The Identification Of A Hidden Long-Term Plastic Damage Stage During
Splitting Tensile Loading Of Concrete
by
Zacharias G. Pandermarakis and Anastasia B. Sotiropoulou
Department of Civil and Structural Engineering Technology Teachers
School of Pedagogical and Technological Education (ASPETE)
“Irene” Train Station, Athens – Greece
[email protected], [email protected]
ABSTRACT
A new method for the determination of fracture toughness KIC, energy released rate GC and critical crack tip opening
displacement CTODC of mortars and concretes with small size aggregates –of at least 8mm diameter- is presented. This could
be achieved using a suitable modification of the conventional splitting tensile test. The values that were obtained coincide with
those given by other researchers. Based on a clear-cut fracture mechanics analysis, we extract a set of equations that
correlates the above quantities with crack rate. The identification of specific plastic damage stages also is exposed. They were
found to accompany the crack opening initiation and propagation processes. A pre crack propagation damage process of a
characteristic length 0.1 to 1m appears first. This could be diffused in many region of loading component, could be localized
or form an active attractor cracking path by a chain accumulation of them. Then another one that comes directly from the
formation of the well-known fracture process zone near the crack tip of the order of 10m must be added. Lastly, a large scale
one that has to do with composition and texture of stressed components and loading and geometry conditions must be
superimposed to the first two. This damage process will come directly from the reordering of the previous mention features, as
the fracture is developed giving a corresponding length of the order of some mm to some meters.
Introduction
The extensive use of concrete for so many years didn't happen accidentally. A lot of factors have been contributed to this.
Factors like its convenience during manufacture process, its reliability and availability -especially when limestone deposits are
easily accessible -the highly achieved strengths and its durability in a wide period of time are crucial. Also its capability to
collaborate with other basic construction materials and mainly with steel, made the concrete exceptional and irreplaceable.
Nevertheless, the continuously increasing requirements about bigger and safer structures combined with the added
experience from all previous years of continuous uses of concrete led to the need for a comprehensive knowledge for its
behavior. The utilization of suitable software for almost all calculations in construction industry, highly simplify the production
processes but simultaneously however requires from us a better description of materials response.
A big effort has been made the last thirty years about constitutive description of concrete. Serious problems arise due to
no-linearity that exposes. This no-linearity could be resulted from the specific way of its fabrication, its composition and of
damage mechanisms at loading state. The appearance of distinct masses of unhydrated cement, coarse aggregates, pore
clusters, microcracks, different phases and constituents with specific texture as round-like, needle-like and plate-like ones
leads to an extensive inhomogeneity that causes high stress concentrations.
The polar nature of atomic bonds of structural elements of concrete constituents has as result the restriction of the
appearance of large strains. As results, concrete appears a brittle response. High or not so high loading stresses externally or
not so imposed, as these come from abnormal curing conditions, dry shrinkage, heat release during cement hydration, creep
etc. could easily lead to its microcracking. Its non-continuous structure and the convenience that concrete is cracked, they
give to it a characteristic non-linearity, which also will be influenced by scale effects. The fracture mechanics approach could
explain with success many of them. As results a lot of investigation starts from this scientific approximation. Criteria that
determine the conditions under of which failure of a loading concrete component could come and criteria that try to attribute
the way and also the conditions under of which a crack will propagate are all in continuous investigation.
As concrete belongs to the class of brittle materials follows the empirical rule that a critical quantity for failure conditions is
this of maximum tensile stress. Due to the fact that unidirectional tensile tests are remarkably time-consuming and ambiguous,
the most of time we follow the way of indirect tensile through bending or splitting tensile tests. Bending test offers better
control, higher influence of loading parameters and also the possibility through a suitable formation of narrower strength
distribution. So, much effort has been given in the research of concrete through manufacturing and loading of concrete beams
that could carry or not a suitable shaped notch.
A much lesser contribution to this investigation comes from the corresponding splitting tensile tests. This ought to the
limited possibilities to influence main feature of the test. Through a pure fracture mechanics approach of its behavior, critical
role poses the type (shape, size, spatial and size distribution) of the inhomogeneities – flaws into the body. The formation of
suitable notches in the edges or on surfaces of concrete specimens or of any other modeling material as polymer one give us
indirectly the influence of cracks in mechanical behavior of concrete.
Nevertheless, the formation of notches on concrete specimens and their followed strain doesn't always lead to secure
conclusions for concrete components with the appearance inside or on their surface natural flaws under real loading
conditions. Certainly, the true response could be approximated much closer if we try to determine all those fracture mechanics
quantities in a common un-notched concrete specimen. In this investigation we managed, with high accuracy, to determine a
number of fundamental fracture mechanics quantities like fracture toughness KIC, fracture energy GF and also critical crack tip
opening displacement CTOD without the formation of notches on specimens. Moreover contrary to the usually followed
practice this will become evident through splitting tensile tests. Due to the specific features that this experiment appears, it was
modified suitably in order to get the wanted quantities. The determination of these quantities will be reduced so, on a linear
elastic fracture mechanics approximation.
A more general consideration demands the development of a micromechanical model of fracture. Many researchers tried
to attribute the real behavior of concrete by developing suitable micromechanics models that try to describe concrete
macroscopic response as results from experimental procedures. Two main tendencies have been developed for
micromechanical modeling of concrete fracture. The first is the so-called fictious crack model that developed by Hilleborg and
his co-workers and the second the crack band model by Bazant and his co-workers. At fictious crack model it is considered
that crack behaves in such a manner that its response is determined by normal and tangential forces that act between crack
tips. From the other hand Bazant's model of crack band it is considered that the crack will propagate in a microfracture area
where will come from the development of a suitable fracture band. In present investigation simple observations from splitting
tensile tests will lead us to a combination and an extension of previous two models. At next, we will arrive to the conclusion
that three different procedure of mechanical plastic damage processes which accompanies the stages of crack initiation and
propagation procedures into loading component seems that could give a satisfactory description of real response of a concrete
component.
Experimental Procedure
The modification in experimental procedure that help to the identification of these not easily obvious stages had to do with
differentiations in geometrical features of using constituents, with differentiations in scale of shaped specimens, with
differentiations of measured quantities and finally with differentiations in measuring arrangements. A totally different character
for splitting tensile test was demanded and planed in order to exclude all secondary effects that entering in the system could
alter it noticeably.
Figure 1: Experimental arrangement
According to the followed plan, concrete was fabricated under control conditions in laboratory, using common limestone
aggregates with maximum size of 4mm in order to not-lose the pure cement and also interfacial zone responses. The type of
cement was I52.5 and water to cement ratio was specified at 0.40. The dimensions of specimens were chosen much smaller:
1/3 of common one, i.e. 5cmx10cmm instead of 15cmx30cm. This was chosen so in order to become achievable the
identification of pre crack initiation stage and is corresponding regions, and also this of fracture process zone during crack
propagation. It must be quoted here that the main features of the obtained results didn't altered even when we used aggregate
with 8mm diameter, when we used marble aggregates and when instead of the specimens from mold we tested bored cores
with the same composition from suitable prepared panels by sprayed concrete. All test were carried out in 3, 7, 14, 28 and 56
days after their preparation and all give similar responses.
Also contrary to the traditionally executed splitting tensile test where only the maximum stress is aimed, here we seek also
the strain state of loading component in various directions. In such a basis, innovating suitably, we used strain gauges in
vertical and horizontal directions in the bases of cylinder specimens. Doing so, we manage to identify easier all these
differentiations in response and also in mechanical features of material in various directions and in real time conditions. Using
smaller specimens, lower loads they were required for their failure, and the same was true for load cell apparatus. But using a
lower load cell a better measurement precision is achieved. Here was in the order of ±0.5kN. For real time data acquisition
suitable software was used. The details of experimental arrangement are shown in Fig.1.
Results
In Fig.2a, a characteristic splitting tensile stress-strain response is appeared. Strain was obtained from a transverse strain
gauges in a flat base of specimens so was every time the tensile strain. Similarly the corresponding tensile stress in the middle
of cylinder base was estimated in each time from the well know relation:
σ=
2P
π DL
(1)
where P the applied load, D the diameter of specimen and L its length. It must be pointed here that simultaneously with tensile
stresses in a splitting tensile test in every moment during loading also compressive stresses are appeared with maximum
value that of –31.
6
60
yielding like
crack propagation
point
5
(a)
(b)
50
plastic damage stages
40
1 (MPa)
1 (MPa)
4
3
30
20
2
linear elastic stage
10
1
0
0
0
500
1000
1500
0 tr
2000
2500
3000
(x10-6)
0
1000
2000
3000
0
4000
5000
(x10-6)
Figure 2: Stress – strain curve after tensile splitting test (a) and
unconstrained compression (b) of a common concrete.
In Fig.2b we exposed for comparison the corresponding stress-strain curve for unconstrained compression test of the same
type of concrete and at the same age. Tested specimens were of the age of 28days and were cast in steel moulds.
The results of Fig.2a, even though non-expected, were repeatedly verified in our laboratory also for many concrete
specimens with various compositions (different aggregate sizes up to 8mm) and prepared conditions (i.e cast or sprayed
concrete). In this curve we could identify three characteristic regions. First, a linear one at the beginning of stress-strain curve
that corresponds to a linear elastic response. This linear elastic stage is followed by a fracture process starting from a
characteristic yielding-like point. At this stage typical crack initiation and propagation are observed which accompanied by the
development of corresponding plastic damage processes.
The most characteristic feature of this behavior is the appearance of a typical yielding like point –just before initiation of
-6
cracking take place-DWDSSUR[LPDWHO\WRV[ ) strain (fig.2a), in a moment where normally a tensile fracture point
would be observed. This point is followed by a long-term yielding-like flowing –that accompanied crack propagation- in an
imposed load P to transverse strain 0tr diagram. During this stage the formation of a characteristic fracture process zone in
front of the crack tip appears. This almost horizontal variation of load, continues until macroscopically fracture of concrete
F\OLQGHUVSHFLPHQVWDNHSODFHDWVWUDLQVZKLFKYDULHGIURPWR VVWUDLQVWKDWDUHDOPRVWWRKLJKHUWKDn
the expected ones. And all these for a severe loading state where intensive tension stresses are generated. This same feature
was obtained also for compression negative strains, but only for a limited cases due to the fact that it is very difficult to identify
a priori the exact point where a fracture will be initiated. Even though a part of these strains could be attributed to crack
formation and propagation, this means that we observe, in un-reinforced concrete,
The region of the cylinder concrete specimen, which is responded for the above behavior, is localized along a vertical
plane that its trace is shown in Fig.3c. This plane passes through circular specimen bases in the same place where the
fracture surface will be formed after the total failure. From this set of experiments it is evident that the energetic and important
role of this fracture surface isn’t limited to the simple initiation, propagation and finally the appearance of fracture cracks at the
end of the test but an intensive plastic damage is grown inside and next to this region. First, through the development of a pre
crack initiation, plastic damage process in distinct areas of specimen is appeared. From the analysis of stress – strain curves,
taking in mind the maximum elastic strain that a concrete specimen of this type, could undertake –approximately 350 to 500sand determine the strain that crack starts to propagate we could identify the size of this initial plastic damage process. Working
so, we found that this plastic stage forms damage areas of approximately 0.1 to 1m size. This damage process seems to
correspond to a pores strain and damage, probably to an limited enhancement and local microcracking of mircopores. Next
and during crack propagation stage, a second plastic damage process take place where ought to the formation of fracture
process zone (3,4) in front of the tip of the cracks. From tension and compression strains that appear during the carried out
tests, this stage is estimated to correspond to the formation of plastic regions of the order of 10 to 50m. This second stage
must correspond to damage of crystal phases through crack propagation. The gradually growing of these two stages (fig.3) is
the reason for the inelastic behavior of the semi-brittle concrete (Fig.2).
Figure 3: a) stress state condition –at the centre of the specimen- under splitting tensile loading where an almost linear
elastic stage is appeared, b) the formation of a damage zone along a vertical plane in the early almost pure plastic damage
stage followed by the formation of a fracture process zone during crack propagation. The initial damage zone could be
dispersed or could form a chain like crack attractor region, and c) the formation of a fracture surface after the crack
propagation with the completion of the fracture process zone stage
Damage regions in specimens, that be formed during the above plastic damage stages, act as crack attractors
leading the cracks to pass through these regions due to a corresponding lowering of their compliance. This damage zones
restore the continuum character of component but with a less resistance and less shift material. The formation of this damage
zone starts already from approximately 30% of the concrete strength. In splitting tensile test the location of the main damage
zone remains at a vertical plane between the loading generatrix of specimen. In this plane appears a gradually high strain that
comes from the diffuse damage process. As load approaches yielding like point the material starts to compliance and a softer
response is appeared. This means that a lesser modulus characterized the material during this process at least locally. But
now in this plane appears a gradually high strain that comes from . diffuse damage process. At the final stages of failure,
large cracks start to propagate in already active damage paths with an accelerated manner. Crack velocity as result from strain
gauge measurements gives rates that fluctuate from 0.5 to 50 m/s. This third plastic damage stage could be attributed to the
cracking of interfacial zones, to transgranular cracking and generally to crack passing from one phase to another. The size of
these regions must be in the rage of some mm to some meters, i.e. of the order of aggregates and components to the order of
construction.
At next we will develop a fracture mechanics procedure in order to verify these deformation processes and extract most of
the crucial fracture mechanics quantities.
A Fracture Mechanics Approach
Due to its fabrication process all cementitius materials include innumerable flaws in form of voids, pores, channels,
anhydrated areas, microcracks, on every aggregate an interfacial transition zone, heterogeneities as aggregates are etc. Their
shape, size, distribution and nature could be varying broadly. From fracture mechanics of view nevertheless the most crucial
feature is their size. If we identify and characterize this critical quantity then, as known, a measure for its toughness will be the
quantity:
K I = σφ π a
ZKHUH
(2)
1WKHQRPLQDOVWUHVVDQGa the size of flaws. 3 is a loading and specimen geometry factor that could take the value 1 in
some cases, but here 3 equals to 2 / 2 . A flaw could have any arbitrary shape. In a general case would have that of an
elliptic shape with a random orientation. From a statistical point of view we could get that the most decisive one would be this
with perpendicular to tension loading orientation. i.e. the vertical ones. So the above relation will turn to:
K IC = f tφ π ac
(3)
where KIC the fracture toughness of concrete, ft is tensile strength and aC a characteristic for this concrete size flaw. Stress
intensity factor KI will be increased with the samePDQQHUDVWHQVLOHVWUHVVLQWKHWLSRIWKLVVWDWLFIODZXSWRDYDOXH LFZKLFK
becomes equal to the specific fracture toughness of concrete. At this stage flaw will turn to a propagated cracking.
But as we referred previously, the impose loading will lead to an extensive damage many areas of the material substrate
for reasons so simple as material inhomogeneities and stress concentrations due to different phase properties and geometries.
This damage process alternates its initial characteristic size ac and leads to its enhancement. The damage areas need not to
be continuous in bulk material. Could correspond to isolated areas that forms crack attractors. From a fracture mechanics point
of view this could be handle as many researchers show before (Dugdale, Barenblatt in ductile materials and Hillerborg, Bazant
in quasibrittle materials as concrete) by a crack with an enhanced effective size: aef., i.e.:
aef = a + ∆a
(4)
We must state KHUH WKDW WKLV HQKDQFHPHQW ûD LQFOXGHV WKH FRQVHTXHQFHV Irom the formation of the diffuse damage before
crack propagation and also microcracking and debonding formations in frond of the crack tip as this propagate inside the
specimen forming the fracture process zone. This damage process could be varied between a more diffuse and non-localize
process to a more path like one that will determine in a high level the mode of fracture and of crack propagation. Also possible
must be a combination of the above pure modes.
So, this effective crack enhancement could be attributed by a pre crack propagation diffused and continuous or distinct
damage, by an extended fracture process zone in front of the tip of the propagated crack and lastly by a diffused damage
process in bulk material due to reordering of stresses during crack propagation, i.e.:
∆a = ∆a pre + ∆aFPZ + ∆areord
(5)
The sizes of these contributors it is expected to be varied according to material and geometry of the specimens and
ûapreWREHYDULHGDPRQJWRPûaFPZ in
WKH RUGHU RI P DQG ûareord to be depended mainly from geometry of specimens and loading conditions and also from
PDWHULDOVSURSHUWLHV7KHYDOXHVRI ûapreDQGRI ûaFPZ comes directly from the inspection and analysis of the diversions from
linearity of stress-strain curves obtain from strain gauges and the fact that a maximum elastic tensile strain could be accepted
WRRFFXUDQGWKDWZDV 0taWRV)RU ûareord we estimate that no evidence for its appearance is obtained by all carried
out test in this project. Nevertheless its existence must be necessarily in order to explain the post peak branches during
ORDGLQJ DQG KLJK FKDUDFWHULVWLF OHQJWK LQ PDQ\ FDVHV LQ LQWHUQDWLRQDO OLWHUDWXUH ûareord contribution seems to correspond and
ORDGLQJFRQGLWLRQVEXWKHUHIRUWKHW\SHRIVSHFLPHQVWKDWWHVWHGZHHVWLPDWH
EHKDYHOLNHWKHPRGHORIILFWLWLRXVFUDFNZKHUHDV
ûapreDQGûaFPZ to crack band model.
From all the above it is evident that finally we will have:
K I = σφ π aef
(6)
If we suppose that this crack length enhancement remains low, geometrically similar to the initial flaw and generally small
UHODWLYH WR VSHFLPHQ VL]H WKHQ ZH FRXOG VXSSRVH WKDW WKLV VXUIDFH FUDFN RSHQLQJ GLVSODFHPHQW
/ DV PHDVXUHG IURP VWUDLQ
gauges will be approximately equal to crack tip opening displacement CTOD and then it could be given by the following
relation:
σ 2φ 2 aef
ft E *
δ =π
(7)
DQHDVLO\H[WUDFWHGUHODWLRQIURPWKHFRUUHVSRQGLQJJHRPHWU\,QWKLVUHODWLRQ
1LVWKHQRPLQDOVWUHVVDQGIt a yield like stress.
/
1
1
1
CTOD
aef
/
Figure 4: Fracture Process Zone during crack extension enhances its length and leads to a
smooth stress reduction near the crack tip. This stress curvilinear variation could be
approximated by a simpler linear one (dotted straight line)
Here the tensile strength of concrete. E =E for plane stress conditions and ü =ü/(1- ) for plane strain. In a self similar crack
propagation the above relation could be differentiated to give the rate of crack propagation or proportionally the rate of crack
tip opening displacement:
*
*
σ 2φ 2
∂δ
δ =
= π aef
ft E *
∂t
or
2
δ f t E *
aef =
π σ 2φ 2
(8a,b)
Also, from eq.3. comes that:
( K IC )
2
= ( f t ) φ 2π aef
2
(9)
So dividing apart equations 8a and 9 comes that:
∂δ f t E *
πσ 2φ 2 ( K IC )
2
=
∂aef
( ft )
2
π aef
(10)
But, from an energy equilibrium approximation we could extract from an Irwin relation the stress released rate G as:
G=
K I2
E*
(11)
2
K IC
= GF E *
or finally
(12)
were GF the fracture energy. So we will have from relation 10 that:
∂aef
∂δ f t
=
σ 2φ 2GF ft 2 aef
(13)
and also from relation 7 that:
GF = δσ ys
or
GC′ =
πσ cr acr
E*
(14a,b)
From these relation we could easily see that fracture toughness KIC and fracture energy GF depends from crack length. But it
seems more reasonable in order to avoid large fluctuation of crack length, these quantities to depend more satisfactory from
the rates of crack da/dt and of crack opening G/GW. So we could construct a map in which we will draw the energy release rate
G and stress intensity factor according to da/dt (and similarly to d//dt).
ÿ
Gc
G (N/m)
da/dt (m/s)
ï
*ï
K Ic
K I (MPa.m^1/2)
Figure 5: From this crack rate via energy release rate and stress intensity factor map could identify two distinct areas that
correspond to a same number of fracture stages of concrete. A third one follows the previous leading to an unstable
catastrophic crack extension.
From fig. 5 we could see that two distinct areas appeared. One for lower crack velocities and one more for higher ones.
Actually the lower region corresponds to the slow crack growth. In this area we expect fracture toughness and fracture energy
to be equal to toughness resistance and fracture energy resistance. Whereas in the higher one a fast unstable fracture
appears. In this region energy release rate and fracture toughness as comes from these measurements 0Œ0! * the
corresponding fracture resistances GR and KIR of concrete. So we could gate a critical rate for crack growth rate
and
 da 
 dt 
 cr
 dδ 
 dt 

cr
that separate the possible states in stable and unstable. Thus from energy release rate we could get a
threshold value that separate so the stable from unstable cases. So, the released energy that comes from fracture of
specimens leads to a faster resulting crack growth. The interesting point here is that K, G and crack rate appear a liner
dependence in a semi-logarithmic diagram. At next we will try to extract a relevant relation.
From relation 13 we will have:
:
∂δ
ft
∂δ σ 2φ 2 ∂a
1
a
=
∂
⇒
= 2
ft 2 a
GF
ft a
σ 2φ 2GF
(15)
But we could suppose that energy released rate is:
δf
GF = ∫ σ d δ
(16)
0
and if we accept that stress state at the tip of the crack varied in such a manner that the area underWKH1/FXUYHFRXOGEH
approximated by the area under a linear one (Fig.4), i.e.:
GF f tδ f
(17)
2
or in a differentiated form:
ft
∂δ f
2
∂GIf =
(18)
so we will have:
2 ∂G σ 2 ∂a
∂G σ 2 ∂a
= 3
⇒
=
ft G
ft a
G 2 ft 2 a
(19)
In the above differentiate form the total length of crack a hardly could give a representative measure of crack propagation. This
happens mainly due to large fluctuations and the arising difficulties for determining an absolute value. It seems easier and also
more accurate to join energy release rate G with a more stable quantity as the rate of crack propagation da/dt seems to be.
So will be:
∂G σ 2 ∂a
=
G 2 ft 2 a
(20)
And after integrations and .2.2.1210" we will have:


G
KI
σ 2  a  σ 2  δ  σ 2  CTOD 
ln
=
ln
ln
 ln  ≈
 ln  =

G0
K I 0 2 f t 2  a0  2 f t 2  δ0  2 f t 2 

 CTOD0 
<
(21a,b,c,d)
<
or rearranging the terms:
<
G
δ
CTOD
a 2 f
G
ln = 2t ln
= ln  
ln ln
δ0
a0 σ
G0
 G0 
CTOD 0
2
<
 2 ft 
 σ 


2
 K 
= ln  I 
 KI 0 
 2 ft 
 σ 


2
(22a,b,c,d,e)
so we will have:
<
δ
CTOD
a  G 
=
=
= 
δ0
a0  G0 
CTOD 0
<
 2 ft 
 σ 


2
 K 
= I 
 K I0 
Thus finally we attain the follow set of equations:
 2 ft 
 σ 


2
(23a,b,c,d)
da
dδ
da
dδ
′ K I2n
= Aa G n ,
= Aδ G n , CTOD = ACT G n ,
= Aa′ K I2n ,
= Aδ′ K I2n , CTOD = ACT
dt
dt
dt
dt
<
<
(24a,b,c,d,e,f)
Acknowledgements
The present work consist a part of research program with code number MIS86473-14096/5/8/04 in the framework of
“Archimedes I”, and is co-financed by the European Social Fund and National Resources from the EPEAEK II Program.
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