Crack
band
theory for fracture
of
concrete
Zdenek
P.
Bazant
Active Member of RILEM. Professor of Civil Engineering and Director. Center for Concrete and Geomaterials. The TechnologJ!;,;:
;i
s:;'ute,
Northwestern Univefsity, Evanston. Illinois,
60201.
B.H.Oh
Graduate Student, Dept. of Civil Engineering, Northwestern University, Evanston, llIinois.
60201.
A fracture theory for a heterogeneous aggregate material which exhibits a gradual strain-
softening due
to microcracking and contains aggregate pieces that are not necessarily small
compared to struttural dimensions
is
developed. Only Mode I
is
considered. The fracture
is
modeled
as
a blunt smeared crack band, which
is
justified by the random nature
of
the
microstructure. Simple triaxial stress-strain relations which model the strain-softening and
describe the effect
of
gradual microcracking
in
the crack band are derived. It
is
shown that
it
is
easier to use compliance rather than stiffness matrices and that it suffices to adjust a
single diagonal term
of
the compliance matrix. The limiting case
of
this matrix for complete
(continuous) cracking
is
shown to
be
identical to the inverse
of
the well-known stiffness
matrix for a perfectly cracked material. The material fracture properties are characterized
by only three
paPlameters
-fracture
energy, uniaxial strength limit and width
of
the crack
band (fracture Process zone), while the strain-softening modulus
is a function
of
these
parameters. A
m~thod
of
determining the fracture energy from measured complete stress-
strain relations
is'
also given. Triaxial stress effects
on
fracture can be taken into account.
The theory
is
verljied by comparisons with numerous experimental data from the literature.
Satisfactory fits
of
maximum load data as well as resistance curves are achieved and values
of
the three matetial parameters involved, namely the fracture energy, the strength, and the
width
of
crack
b~nd
front, are determined from test data. The optimum value
of
the latter
width
is
found to be about 3 aggregate sizes, which
is
also justified as the minimum acceptable
for a homogeneous continuum modeling. The method
of
implementing the theory in a finite
element code
is
al$o
indicated, and rules for achieving objectivity
of
results with regard
to
the
analyst's choice
of
element size are given. Finally, a simple formula
is
derived to predict from
the tensile strength and aggregate size the fracture energy,
as
well
as
the strain-softening
modulus. A statistical analysis
of
the errors reveals a drastic improvement compared to the
linear fracture
th~ory
as
well as the strength theory. The applicability
of
fracture mechanics
to concrete
is
thz4 solidly established.
INTRODucnON
The structural
size
effect
is
the central problem
in
predictions of fltacture. Fracture tests are normally
conducted on relatively small specimens, and fracture
theories are
usual~y
verified in the laboratory
by
testing
relatively small ,*ams or panels. In practice,
we
then
dare to
extrapola~e
this information to structures which
are often far larger than anything tested. This, of course,
cannot
be
done I reliably without a sound, realistic
fracture theory. '
For the initiation of cracks in a body
without,cks
and stress concentrations, the concept of
stfi:,'
:1
is
acceptable. Not so, however when a sharp crack,
.:ady
exists. The elastic analysis then yields infinite
strf'~S
at
the crack front, and the strength criterion
inc,'
c~ctly
predicts the crack to extend at an infinitely
sma)
,oad.
When a finite element mesh
is
refined, the load
'1i.:edc!d
to reach the strength limit strongly depends u: the
choice of element
size
and incorrectly converges tc
le:-o.
Thus, the elastic finite element analysis of cracking
based on the strength criterion, as currently used
in
0025-5432/1983/155/$ 5,00/
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BORDAS-DUNOD
155
Materials and Structures 16: 155-177, 1983
Vol.
16 -
N°
93 - Materiaux et Constructions 1981
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ELASTICITY
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PLASTICITY
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I~ST
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NONLINEAR
Z FRACTURE
MECHANICS
(a)
LINEAR
FRACTURE
//
MECHANICS
,
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, 2
......
---.,
"ll
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(SIZE
Hl
(b)
Linear
Fracture
(C)
Melal.
CF<WX;'<P"
8M
4M
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or
b"2
Pi
''"''''"
" /
V
1.5v
V
(TN:I
bit
or bit
(d)
Cancr.'e
Fig.
1.
-
(a)
Illustration of structural size effect
in
failure;
(b-d)
Relative sizes of fracture process zone (Fl, nonlinear hardening
zone
(N) and linear zone (L).
computer codes,
is
unobj~tive
in that it strongly
depends
on
the analyst's
chC!>ice
of
mesh. Neglect
of
the
infinite stress concentration
at
the crack front.
e.
g.,
when the bending theory is applied
to
the ligament
section, does
not
make the strength criterion correct.
This may
be illustrated in figure 1 a where
we
consider
any type
of
geometrically similar specimens
or
structures
of various sizes, with
geomCl!trically
similar cracks,
and
plot the logarithm
of
the nominal stress
UN
at
failure
(calculated
e.
g., by
applyinl
the bending theory
to
the
ligament) versus the logaritbm
of
the size. The strength
criterion predicts
UN
at
failure to be independent
of
size,
while all tests indicate a decrease of
UN
with
an
increase
In
SIZe.
These difficulties can
be
circumvented only by
fracture mechanics, in which the basic criterion
is
that
of
energy release needed to create the crack surface.
According to the classicallin¢ar fracture mechanics
[30],
UN
in figure 1 a
is
then proportional to (size)
-1/2,
i.
e.,
the plot
of
log
UN
versus
log:
(size)
is
a straight line
of
downward slope
-1/2
(fig.
1
a).
However, with the
exception
of
very large structures, this slope appears to
be too steep in comparison with most existing test
data
[49].
The reality seems
to
be a gradual transition
from the horizontal straight line for the strength
criterion to the inclined straight line
of
slope -
1/2
(fig.
I
a).
Failure analysis in this transition range is
more difficult
than
it
is
for the limiting cases
of
strength
criterion
or
linear fracture mechanics,
and
most concrete
structures unfortunately fall in this category. In the
work which follows (based
an
report
[10]),
we
develop
a similified theory which
apptars
to give realistic results
over the complete range
of
sizes, including the transition
range, and can
be brought in a good agreement with
156
essentially all available fracture test
data
for concrete,
a goal not yet achieved in the existing literature.
The reason for the deviations from linear fracture
mechanics observed in concrete (as
well
as rocks)
consists in its heterogeneity
of
the material, causing that
it behaves nonlinearly within a relatively large zone
adjacent to the fracture front, while the linear fracture
mechanics requires this zone to
be
small. This behavior
is
similar to ductile fracture
of
metals
[40],
but there
is
one significant difference. The fracture process zone,
representing that part
of
the nonlinear zone in which
the material undergoes progressive microcracking
manifested by strain-softening
(a
decrease
of
stress
at
increasing strain)
[18],
is
still usually small
in
ductile
fracture
of
metals, but
in
concrete it
is
often very large
(compared to the cross section of the structure), due to
the large size
of
aggregate
(fig.
1 b-d).
For
this reason,
the non linear fracture theories developed for metals
cannot
be indiscriminately transplanted to concrete.
Furthermore, since the plastic deformation
of
concrete
in tension
is
negligible and the strain-softening in a
tensile test
[18]
is
not
preceded by a horizontal plateau,
the boundary
of
the fracture process zone may
be
considered to be nearly identical to the boundary
of
the
nonlinear zone, whereas in metals these boundaries are
far
apart
(fig.
1
c,
d). Thus. it appears
that
we
do not
need to analyse fracture by means
of
the J-integral,
and
can directly use the fracture energy
of
the crack band.
The J-integral
is,
anyway, inapplicable for contours
inside the strain-softening region,
and
for contours
within the linear exterior region it reduces to the
fracture energy.
Because
of
computational convenience as
well
as
resemblance to reality, the cracking
of
concrete (and
also rock) has long been modeled in large finite element
programs as systems
of
parallel crack that are
continuously distributed (smeared) over the finite
elements, as introduced by Rashid
(cf. Ref.
[1]).
As
the
cracking criterion, the strength has been used.
It
was
shown, however, that the strength concept
is
unobjective
in
that
the results
of
analysis can be strongly affected
by the analyst's choice
of
finite element size
[6,
7,
16].
A remedy
is
to use as the cracking criterion the fracture
energy. This concept was worked
out
in detail for the
case when the stress can
be assumed to
drop
suddenly
to zero as the fracture forms [5-7,
16].
The assumption
of
an abrupt stress
drop
is,
however, inadequate for
cross section dimensions
that
are not sufficient! y large
compared to the aggregate size. In this case a gradual
strain-softening due to progressive microcracking must
be considered,
and
extension
of
the previous work
[6,
7,
16]
to cover this case will be the purpose
of
this
study, which
is
based
on
a
1981
report
[10].
We
restrict
our
attention to Mode I cracks, i. e.,
cracks (straight
or
curved) which have no shear stress
at
their front. This does not detract much from practical
usefulness since cracks in concrete seem to propagate in
most situations along such a path that Mode I prevails
at
the front.
THE
HYPOTHESIS
OF
BLUNT CRACK BAND
Central to the .analysis which follows
is
the hypothesis
that
fracture in a heterogeneous material can
be
modeled as a band
of
parallel, densely distributed
microcracks with a blunt front. This hypothesis may be
justified as follows.
Justification
I. Rlepresentative volume
When a heterpgeneous material
is
approximated by
an
equivalent homogeneous continuum (without couple
stresses), as
is
standard
for concrete structures, one
must distinguish the continuum stresses
and
strains
(macrostresses
and
macrostrains) from the actual
stresses
and
strains in the microstructure, called the
microstresses
and
micros trains. In the theory
of
randomly inhotnogeneous materials, the equivalent
continuum
stre~es
and
strains are defined as the
averages
of
the tnicrostresses
and
microstrains over a
certain representative volume. The cross section
of
this
volume should ideally be taken to be much larger
than
the size of the inhomogeneities,
and
even
for
a crude
modeling must
b¢ considered to be
at
least several times
their size.
i. e., several times the maximum aggregate
size in case
of
concrete.
Consequently,
in the usual analysis, in which only the
average elastic
(or
inelastic) material properties are
considered
and
the geometry
of
the microstructure with
the differences in the elastic constants between the
aggregate
and
the cement paste
is
not
taken into
account, the di$tribution
of
stress
or
strain over
distances less tllan several aggregate sizes has no
physical meaning. Only the stress resultants
and
the
accumulated
st$n
over the cross section
of
the
characteristic Volume do. In the finite element context,
this means, therefore, that it makes no sense to use
finite elements smaller
than
several aggregate sizes (and
also
that
it
makeS!
no sense to use in such smallest finite
elements distributlion functions
of
higher order). In case
of
fracture, this further means
that
if
an
equivalent
homogeneous continuum
is
assumed, it makes no sense
to consider
conc~ntrations
of
stress
(or
of
microcrack
density) within volumes less
than
several aggregate sizes.
A similar
conclu~ion
follows when
we
realize
that
the
actual crack
path
in concrete
is
not
smooth
but
highly
tortuous. Since the crack tends to pass
around
the
hard
aggregate pieces
and
randomly sways to the side
of
a
straight
path
by distances roughly equal to the aggregate
size, again the actual stress (microstress) variation over
such distances can
be
relevant for the 'macroscopic
continuum model.
According to the foregoing justification, one should
not
attempt
to
s~bdivide
the width
of
the crack band
front into several finite elements. There
is
however also
another
reason. : The strain-softening continuum
is
unstable
and
a strain localization instability, in which
the deformation would localize into
one
of
the elements
in
the
subdivision~
would take place.
z.
P Bazan! -
B.
H.
Oh
Justification II. Equivalence of results
For
an elastic material in which the stress drops
suddenly to zero
at
the fracture front, it was found
[5,
16]
that a
sharp
interelement crack
and
a smeared
crack band give essentially the same results for the
energy
re!e~se
rate
and
agree closely (within a
few
percent) with the exact elasticity solution, provided that
the finite element is not larger than
about
1/15
of
the
cross section dimension (square meshes without any
singularity elements were used). This
is
true regardless
of
the aggregate size.
To
demonstrate it here, figure 3 shows some
of
the
numerical results for a line crack (left)
and
crack band
(right) extracted from Reference
[5].
The finite element
mesh covers a cut-out of infinite medium loaded at
infinity by uniform normal stress
Ii
perpendicular to a
line crack
of
length 2
a.
The nodal loads applied
at
boundary are calculated as the resultants (over the
element width)
of
the exact stresses in infinite medium
at
that location. The solid curve is Westergaard's exact
solution. The calculated points are given for the square
mesh shown (mesh A), as
well
as meshes
Band
C (not
shown) with elements reduced to 1/2
and
to 3/8. (Each
element consisted of
2
constant
- strain triangles,
and
calculations were mane for 1i=0.981 tX(MPa),
E
c
=2256MPa,
v=0.1,
and
stress intensity factor
0.6937
MNm
-3/2.)
The same equivalence
of
line cracks
and
crack bands
may be expected when a gradual stress
drop
is
considered (and fig. 5 confirms it).
The
reason for this
equivalence
is
the fact
that
fracture propagation depends
essentially
on
the flux
of
energy into the fracture process
zone
at
the crack front,
and
this flux is a global
characteristic of the entire structure, depending little
on
the details
of
the stress
and
strain distributions near the
fracture front.
Furthermore, the fact
that
the results for the stress
intensity factor
[30]
obtamed with nonsingular finite
elements agree closely (within
1%
for typical meshes)
with the exact elasticity solution also confirms
that
there
is
no need
to
use singularity elements for fracture
analysis. In any event, the non-uniform stress distribu-
tion implied in a singularity element
is
meaningful only
if
its size
is
many times the representative volume,
i.
e.,
at
least 20-times the aggregate size, which is too large
for most applications.
Computational advantages
Since the line crack
and
the crack
band
models are
essentially equivalent, the choice
of
one
or
the other
is
basically a question
of
computational effectiveness.
From
this viewpoint, the line crack model appears to be
disadvantageous. When the crack extends through a
certain node, the node must be split into two nodes,
increasing the total number
of
nodes
and
changing the
topological connectivity
of
the mesh.
Unles~
all nodes
are renumbered, the
band
structure
of
the structural
stiffness matrix is destroyed. All this complicates
157
Vol. 16 • N" 93 • Materiaux et Constructions
(a)
,z
i
(e)
I-
a-
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representOflYe
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(b)
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l
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't
-
Jilt.
2.
- (a) Actual crack morpholpgy.
(b)
Actual stresses and then
smoothing,
(e) Line crack model.
(d)
Crack band model
used
here.
programming. When the direction in which a Mode I
crack should extend is not known in advance, one must
make calculations
for
various possible locations of the
node ahead
of
the crack front, through which the crack
should pass, in order to identify the location for which
the energy release rate
is
maximum.
These difficulties are
aVQided
by the crack band
model. Here the crack
is
modeled by changing the
isotropic elastic moduli matfiix to
an
orthotropic one,
reducing the material stiffne$s in the direction normal
to the cracks in the band. This
is
easily implemented in
MESH A
.(,
.....
6,
n.
7)
MESH
B
("
..
12.".13)
MESH
C
(m.10.
".25)
m elements
a
a finite element program, regardless of the direction of
the crack with respect to the mesh lines. A Mode I
crack propagating in an arbitrary direction with respect
to the mesh lines,
or
a Mode I crack following a curved
path, may
be
easily modeled as a zig-zag crack band
(see
fig. 4
e)
whose overall direction in the mesh
approximates the actual crack direction. (Numerical
applications for skew, curved
or
asymmetric cracks are
however beyond the scope of this study.)
A further advantage of the crack band model
is
that
the information obtained in studies of stress-strain
relations and failure envelopes can
be
applied to
fr
..
cture
(e.
g., the effect of the compressive normal
stress parallel to the crack). Still another advantage
is
the fact that with the crack band model one can treat
the case when principal stress directions in the fracture
process zone rotate during the progressive fracture
formation,
i.
e., during the strain-softening. This case
arises,
e.
g., when first a vertical tensile normal stress
produces only a partial cracking, and failure
is
subsequently caused by horizontal shear stresses. The
line crack models do not seem suited to handle this
situation.
Actual pattern of microcracks
Recently, various measurements are being made to
observe the formation of microcracks
at
the fracture
front
[37,
41].
From
these observations it seems that the
larger microcracks that can be seen are not spread over
a band
of
a large width but are concentrated essentially
on a line. However, the line along which the microcracks
are scattered
is
not straight
(or
smoothly curved) but
is
highly tortuous (fig.
2),
deviating to each side of the
p Q
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I;
l
L
A
b-c:
b.~
x,
-
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'IV
25
wS
A
o
o •
o
6 7 8
Fic.
3.
- Finite element results for crack band (right) and for line crack (left).
compared to exact linear fracture mechanics solution (after Bazant-Ccdolin.
1979,
Ref.
[5]).
158
I,
(a)
crt
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t
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Y
(g)
~
(b)
l
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E
f
Eo
E
f
(e)
crt
f'
I
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P.
Bazant - B
H.
Oh
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(d)
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1,2'"
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~,
• area OP3A
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OP48
5,6,7'
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OPe
Fig.
4.
-
(a-d)
Stress-strain diagrams for fracture process zone; (e) Zig-zag crack band;
(f-g)
Stress distributions
in
fracture process zone.
straight line extension
by
a distance equal to about the
aggregate
size,
as the crack
is
trying to pass around the
harder
aggreg~te
pieces. In the equivalent, smoothed
macroscopic cQntinuum which
is
implied
in
structural
analysis, the scatter
in
the locations of visible
microcracks
re~ative
to a straight line
is
characterized
by
a microcracJc band better than
by
a straight row of
microcracks.
At the same
~me,
we
should realize that the boundary
of the fracture process zone should not
be
defined
as
the boundary of visible microcracks but
as
the boundary
of the strain-sqftening region,
i.
e.,
the region
in
which
the maximum stress decreases with increasing maximum
strain. Since the strain-softening
is
caused not only
by
microcrackingbut also by any bond ruptures, the
fracture proces$ zone could
be
much wider (as
well
as
longer) than
th~
region of visible microcracks.
101
PI
, y
Ibl
T
"
del
0:0
da
, "_b:.::::t:.
i
i'
+
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~
_ D ...
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p,
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Ie}
i
These questions are, however, unimportant for the
macroscopic continuum modeling because of the
foregoing Justifications I and II. They would matter
only for micromechanics analysis, aimed,
e.
g.,
at
calculating the fracture energy from the constituent
properties and geometry of the microstructure.
Previous works
According to Justifications I and II, it cannot make
much difference whether the fracture
is
modeled as a
line crack (a sharp interelement crack) or as a band of
continuously distributed (smeared) parallel cracks.
Thus, if the relation of the normal stress
(1.
and the
relative displacement
J / across a line crack
is
identical
to the relation of
(1:
to the displacement J / = e /
We
obtained
by
accumulating the strains
e/
due to
lal
0.30
-Eaact
a
M.,,,
A
025
4
Mn"
a
.
M.,II
C
---
M.,,,
C
.,
,
C,
It
KYne
Ot
020
for Me,,, A
,
0
,
0.
.--
.....
0.
015
,
010
005
00
02
04
06
08
Rol
Crack
Lonqth 101 bl
Fi
..
5.
- (a) Center-cracked plate; (b) Finite element mesh refinements; (e) Crack front for mesh
A.
B.
and
C;
(d)
Comparison
of
numerical results.
159