Ba~ant,
Z.P., and Cedolin,
L.
(1984). "Approximate linear analysis
of
concrete fracture by R-
curves.·
J.
of
Structural Engineering, ASCE, 110, 1336-1355.
ApPROXIMATE LINEAR ANALYSIS OF CONCRETE
FRACTURE
BY
R-CURVES
By
Zdenek
P.
Batant,'
F. ASCE
and
Luigi
Cedolin;
M.
ASCE
ABSTRACT: Using linear elastic fracture analysis,
the
energy
consumed
per
unit
length of fracture (fracture energy) varies
with
the
crack length,
as
described
by
the
resistance curve (R-curve). This concept, originally
proposed
for metals,
is developed
here
into a practical, applicable form for concrete.
The
energy
release rate is determined
by
an
approximate linear elastic fracture analysis
based
on
a certain equivalent crack length,
which
differs from
the
actual crack
length,
and
is solved
as
part
of structural analysis. It is
shown
that
such
an
analysis,
coupled with
the
R-curve concept, allows achieving satisfactory fits of
the
pres-
ently existing fracture
data
obtained with three-point
and
four-point
bent
spec-
imens. Without
the
R-curve,
the
use
of
an
equivalent crack
length
in
linear
analysis is
not
sufficient to achieve a satisfactory agreement with these data.
The existing
data
can
be
described equally well with various formulas for
the
R-curve,
and
the
material parameters
in
the formula can vary over a relatively
broad range
without
impairing
the
representation of test data.
Only
the
overall
slope of
the
R-curve,
the
initial value,
and
the
fina1 value ,are important. A
parabola seems to
be
the
most
convenient
shape
of R-curve because
the
failure
load
may
then
be
solved from a quadratic equation. For
the
general case, a
simple algorithm to calculate
the
failure load is given. Deviations from test
data
are analyzed statistically,
and
an
approximate relationship of
the
length
param-
eter
of
the
R-curve to
the
maximum aggregate size is found.
INTRODUCTION
Due
to
the
large size of
the
fracture process
zone
at
the
crack front,
concrete structures
do
not
follow linear elastic fracture mechanics, ex-
cept
when
the
cross section is extremely large compared to
the
aggregate
size. Nevertheless, engineers
need
to
be
able to
use
linear elastic fracture
mechanics
at
least
in
some approximate, equivalent sense because non-
linear fracture analysis is
much
more complicated. Since concrete
does
not behave plastically
under
tensile situations,
the
exterior of
the
frac-
ture process zone is essentially elastic. Therefore,
the
stress field farther
away from
the
fracture process zone
should
be
dose
to
that
correspond-
ing to a linear fracture mechanics solution for a certain equivalent crack
length.
As it
turns
out, however, this does
not
suffice to achieve
good
agree-
ment with fracture test results. Evaluating these results
by
linear elastic
fracture mechanics,
one
finds
that
the
fracture energy, i.e.,
the
energy
consumed
by
fracture
per
unit
crack length, is variable. Thus,
in
addi-
tion
to
considering a certain equivalent crack
length
instead
of
the
actual
crack length,
one
must
also take into account
the
variation of
the
fracture
energy. The situation for concrete
happens
to
be
the
same as for ductile
lProf. of Civ. Engrg.
and
Dir., Center for Concrete
and
Geomaterials, Tech-
nological Inst.,
Northwestern
Univ., Evanston,
m.
60201.
ly'isiting Scholar,
Northwestern
Univ.; Prof.
on
leave from Dept.
of
Struct.
Engrg., Politecnico
di
Milano, Italy.
Note.-Discussion
open
until November 1, 1984. To extend
the
closing
date
one
month,
a written request
must
be filed
with
the
ASCE Manager
of
Technical
and
Professional Publications. The manuscript for this
paper
was
submitted for
review
and
possible publication
on
August
25, 1983. This
paper
is
part
of
the
Journal
of
Structural Engineering, Vol. 110,
No.6,
June,
1984.
©ASCE, lSSN
0733-
9445/84/0006-1336/$01.00.
Paper
No. 18954.
1336
fracture of metals, for which the fracfure energy variation
has
been
Sttid·
ied extensively
(6).
This variation is described
by
the
plot of fracture
energy (or fracture toughness) versus the crack extension,
c,
from a notch
or smooth surface.
This
plot is
called
the resistance curve
or
R~e.
As is well known,
the
R~e
for
any
given material cannot
be
unique
unless
the
crack length,
c,
is negligible compared
to
the
dimensions
of
the cross section,
the
li~ament,
and
the
distance
to
the
nearest applied
load. Otherwise, the
shape
of
the
R~e
depends
on
these parameters
and
on
the geometrical shape of
the
stru~
and
the
nature of loading.
The shape of
the
R~e
can
be
approximately calculated
by
various
methods (for concrete, see Ref. 28, 3).
As it appears, however, the shape of
the
R<UrVe
does
not
vary strongly
from one
type of structure (specimen)
to
another,
and
also
it
does
not
have a strong effect
on
calculation results,
as
will
be
seen
later. Thus,
one may postulate a priori a certain suitable fixed
shape
of
the
R~e
for all situations, which allows a great simplification of analysis. This
was
proposed for metals by Krafft,
et
al. (6,16)
and
has
been
widely
used
in
ductile fracture.
The present
study
shows
that
the
R~e
approach, combined
with
linear elastic fracture analysis for a certain effective crack length,
which
differs from
the
actual crack length, allows achieving a goOd agreement
with the available fracture test data for concrete.
REVIEW
OF
R-CuFlVE
CoNCEPT
Let c = a -
Ilo
in which
Ilo
= length of
the
notch [Flg. l(a»;
and
a =
total length of crack
and
notch. Consider
that
the
fracture energy, G
c
,
is a certain given function of crack extension,
c,
i.e., G. =
Gc(c).
The
energy that
must
be supplied
to
the
structure to produce
the
crack is U
=
JGc(c)da
-
Weal
if
the
thickness of
the
structure
in
the
third dimen-
sion is considered
as
unity; here W is
the
total release of strain energy
from the structure (or specimen).
An
equilibrium state of fracture occurs
when
no
energy needs. to be supplied .to change a
by
8a
and
none
is
released, i.e.,
when
8W
=
O.
Since
8U
= (au/aa)8a =
0,
in
which
au/
aa
= G
c
- W' = 0,
and
W'
=
aw
laa, it follows
that
fracture equilibrium
occurs
when
W'
(a)
= G.(c) (equilibrium)
....................................
(1)
The equilibrium fracture state is stable
if
the
second variation 8
2
U is
positive. Since
8
2
U = (alu/aal)8a
2
and
a
2
u/aal
=
aGc/aa
- aW'/aa,
the
cOi\dition for stability of fracture
and
the limit of stability, i.e.,
the
fail-
ure, are given
by
aGc(c)
_
aw'
(a)
> 0 (stable)
...................................
(2a)
ac
iJa
= 0
(critical)...................
. . . . . . . . . . . . .
..
(2b)
For most structures,
the
strain energy release
rate
increases as
th~
crack grows, i.e., W'
(a)
>
O.
By elastic structural analysis, one can
cal~
133'J
(G) W'(c)
(6)
W'(c)
"
~---
'-,
,
-~
......-.
......
_ - P .. constant
......
_-
c
c
FIG.
1.-R-CUrva
and
Dlagrama
of
Energy
R.'
....
Rate
late the curve
WI
(a)
corresponding to a
unit
load, P =
1.
Then, for
any
load P
W'(a) = p
2
W
I
(a)
.................•.............................
(3)
as can be deduced
by
dimensional analysis. Fig. l(a) shows the curves,
W'
(a),
for a succession of increasing P-values,
PI,
P
2
,
P
3
,
••••
According
to
Eq.
1,
the
equilibrium states of crack extension for various load values
are given
by
the intersections of these curves with
the
given curve
Gc(c).
~cco~g
to
Eq.
2, these equilibrium states are stable if
at
the
point of
mtersection the slope of
the
Gc(a)-curve is larger
than
the
slope of
the
W' (a)-curve (see Fig. 1). As
the
crack grows, the difference between
the
slopes,
8G
claa
and
aW'laa, gradually diminishes qntil,
at
a certain point,
the slopes become equal (Fig.
1);
this is
then
the critical state
at
which
the structure fails. Beyond this point the crack extension is unstable
and
occurs dynamically since there is
an
excess of energy release
that
must
go into kinetic energy.
In
the rare case
that
W' < 0 for all
a,
Eq. 2 is always satisfied,
and
the crack is
then
stable for all a [Fig. l(b». .
In
the case that G
c
is constant, Eq. 2 reads
0>
aW'laa. This condition
can never be satisfied
if
W'
increases
with
a [Fig. l(c)]. Thus, if a stable
crack growth from a notch is observed in experiments, it implies
that
G
c
cannot
be
constant
but
must
increase, provided
the
test specimen ge-
ometry is such that W' increases with
a.
Comparing structures that are geometrically similar (including their
notches)
but
of different sizes,
the
curves of W'(C) are also of similar
1338
shape (i.e. related
by
affinity transformation), while the curve
Ge(c)
re-
mains the same. This causes failure to occur
at
a larger C for a larger
structure, as shown
in
Fig.
l(d).
CALCULATJON OF FAILURE
LOAD
AND ANALYSIS
OF
FRACTURE
DATA
The energy release rate,
W',
to
be
used
in Eq. 2 may be determined
by linear elastic fracture analysis using various methods. For typical frac-
ture specimens, highly accurate approximations are available (see Refs.
6,
15, 19,
24).
For example, for
an
infinitely long strip of
width
Land
unit thickness, containing a symmetric crack of length
2a
normal to the
strip sides,
and
loaded at infinity by axial load P, the stress intensity
factor
is
K1
= f
(L
tan
~a)
1/2
............................................
(4)
from which
W'(a)
=
KUE
in which E = Young's modulus
(15).
For the
three-point bent specimen
and
the four-point bent specimen (Fig.
2)
..
r-
PL
K1
= V
'Ira
bd
2
/3(o.),
/3(0.)
= 1.635 -
2.6030.
+
12.300.
2
-
21.270.
3
+
21.860.
4
•••••••••••••••
(5}
..
r-
PL
K1
= V
'Ira
bd
2f4
(o.),
/4(0.)
= 1.12 -
1.390.
+
7.320.
2
-
13.070.
3
+
13.990.
4
••••••••••••••••••
(6)
in which a = aid; d = beam depth; L = beam span, b = beam width;
and for the four-point bent specimen, the loads are applied symmetri-
cally at distances
L/3
from the supports (15,20,24). These expressions
have been used
in
the analysis of the test data described later.
For structures of arbitrary geometry,
one
may determine the values of
W
I
(a)
for various small a
by
finite element analysis. Highly accurate re-
sults can be obtained with the use of singular elements, however, for
concrete it makes
no
sense to strive for errors less
than
about 1 %.
Then
it suffices to use a regular grid of nonsingular elements, for which the
crack may be modeled with about the same accuracy either as a sharp
interelement crack or as a
band
of cracked elements of a single-element
width (1-5,19), the latter being usually easier for programming.
W'(a)
may be estimated either from the difference between the total potential
FIG.
2.-
three-Point
Bent
and
Four-Point
Bent
SpecImene
1339
energies of the structure for two adjacent values of
a,
or
from the field
of displacements or stresses near the crack front.
Further, one needs to choose a priori
a suitable formula for the frac-
ture energy,
G
e
,
of the material. Experiments show (25-28)
that
for con-
crete, similarly to other
materials,G
e
increases as a function of C
and
seems to approach a certain asymptotic value, G
f
.
Existence of the
asymptotic value is also indicated theoretically
(28).
Measurements are,
however, quite scattered
and
do
not permit distinguishing too well be-
tweel': various possible expressions for
Ge(c).
The following three for-
mulas have been examined in calculations:
Ge(C)
= G
f
(1
-
~-e/em)
.
..........................................
(7)
Ge(c)
= G
f
[ 1 -
~
(:m _
1)
2]
for O:s C :s C
m
,
Gc(c)
= G
f
for C
~
C
m
••••••••••••••••••••••••••
(8)
for O:s C :s C
m
,
Ge(c)
= G
f
for C
~
C
m
••••••••••••••••••••••••••
(9)
in which G
f
,
~,
and
C
m
= material parameters to
be
found empirically.
A more complicated formula with two additional parameters, q
and
r,
Ge(c)
= G
f
{1
-
~
exp [-(clcm)q]}" was also tried; however,
no
appreciable
improvement in the fits of test data could
be
achieved.
Parameter
C
m
characterizes the length over which G
c
approaches its
final value,
G
f
•
Since the dimensions of the fracture process zone in con-
crete appear to be in a certain fixed ratio to the maximum aggregate size,
d.,
as suggested by various recent analyses (1,3), it seems reasonable to
assume that
C
m
=
md
•
.....................................................
(10)
in which m may be considered the same for all concretes.
When
C
m
is fixed, the formulas in Eqs.
7-9
may be written in the form
Ge(c)
= G
f
- bx
......................
"
..........................
(11)
in which b =
Gf~
and
x = exp
(-clc
m
)
for
Eq.
7;
x = (clc
m
-
1)2
or
x =
(clc
m
-
1)
for c < c
m
;
and
x = 0 for c
~
C
m
in the case of
Eq.
8
or
9.
Since
Eq.
11
is linear, linear statistical regression analysis may be used to ana-
lyze test data
on
the R-curves reported in the literature (9,23,25,27). Such
analyses have been carried
out
first for each data set taken individually,
and the coefficients of variation
Wi for the deviations of measured data
from the regression line
(Eq.
11)
have been evaluated for each data set
by a computer for a series of values of
C
m
ranging from 1 to 50. A few
examples of such plots are demonstrated
in
Figs.
3-5,
respectively, fo'
Eqs.7-9.
Subsequently, for each C
m
value, the combined coefficient of variation
for all data sets was calculated as
Ii>
=
~iwf
In)
1/2
in which i = 1,
2,
...
n are the data sets used.
It
appeared tnat the smallest
Ii>
occurs for the
exponential formula
(Eq.
7)
roughly for C
m
=
12,
and
for the parabolic
1340
shape (i.e. related by affinity transformation), while the curve
Gc(e)
re~
mains the same. This causes failure to occur at a larger c for a larger
structure, as shown in Fig.
l(d).
CALCULATION
OF
FAILURE LOAD
AND
ANALYSIS OF FRACTURE DATA
The energy release rate,
W',
to be used in
Eq.
2 may be determined
by linear elastic fracture analysis using various methods. For typical
frac~
ture specimens, highly accurate approximations are available (see Refs.
6, 15,
19,
24). For example, for
an
infinitely long strip of width
Land
unit thickness, containing a symmetric crack of length
2a
normal to the
strip sides,
and
loaded at infinity
by
axial load P, the stress intensity
factor is
KI
==f
(Ltan
~a)I/2
............................................
(4)
from which W'(a) =
KVE
in which E = Young's modulus (15). For the
three-point bent specimen
and
the four-point bent specimen (Fig.
2)
..
r-
PL
KI
==
V
1ra
bd
2h
(a),
Mo.) = 1.635 - 2.6030. + 12.300.
2
-
21.270.
3
+ 21.860.
4
...............
(5)
..
r-
PL
KI
==
V1ra
bd
2
/
4
(a),
/4(0.) = 1.12 - 1.390. + 7.320.
2
-
13.070.
3
+ 13.990.
4
..................
(6)
in which a = aid; d = beam depth; L = beam span, b = beam width;
and for the four-point
bent
specimen, the loads are applied symmetri-
cally at distances
L/3
from the supports (15,20,24). These expressions
have been used in the analysis of the test data described later.
For structures of arbitrary geometry, one may determine the values of
W'(a) for various small a by finite element analysis. Highly accurate re-
sults can be obtained with the use of singular elements, however, for
concrete
it
makes
no
sense to strive for errors less
than
about 1%. Then
it suffices to use a regular grid of nonsingular elements, for which
the
crack may be modeled with about the same accuracy either as a sharp
interelement crack or as a band of cracked elements of a single-element
width
(1-5,19),
the latter being usually easier for programming. W'
(a)
may be estimated either from the difference between the total potential
FIG.
2.-
Three-Point
Bent
and
Four-Point
Bent
Specimens
1339
energies of the structure for two adjacent values of
a,
or from the field
of displacements or stresses near the crack front.
Further, one needs to choose a priori a suitable formula for the
frac~
ture energy, G
c
,
of the material. Experiments show
(25-28)
that for con-
crete, similarly to other materials,
G
c
increases as a function of e
and
seems to approach a certain asymptotic value, G
f
.
Existence of the
asymptotic value is also indicated theoretically
(28). Measurements are,
however, quite scattered
and
do not permit distinguishing too well be-
tween various possible expressions for
Gc(e).
The follOwing three for-
mulas have been examined
in
calculations:
Gc(e)
= G
f
(1
-
~e-c/c
..
)
...........................................
(7)
Gc(e)
= G
f
[1
-
~
(:m
_
1)2]
for 0 s e s
em,
Gc(e)
= G
f
for e
~
em
..........................
(8)
Gc(e)
= G
f
[
1 -
~
(
1
-
e:)
]
for 0 s e s C
m
,
Gc(c)
= G
f
for c
~
C
m
••••••••••••••••••••••••••
(9)
in which G
f
,
~,
and
em
= material parameters to be found empirically.
A more complicated formula with two additional parameters,
q
and
"
Gc(c)
= G
f
{1
-
~
exp
[-(ele
m
)4JY,
was also tried; however,
no
appreciable
improvement in the fits of test data could be achieved.
Parameter
C
m
characterizes the length over which G
c
approaches its
final value,
G
f
•
Since the dimensions of the fracture process zone in con-
crete appear to be in a certain fixed ratio to the maximum aggregate size,
d.,
as suggested by various recent analyses (1,3), it seems reasonable to
assume that
em
=
md
•
.....................................................
(10)
in which m may be considered the same for all concretes.
When
em
is
fixed, the formulas
in
Eqs.
7-9
may be written
in
the form
Gc(e)
= G
f
-
bx
................................................
(11)
in which b =
Gf~
and
x = exp (-clem) for
Eq.
7;
x =
(clem
-
1)2
or x =
(clem
-
1)
for e <
em;
and
x = 0 for c
~
em
in the case of Eq. 8 or
9.
Since
Eq.
11
is linear, linear statistical regression analysis may be used to ana-
lyze test data on the R-curves reported in the literature
(9,23,25,27). Such
analyses have been carried
out
first for each data set taken individually,
and the· coefficients of variation
Wi for the deviations of measured data
from the regression line
(Eq.
11)
have been evaluated for each data set
by a computer for a series of values of
em
ranging from 1 to 50. A few
examples of such plots are demonstrated
in
Figs.
3-5,
respectively, for
Eqs.7-9.
I
Subsequently, for each
em
value; the combined coefficient of variation
for all data sets was calculated as
w =
(IiW~
In)I/2
in
which i =
I,
2,
..
,
n are the data sets used.
It
appeared that the smallest w occurs for the
exponential formula
(Eq.
7)
roughly for
em
= 12,
and
for the parabolic
1340
.eo
1.00
.eo
1.00
Wecharatana
ond
Shah I
...
-8
Wechoratona
and
Shah 2
...
=8
WMha
.....
na and
Shah
1
..
=23
Wecharatana
.ltd
Shah
c.~23
e
.75 .10
;JD
...
.70
.80
Gc
1J
GC
U
GC
Gc
.15
.70
.15
1J
.70
LJ
•
ID
...
.ID
.ID
0.0
..
.4
.1
..
1.0 0.0
.2
A
..
..
1.0
0.0
..
.4 .1
.B
1.0
0.0
.2
.4
..
.1
1.0
z_.-·/c.
s_.-c/c.
.,=(c(c
..
-l)·
.,-(c(c
..
-l)·
I.DD
1.25
I.DD
1.25
Vllal'IQnloh
and
Naoman
I
...
-8
VIMIvontoh
and
Noaman 2
...
-8
,VflOlvanlch and Naoman
I
0",=23
VJHtvanlch and Naoman 2
0",-23
•
.75
e
1.00
1.00
.so
.75
.50
G
c
"
G
C
GC
G
C
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z=e-c/c..
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%=(0(0",-1)'
FIG.
3.-Llnear
Regression
of
R-Curve Data
from
Literature According
to
Eqs. 7 FIG.
4.-Llnear
Regression
of
R-Curve Data
from
Literature According
to
Eqa. 8
and
11
and
11
1341
1342