Fracture Mechanics of Concrete Structures,
Proceedings FRAMCOS-2, edited by Folker H. Wittmann,
AEDIFICATIO Publishers, D-79104 Freiburg (1995)
LINEAR OR NONLINEAR FRACTURE MECHANICS
OF CONCRETE?
G. A. Plizzari
Dept. of Civil Engineering, University of Brescia, Brescia, Italy
V. E. Saouma
Dept. of Civil Environmental & Architectural Engineering, University of Colorado, Boulder, CO, USA
Abstract
This paper addresses a fundamental problem that is at the root
the applicability of fracture mechanics to concrete. On the basis
of theoretical considerations, it quantifies the error introduced by
a simple LEFM analysis vis a vis of a more comprehensive (albeit
complex) NLFM one. The error is expressed in terms of peak load,
which is the primary metric of practical interest to engineers.
1 Introduction
In the early days, fracture mechanics of concrete borrowed extensively from the linear elastic fracture mechanics (LEFM) of metals.
However, it was soon discovered that due to the presence of
fracture process zone (FPZ) at the tip of a crack, LEFM may
necessarily be applicable. In particular, when the size of the
is
appreciably less than the characteristic dimension D of the structure, crack extension can be quantified using LEFM procedures,
otherwise, a nonlinear fracture mechanics (NLFM) analysis should
1377
However, computational tools for NLFM analyses
widely available, and yet engineers might be confronted
of
a cracked concrete structure.
Subsequently, various guidelines have been proposed to determine modified LEFM parameters or the conditions under which
LEFM is applicable (Carpinteri, 1982; Hillerborg, 1985; Shah and
Jenq, 1985; Bazant et
, 1986).
Whereas fracture
of concrete has
under continuous and active investigation for over fifteen years, there is little
doubt that it is of most relevance in those structures where there
is little or no reinforcement. In fact, properly engineered reinforced
concrete structures are designed in such a way that reinforcement
most, if
the load originally carried by concrete
will
Prime examples of unreinforced concrete structures
and bulk-heads, which are large concrete structhe authors' opm10n
those structures are indeed
for
application of fracture mechanics, be it
vUJJC.L'-1..t,'-.i."-"UvU
In
structures, because of
smaller size of the FPZ, limerrors should occur under the assumption of LEFM and, in
engineering analyses, limited errors are acceptable as long as they
Therefore, it is useful to develop a procedure that would
an approximate value of
error
a
compare it with the accepted errors reanalysis,
lated to
basic properties such as compressive strength and
applied load.
In this
through theoretical considerations based on size
effect
LEFM
NLFM, a simple geometry independent
equation proposed to assess the error encountered in performing a
LEFM analysis of a large concrete structure. The theoretical curves
are first
and
contrasted with results of several finite
element analyses.
2 Theoretical considerations
According to LEFM,
stress singularity at the crack tip of a solid
with a
of length ao is characterized by the stress intensity
factor
where:
(1)
is the characteristic dimension of the structure, f (ao)
1378
is a dimensionless geometry shape
mensionless crack length and a N is
dimensional structures subjected to a point
can be generically expressed as a N == P /bD where b is
specimen thickness.
maximum load is attained when K1 reaches
ness of the material Jc= JE'G F
it can
nominal stress
LEFM
aN,D
(
E' G F
Dg(ao)
==
)
~
lS
plane stress and
Young's modulus,
(]"LEFM
==
N,D
script
to
acteristic dimension; also,
for plane strain
similar procedure).
Eq. 2 can
rewritten
..l..LL'-.&...l.v4l..4;Uv
(]"LEFM
N,D
_
---
1
(
D
lch
)
a
~~~~~~-L~~~
as
1
~
2
g( ao)
LEFM geometry
where D* == Dg( ao) is
EGF.
teristic dimension of the
== 7"{2 lS
length of the material (Hillerborg
al., 1976).
sents the scale law
structures
it shows that
proportional to
the structure.
Let us consider now an elastic-softening
geometry (i.e. the shape
sion), containing a crack with an
D; according to Planas
size effect law (Bazant, 1984; Bazant
etry independent nominal maximum stress a-:i:M is
0-NLFM
N,D
== )g'(aQ
)aNLFM
N,D
==
E'GF
( Cf+
P(:o})D
1379
)~
1
2
where aNLFM
== P max
NLFM /bD
g'(o: 0 ) is the first derivative of g(o:)
N,D
'
evaluated at o: == o:o, CJ is the elastically equivalent crack extension
at
load in an infinitely large structure (Bazant and Kazemi,
Planas and Elices, 1991 ), and D** == Dg( a 0)/ g'( a 0 ) represents
geometry independent characteristic dimension of the structure.
4 is strictly applicable when CJ is small compared with
(as
large structures)
gives an approximation of the maximum
nominal stress for larger CJ/ D ratios (Smith, 1995).
Since the applicability of the NLFM to concrete structures depends essentially on
size of the specimen relative to the
size, which in turn depends on the material characteristic length
Zeh, it is worth to
the structure size to Zeh· This can be
. . . . ,. . . . . "-'.__._ from Eq. 4 by normalizing the nominal stress to the tensile
ff
a-NLFM
1
~
( !::1_
ff
lch
+ D**
)
~
(5)
lch
Since CJ is a material property (Bazant and Kazemi, 1990), and
depends on the shape
the softening curve (Planas and Elices,
as
, it could be
(6)
Zeh takes into account the mechanical properties of the maand ""f is a dimensionless parameter which takes into account
shape of the softening law .
....... ~. . . . . ~~- Eqs. 5 can be rewritten as
~
1
)
(7)
- (
D**
""J + z;;:
Revisiting Eq. 3, it can also be expressed in terms of D**, and
nominal stress a- N n
ft'
)
a-LEFM
_!'L_
O"LEFM
=
,jg' (eto) j £
(
=
1 )
f :·
~
(8)
the ratio between the maximum loads obtained from a
and NLFM analyses (Eqs. 8 and 7), which varies with the
structure dimension, can be expressed by
LEFM
O"LEFM
~
(5-NLFM
N,D
aNLFM
N,D
p
(9)
NLFM
max
1380
Eq. 9, valid for all positive specimen geometries, shows that the
load ratio depends on the material by means of cf
is independent of the specimen geometry if the characteristic dimension of the
structure is expressed as D**. Furthermore, when
is normalized to Zeh, the load ratio depends only on the
K f which
is related to the shape of the softening curve.
Following Planas and Elices ( 1992), a lower
cf is related
to the critical crack opening We
7r
CJ> -w 2-
E'
(
- 32 eGF
and expressing the critical crack opening in
of characteristic crack opening as We == c2Weh == c2G F /ff, where c2 is a dimensionless parameter, Eq. 10 yields
CJ
7r
2E'GF
1)
>-c2-JI2-
- 32
and thus
(12)
Using the lower bound value of KJ from Eq. 12, Eq. 9 can be rewritten as
p
LEFM
_m_ax_
p
>
NLFM
max
3 Comparison between theory and numerical results
The theoretical relations previously obtained are now contrasted
with the results of a numerical parametric study on CT (Fig. la,b)
and TPB specimens (Fig. 2) performed by Plizzari and Saouma
(1995b) using MERLIN (Reich et al., 1994). The study referred
large specimens with different dimensions characterized by a bilinear
softening law where the values of the governing parameters were
changed, as summarized in Figs. lc,d,e,f.
Fig. 3a shows the relationship between KJ and D** /Zeh as obtained
by all the FE analyses with c2 == 5 that allowed a good fitting of
several experimental results on concrete specimens (Plizzari
Saouma 1995). We observe that the numerical results diverge
increasing values of the D** /Zeh ratio; this dispersion is in great
part caused by the fact that, in all cases, the mesh topology was
1381
Oc'.=15°
F~ ~F
6'1
b
• •
FE mesh
b)
a)
2.5
25 ft.
50 ft.
6.86
1.52
7.62
Effect of
en 1.5
en
1.08
E
1n 1.0
13. 72 3.04 15.24 2.16
c,=0.75
f\=2.25 MPa
C:z=5.QQ
J!2
"(ij
100 ft. 27.44 6.08 30.48 4.32
~ 0.5
I-
4.88 12.16 60.96 8.64
o.o
0.2
0.4
0.6
0.8
Crack opening (mm)
3.0
.2.5
Effect of
- - - - Wc=0.66 mm
- - Wc=1.00 mm
-------- Wc=1.33 mm
02.5
5.0l
7.5
0.0
f't
- - - - f't=1.90 MPa
- - f't=2.25 MPa
-------- f\=2.60 MPa
CL
~2.0
Cl)
(/)
C1=0.75
e 1.5
-+-'
f't=2.25 MPa
(/)
J!2 1.0
G,=300 N/m
C1=0.75
C:;i=5.00
'(ii
c::
f)
~0.5
1.6
0.0
0.2
0.4
0.6
0.8
Crack opening (mm)
1. Mesh and '""'" . . . . . . . . ._, . . . . . . . . . ., . . . . . . adopted for
numerical parametric
study of CT
(a,b,c), bilinear concrete softening law
for
three
values of fracture energy G F ( d),
w c ( e) and tensile strength f; (f)
1382
0
s
D
Oo
s
[m]
[m]
[m]
vs
6.35
2.0
25.4
s
12.70
4.0
50.8
L
VL
25.40
8.0 101.6
Specim.
50.80 16.0 203.2
2. Meshes adopted for the numerical parametric study of
Point Bending models
kept identical whereas the size was linearly scaled to u~•_,vJ.J.J.J...l
larger D (Fig. lb). As a result, larger models may not have a
enough mesh at their crack tip to capture the FPZ and the
;;, f adequately. However, it should be noted that, for large .. "''Lu.,.,,.
Kf varies between 1.5 and 2.0 (the value KJ == 1.8 was chosen as
reference value). For c2 equal to 7.5 and 10, KJ is larger but its
was not determined in that study since only few numerical
were available. From Eq. 12, a lower limit of ;;, f corresponding
c2 == 5 is 2.45, which is higher than the numerically
one, while for c2 == 7.5 and 10, the values of "'J are 5.52
respectively.
Fig. 3b contrasts the theoretical prediction of the normalized
ometry independent stress ffN,D/ ff, as obtained by Eqs.
8
NLFM and LEFM respectively, with all the numerical results
tained for both CT and TPB specimens. The theoretical
from Eq. 7 are plotted for different values of Kf : 1.8, 2.45, 5.52
9.82. We observe that the theoretical LEFM curve matches
numerical LEFM results very well, and that the theoretical
curve for ;;, f == 1.8 matches
numerical NLFM results
..l..V'-..1..<AllJ•...,
1383
2.2
0
D
1.8
0
~ 1.4
1.0
D
§
0
o
0 ~~
0
0
0.7
0
~ 0.6
goo
1
'0z o.5
0.4
OCT Model-c2=5
OTPB Models-c2=5
0.6
0.3
0. 2
a
4
6
8.., 10 12 14 16 18 20
D I
OCT Models-LEFM
OTPB Models-LEFM
t:i,. CT Models-NLFM-c2=5
VTPB Models-NLFM-C2=5
0 CT Models-NLFM-C2=7.5
1111 CT Models-NLFM-C2=10
Theory-LEFM
Theory-NLFM-Kf=1.25
Theory-NLFM-1Cf=2.45
..,.._.,LJ!"llo...,~Theory-NLFM-1Cf=5.52
0.8
-
'
=~~~T::heo~ry-NLFM-Kf=9.82
._._....._.__.__.__.__..--'-''---'-~~......___.-==L.-...c..J
0
2
4
6
p LEFM I p NLFM
lch
8... 10 12 14 16 18 20
D I
lch
1.8
1.7
1.6
1.5
1.4
..._
1.3
1.2
u
1.0
0.9
0
2
4
6
8
10
D I lch
**
12
14
16
18
20
Fig.3. Comparison of the theoretical predictions with the numerical
results
Also, the theoretical curve for ""f = 2. 45, as obtained from
Eq. 12, is very close to the numerical results. On the other hand,
curves for ""! = 5.52 and 9.82 underestimate the numerically
determined maximum load; however, it should be noted that the
adopted meshes probably become too coarse when a large We (and
thus a large FPZ) is adopted since the FPZ develops in a very small
number of elements.
Fig. 3c contrasts the theoretical predictions of the P ~::M P :~xFM
(Eq. 9), with all the numerical results. Again, it should be noted
that the theoretical curve matches well to all the numerical results
obtained with c2 = 5 for both ""! = 1.8 and 2.45 when the D** /Zeh
ratio is larger than 3, and overestimates the error for smaller ratios. The same good match is not obtained with c2 = 7.5 and 10
but, as mentioned, the coarseness of the adopted mesh could have
influenced the numerical results.
c2
= 5.
/
1384
4 Concluding remarks
On the basis of the LEFM and NLFM scale laws for concrete, a
theoretical expression to assess the error encountered in performing
a LEFM analysis of a large concrete structure was proposed (Eq. 9).
This very simple equation, valid for positive specimen geometries,
P :~:M which varies with strucshows that the load ratio P
ture dimension, depends only on the material and is independent
of specimen geometry if the characteristic dimension of the
ture is expressed in terms of D**. Also, if D** is normalized to
characteristic length of the material Zeh, the load ratio depends
on the parameter K, f which is related to the shape of the softening
curve.
A comparison with numerical results obtained by the authors
(Plizzari and Saouma, 1985b; Fig. 3c), shows that the theoretical
P
P :~:M ratio matches all the numerical results well when the
D** /lch ratio is larger than 3, and overestimates the error for smaller
ratios.
!::M /
,
!::M /
Acknowledgments
This research has been funded by the Electric Power Research
tute, Palo-Alto, through Research Contract No. RP-2917-08. The
support of its program manager, Mr. Doug Morris, and its project
monitor, Mr. Howard Boggs is gratefully acknowledged.
The first author would like to acknowledge the Commission for
the Cultural Exchanges between Italy and the United States (Rome)
and the Council for International Exchange of Scholars for
support for his stay at the University of Colorado at Boulder.
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...... LJLJ"-".......
1386