ARTICLE IN PRESS
International Journal of Impact Engineering 34 (2007) 609–626
www.elsevier.com/locate/ijimpeng
Tensile failure of concrete at high loading rates: New test data on
strength and fracture energy from instrumented spalling tests
J. Weerheijma,b,, J.C.A.M. Van Doormaalb
a
Faculty of Civil Engineering and Geosciences, Delft University of Technology, P.O. Box 5048, 2600 GA Delft, The Netherlands
b
TNO Prins Maurits Laboratory, P.O. Box 45, 2280AA Rijswijk, The Netherlands
Received 2 March 2004; received in revised form 25 January 2005; accepted 6 January 2006
Available online 3 April 2006
Abstract
For the numerical prediction of the response of concrete structures under extreme dynamic loading, like debris impact
and explosions, reliable material data and material models are essential. TNO-PML and the Delft University of
Technology collaborate in the field of impact dynamics and concrete modelling. Recently, TNO-PML developed an
alternative Split Hopkinson Bar test methodology which is based on the old principle of spalling, but equipped with up-todate diagnostic tools and to be combined with advanced numerical simulations. Data on dynamic tensile strength and,
most important, on fracture energy at loading rates up to 1000 GPa/s are obtained. The paper describes the test and
measurement set-up, presents the new test data and the analysis of the test results. In addition, a rate-dependent softening
curve is given which is based on the integrated findings so far.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Dynamic loading; Concrete testing; Instrumentation; Fracture energy; Tensile strength
1. Introduction
It is well known that the mechanical material response of concrete depends on the loading rate. Research in
the last 20–30 years has shown that the rate dependency of the strength tends from a moderate rate
dependency to excessive rate effects at high loading rates [1–6]. Especially the tensile strength exhibits a strong
increase beyond loading rates in the order of 10 GPa/s. Most materials exhibit a certain degree of rate
dependency, but for concrete the rate dependency occurs at relatively low loading rates. The cause is the scale
size of the heterogeneity [7,8]. Due to the composition of aggregates embedded in the mortar matrix, the
dominant defects and discontinuities in concrete occur at a length scale of 103 m, while, e.g. in ceramics the
defects are at the scale of 106 m. For ceramics the steep strength increase in tensile strength is observed at
loading rates of 105 GPa/s, significantly higher than the 10 GPa/s mentioned for concrete. The rate dependency
of the strength is governed by the initial damage state of the material and the contribution of the inertia effects
Corresponding author. Faculty of Civil Engineering and Geosciences, Delft University of Technology, P.O. Box 45, 2280AA Rijswijk,
The Netherlands. Fax: +31 15 284 9354.
E-mail address: jaap.weerheijm@tno.nl (J. Weerheijm).
0734-743X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijimpeng.2006.01.005
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to the extension of the (initial) damage. Based on the principle of damage extension a model was built that
enables the prediction of the strength increase in the whole range from static loading up to loading rates of
1000 GPa/s. The results correspond very well with the reported experimental data in literature, see Fig. 1.
These were promising results, but only one step forward. For practice like the design protective structures, the
prediction of damage levels under explosive and impact loadings or to develop countermeasures for terrorist
attacks, more knowledge about concrete behaviour is required. For tensile loading the current situation and
problem definition are:
– Strength is rate dependent and depends on moisture contents, concrete quality and concrete composition.
The latter dependencies are quantitatively not well understood;
– Moderate strength increase up to loading rates of 10 GPa/s. For engineering purposes and design methods a
strength increase of 1.2 is used;
– The strength increases extensively beyond loading rates of 10 GPa/s. Almost all data in the high loading
regime are obtained from indirect tension tests (the Brazilian splitting test), no direct tension tests are
available;
– No test data are available on the dynamic fracture process and energy dissipation at high loading rates.
Quite often the assumption is made that the rate effect on the fracture energy equals the rate effect on
strength. This is incorrect and not based on experiments. For loading rates up to 10 GPa/s the fracture
energy is not rate dependent [7,9]);
– Numerical code application demands reliable material models for the high loading rate regime because
detailed calculations result in these rates. No reliable models are available that deal with rate dependent
strength and fracture energy or softening [10].
The Prins Maurits Laboratory of TNO and the Delft University of Technology aimed themselves to develop
a (numerical) material model that covers the dynamic response of concrete up to complete failure. The first
focus is on tensile loading and especially the rate effects on the failure process and fracture energy. The current
paper deals especially with the first development phase of an instrumented spall test. An old technique but it
offers new perspectives when it is combined with ‘‘up-to-date’’ diagnostic techniques, the physics of wave
propagation, the material characteristics of concrete and advanced numerical simulation. New data is
gathered on the dynamic tensile strength and fracture energy for loading rates at 1000 GPa/s. The topics of this
paper are the spall test method, the applied instrumentation, the obtained data and analysis method. Based on
the new data a rate-dependent load–deformation curve is suggested.
8
7
ftdyn /ftstat [-]
6
5
4
3
Mc Vay
Birkimer
Ross
Ross
Zilienski
Takeda
Hatano
Komlos
Kivirikadse
Sneikin
Heilmann
Cowell
Bachmann
Zheng
Weerheijm
Schuler
-07
-05
2
1
0
-09
-03
-01
+01
strain rate [1/s]
Fig. 1. Relative tensile strength increase for different loading rates [11].
+03
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2. Hopkinson spall bar
2.1. Spall principle
Spalling is defined as the material failure in tension due to (partly) reflection of a pressure wave at the
material transition to a material with lower acoustic impedance. Commonly, spalling is related to the fracture
in a plate of material which occurs near a free surface remote from the impulsively loaded area.
The principle of spalling was studied and reported by Hopkinson. He presented his ideas of measuring the duration and amplitude of a pressure pulse generated by the impact of a rifle bullet, or a
small detonating charge at the end of a cylindrical steel bar [12,13]. Hopkinson used the spalling phenomenon to capture the loading pulse in an end segment that was in direct contact with the incident bar
but the seam had zero strength. By varying the length of the end segment he determined the impulse
of the load. Using the spall principle for studying dynamic material failure, the loading pulse must be
known.
In most textbooks the failure initiation is coupled to complete failure. Only the strength criterion is
mentioned and the energy criterion is not discussed. However, both criteria have to be fulfilled. Besides the
strength and energy criteria one has to realise that failure requires time. The fracture time is not an additional
criterion but a consequence due to the fact that energy dissipation takes some time. This ‘‘time to failure’’
or ‘‘incubation time’’ will affect the load history of the material next to the failure plane or fracture zone.
When a crack is formed elastic release waves emanate from the newly formed surfaces. When the fracture time
equals zero the release waves have sharp profiles but when failure is a more gradual process the release waves are
more diffused. In the test set-up it is intended to record the loading and failure process by means of strain
measurements on the specimen. The ‘‘time to failure’’ has to be incorporated in the interpretation of the
measurements.
2.2. The experimental spall test set-up
In order to examine the possibilities of the old spall technique, combined with measurements and numerical
simulations, it was decided to apply small detonation charges (up to 20 g). The test idea is as follows. The
charge at short distance from the steel bar loads the end of the bar and the pressure propagates and impinges a
concrete specimen that is situated at the other end of the bar. The transmitted pressure pulse reflects at the free
end of the concrete specimen as a tensile pulse and, if the resulting stress exceeds the dynamic strength and the
energy criterion is fulfilled, the concrete fails.
In order to determine material properties from this test set-up, the minimum requirements are that the
loading pulse is known and can be reproduced. On the other hand, the response of the concrete has to be
recorded. The test set-up, without the concrete specimen, is given in Fig. 2. In order to contain the explosion
Stress bands
500
Detonator
75
Metal box
S1 Above
S2 Above S3 Above
S4 Left
S4 Above
S2 Under S3 Under
S4 Under
S4 Right
Rubber sheet
S1 Under
Stand
Floor
500
1300
600
200
300
2000
Fig. 2. Schematic overview of the test set-up with the locations of the strain gauges (dimensions in mm).
200
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40
20
Ptot(xx10, tt)
Ptot(xx40, tt)
0
Ptot(xx80, tt)
σfail
Ptot(xx100, tt)
Ptot(xx160, tt)
-20
-F
Ptot(xx200, tt)
Ptot(xx240, tt)
-40
-60
0
0.05
0.1
tt
ms
0.15
0.2
0.25
40
Ptot(xxi, t0)
Ptot(xxi, t0)
20
Ptot(xxi, t1)
Ptot(xxi, t2)
Ptot(xxi, t3)
0
0
Ptot(xxi, t4)
−10
Ptot(xxi, t5)
Ptot(xxi, t6)
-20
xxi
thick
Fig. 3. Results of incident and reflected waves as a function of time at various cross sections (top) and along the specimen in time sequence
(pulse travels from right to left).
effects and protect the instrumentation, a steel box (open at one side) was placed around the charge. The strain
gauges were shielded with aluminium foil and grounded.
In the first phase of the development programme, a test procedure was established to generate a
reproducible loading pulse of sufficient amplitude and strain rate. It was decided to continue the research with
the load generated by an explosive charge of 10 g pressed Hexolite and a standoff distance of 20 mm. The
characteristic values of the generated pressure pulse in the incident steel bar are:
Amplitude pulse/(7D90% reliability interval1): 68 MPa/(1.5 MPa)
Strain rate/(7D90% reliability interval): 13.8 l/s/(1.74 1/s)
Energy ascending branch: 1400 J/m2.
Fig. 4 gives an example of the four strain records at location S4, at the end of the steel bar.
1
Reliability interval based on normal distribution of data.
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400
300
200
μStrain
100
0
-100
S4so
S4sb
S4sl
S4sr
-200
-300
-400
280
300
320
340
360
380 400
μSec.
420
440
460
480
Fig. 4. Example of generated loading pulse, recorded at the end of the steel incident bar.
2.3. The applicability of the loading pulse
Besides the reproducibility, amplitude and strain rate of the loading pulse, other features have to be
considered in testing concrete.
The first and obvious requirement is that any concrete specimen should fail under tension and not under
compression. Since first a compressive wave enters the specimen, this wave may not exceed the level at which
damage is initiated. This level is about 0.3fc for static loading, but no data are available for this damage
threshold at dynamic conditions. Consequently, the amplitude of the loading pulse, and so the charge weight,
had to be limited.
Furthermore, it should be noted that the residual stresses in the bar are a result of the incident compressive
wave and reflected tensile wave. It depends on the shape of the loading pulse when and where in the
specimen the maximum tensile stress occurs. The MathCad-program was used to examine the interaction
process and the residual stress distribution in a 240 mm long linear-elastic specimen. Fig. 3 illustrates the
interaction process. For the 10 g charge at 20 mm standoff distance, the calculations learned that the (tensile)
strength criterion can be first fulfilled at a distance of about 40 mm from the free end. Beyond 100 mm
the completely reflected wave determines the stress condition. This information was used to instrument the
specimen.
The last aspect that is mentioned in this section is the tensile tail in the loading pulse. The compression
phase is followed by a tensile phase (see Fig. 4). This tensile tail develops during wave transmission in
the steel bar. From the strain records it became clear that when the tensile phase is transmitted to the concrete
specimen probably it would lead to complete tensile failure. The solution to this problem is to abandon the
tensile tail by creating a sufficiently weak seam between the incident steel bar and the concrete specimen. This
seam has to transmit the compression phase and has to fail brittle in tension at stress levels below
approximately 1 MPa. Various seam materials were tested resulting in a prescribed preparation procedure
using model plaster.
3. Tests on instrumented concrete specimen
3.1. Instrumentation of concrete specimen
The incident loading pulse can be recorded easily with strain gauges on the steel bar. The transmitted pulse
to the concrete specimen can be calculated theoretically when the density and Young’s modulus are known.
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PML01t008
P2
P1
S8
S7
S4
S6
S2
74
S1
S3
S5
35
60
240
85
110
25
65
Fig. 5. Instrumentation of the concrete specimen.
Here we are confronted with the first problem because the Young’s modulus is a function of the loading rate.
Data on this dependency are scarce and only related to loading rates up to 10 GPa/s [14,15]. Based on the
given data it might be expected that the stiffness would increase with a factor 2 for loading rates in the order of
100–1000 GPa/s.
Strain measurements at the surface of the specimen learn about the deformation but can only be
related to stresses in the linear-elastic response phase by means of the ‘‘unknown’’ Young’s modulus.
To overcome this problem it was decided to apply the embedded carbon stress gauge developed by the
WTD 52 for a direct recording of the compression pulse in the concrete [16,17]. The problem of local
stress measurements in a very heterogeneous material like concrete was recognised, but accepted. The
size of the gauges and difference in acoustic impedance were studied numerically and calibration factors were
derived [20].
By recording the incident pulse, the transmitted pulse in the concrete and the deformation at multiple
locations at the surface of the specimen, a data set is gained which should be sufficient to reconstruct the
material response and derive material parameters. It is foreseen that the parameters can only be quantified
accurately when the experiments are combined with numerical simulations in order to extract mechanical
inertia effects from the measurement data [7,9,18,19].
The instrumentation scheme of the specimen is given in Fig. 5. Two carbon gauges are applied near the seam
with the steel bar. At these locations the deformation is recorded with 4 strain gauges.2 Four additional strain
gauges are equally distributed along the rear half of the specimen in order to record the response and are
intended to capture the failure zone.
3.2. Performed tests
Exploring tests showed that the most probable location of failure was about 70 mm from the free end,
which corresponds quite well with the location prediction with the analytical Mathcad-calculations
(see Section 2.3). To predetermine the location of failure it was decided to apply a notch of 0, 2 and 4 mm
deep at 70 mm from the end. During the test series this decision did not have to be adjusted. All specimens
were loaded up to final failure except one specimen with ‘‘0-notch depth’’. The static concrete properties are
given in Table 1.
2
The applicability of the carbon gauges in the current test set-up and the interpretation of the signals are discussed in Weerheijm et al.
[20]. In the current paper, no further attention is paid to this specific aspect of the measurement set-up.
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Table 1
Static concrete properties of test specimen
2350 kg/m3
37 GPa
40 MPa
3 MPa
0.2
8 mm
Density, r
Young’s modulus, E
Compressive strength, fc
Tensile strength (direct tension), ft
Poison ratio, u (assumed value)
Maximum aggregate size
400
300
200
μstrain
100
S4sb, 01spal5
S4so, 01spal5
S4sl, 01spal5
S4sr, 01spal5
zero
0
-100
-200
-300
-400
280
300
320
340
360
380 400
μSec.
300
300
200
200
100
100
0
0
-100
-100
μStrain
μStrain
(a)
-200
S1
S2
S3
S4
-300
-400
-500
460
480
-200
S5
S6
S7
S8
-300
-500
-600
400
450
500
550
600
μSec.
(b)
440
-400
-600
-700
350
420
-700
350
400
450
500
550
600
μSec.
Fig. 6. (a) Strain records incident pulse in steel bar. (b) Strain records transmitted pulse (locations 1–4 and 5–8) in specimen.
4. Test results
4.1. Incident and transmitted pulse, young’s modulus Edyn
In this section, the results concerning the damping in the plaster seam and the dynamic Young’s modulus
are discussed. The transmitted pulse into the specimen is determined from the strain records at positions 1–4
and the wave velocity. The incident loading wave consists of a pressure phase followed by a tensile phase. A
weak and brittle seam of plaster was applied. Comparison of the records of the incident and transmitted pulse
clearly shows the effectiveness of the applied seam. Fig. 6 gives an example of a set of strain records.
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For all the tests, the wave velocity was determined using the transmission time between locations 1/3–2/4, 1/
3–5 and 2/4–5. The records at locations 6–8 were not used because the ascending branch of the records was
already affected by the reflected wave at the free end. Comparison of the strain records 1–8 learned that the
response in the compression phase was linear-elastic in most cases. Therefore, the dynamic Young’s modulus
could be derived from the wave velocity data. The results are given in Table 2.
Because the dynamic Young’s modulus is known, the amplitude of the transmitted stresses can be calculated
from the strain records. But it also enables a cross check. In a linear-elastic response without damping the
transmitted pulse can be calculated from the incident pulse. The amplitude of the incident pulse is:
sinc ¼ 67 MPa. Without damping, the amplitude of the transmitted pulse is given by the transmission factor atr
that depends on the acoustic impedance values (I ¼ rc) of the steel bar and the concrete specimen:
atr ¼
2
¼ 0:413,
1 þ I steel =I concr
(1)
strans ¼ 0:413 67 ¼ 27:7 MPa:
The theoretical value equals the value of 27.4, emerging from the strain gauges directly. The conclusion
from this result is
– the damping in the plaster seam can be neglected
– Edyn ¼ 46.2 GPa.
With this firm result for the dynamic Young’s modulus, it is interesting to make the comparison with the
results published in literature. The strain rate in the test series is 22.4 1/s. The observed rate effect is (46.2/
37) ¼ 1.25, while the CEB formula [15] predicts a rate effect of 1.4. Both numbers fit into the branch of
moderate increase as given in Fig. 7 and not the suggested steep increase in this figure.
Table 2
Acoustic wave velocity and Young’s modulus derived from strain measurements
Cp
Stdev Cpa
Edyn
Standard deviation based on normal distribution of data.
1.6
Ed/Es
a
4388 m/s
255 m/s
46.2 GPa
1.2
0.8
10-4
10-8
.
ε [static]
10-2
1
101
.
ε [1/s]
Fig. 7. Young’s modulus as a function of the strain rate [14].
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4.2. Reconstruction of the load– deformation relation
The mechanical material behaviour is characterised and given by the stress–strain relation. In the current
test set-up the loading and response recordings are combined or even integrated. In this section the
possibilities are described to reconstruct the stress–strain relation from the recordings.
Step 1: The transferred pressure pulse into the concrete specimen is defined and given by the average of the
strain recordings at locations 1–4. The total duration of the pressure pulse is in the order of 70 ms. The rise time
is about 50 ms and the descending branch takes 20 ms. With Cp ¼ 4388 m/s the corresponding wavelength is in
the order of 300 mm. The failure process takes place while the pressure and tensile wave interfere. To describe
and quantify the resulting stress–strain distribution in the specimen it is assumed that the concrete remains
linear-elastic and does not fail.
Step 2: The pressure pulse propagates through the linear-elastic specimen, reflects at the free end as a tensile
pulse. Location 5, beyond the tensile failure zone is taken as a reference, and the resulting elastic strain history
is determined. Fig. 8 gives the result of these interfering waves under the condition that the whole specimen is
linear-elastic and no failure occurs.
Step 3: The next step is to combine the theoretical loading pulse at the specific location with the recorded
deformation. For location 5 the result is given in Fig. 9 with the stresses using the Edyn value of 46.2 GPa on
the vertical axis.
From the reconstructed relations the following conclusions are drawn:
– Damage due to the incident compression pulse is negligible, the material exhibits a linear-elastic behaviour
although the amplitude of the residual compression pulse (25 MPa) is about 0.6fc and exceeds the static
threshold of non-linearity in compression (0.3fc). This observation counts for all tests. Only a few signals
showed some non-linear effects;
– The strain recordings at location 5, beyond the failure zone, can be used to quantify the tensile strength. No
evidence was found that non-linear tensile response occurred. Therefore the Edyn and the maximum strain
can be used to quantify the dynamic tensile strength;
– Because the deformation of the failure zone itself was not recorded, no load–deformation curve could be
reconstructed.
700
600
500
400
300
μStrain
200
100
0
-100
-200
-300
S5Bk
Sres5
SAs5
-400
-500
-600
-700
350
375
400
425
450
475 500
μSec.
525
550
575
600
Fig. 8. Reconstructed wave propagation in specimen. The incident pressure wave (dotted), the resulting compression and tension pulse at
location 5 (dashed) and the recorded strain at location 5 (solid).
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30
25
20
15
10
MPa
5
0
-5
-10
-15
-20
-25
-30
-550 -500 -450 -400 -350 -300 -250 -200 -150 -100
-50
0
50
100
150
200
250
300
μrek
Fig. 9. Reconstructed stress–strain curve (test 4, location 5, beyond the failure zone).
4.3. Reflected tensile pulse, tensile strength ft,dyn
In the previous sections, attention was focussed on the transmitted compression pulse in the concrete. Using
the strain recordings and the derived value of 46.2 GPa for the Young’s modulus, the compression pulse is
known and the reflected tensile pulse can be determined theoretically. Comparison with the strain
measurements just outside the failure zone gives information on the dynamic tensile strength.
The reflected pulses caused failure in all tests except one. For the un-notched specimen failure occurred in
the zone of the measuring locations 6 and 7. Also, for the notched tests permanent deformation was recorded
at these locations. Most reliable information about the tensile strength follows from strain record number 5
when no residual deformation is recorded. For the notched specimen the area reduction has been taken into
account. The dynamic load conditions and observed tensile strengths are given in Table 3. The overall
observed rate effect is
f dyn =f stat ¼ 5:3
at the rate of 22:5 1=s or 1050 GPa=s:
It should be noted that due to the application of the geometrical discontinuity at the notch,
stress concentrations are introduced leading to early crack initiation and reduced specimen strength. The
effect of the notches on the dynamic strength cannot be quantified at this stage. It is possible that at
the high loading rates the inertia effects at small scale, leading to the steep strength increase, dominate
the failure process and stress concentrations have a minor effect on the observed specimen strength.
Detailed numerical modelling is needed to quantify the effect. Comparison of the few data that were
gathered in the current series indicates that the dynamic strength of the 4-mm notched specimens was
affected by the notches and is not representative for the material strength. It is concluded that the rate effect
of 5.3 on the tensile strength is a lower bound value for the true dynamic material response. Fig. 10
gives the rate dependency from literature and previous research and shows the consistency with the
current data.
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Table 3
Strain amplitudes at location 5 and the dynamic strength for the tests without residual deformation
Notch depth (mm)
Number of tests
0
Average
Stdev
3
2
Average
Stdev
3
4
Average
Stdev
4
All tests
Average
Stdev
10
Sb-5 (mstrain)
ftension (MPa)
Strain rate (1/s)
399
31
18.4
1.4
20.0
2.6
920
103
343
63
17.7
3.2
21.3
3.2
988
152
251
26
14.5
1.5
25.5
6.5
1166
299
315
73
15.8
3.2
22.5
3.8
1039
177
Load rate (GPa/s)
5
Mix A; gam = h (p')
psi (=ftd/fts)
4
3
Mix A; gam = C
Mix B; gam = h (p')
Mix B; gam = C
+ experimental data
2
1
0
10-3 10-2 10-1 100 101 102 103 104 105 106 107
loading rate [MPa/s]
Fig. 10. Rate effect on uni-axial tensile strength, data and modelling results [7].
4.4. Fracture energy, Gf,dyn
Besides the dynamic tensile strength, the aim of the current test-set up is to gather data on the dynamic
fracture energy. Failure occurs when the strength criterion is fulfilled and sufficient energy is available, and can
be released into the fracture zone to establish complete failure. From previous research it is known that the
fracture energy of the applied concrete for static loading is in the order of 100 J/m2. From the current tests, no
direct information about the fracture energy is available because the deformation of the failure zone was not
measured like in the previous research with the gravity driven SHB [7]. The measurement data have been
analysed in several ways in order to learn about the dynamic failure process and to see what quantitative
information can be derived from the strain and velocity measurements. These analyses have been reported in
Weerheijm et al. [21]. Because the quantitative results for the analyses were about the same, in this paper only
one method is given.
The first step in the analysis is to make the following assumptions:
– Only one fracture zone occurs, which is located at the notch;
– The concrete specimen is undamaged, and is linear-elastic except in the fracture zone. Consequently, all
energy absorption takes place in the fracture zone;
– The width of the fracture zone, lfrac, is in the order of 25 mm (3 times the maximum aggregate size).
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During the failure process, deformation and kinetic energy of the surrounding material is released into the
fracture zone and cracks are formed. The failure zone is defined as the zone in which the material is involved in
the energy exchange process. It should be noted that the width of the failure zone, lfail, is larger than lfrac.
One of the options to estimate the dynamic fracture energy from the tests is given by the gross energy
balance of the specimen. The energy balance can be made based on the compression pulse, the energy
trapped in the spall debris, the tensile pulse beyond the failure zone and the unknown fracture energy.
In symbols:
E compr ¼ E debr þ E trans;tens þ E frac .
The assumption is that all the material is linear-elastic except the failure zone. All energy dissipation is
concentrated in the failure zone and contributes to Efract ( ¼ Gf). The energy terms, Evv+Ekin, for the stress
pulses are determined using the dynamic Young’s modulus value of 46.2 GPa and the positive phase of the
recorded pulses at locations 1–4 and 5 for the compression and tension pulse, respectively:
Z
1
e2 ðxÞ dl pulse ,
E vv ¼ E
2
E kin ¼
1
2
Z
r v2 ðxÞ dl pulse
pffiffiffiffiffiffiffiffiffiffiffiffi
and because v ¼ ðE=rÞ e, E kin ¼ E vv for LE materials.
The kinetic energy of the spall debris follows from displacement measurements and the spall mass.
The displacement measurement signals initially show a strong fluctuation, the velocity is derived after about
10 ms which leads to an underestimation of the initial velocity. It is assumed that the displacement
measurements are still representative for the first moments after final failure when stress waves still run
through the spall debris. Because the spall debris is linear-elastic, Edebr is given by twice the calculated kinetic
energy. Fig. 11 gives the results for the energy terms while the derived Gf values for the gross and nett section
are given in Fig. 12.
The results show that:
– Gf for the notched specimen o250 J/m2, except for one test;
– The results obtained from the gross energy balance are quite sensitive to the displacement, velocity
measurements of the spall debris. In the current set-up the velocity and the kinetic energy are
underestimated, resulting in an overestimation of the fracture energy;
– The results show that the 2-mm notch tests give consistent results for the fracture energy;
– Using the 2 mm notch results and the assumption that all the dissipated energy contributes to the fracture
energy of the single failure zone, a new upper bound for the fracture energy is obtained: Gf, dyno250 J/m2 at
loading rates of 1000 GPa/s.
Energy terms for C series
8.0
E incident
[Nm]
Energy [Nm]
7.0
6.0
5.0
E residual
[Nm]
4.0
3.0
E debris
[Nm]
2.0
1.0
E fracture
[Nm]
0.0
4
4
4
2
2
2
notch depth [mm]
0
0
Fig. 11. Energy terms of gross energy balance for the various notch depths.
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Gf for C series
400
Gf [N/m]
350
Gf gross section
[N/m]
300
250
Gf pernett section
N/m]
200
150
100
4
4
4
2
2
2
notch depth [mm]
0
0
Fig. 12. Fracture energy related to the gross and nett cross section.
ft
Eel ; δel
εfrac ; δfrac
Fig. 13. Load–deformation curve.
5. Suggested load–deformation curve
Although the information on the dynamic fracture process is still limited, it is worthwhile to combine all the
current knowledge and ‘‘define’’ the best possible load–deformation curve for concrete under dynamic tensile
loading.
5.1. Problem definition and available information
In FE-material models the load–deformation relation is commonly characterised as given in Fig. 13. Up to
maximum strength (ft) a linear-elastic behaviour is assumed while the softening branch is schematised
(bi)linearly or with a power function. Extensive experimental research to the softening behaviour has been
performed at Delft University of Technology. It resulted in the definition of the softening curve as a power
function (‘‘Hordijk-Reinhard expression’’) [22]. It is proposed to use the combined linear-elastic ascending
branch and the ‘‘Hordijk expression’’ as the static reference relation. Characteristic points to judge the
response and damage stage of the material are:
–
–
–
–
The
The
The
The
maximum strength ft;
strain, and deformation of the response zone at maximum strength, eel and del, respectively;
maximum deformation of the failure zone at complete failure, dfrac;
fracture energy Gf that equals the area ‘‘below’’ the load–deformation curve.
The deformation, d, gives the elongation of the failure zone. The strain is defined as e ¼ d=l frac .
The softening curve, the residual strength sres as a function of the non-elastic strain, en, is given by
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the expression
sres
¼
ft
!
en 3
en
en
1 þ c1
ð1 þ c31 Þ expðc2 Þ.
exp c2
efrac
efrac
efrac
The coefficients c1 and c2 determine the curvature of the softening curve. Hordijk and Reinhardt defined
these values for static loading as c1 ¼ 3 and c2 ¼ 6:93. With these values the relation between the characteristic
parameters is given by efrac ¼ 5:136ðG f =l frac f t Þ. The parameters Gf and lfrac are directly related because the
energy required to fail the concrete is dissipated within the failure zone with length lfrac. Deformation
controlled tension tests give data on the dissipated energy [22,23]. But hardly any quantitative information is
available on the size of the fracture zone. According to the recommendation of Bazant for static loading
conditions, the size of the fracture zone is related to the maximum aggregate size Fmax, as lfrac ¼ 3Fmax. This is
consistent with the few data the authors obtained from the spall tests. Fig. 14 gives examples of the fracture
pattern of a spall test with 2 mm notch and the pattern in a test without a notch.
The static and dynamic tests (up to loading rate of 15 GPa/s) at the Delft University of Technology show
that dfrac is in the order of 0.1 mm. It should be noted that these experiments were performed on concrete with
Fmax of 8 mm. The scale effect on dfrac still has to be examined.
5.2. Including rate effects on tensile properties
5.2.1. The ascending branch
_ The ‘‘moderate regime’’ (o10 GPa/s) and
The rate effects on concrete properties depend on loading rate, s.
the ‘‘steep regime’’ (410 GPa/s) have to be distinguished. For the moderate regime the CEB relations are
adopted [14]:
f dyn
s_ dyn a
¼
with s_ stat ¼ 0:1 MPa=s
f stat
s_ stat
and
a¼
1
¼ 0:033.
10 þ 0:5f c; stat
For the rate effect on the Young’s modulus the CEB proposed
E dyn
s_ dyn 0:016
¼
.
E stat
s_ stat
Consequently, the strain at maximum strength also increases with increasing loading rate. The ratios of
dynamic and static values for ft, E and eel are given in Fig. 15 in which also the experimental strength data for
Fig. 14. Examples of crack patterns in dynamic spall test with 2 mm notch (left) and without notch (right).
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Rate effects
5.00
4.50
fdyn
Edyn
strain dyn
4.00
Dyn/static value
3.50
3.00
2.50
2.00
1.50
1.00
0.50
0.00
-04
-03
-02
-01
+00
+01
+02
+03
Loading rate GPa/s
Fig. 15. Rate effects for ft, E and eel.
the high loading rate regime are included. The instrumented spall tests showed that the rate effects on the
Young’s modulus does not exhibit a steep increase for loading rates up to 1000 GPa/s. Consequently, the rate
effect on eel,dyn is predicted as a steep increase in the proposed schematisation.
5.2.2. The softening branch
The research at TNO and Delft University of Technology to the rate effects on the failure mechanisms
showed:
– A pronounced rate effect on strength occurs beyond loading rates of 10 GPa/s. The rate effect has been
quantified experimentally;
– The fracture energy Gf for a single failure zone is rate independent up to 10 GPa/s;
– The upper limit for the rate effect on fracture energy is 2.5 at a loading rate of 1000 GPa/s;
– Numerical material models and material data have to be related to single failure zones;
– Observations confirm the statement that the size of the failure zone does not exceed 3Fmax;
– For static loading the length of the softening tail, dfrac, is determined by the largest aggregates. Crack
bridges are formed at the final failure stage (van Mier [23]). It is assumed that this also counts for dynamic
loading. The crack patterns in the spall tests support this assumption.
These observations are used to describe the dynamic softening behaviour according to the format of the
‘‘static, Hordijk relation’’. The basic assumption is that the fracture energy Gf is constant and is dissipated in
the formation of ‘‘n-(micro) cracks’’
X
ð2g cracksurfaceÞ with g the specific surface energy:
Gf ¼
n
For loading rates up to 10 GPa/s counts: dfrac,dyn ¼ dfrac,stat; Gf,dyn ¼ Gf,stat; ft,dyn4ft,stat. The authors
assume that the size of the fracture zone, lfrac, is constant and equals 3Fmax. Consequently, the softening curve
becomes more brittle, the descending branch just beyond maximum strength becomes steeper, while the long
tail remains unaffected.
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The physical explanation can be:
– With increasing loading rate, more deformation energy is stored in the failure zone during the ascending
branch; the critical stress intensity is reached for a larger number of ‘‘potential cracks’’ at maximum stress
level.
– The number and kind of cracking is not affected by the loading rate, only the number of cracks that are
activated ‘‘just’’ beyond the peak strength increases with increasing loading rate. The number of cracks
remains the same, because when more cracks are activated, the total deformation should also increase and a
less brittle behaviour would occur.
For loading rates 410 GPa/s most probably the fracture energy increases. It is assumed that the increase is
related to the mechanism that leads to the steep increase in tensile strength. So, Gf increases for loading rates
410 GPa/s and the maximum enhancement factor is 2.5 at 1000 GPa/s. The steep strength increase is mainly
caused by the ‘‘activated’’ inertia of the material around the crack tips. The stress singularities decrease and
the stress distribution becomes more equally distributed. For a proper and thorough study and description of
the phenomena see Weerheijm [7]. Due to the more equally distributed stresses, the local stress maxima will
Softening curves for dynamic loading
1.80
Gpa/s 1.00
Gpa/s 1.00
Gpa/s 1.00
Gpa/s 7.00
Gpa/s 2.20
Gpa/s 1.00
1.60
stress level [Pa]
1.40
1.20
1.00
8.00
6.00
4.00
2.00
0.00
0
0.001
0.002
0.003
0.004 0.005
strain [-]
0.006
0.007
0.008
Softening curves for dynamic loading
5.00
Gpa/s 1.00
Gpa/s 1.00
Gpa/s 1.00
Gpa/s 7.00
Gpa/s 2.20
Gpa/s 1.00
4.50
4.00
stress level [Pa]
3.50
3.00
2.50
2.00
1.50
1.00
5.00
0.00
0
0.001
0.002
0.003
0.004 0.005
strain [-]
0.006
0.007
Figs. 16. Dynamic softening curves at two amplitude scales.
0.008
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depend less on the crack size. Consequently, at the same load level, more cracks become critical and can be
activated. Combined with the increased strength, and thus the larger amount of energy stored in the failure
zone during the ascending branch, probably the number of micro cracks will increase and so the fracture
energy Gf does increase. Conclusion for loading rates 410 GPa/s, the fracture energy increases due to more
extensive micro cracking, while dfrac remains constant (dfrac,dyn ¼ dfrac,stat).
Proposal for dynamic softening branch:
– It is proposed to use the reference relation as derived by Hordijk for static tensile;
– The rate effects on the tensile strength are applied (see Fig. 15);
– The fracture energy Gf is constant for loading rates up to 10 GPa/s, for higher loading rates Gf increases
proportional to the rate effect on tensile strength;
– The length of the fracture zone lfrac is constant and equals 3Fmax.
Following the proposal, the softening curve can be determined consistently. For the concrete tested in the
spall tests, with a static tensile strength of 3 MPa and Gf,stat of 100 N/m, the rate dependent softening curves
have been determined and depicted in Fig. 16 at two amplitude scales.
6. Concluding remarks
The paper presented the research performed at TNO-PML and the Delft University of Technology
concerning the rate-dependent material behaviour of concrete under dynamic tensile loading. The main results
obtained are:
– An experimental set-up has been accomplished to quantify dynamic tensile properties of concrete at high
loading rates. New information and data are obtained for the regime of loading rates of 1000 GPa/s for
which hardly any data were available;
– The old Hopkinson bar technique combined with sophisticated instrumentation offers good perspective to
study the dynamic failure process in tension;
– For the selected test conditions loading rates in concrete are in the order of 1000 GPa/s. The observed
enhancement factors for the tensile strength and the Young’s modulus (compression) were 5.3 and 1.2,
respectively. The results are in accordance with trends reported in literature previously;
– The measurement set-up enabled the quantification of all terms of the gross energy balance of the test setup and an upper limit for the dynamic fracture energy was derived. The maximum enhancement factor for
the fracture energy is 2.5 at loading rates of 1000 GPa/s. Previous research learned that for loading rates up
to 15 GPa/s the fracture energy does not increase;
– The current knowledge and data have been combined and a rate dependent softening curve has been
suggested. The proposed softening relation is based on the commonly used static ‘‘Hordijk’’ relation and is
consistent with all experimental data available.
Acknowledgements
The authors thank the support of the Dutch Ministry of Defence under the research programme Veiligheid
Defensie Infrastructuur (Safety of Defence Infrastructure).
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