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Correlation between the internal length, the fracture
process zone and size eect in model materials
Khalil Haidar, Gilles Pijaudier-Cabot, Jean-François Dubé, Ahmed Loukili
To cite this version:
Khalil Haidar, Gilles Pijaudier-Cabot, Jean-François Dubé, Ahmed Loukili. Correlation between the
internal length, the fracture process zone and size eect in model materials. Materials and structures,
Springer Verlag, 2005, 38 (2), pp.201-210. �10.1007/BF02479345�. �hal-00351757�
Correlation between the internal length, the fracture
process zone and size effect in model materials
K.
Haidar
1
,
G.
Pijaudier-Cabot
1
,
J.F.
Dube
2
and
A.
Loukili
1
(1) R&DO- Institut
de
Recherche en Genie Civil et Mecanique (GeM), UMR CNRS 6183, Ecole Centrale
de
Nantes, France
(2) LMGC, UMR5508, Universite Montpellier, France.
ABSTRACT
In this paper, we examine the correlation between the width
of
the fracture process zone, the parameters entering in the description
of
size effect
(related to the dimension
of
the specimen especially), and the internal length in non local constitutive relations for a model mortar material with a
controlled macro-porosity. Experimental investigations on this material
in
compression, bending, acoustic emission measurements and their
analysis are detailed. The experiments show a good agreement between the evolution
of
Ba2ant' s size effect parameter d
0
and the evolution
of
the
width
of
the FPZ. The internal length obtained with the help
of
inverse finite element analysis
is
also proportional to these quantities. This
correlation provides a reasonable approximation
of
the internal length, from an experimental test on specimens
of
a single size directly, equipped
with acoustic emission localization devices.
RESUME
Dans cet article, nous examinons les com?lations entre la largeur de la zone de microfissuration (FPZ), les parametres entrant dans la description
de l'effet d'echelle et la longueur interne du modele d'endommagement non local
pour
un mortier a macro-parasite controtee. Des resultats
experimentaux
sur
ce materiau en compression, enjlexion ainsi que des mesures d'emission acoustiques
et
leur analyse sont presentes. Les resultats
d'essais montrent une bonne correlation entre !'evolution du parametre
do.
parametre de la loi d'effit d'echelle de Bcdant, et la largeur de la FPZ.
La
longueur interne obtenue numeriquement
par
analyse inverse est aussi proportionnelle a ces parametres. Une bonne approximation de la longueur
interne
a partir d'essais sur une seule taille d'eprouvette equipee d'un systeme d'emission acoustique est aussi obtenue.
1.
INTRODUCTION
In quasi-brittle materials, fracture exhibits a fmite size
fracture process zone (FPZ). Macro-cracking
is
the result
of
progressive material damage
in
which micro-cracks appear
first in a rather diffuse way, and then coalescence occurs in
order to form the macro-crack. The size
of
the resulting
fracture process zone
is
not dependent on the structural size,
provided
it
does not interfere with the boundaries
of
the
considered body.
It
is
controlled by local heterogeneities, and
by the state
of
stress as well. A consequence
of
the existence
of
such a FPZ
is
structural size effect [I]. Since the ratio
of
the
volume
of
the FPZ to the total volume
of
the structure changes
for geometrically similar specimens, a size effect on the
structural strength
is
observed experimentally.
It
is
milder than
the structural size effect observed
in
linear elastic fracture
mechanics (LEFM) but often more pronounced than the
statistical size effect,
e.g. described by Weibull's theory. In
fact,
it
is
a transitional size effect between strength
of
material
theories and LEFM. From the modeling point
of
view, the
description
of
the FPZ has to involve the introduction
of
an
internal length in the governing constitutive equations.
It
can
be in the form
of
a characteristic length which
is
related to the
length
of
the process zone, Irwin's length or the size
of
the
cohesive zone
in
Hillerborg's model
[2],
or
in
the form
of
an
internal length in non local constitutive relations as defmed
among many others
[3].
This internal length can be related to
the width
of
the fracture process zone [4-6]. Those constitutive
theories
do
predict size effect on the structural strength. In fact
a proper description
of
the response
of
geometrically similar
specimens
can
serve as a good method
of
calibration of the
mo
del
parameters,
inclu
d
ing
the
internal Length
of
the
ma
terial
[7,
8].
One
of
the open issues in non local
mod
eling
is
direct
measurement
of
this internal length or,
in
other wor
ds
,
experimental techniques which aim at observ
in
g the
frac
ture
process zone. Measurements
of
displacement
fiel
ds [9,
IO
]is a
possibility but st
ill
it
is
an
indirect
one
since cracking
is
deduced
from
large strain gradients. In some polymers for
instance, cracking
can
be directly observed with optical
tec
hni
qu
es that can hardly
be
devised
if
the
specimen
is
not
trans
lu
ci
d.
X-ray
techniques have also been considered with
success
[11
]. For cementitious materials, this method
is
quite
demanding in term
of
ene
rg
y
of
X-rays
if
one wants
to
achieve
the resolution
tha
t is required
in
o
rd
er to detect small micro-
cracks (with
an
ope
ning that is
of
the
order
of
microns). Other
research efforts have
been
reported
to
estimate the e
xte
nt
of
the
FPZ
in
concrete using
th
e ultrasonic
pu
lse velocity techniq
ue
[1
2,
13
].
Acoustic e
mi
ss
ion
(A
E)
is
also
an
expe
rim
ental t
oo
l
well su
it
ed
for
monitoring
frac
ture processes. The elastic wave
generated
by
cracking
even
ts can
be
measured
and
processed
using seismic
an
alysis techniques. The strength
of
AE
techniques is the
ab
il
ity to
mon
itor microscopic damage
occurring inside
the
mater
ial.
Several works have focused on
relat
in
g acoust
ic
emission characteristics
to
the properties
of
the fracture process zone [14),
and
AE
source localization
analysis to the damage distribution [lS
].
Mihashi et
al.
[16]
applied three-dim
ens
i
onal
acoustic emission techniques
to
stu
dy
the
FPZ.
Th
e results revealed that micro-cracking occurs
randomly around
the
macro-crack and that
the
FPZ expands
after peak
loa
d due
to
the
presence
of
aggregate
s.
O
ts
uka
et
al.
[
17
] s
bo
w that the loca
tion
of
cracks, obtained
by
the
AE
method, has a cl
ose
relation to
the
extent of
the
fracture
process
zo
ne
s observed with X-rays.
The aim
of
this contribution is to examine the experimental
correlation between
the
width
of
the
fracture process zone
observed with AE analysis, the parameters entering in
the
descrip
ti
on
of
size effect, and the
inte
rnal length
in
non
local
constitutive relations.
In order
to
examine such a correlation,
an
experimental program on mortar and model mate
ri
als with
controlled (macro) porosity
is
presented
in
the next section.
Resu
lts
are discussed
in
section
3.
The correl
at
ion
s between
size effect parameters,
the
wi
dth
of
the FPZ
and
the
in
ternal
le
ngth
of
the material obtained
fro
m inverse analysis
of
size
effect tests are discussed
in
section 4.
2. EXPERIMENTS ON MORTAR AND
MODEL MATERIALS
The experimental program has been des
ign
ed in order to
measure the mechanical cha
ra
cteristics of mortar spec
im
ens
with a controlled microstructure
(18
]. This control was
achieved by adding inclusions of weak mechanical
characteristics (polystyrene beads) in a mortar matrix.
Depending
on
the amount
of
inclusions
ad
ded
in
the
cementitious matrix, variations of the elastic properties,
of
the tensile and compressive strengths and of the fracture
energy are expected. Data
on
this simple model material are
also expected to
be
correlated
wi
th homogenization
theori
es.
All test specimens were made with a mix which consists of
ordinary Portland cement CPA-CEMI
52.5
, polystyrene beads,
normal density
fine
s
an
d wi
th
a maximum size
of
2
mm,
a
superplasticizing agent
(G
leni
um
51)
and
water. Expanded
polystyrene spheres
of
3-7
mm
mean diameter were used as
aggregate in the mix desi
gn.
The spheres have a
mass
d
ens
ity
of
20 kg/m3, a Young's
modul
us
of7
MPa
and
a co
mp
ressive
strength of 80 kPa. Expanded
po
lystyrene consists essen
ti
al
ly
of
air 9
8%.
It
is hydrophobic [
19
).
Mi
x preparati
on
is
particularly
impo
rtant when using such very lightweight
aggregat
es.
Expanded polystyrene
te
n
ds
to
flo
at on t
he
top
of
the mo
uld
s during vibration, increasing
th
e risk
of
poor
mi
x
distribut
ion
and segrega
ti
on
.
Based
on Y arnura
an
d Y arnuchi
[20) and
Pe
rry
et
al.
[21
], controlled
mix
conditi
on
s were
achieved by
mi
xing the dry sand
and
cement before adding
the
water
and
th
en the superp
las
ti
cizer. After su
ffic
ient
mi
xing
time (2- 3 minutes), the polystyrene beads were
added
and
thor
ou
ghly
mixed
in
to
the
mortar.
In
order
to
minimize
segregation, vibration
was
avoided, and
all
test specimens
were compacted by hand tamping. F
ig
. l sh
ow
s the
distribution
of
polystyrene beads achieved for the mix with the
lowest density
(1
.4
).
Four d
iff
erent mixes
of
densities 2.0, 1.8,
1.6
and 1.4,
having polystyrene content g of
13,
22
,
31
and 39%
respectively, were achieved for the present test program
(Table 1 ), in addition to t
he
reference material (mortar
wi
thout inclusions). A
ll
mi
xes have a cement/sand ratio of
0.46 and a water/cement
ra
tio of 0.4. The polystyrene
con
te
nt g % (in volume) which must
be
incorporated to
obtain the desired
den
sity is calculated from the
form
u
la
:
g d
m-
db (
3
)
d
m-
dp
where
db
is the density of the desired mixture, d., is the
density
of
the reference mortar, and
dp
is the density of
th
e
polystyrene beads.
Cylindrical concrete specime
ns
of
diameter 11 cm and
Fig. 1 - Distribution
of
polystyrene beads in
1.
4 density mixture.
Table 1 - Mortar mixture proportions
Mi
x
EC
HO
ECHI
ECH2
ECH3
EC
H4
De
nsi
ty
2.3
2.0
1.8
1.
6
1.
4
Cement (kg!m
3
)
640
558
502 446 389
Sand (kg/m
3
)
1395
1215 1096
969
846
polystyrene (kglm
3
)
0.0 2.
16
4.31
5.
16 6.65
Superplastizer (kglm
3
)
6.4
5.58
5.02 4.46 3.89
Water
(Vm
3
)
250
21
9 197
175
153
Slump (cm) 12.2 9.45 7
.5
5.15 2.15
len
gth 22 cm were used
for
co
mpr
ess
iv
e tests. For
th
e bending
tests,
four
different s
iz
es
of geometrica
ll
y
no
tched concrete
specimens
were
used
. The
de
p
ths
were D = 40, 80,
160
and
320
mm
while the
thi
ckness
was
kept
co
nstant for all
the
speci
mens
b = 40
mm
. The length
to
dep
th
ratio
was
LID
= 8:3
and
the
span to
depth
ratio was
1/D
=
2.5
for
a
ll
specime
ns
.
One no
tch
of de
pth
D/6
and
thi
ckness
1.5
mm
(sam
e
for
all
di
mens
ion
s)
was
pla
ced
in each bend
ing
specimen
by
putting
steel plates in
the
mou
ld
s before c
as
ting (Fig. 2). For
th
e
acoustic emissi
on
meas
u
re
ments, only specimens
of
depth D
==
160
mm
were
tested.
Three identical spec
im
ens we
re
cast
s
imultaneousl
y
fr
om each successive
bat
ch,
for
eac
h type of
tes
t.
Curing conditions were
28
da
ys
at 100% RH
and
20° C.
2.1 Elastic properties and compressive strength
The compression
tes
ts were
pe
rformed
usi
ng 300
kN
capa
ci
ty
hydraulic testing machine at a loading rate
of
0
.5
MP
a/s
un
til
fa
ilure. The compressive strength f., tensile
strength/, and the Yo
ung
's modul
us
Ec change significantly
wi
th
the polystyrene content (Table 2). Mixtures ECHl -
EC
H2 -
EC
H3 and
EC
H4 sh
owe
d
no
min
al elastic modulus
(a)
(b)
L
(8
/
3)0
Dcllccuon
mcasur..:mcnl
(laser sensor)
t t
CMOD
measurement
Table 2 - Mechanical properties
of
test mix
Mix E
CHO
ECH1
EC
H2
ECH3
ECH4
fc (
MP
a) 59.8
36
.8
27.6 23.7
16
.1
ft
(
MP
a)
4.27 3.
93
3.05
2.35
1.82
Ec(GPa)
33.4 25.9 23 17.2
14
val
ue
s, measured at one
thi
rd
of
th
e
fa
ilure stress
of
25900 - 23000 - 1
72
00
and 14000 MPa respec
ti
ve
ly.
Compared
to
the Young's modulus
of
the reference
materi
al
ECHO
, the decrease is
22
-
31
-4
9 and
58%
respectively. Concerning the compressive strength, the
decrease is about 40, 54, 61 and 72% compared
to
th
e
compressi
ve
strength
of
the reference materi
al.
For the Young's modulus at least, these data can be
compared with results from homogenization theories. A
possib
le
candidate, among many
oth
ers, is the well known
formula
[22
,
23
)
E
c=
(l
-gy
Eo
(2)
whe
re
Ec
is the Young
's
modul
us
of
the material with
polystyrene content
g and E
0
is the Young
's
modu
lu
s of
th
e
reference materi
al.
Fig. 3 com
pa
res
th
e measured values
of
elastic modul
us
with
th
e above equation. A very good
agreement
is
obtain
ed.
2.2 Size effect tests
The size effect tests
foll
owed
the
gu
id
elines established by
RIL
EM
[2
4] usi
ng
a closed - loop testing
mac
hin
e,
(a 160
kN
cap
a
ci
ty
INSTRON machine). The
te
sts
were
notch opening
contro
ll
ed
with a constant C
MOD
rate
of
0.1
J.l.m/s
forD = 40-
80
mm;
0.20
(l
miS
for
D =
16
0
mm
an
d
0.25
(lmls
for
D
==
320
mm.
For each size
and
each
mix,
t
he
di
spe
rsion over
th
e
three
specimens
tes
ted
is
qui
te small and in
th
e
following we
are
going
to
show avera
ge
curves
only.
Fig. 4 sh
ows
the
response
of
medium
size
spec
imen
(D =
40
mm) for each m
aterial
density. Note
th
at
the
de
fl
ection at peak is almost
in
depend
en
t
fro
m
th
e
density
of
the
mat
eria
l.
The material
de
nsity
influences st
ill
the
mechanical behavior
of
b
eams;
the
lo
wer
th
e densi
ty,
the lower
the
stiffuess and
the
peak
loa
d.
Fig. 5 shows
the
average
load
- deflection p
lo
ts
for
the
fou
r sizes
of
b
eam
and
for
th
e
five
different
dens
iti
e
s.
Th
es
e data are going
to
be
interpreted with
th
e
h
el
p of
Ba2an
t's size effect
law.
This
th
eo
ry
is
here
restricted
to
the
depe
n
dence
of
the nominal stress at
failure
ON
on
th
e charac
ter
is
ti
c dimension D
of
2D
geometrically similar
specimens.
For stress -
based
fai
l
ure
theories such
as
the
plastic limit analysis or
elast
ic
allow
able
strength des
ign,
th
ere
is
no
size
effect,
ON
is constant.
Fo
r c
lass
ical linear cl
as
tic
fractu
re
mechanics,
ON
a D"
112
•
Du
e to
th
e
influ
ence
of
a r
el
atively lar
ge
micro-cracked
zone
that b
lun
ts
Fig. 2 - (a) Description
of
mortar samples; (b) Descripti
on
of
instrumentation
fo
r the bending tests.
th
e crack
front
in concret
e,
s
iz
e effect is intermediate
between strength
of
ma
teria
ls
and
linear
fractur
e
40
-;;s-
Q..
30
s
V>
::I
~
20
E
()
't; 10
~
t:ij
A experimental
- homogenisati
on
0
~----~----~-----r----~----~
0.0 0.1 0.2 0.3
0.
4 0.5
Polystyrene content g
F
ig
. 3 -
Variation
of
elastic
mod
ul
us
with polystyrene
con
te
nt.
200
JOx40x I 07
mm
1
he-am
160
120
z
"'
~
80
~
~
_,
40
0
0
50
100
150 200
Ddl<ctwn
(~111)
Fig
. 4
-I
nfl
uence
of
density
on
structural
behav
i
or,
average
l
oad-
deflectio
n
curves
for
different
material
density
on
40 x 40 x 1
07
nun
3
beams
.
mechanics and it
re
presen
ts
a transition from the former to the
latter design criteria
as
the size increased. This transition is
described by the approximate formula:
a N =
~
with
fJ
=
DId
0
1 +
fJ
(3)
where
j;
is the tensile strength
of
th
e material, d
0
is a
characteristic size that
co
rrespon
ds
to a change
of
mechanisms between plastici
ty
ph
en
omena and fracture
mechanics, and
B is a material parameter which is a
function
of
th
e geometry
of
the sp
ec
im
en a
nd
ap
plied load.
cr
N is calculated according to the classical formula for a
beam
of
unit
th
ickness:
/'T'
_ 3
FL
V N - 2 0.
83
D
2
(4)
where F is the maximal load and L is the length between
supports. The peak forces are obtained from load - deflection
cu
rv
es
in Fig. 5, for e
ac
h size and for each density.
Th
e
nominal strengths (calculated from Equation
(4)) are given in
Table 3.
The values
of
BJ;
and d
0
in Equation (3) are
ob
tained
fr
om
a linear regression:
1000
800
z
"'
600
"C
'-
] 4
00
200
1
000
800
z
"'
~
600
"C
<:
400
0
...J
200
0
0
1
000
600
:z
600
"'
:::::,
"0
"'
.s
400
200
0
0
1000
800
z
"'
~
600
"C
~
-l
400
200
0
0
1000
800
z
~
600
"C
13
-l
400
200
0
0
100
100
100
1
00
100
density
~
2.3
(a)
200
300
400
500
Deflection
(~m)
density = 2.0
(b)
200
300
400
500
Deflection Cum)
density=
1.8
(c)
200
300
400
500
Detlection
(IJm)
den
s
ity
~
1
.6
(d)
200 300 400
500
Deflecti
on
(~-tm)
density m 1.4 (e)
200
300
400
500
Deflection
(~-tm)
Fi
g.
5 -
Cu
rv
es
r
ep
resen
ti
ng
for
each
size
the
average load-
deflec
tio
n r
esponses
for
each
mate
ria
l
densi
ty
tested