Acta Polytechnica


doi:10.14311/AP.2014.54.0358
Acta Polytechnica 54(5):358–362, 2014 © Czech Technical University in Prague, 2014

available online at http://ojs.cvut.cz/ojs/index.php/ap

EFFECTIVE FRACTURE ENERGY OF
ULTRA-HIGH-PERFORMANCE FIBRE-REINFORCED CONCRETE

UNDER INCREASED STRAIN RATES

Radoslav Sovják∗, Jana Rašínová, Petr Máca

Czech Technical University in Prague, Faculty of Civil Engineering, Experimental Centre, Thákurova 7, 166 29
Prague, Czech Republic

∗ corresponding author: sovjak@fsv.cvut.cz

Abstract. The main objective of this paper is to contribute to the development of ultra-high-
performance fibre-reinforced concrete (UHPFRC) with respect to its effective fracture energy. This
paper investigates the effective fracture energy, considering various fibre volume fractions and various
strain rates. It is concluded that the effective fracture energy is dependent on the strain rate. In
addition, it is found that higher fibre volume fractions tend to decrease the sensitivity of UHPFRC to
increased strain rates.

Keywords: Effective fracture energy; UHPFRC; quasi-static loading; increased strain rates; micro
fibres; fibre volume fraction.

1. Introduction
Ultra-high-performance fibre-reinforced concrete (UH-
PFRC) is an advanced cementitious composite with
enhanced mechanical and durability properties. Nowa-
days, UHPFRC is not considered as an alternative to
conventionally used materials, but it outperforms con-
ventionally used concretes. The reasons for the present
situation are the higher initial costs, and also a certain
inertia in the present-day building industry that has
led to the continued use of conventional concretes.
The application of UHPFRC is therefore limited to
special applications such as energy absorption facade
panels and key elements of building structures that
may be exposed to increased loading rates resulting
from earthquakes, impacts or blasts [1, 2]. In the
event of increased loading rates, a large deformation
of the structural member is expected, while the ex-
posed member is required to continue to possess some
residual capacity to carry the load. The capacity of
the member to absorb the energy can be quantified
via the effective fracture energy, which determines the
overall energy that a material can absorb per square
meter. The energy absorption capacity is the main
material property that benefits from fibre reinforce-
ment. The effective fracture energy (Gf ) is a key
parameter for evaluating the ability of a material to
withstand an increased loading rate and also to re-
distribute the load from the exposed structure to its
surrounding parts. In addition, different behaviour of
UHPFRC in terms of Gf can be expected at higher
loading rates, as the action is shorter by a magnitude
than for quasi-static loading (see Figure 1).

2. Effective fracture energy
The effective fracture energy (Gf ) of a material is
defined as the energy required to open a unit crack

Figure 1. Strain rates for various load events.

Figure 2. Effective fracture energy of a notched
specimen.

surface area. Gf is governed by the tensile mechanism
of the material, and represents the amount of energy
consumed when a crack propagates through the beam.
The fracture energy is expressed as the work of exter-
nal forces acting on the beam related to the actual
depth of the crack. The overall work of the external
forces related to the final crack depth is considered
as the average fracture energy, the so-called effective
fracture energy (Figure 2).
The effective fracture energy (Gf ) is strain rate

dependent, and it is assumed that Gf also increases
with increasing strain rate. This dependence is usually
described by the dynamic increase factor (DIF), which
expresses the ratio of Gf measured under increased
strain rate loading conditions to Gf measured under

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vol. 54 no. 5/2014 Effective Fracture Energy of UHPFRC under Increased Strain Rates

Fibre content 1 % 2 % 3 %
Cement CEM I 52.5R 800
Silica fume 200
Silica powder 200
Water 176
Superplasticizer 39
Fine sand 0.1/0.6 mm 336
Fine sand 0.3/0.8 mm 720 640 560
Fibres 0.22 × 13 mm 80 160 240

Table 1. Mixture design of the UHPFRC used in this
study (all values are in kg/m3).

Figure 3. Micro fibres used in this study.

quasi-static loading conditions:

DIFGf =
Gincreased strain ratef

G
quasi-static strain rate
f

.

Gf was determined in this study on the basis of recom-
mendations given by the RILEM Technical Committee
[3] and also by other studies [4, 5]:

Gf =
Wf + mguu

b(h − a0)
,

where Gf is the effective fracture energy, Wf is the
work of external forces (i.e., the area beneath the L–D
diagram), and mguu is the contribution of the weight
of the beam. In detail, m is the weight of the beam, g
is gravity acceleration, uu is the ultimate deflection of
the beam, b is the width of the beam, h is the height
of the beam, and a0 is the height of the notch.

3. Material
The UHPFRC tested in this study was developed on
the basis of components widely available in the Czech
Republic (see Table 1). The material design process

Fibre content 1 % 2 % 3 %
Compressive strength 150 152 150
Tensile strength 7.8 9.9 11.7
Modulus of rupture 15.8 25.6 33.8
Splitting tensile strength 14.9 20.5 26.6
Modulus of elasticity (in GPa) 45.1 56.3 51.5

Table 2. Mechanical properties of the UHPFRC used
in this study (all values are in MPa).

Figure 4. a) Pull-out failure mode. b) Fibre failure
mode.

has been fully described elsewhere [6–8]. Briefly, the
UHPFRC was mixed in conventional mixers, and the
beams were cured in water tanks. The mixture con-
tained a high volume of cement and silica fume, and
the water-to-binder ratio was 0.18. In this study the
strain rate and the fibre volume fraction (i.e., the
fibre content) were selected as the main test variables.
The high-strength steel fibres used in this study were
13 mm in length and 0.22 mm in diameter (see Fig-
ure 3). The fibres were straight, with tensile strength
of 2800 MPa. The high tensile strength of the fibres
was chosen in order to achieve the pull-out failure
mode. The pull-out failure mode (see Figure 4a) is
a much more energy-consuming mode than the fibre
failure mode (see Figure 4b). Straight fibres also pro-
vided a good trade-off between workability and the
mechanical properties of the resulting mixture.

As shown in Table 2, the compressive strength mea-
sured on cylinders 200 mm in height and 100 mm in
diameter was around 150 MPa. The compressive
strength did not vary with increasing fibre content.
However, the uniaxial tensile strength, the modulus
of rupture and the splitting tensile strength showed
linear dependence on the actual fibre content (see
Table 2). The maximum tensile strength was deter-
mined to be 11.7 MPa when the fibre content was 3 %
by volume [9].

The UHPFRC mixture was placed in moulds along
the length of the beam, and this caused the fibres to
be aligned along the length of the beam [10]. This led
to fibre alignment in the direction of the tensile stress.
No other technique was used to align the fibres. All
beams were tested after 28 days from casting in order

359



R. Sovják, J. Rašínová, P. Máca Acta Polytechnica

Figure 5. Experimental setup.

Figure 6. Load–deflection diagrams for UHPFRC
beams under various strain rates and with various
fibre contents.

to avoid the effect of ageing, which may also influence
the results [11].

4. Experimental program
Experiments were performed on beams 100 × 100 ×
550 mm in size with a clear span of 500 mm. The
beams had a notch in their bottom edge which was
30 mm in height and 5 mm in width (Figure 5). Three
different fibre volume fractions were tested covering
1 %, 2 % and 3 % of the fibre volume content. Each
fibre volume fraction was tested under quasi-static
conditions and under increased strain-rate conditions.
Quasi-static conditions were simulated by a deforma-
tion controlled test with a speed of the cross-head
of 0.2 mm/min. This speed corresponded to a strain
rate of 5.6 × 10−6 s−1, which is considered as the
quasi-static strain rate [12, 13]. An increased strain-
rate was simulated by the greatest possible motion
of the cross-head of the hydraulic testing machine
used in this study. The cross-head developed a speed
of 200 mm/min, corresponding to a strain rate of
5.6 × 10−3 s−1. This level of strain rate is typical for
dynamic loading, e.g., for earthquakes. During the ex-
perimental program, the force acting on the beam and
the deflection measured by two LVDT (linear variable
differential transformer) sensors was recorded with
5 Hz and 1 kHz frequency during quasi-static loading

Figure 7. Development of the effective fracture energy.

Figure 8. Development of the dynamic increase factor.

and increased strain rate loading, respectively. Steel
yokes were implemented in the experimental setup as
mounts for the LVDT sensors, in order to subtract
the settlement of the supports from the measured
deflections [14].

5. Results and discussion
Load–deflection (L–D) diagrams were plotted for all
beams, including various fibre volume fractions tested
under various strain rates. Three beams were tested
for all fibre contents and for both strain rates, making
a total of 18 tested beams. The ZUZ-200 hydraulic
testing machine with a closed loop deformation control
system with a maximal capacity of 100 kN was used
for both loading rates. The tests were deformation
controlled, based on the movement of the crosshead,
which was either 0.2 mm/min for quasi-static load-
ing or 200 mm/min for the increased strain rate (see
Figure 6).
Several authors have suggested that for low strain

rates the fracture energy is constant, and is therefore
not dependent on the loading rates [15, 16]. However,
in our study it was found that for increasing fibre
contents and also for increasing strain rates the ef-
fective fracture energy also increases (see Figure 7).
This is because in the case of quasi-static loading the
crack propagates along the line with the least resis-

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vol. 54 no. 5/2014 Effective Fracture Energy of UHPFRC under Increased Strain Rates

Gf [J/m2]

Strain rateFibre
content 5.6 × 10−6 s−1 5.6 × 10−3 s−1 DIF

1 % 12000 (2100) 15300 (3200) 1.28
2 % 17900 (1200) 20300 (1500) 1.13
3% 25300 (700) 26900 (1900) 1.06

Table 3. Effective fracture energy of the UHPFRC.

tance, which leads to minimal fracture energy. In the
case of increased strain rates, the crack does not have
enough time to seek the lowest energy consumption
path, and goes straight through the beam, which is a
more energy-consuming procedure [17]. Other authors
have suggested that the rate effect for low strain rates
can also be attributed to viscous effects, which mainly
originate due to the presence of free water in voids and
in porous structures [18]. For higher fibre contents,
the sensitivity to strain rates decreases due to the
group effect of fibres that interact together. Denser
fibre concentration decreases the pull-out capacity of
the fibres, because the matrix surrounding the fibre
will not be sufficient to keep the interfacial bonding
as strong as in the single pull-out case [17]. When the
pull-out capacity is lower, the sensitivity to loading
rates is also lower. This is indicated by the dynamic
increase factor (DIF), as shown in Table 3 and Fig-
ure 8. The effective fracture energy values indicated
in Table 3 are averages from three beams. The value
in parentheses gives the standard deviation from the
tested beams.

6. Conclusions and further
outlook

The effective fracture energy was determined on a total
of 18 beams, which were tested under various strain
rates and with various fibre contents. The fibre volume
fraction ranged from 1 % to 3 % by volume, and the
strain rate was either 5.6 × 10−6 s−1 or 5.6 × 10−3 s−1.
The following conclusions can be drawn on the basis
of the experimental outcomes derived from this study:

(1.) The effective fracture energy (Gf ) increases as
the fibre volume fraction increases. Higher scatter
in the experimental outcomes was observed for lower
fibre contents. The pull-out capacity of the fibre
in a higher fibre volume content is lower than the
pull-out capacity of the fibre in a lower fibre volume
content. Each fibre plays a more significant role in
terms of Gf in beams with a lower fibre content,
because its pull-out capacity is higher. Thus each
mismatch in the fibre distribution in lower fibre
contents will scatter the results more.

(2.) Gf is dependent on the strain rates. It was veri-
fied experimentally that Gf increases as the strain
rate increases. In addition, it was found that higher

fibre volume fractions attenuate the dependence of
Gf on the strain rate. This is because when there
is a higher fibre volume fraction the maximum pull-
out capacity on each individual fibre is reduced; its
sensitivity to higher strain rates is therefore also
reduced.

(3.) Two strain rates were considered in this study,
which can both be classified as low strain rates. An
increased strain rate was simulated as the maximum
speed developed by the hydraulic testing machine.
It is important to note that it is a fairly complex
and highly time-consuming task to make proper
measurements of the effective fracture energy. This
is the main reason why no experimental work was
performed on other strain rate levels. It is therefore
highly desirable to extend the scope of our study
presented here, and to verify the effective fracture
energy of fibre-reinforced cementitious composites
under higher strain rates above 10−2 s−1.

Acknowledgements
The authors gratefully acknowledge the support provided
by the Czech Science Foundation under project number
GAP 105/12/G059. The authors would like to acknowl-
edge Michaela Kostelecká, from the Klokner Institute, for
her assistance with the microscopic investigation of the
fibres used in this study. The authors would also like to
acknowledge the assistance given by the technical staff
of the Experimental Centre, Faculty of Civil Engineering,
CTU in Prague, and by students who participated in the
project.

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	Acta Polytechnica 54(5):358–362, 2014
	1 Introduction
	2 Effective fracture energy
	3 Material
	4 Experimental program
	5 Results and discussion
	6 Conclusions and further outlook
	Acknowledgements
	References