,
but for a general path the damage yield surface belongs to a mixed stressstrain space. The
coefficients
a,
and
a,
remain constant when radial stress paths are followed.
The uncoupling guaranteed by the introduction of
E;
=
E~

E;
implies a mixed hardening of the
F,
U
F2
U
G
domain in the principal strain space (Fig.
3),
and it may be noted that the model is non
associated for a general path.
4.
IDENTIFICATION PROBLEMS AND COMPUTATIONAL ASPECTS
The model is based on the description of the evolutive laws for damage related to the three failure
modes taken into account:
D$+),
or'(;+),
and
Dt(Ei.).
Knowing the constitutive relationships
a,
=
E,
in uniaxial tension and compression and in biaxial
compression tests
(a,
=
a2;
a3
=
0),
suitably assumed to be homogenous (thus
E+
=
E+),
it is
possible to identify the damage curves
Di(E+)
considering the evolution of irreversible strains owing
to crushing and damage.
CRUSHCRACK
329
The required constitutive curves can be assigned by points or by choosing any kind of analytical
curve with different numbers of parameters; in particular, it is possible to adopt the same expressions
as Mazars:
There are eight parameters in total besides the two elastic constant
E,
v,
including the
two
thresholds
and which limit the elastic behavior. If the experimental constitutive curves
are
reliable and
available as a point set, the model allows a precise description of the concrete behaviour.
The description of the damage evolution laws can be computed point by point owing to the
integrated form for these particular paths
(a,,u,
constants, see Appendix 1). More refined
interpolation tools can be used if the input experimental curves are not smooth enough.
In addition, the six parameters
fi
,f2,
g,
,
g,, M,
n
(equation
(13),
(18),
(21)) which describe the
potential functions associated to cracking and crushing should be identified, provided that the default
values are not chosen.
and the
increments
A E ~,
the algorithm computes
As,
A&;,
AD,
AB,
and only if
AD
or
A6
are positive, the
loading condition is active, and if
AD
>
0,
A&;+
are computed. The algorithm is not explicit because
D
depends on
a,
and
ut
which are related to the stresses
ay
and
6.
If the internal convergence is not reached in a selected number
of
iterations
(50
as a default), the
coefficients
a,
and
U,
are assumed to be equal to those of the previous step. It is interesting to notice
that for several paths
U,, U,
coefficients do not change and the algorithm becomes explicit and can be
computed very rapidly.
For the general path the convergence test is based on the vaiues
AD,
A6
which depend on
a,, at
and
q.
Furthermore, having established
E ~,
A E ~,
the nonlocal variables
E+,
aF/aEk,
are evaluated only
once for each step.
Having established the internal state variables
E;,
D, 6,
s,
E;+,
the total strains
5.
NUMERICAL SIMULATIONS
In the following, several experimental tests are simulated. In order to show the reliability of the cc
model and its sensitivity to the various parameters introduced,
two
groups of problems are
investigated.
The first group deals both with tests which can be regarded as reasonably homogeneous (such as
PIED
tests, cube or cylinder specimens where any shear band was observed) and tests which have
been homogenized.
The second group deals with boundary value problems in a plane strain condition, such as uniaxial
tension and compression tests on parallelepipeda with an internal imperfection and shear on four
notched specimens.
For the whole set of tests referred to above, the parameters which describe the inelastic potential
functions
(fi
&,
g,
,
g2)
as well as those describing the evolution of the internal variable
6
( M,
n)
are
assumed to be fixed. The chosen values are the following:
fi
=
0.3;h
=
1.6; gl
=
130;
g2
= 14;
M
=
0.45;
n
=
1.1.
5.1
Homogeneous tests
Uniaxial tension
Let us first consider the
PIED
test briefly described before in Fig. 2. The limits of the homogeneous
stress field assumed have been highlighted by F ~ k w a.~ ~ Taking into account some results obtained on
330
M.
DI
PRISCO AND J. MAZARS
low strength concrete,33 it is possible to follow the mechanical behaviour by directly introducing the
experimental points of the uniaxial tension curve as an input for the model (Fig. 4a). The unloading
branches are affected by the two parametersfi andf2 which appear in the function f(D) (equation
(13)): the parameter sensitivity is not high and the comparison shows the inability of the model to
describe the unilateral behaviour of concrete. Figure 4b highlights the evolutive trend of positive
elastic and irreversible strains, and damage.
Uniaxial
compression
With reference to a uniaxial compression test on cylinder specimens, where only cracks parallel to
the loading direction were observed33, Figure 5a shows the perfect fitting of test results with reference
to the envelope curve.
As
in the previous case the points are taken into account as input data and a
simple sensitivity analysis is proposed considering a small variation of the parameters
gl
and
g2
which control the evolution of the irreversible strains in compression
E;.
The evolutive trends of
damage, elastic and inelastic strains in compression are shown in Figure 5b. Figure 5c performs a
comparison in terms of reversible transverse strains: Poisson's coefficient fits the experimental
evolution up to failure quite well only if a drastic softening can be taken into account.37 It is
important to examine the transverse irreversible strains: they are the most significant component for
modelling the dilatancy effect in geomaterials like concrete. Figure 5d compares numerical and
experimental results: even here only a drastic softening allows a
fit
of the experimental results up to
large strain values, avoiding the arrest of the numerical curve at values which are too small (less than
15
x
lOW3).
A
variation off2 (orfi) can also seriously affect the comparison, providing large strains
even if no drastic softening is taken into account.
The good reliability of the proposed model is shown also in Figure 6 with reference to Van Mier's
tests on cube specimens;*' the damage evolution laws are described by means of Mazars' analytical
expressions
E,'
=
1
x
l OP4,
E,'
=
5
x
104;
E
=
40000 MPa,
v
=
0.2).
The uniaxial compression curve
(a,

E ~ )
assumed as
an input is quite close to the envelope given by Van Mier and corresponds to a maximum strength
equal to 44 MPa. The model fits the test envelope of the transverse strains reasonably both in the
curve
(al
c2;
Fig. 6a) and in the evolution of the volumetric strain
( E ~

eVol;
Fig. 6b).
(A:
=
1,
BL
=
750;
,I:
=
1.4,
B:
=
1750;
A,
=
0.8;
B,
=
20000;
1.2
0.8
0.4
0.0
0.4
0E
304
Figure
4.
Uniaxial tension according to
PIED
test:33 (a) fitting of test results and sensitivity analysis forj; parameters;
(b)
strain
components and damage versus normalized tensile
stress.
CRUSHCRACK
33
1
501
,
,
,
E
1

60

5.0E 003

2.55
003 0.0€+000
0.5
Y
0.4
0.3
0.2
0.1
 0
a...
S
a,
':
S
?,
\
E

damage
0.2

4E
003

2E
003 0E+ 000
0.002
I/
,
,
, ,
,
, ,
t2
0.080
.000 0.0@2 0.004 0.1
I
06
(b) strain Figure
5.
Uniaxial compression: (a) fitting
of
Ramtani's test results and sensitivity analysis
for
gi
parameters;
~
,
components and damage versus normalized compressive stress; (c) evolution
of
reversible transverse strains; (d) evolution
of
irreversibie transverse strains: only the numerical investigation withfi
=
I
.6
is extended increasing
E,
according with the
proposed softening shown in Figure 5c.
Biaxial and Pluriaxial behaviour
The biaxial domain for radial stress paths which initiate at the origin of the principal stress space is
shown in Figure
7.
The concrete mix used for these tests is the same as in the uniaxial compressive
test, even if the maximum strength has a small scattering (perhaps owing to a different loading age).
Nevertheless, exactly the same material constitutive parameters adopted in the simulation of the
uniaxial compression tests investigated by Van Mie?' reported above (Figure
6 )
are considered. The
domain is not convex, but this feature is irrelevant to the cc model: the three failure modes
associated to cracking and damage evolution are easily recognizable. The domain fits the
experimental tests well without the introduction of any specific parameter and is very close to that
proposed by Kupfer. It is also interesting to note that it is pathdependent, because the damage growth
is controlled by the evolution of
a,,
a,
and
q.
If the third stress
a3(a3
=
pal
>
a2
=
aaI
>
al)
becomes negative, obviously maintaining the same constitutive parameters, the biaxial domain in the
332
M.
DI
PRISCO
AND
J.
MAZARS
Figure
6.
Uniaxial compression: (a) fitting of Van Mier's'' test results (tests 3A/3B) in the loading and transversal directions;
(b) fitting
of
volumetric strain.
principal stress plane
0,

oZ
grows reasonably larger (Figure
8)
as highlighted by Van Mier.20 The
computed domains for
p
#
0
are not symmetric because the planes at constant
,!?
are not at right
angles to the hydrostatic axis. Even the stressstrain curves for constant stressratio appear realistic as
shown in Figures 9a,b.
At the end, to check the reliability of the cc model for highhydrostatic compression stresses too, a
comparison with tests by Bazant
et
~ 2 1.~ ~
was performed. The damage evolution laws are
described, once again, by means of Mazars' expression
(A:
=
1.2,
Bf:
=
800;
A:'
=
1.5,
l?:'
=
1700; A,
=
0.9,B,
=
20000;
ct0
=
9.35
x
105,
cC0
=
1.2
x
103; E
=
31000 MPa,
v
=
0.2)
starting from the cylinder compressive strength
(f,
=
35 MPa) and the
EC2
formulae for Young's
modulus and the tensile strength. Figure 10 fits the test results reasonably and highlights the
sensitivity to the parameters
g,
and
g2.
10
I
I
30
40
50
60
(MPa);
 7  7
I I
8
i

70 60
50
40
30
20 10
0
1
3
Figure 7. Biaxial domain: fitting of Van Mier's tests and comparison with Kupfer equations (test series 5A/5B/6A)
CRUSHCRACK
120
160
333
.....
test a=OO
a=O
1
*a*+.
test
a=O
10
...
 
a = ~
33
model
..~AA
test
a=o.33
(52

Gc
E1
~ t ~ r ~ i ~ ~ m i m

I
0

25


50

75

100
3
0.1
( MP ~ )
fC=46.6MPa
A
exp.
tests
.
p=o
3
Figure
8.
Biaxial domain with confinement in the third direction: fitting
of
Van Mier’s tests (series 8A/SB/9A/9B)
Shear behaviour
Mixedmode fracture in concrete has been investigated by several authors in the literature with
reference to either crack propagation and aggregate interlock. Three broad categories accept the
smeared crack concept: the fixed, fixed multidirectional and rotating crack approaches. The problems
related to these approaches were well analysed and discussed by de Borst and N a ~ t a ~ ~ and de B o r ~ t.~ ~
A
specific test in order to compare the proposed model with these three broad approaches is the
tensionshear test originally suggested by Willam and further investigated by Weihe and Kroplin.”
The primary crack in the homogeneous test is initiated through uniaxial loading under displacement
control. Subsequently the loading is applied in terms of strains with a continuous rotation
of
their
principal component directions according to the following ratios
E,
:
iYy
:
i,..
=
0.5
:
0.75
:
0.5.
In
F/
p=o
1
f

 *
E3
160
8
I I
0.00
0.01
0.02
0
3
Figure
9.
Triaxial constitutive behaviour for constant stressratio experiments
a,/a,
= 0.
I,
and
oz/a,
=
a,
with
a
=
0,O.
1
and
0.33:
fitting of Van Mier’s tests (SB14,8A25,9B15).
334
M.
DI
PRISCO
AND J. MAZARS
0.

0.
 1.
 1.
 2.
gl =l 63;g1=10.
I
30
Figure
10.
Hydrostatic compression: fitting
of
Bazant, Bishop and Chang's4'
test
results and sensitivity analysis
for
g,
parameters.
this numerical simulation, the damage evolution laws are described by means
of
Mazars' equations,
based on the data provided by Weihe and Kroplin50
(A:
=
1,
BE
=
1000;
A:'
=
1.0
BY
=
1850;
A,
=
0.9,
B,
= 20000; =
1
x
104,
cC0
=
1
x
103;
E
=
10000
MPa,
v
=
0.2 which
correspond to a maximum tensile strength
f;
of
1
MPa). The crushcrack (cc) model shows
a
behaviour (Figure 1 la,b) which can be regarded as a
sort
of
fixed multidirectional crack approach,
without the introduction
of
any specific parameter or angle. It is interesting to note that the model
is
capable
of
describing induced anisotropy caused by cracking; thus the principal directions
of
stress
tensor do not generally correspond to the principal directions
of
global strain tensor. However, in the
test under discussion, the principal directions, stress and strain, do not rotate significantly compared

proposea model
carnage
. . . ... .. . ..
'a
*a
fixed
crock
E
XY
0@?+=~7
2E
004
4E
0.8
1.5
0.6
1.0
0.4
0.2
a.
5
0.0
04
O@?.
no,
x
3
E,
O1
D

Drooosed
model
1.0
0.5
0.0
04
Figure
1
I.
Willam's test: (a) shear behaviour and damage evolution;
(b)
principal stress versus principal strain according to the
proposed model and
to
fixed and rotating crack models as suggested by Weihe and KroplinSo and scalar product between the
maximum principal strain and stress rotating components.
CRUSHCRACK
335
with each other (as may be seen by their scalar product in Figure
1
lb), even if they do rotate with
reference to a fixed coordinate system.
Examining Hassanzadeh’s tests performed on a foursideedge notched ~pecimen?~ with reference
to parabolic paths, the reliability of the crushcrack model is demonstrated once again. The specimen
is first tensioned up to the peak strain; then a parabolic path in terms of displacements is imposed
(6,
=
PK).
Assuming a smeared softened region whose depth is assumed to be equal to the
characteristic length
I,
= 3da
(d,
=
8
mm,
the maximum aggregate size), a comparison can be
performed in terms of homogenous strain (Figure 12a). Once again Mazars’ expressions are adopted
lOP4,
= 103;
E
=
32900 MPa;
U
= 0.2) to reproduce the experimental cube compressive
strength
(S,
=
50
MPa); the other material characteristics were estimated by means of EC2
formulae. The cc model shows a stiffer behaviour and a greater shear capacity compared to the
homogenized tests (Figure 12a). In any case, the shear constitutive relationship remains comparable
with other empirical relations suggested in the literature. Figure 12b shows the evolution trend
of
elastic, inelastic strains in tension and compression, damage and Poisson’s coefficient.
(A:
= 1.2, Bf
=
800;
Afl
=
1.5,
Bil = 1.5, BL’ = 1700;
A,
=
0.9,
Bt
=
0.9Bt
=
20000,
Em
=
8.2 6 ~
5.2 Boundaty value problems
To highlight the limit
of
the nonlocal approach adopted, let
us
now examine the behaviour
of
a
concrete strip in plane strain subjected to uniaxial tension. While in the previous tests the nonlocal
strain invariant
E+
is equal to the local one
E+,
the strip mechanical response is characterized by the
appearance of a softened region with an inhomogeneous strain pattern and thus
E+
#
E+.
Different
equations exist in the literature to describe tensile softening in concrete (Figure 13). Usually they
need one or two parameters, which have to be added to the characteristic length. The input data are
the two elastic constants
(E,
U),
the tensile peak strengthA, which can be used in order to identify the
threshold if an elastic behaviour is assumed up to the peak, the fracture energy and the maximum
Figure
12.
Hassanzadeh’s tests43 assuming
an
homogeneous behaviour in the localization band: (a) proposed model compared
with empirical equations proposed by different authors; (b) strain components, damage and Poisson’s coefficient versus
normalized shear stress according
to
the proposed model.
336
M. DI
PRISCO AND
J.
MAZARS
aggregate size. It is interesting to observe the strip response (Figure 14), if Mazars’ equations are
taken into account (A,
=
0.8,
B,
=
20000,
E
=
32860 MPa,
v
=
0.2;J;
=
3.1 MPa;l,
=
12
mm)
and
a narrow central region (with a length
t =
1 mm) characterized by a threshold
&To
=
0.8 ~ ~ ~
is
considered in order to introduce a defect. Figure
14
shows
also damage and strain evolution versus
vertical displacement increase. The introduction of the nonlocal invariant
E+
prevents mesh
sensitivity (Figure 14a,b), but the increase of the vertical displacement also induces an increase in the
softened region’s depth. The unrealistic diffusion occurs when the second flat branch of the
constitutive relationship (which describes the crackopening) is reached. The same feature is shown
by other softening constitutive relationships, a5 Figure
15
shows analysing the increase of the
damaged Gauss point number versus the vertical displacement
U,
with reference to the same strip
mesh (the fine one) and the constitutive behaviour pointed out in Figure 13. This phenomenon also
causes some problems in the mixedmode crack propagation as specified in the following.
As
suggested by Carmeliet and de Borst,” the most reliable means
of
identifying the best tension
relationship remains perhaps the evaluation of the structure response in order to fit the experimental
test of pure tension and,/or bending tension. Figure 16 shows the modelling of one of Hordijk’s testss2
in tension; Carmeliet’s equations’ (A,
=
0.98,
B,
=
2400) has been adopted for the description of the
uniaxial tension curve, while the material parameters are the same as in the strip example. The
characteristic length 1, was evaluated according to Carmeliet’s proposal in order to maintain the same
localization zone depth 1,
=
6.93
mm).
The strip and the 2Dstructure show a similar mechanical
response. Both structures cannot be assumed to represent a real modelling of Hordijk’s test: the strip
assumes a different geometry and a different defect and neglects the biaxial behaviour, while the 2D
structure, assuming a different total length and a vertical symmetry, changes the real boundary
conditions. Nevertheless they fit the test results quite well. The mesh does not significantly affect the
results of the 2Dstructure either in terms of global response (Figure 16a) or in terms of damage or
Mozars
1
(A,fo.8;
El=7000)
bCr M4Mozar s
2
A0.8:
Bl=20000)
Hordirk
(G,=llON/m;
w=12rnm)

C o k e l i d
’
&=0.90;
8,=6CO)
QwoEl
Capmeliet
2
[&=0.98;
81=2400)
t*tt1;
Lirieor
0.000 0.005 0.010 (50.’
15
Figure 13. Concrete
strip
subjected
to
uniaxial tension in plane strain: (a) different suggested equations from literature.
CRUSHCRACK
337
I.
1.0
1.0
.O
5.
10.
15.
20.
25.
30. 35.
40.
.2
u=10
(p)
.O
5.
10.
15.
20.
2s.
30.
35.
40.
0.
Figure
14.
Concrete strip subjected to uniaxial tension in plane strain according to Mazars' equation
[2]
(Figure
13):
(a) tensile
load versus longitudinal displacement
U;
(b)
damage evolution for coarse (123 d.0.f.) and
fine
(235
d.0.f.) meshes (qua8 9 Gauss
point element); (c) longitudinal strains compared
to
nonlocal invariant strains and
(d)
longitudinal irreversible strains for h e
mesh.
stresses (Figure 16c,d). It is interesting to note that the local stresses
oW
at the notch are larger than
the peak tensile strength owing to the nonlocal approach followed.
Figure 17 analyses the structural behavior in uniaxial compression (plane strain), with the same
parameters adopted by Mazars to perform Van Mier's tests, when a defect is introduced. The
characteristic length
( I,)
is taken as equal to 12 mm to keep a reasonable value. The defect, with a
nonsymmetric location (Figure 17d), is obtained by a reduction in the compressive strength
(
10%).
The deformed shape is asymmetric and it is possible to observe the onset
of
one shear band which is
initially zigzag, and then skew, and is concentrated in the upper region. In fact, while the defect
seems to pilot the evolution of damage up to the point
(B)
just after the peak, the softening branch
seems controlled by the boundary condition of the point
P
(Figure 17d). This boundary condition
338
M.
DI
PRISCO
AND J. MAZARS
P
Plane
strain
cio
=
o.8ct0
Isoparametric 8node
elements
(9
Gauss
points)
I
Gauss
300
Damaged
Gauss Point Numbe.

200

150
100
Figure
15.
Concrete strip subjected to uniaxial tension in plane strain: extension
of
the damaged zone according
to
Hordijk and
Carmeliet’s constitutive relationships quantified
by
means
of
the number
of
damaged gauss points.
imposes a symmetry along a vertical axis, only in the bottom region of the specimen allowing the
formation of the diagonal asymmetric band
in
the top region. Figure 17e also shows the negligible
influence on the damage in the postpeak region of the finite element type: two different meshes with
different elements and the same number of nodes present in the same mechanical behaviour, even if
the number of integration points in the quadrilateral, 4node elements is twice that of those in the
triangular, 3node elements.
With reference to mixedmode fracture, we now analyze Hassanzadeh’s tests (described in the
previous chapter) as a structural problem, by means of the mesh drawn in Figure 18a and using
Mazars’ constitutive equations with the same parameters adopted in the test considered previously.
The characteristic length is chosen as 12 mm because
it
is a reasonable value which prevents
a
meaningless onset
of
damage.53 The initial stiffness of the specimen changes drastically compared to
the homogenized tests (Figure 12a), fitting the parabolic path characterized by
fl
=
0.6
mm”’
very
well, and reasonably for
fi
=
0.4 mm”’ (Figure 18b). In fact, though the initial stiffness is quite
different, only a few microns in the horizontal displacement
(U
component) are sufficient to reproduce
the same trend. It is important to highlight that the comparison curve, chosen as an experimental
reference, interpolates the test results reasonably, but the curve is not experimental and the domain is
very close to the origin. Damage distribution and shear stress pattern are shown in Figure 18c,d. An
horizontal damaged zone highlights the onset of the crack owing to the vertical displacement. The
subsequent parabolic path shows a stress concentration close to the notch. For larger displacement
values, some problems investigated using the nonlocal approach chosen were highlighted by the
same authors before53 and research on this topic is in progress.
CRUSHCRACK
339
4
P

(MPo)
I c
=
6.93rnm
A
1
PP
I1
,
Uniaxial
tension
S
A
= Cross section
orea
1
0
3
JI
8.50
0.03
25
mm
5
0* 05
(a)
"t
(a
coarse
mesh
A B C D E F
00
80
1 6 2 4 3 2
40
A B C D E F G H I J h L M
00
OS
10
I5
20 25
3U JS
40
45
511
5S
W
G H
I J K
4 8 5 6 6 4 7 2
8 0
fmc
mesh
Figure
16.
Uniaxial tension in a structure according
to
Carmeliet's equatlon
121:
(a) normalized tensile load versus imposed
vertical displacement (thickness= 1 mm); (b) fine
( 1
100
d.0.f.) and coarse
(644
d.0.f.) meshes adopted in the numerical
analysis; (c) damage
D
and vertical
stresses u.~
at impending localimtion
(t.
=
3
pm);
Id)
damage for
i.
= 20
pm.
6.
CONCLUDING
REMARKS
The proposed model, based on the uncoupling of cracking and crushing phenomena, can be
considered a reliable tool
for
the analysis
of
complex structural behaviour
in
concrete
for
prevalently
monotonic loading paths.
Damage is assumed to be related only to cracking: an equation affected by two parameters
correlates it with the irreversible strain rates.
As
suggested by some experimental tests three failure
modes are taken into consideration: each failuremode is associated to an evolution damage law. The
model describes the triaxial behaviour of concrete quite well, and combines the three damage
evolution laws by means of suitable coefficients dependent on stress and strain states. The
340
M.
DI
PRISCO
AND
J.
MAZARS
1,=
12
mm
Damage
at point
A
Uniaxid campression
D
A 24
B
29
c
33
D
37
E
41
F
45
G
50
H
S4
J
62
L
71
M
75
N
79
I
sa
a
I
0
defect
 0
A
.23
(a)
A
.I9
B
.25
B
.22
c
2 7
C
.25
D
.30
D
.28
E
.32
E
.31
F
.34
F
.33
G .36
G
.36
H
.3a
H
.39
I .41
D
I
.42
K
.45
K
.48
L
.47
L
.50
M
.so
U .53
N
.52
(e)
N
3 6
Damage
at point
B
Figure 17. Uniaxial compression in a structure according
to
Mazars' equation: (a) global response in plain strain
(A
=cross
section area);
(b)
deformed shaped
for
point
A
and
B
respectively (c) final damage (qua4 element mesh); (d) geometry
(thickness
=
1
mm), boundary conditions and meshes adopted; (e) influence of the element type on damage in the postpeak
region (point
B).
identification of these evolution laws can be done by adopting different expressions proposed in the
literature or the experimental test results themselves. The calibration procedure can be performed
easily, if three tests (uniaxial tension and compression and biaxial compression) are available.
Otherwise, reasonable results can be obtained adopting only the uniaxial compressive tests, and
estimating the other material parameters from it. The model is able to reproduce pathdependency,
dilatancy and inducedanisotropy due to cracking. The nonlocal approach followed allows
us
to
avoid meshsensitivity owing to softening.
The reversible change in volume
is
associated to another internal variable which is correlated only
to damage in compression: the model fits test results reasonably well.
CRUSHCRACK
34
1
Figure
18.
Hassanzadeh's tests (1991): (a) mesh adopted in plane strain (thickness= 1
mm);
(b)
shear stress versus shear
displacement at very small displacement values; (c) damage and (d) shear stress pattern
for
v
=
20
pm.
Crushing is associated with an internal variable which evolves with a strain invariant through the
compressive total strains only.
A
simulation of high hydrostatic stress states is thus possible.
Shear behaviour is studied with reference to mixedmode crack propagation and aggregate
interlock for large crack openings: although research is still in progress, the model shows a behaviour
similar to a fixed multidirectional approach, without any introduction of specific empirical laws or
suitable coefficients like the retention factor.
Owing to its general formulation, the crushcrack model might also be able to fit HSC and FRC
structural behaviour, thus becoming a powerful tool for the investigation of concrete structures.
ACKNOWLEDGEMENT
The financial support of the European Community to Alliance of Laboratories in Europe for
Research and Technology

Geomaterials of C.E.C. in the framework of the programme Human
Capital and Mobility
is
gratefully acknowledged. We also
thank
Pierre Pegon for his invaluable
assistance in the implementation of the model in the Finite Element Code Castem
2000
and the Joint
Research Center in Ispra for its kind support.
M.
DI
PRISCO
AND
J. MAZARS
NOTATION
confinement coefficient
crushcrack model
maximum aggregate size
scalar, internal variable damage; functions describing damage in compression and
in
tension
damage in biaxial compression and in uniaxial compression
Young’s modulus
functions describing the evolution of irreversible strains in tension and compression
uniaxial, cube and cylindrical compressive and tensile strength
cracking and crushing yield functions in the strain space
yield functions describing damage and reversible volume change
characteristic length according to nonlocal damage approach
internal variable associated to crushing
weighting coefficients describing the evolution damage law
internal variable describing the irreversible volume change; function describing the
evolution of
6
in compression
finite increment
total strain and strain rate
elastic and irreversible strains
strains affecting cracking
irreversible strains related to cracking and crushing respectively
strain invariants
strain thresholds bounding the elastic domain
weighting coefficient describing the evolution of damage in compression
weighting function introduced in the nonlocal approach
Poisson’s coefficient; initial and asymptotic values
stress tensor
APPENDIX. IDENTIFICATION PROCEDURES ALONG THE ‘THREE FAILURE MODES
INDUCING CRACKING
In the following tables the identification procedures for the evolution damage laws
Dr(E)
are briefly
resumed, performable when the constitutive relationships along the three failuremodes related to
cracking are
known.
It is interesting
to
notice that while uniaxial compression shows an explicit
algorithm, biaxial compression and uniaxial tension have to be solved iteratively. Also in these cases
the computation is very fast.
CRUSHCRACK
343
no
I
11
Newton's method
(kth iteration)
no
fi
U
yes
344
M.
DI PRISCO
AKD J. MAZARS
biaxial compression
(01
=
0 2 )
0
E l
>
E&
yes
I
U
fi
=3
Newton's
method
(kth
iteration)
CRUSHCRACK
345
uniaxial
compression
I
I
I
I
REFERENCES
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J.
Sluys and R. de Borst, ‘Solution methods for localisation in fracture dynamics’, In
Proc.
Conf
on
Fracture Processes
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and Ceramics,
eds,
J.
Van Mier,
J.
G.
Rots and A. BaLer, Chapman and Hail, London, pp. 661471,
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L.
J.
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deformation’. In
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P. Bazant, ‘Nonlocal damage theory’,
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I),
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J.
Engng. Mech.,
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J.
Mazars and G. PijaudierCabot, ‘Continuum damage theory. Application to concrete’, ASCE
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Engng. Mech.,
89(2),
345365 (1989).
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J. Mazars, ‘Damage models for concrete and their usefulness for seismic loadings’, In
Proc.
Conf: on
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Numerical Methods
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Earthquake Engineering,
eds.
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DoneB and P. M. Jones, ECSC, EEC, EAEC, Brussels and
Luxembourg, pp. 19922 1. 1991.
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P. Bazant, ‘Why Continuum damage is nonlocal: micromechanics arguments’,
ASCE
J.
Engng. Mech.,
117(5),
1070
1087 (1991).
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DI
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