Rebound Surface Hardness of Concrete: Introduction of an Empirical
Constitutive Model
Paper published in Construction and Building Materials 25 (2011) 2480-2487
Authors: Katalin Szilágyi PhD; Adorján Borosnyói PhD; István Zsigovics PhD
Abstract: Surface hardness
testing of concrete is a long established NDT method for in-situ strength
estimation. Nowadays, the rebound hammer is the surface hardness testing device
for concrete of the most widespread use. Based on a comprehensive literature
review it was realized that no general theory was developed in the last more
than 50 years that could describe the relationship between surface hardness and
compressive strength of concrete. The diversity of the numerous empirical
proposals that can be found in the technical literature needs to be explained.
It can be even found in some publications that the method is suitable only for assessing
the uniformity of concrete. There is long a time need for a model that can
clarify the rebound surface hardness of concrete. Present paper introduces a
phenomenological constitutive model (SBZ-model) that
can be formulated for the surface hardness of concrete as a time dependent material property. The generating
functions of the model are based on the time dependent development of the
capillary pore system of the hardened cement paste in concretes that is
characterised by the water-cement ratio as a practical simplification. The
modelling assumptions and the use of the model would add to the fundamental
understanding of the rebound surface hardness of concrete. An extensive
experimental verification study clearly demonstrated the reasonable application
possibilities of the SBZ-model.
Keywords: concrete; compressive strength;
nondestructive testing; surface hardness; rebound index
1. INTRODUCTION
In-situ surface hardness testing of materials
is an accepted method for strength estimation. Development of surface hardness
testing devices goes back more than 100 years. Hardness research was
initialized by the pioneering work of Hertz
in the 1880’s [1]. Nevertheless, hardness testing was also the first material
testing effort from the 1600’s in engineering through scratching hardness
testing (1640, Barba; 1722, Réaumur; 1768, Kvist; 1801, Haüy; 1812, Mohs)
[2].
In-situ testing of concrete structures was
started in the 1930’s. The testing methods at that time covered chisel blow
tests, drilling tests, revolver or special design gun shooting tests, splitting
tests, pull-out tests, strain measurements from loading tests and the
adaptation of the Brinell hardness
testing method [3]. This latter technique was found to be the most popular in
the European testing practice for decades according to its relatively simple
and fast operation [4,5].
Researchers adopted the Brinell method to
cement mortar and concrete to find correlations between surface hardness and
strength of concrete in the four decades following that Brinell [6] introduced
his ball indentation method for hardness testing of steel [7,8,9,10,11]. The
first NDT device for in-place testing of the hardness of concrete was
introduced in Germany UK
Nowadays the most widespread method for the
surface hardness testing of concrete is the rebound hammer method that is
appeared in the 1950’s through the Schmidt
rebound hammer (also known as Swiss hammer) [15].
In 1950, Ernst Schmidt developed in Switzerland
2.
EXPERIENCES OF THE SURFACE HARDNESS OF CONCRETE
Rebound surface hardness measurements were
found to be very popular in the in-situ material testing due to the inexpensive
testing devices and their relatively simple use. Numerous publications are
available in the technical literature concerning experimental results and
analyses.
Aim of rebound hammer tests of concrete is
usually to find a relationship between surface hardness and compressive
strength with an acceptable error. For the rebound method no general theory was
developed that can describe the relationship between measured hardness values
and compressive strength, nevertheless, it is deemed in some technical papers
that the behaviour is commonly understood. The existence of only empirical
relationships was already considered in the earliest publications [19,25] and
also recently [26]. In the opinion of the authors of present paper, finding a
good fit regression curve does not necessarily mean the understanding of the
behaviour.
It should be emphasized that the concrete construction
practice needs in-place NDT equipment provided together with simple,
easy-to-use, generalized relationships (in the form of equations, graphs or
tables) which express the measured value (e.g. rebound index) as a value of the
concrete compressive strength of standard specimens. Such generalized relationships,
however, usually could not accurately characterize the concrete in the
structure being tested.
Generalized relationships are allowed to be
used only if their validity has been
established by tests carried out on concrete similar to that being
investigated and with the same type of testing device that is intended to be
used in the investigation.
Fig. 1. Rebound surface hardness vs. compressive strength relationships from the
technical literature (represented as standard cube strength).
In Fig. 1 a demonstration is presented regarding the empirical
relationships found by several researchers for concrete strength estimation with
rebound hardness method in the last 60 years. Due to space limitations only 30
of the available curves are shown, however, more than 60 can be found in the
technical literature. Data is given in the graphical representation of Fig.
1 with a correction to provide results for 150 mm standard cubes. For
the sake of better visualization results are separated by their relation to the
strength estimation curve that is recommended by the manufacturer of the original
Schmidt rebound hammers for the N-type rebound hammers [43], as follows:
- Proposed curves running continuously
over the manufacturer’s curve (Fig.
1a),
- Proposed curves running continuously
under the manufacturer’s curve
(Fig. 1b),
- Proposed curves intersecting the manufacturer’s
curve coming from below (Fig.
1c),
- Proposed curves intersecting the manufacturer’s
curve coming from above (Fig.
1d).
Compositions of the proposed empirical
relationships are linear relationships, power function relationships,
polynomial relationships, exponential relationships or logarithm relationships.
The proposed curves are usually valid for 28 to 365 days of age, conventional,
normal weight concretes under air dry moisture condition. It can be realized
that the concrete strength can be estimated at certain rebound indices by a
±40-60 N/mm2 variation. Results clearly demonstrate that the
validity of a proposal should be restricted to its testing conditions and an
extension of the validity to different types of concretes or testing circumstances
is impossible. It is also worth to mention that several linear estimations can
be found among the proposals. It is possible when the strength range is
chosen to be narrow in the
experimental tests, e.g. [53]. Rigorous experiments always resulted in nonlinear
relationships since the very beginning of rebound surface hardness testing of
concrete. The formulation of a phenomenological constitutive model for the rebound
surface hardness of concrete is therefore of high interest.
3. HETEROSCEDASTIC DATA OR MISCOMMUNICATED DATA
The rebound index vs. strength relationship can
be determined if the experimental data are available. The usual practice is to
consider the average values of the replicate compressive strength and NDT results
as one data pair at each
strength level. The data pairs are presented in a way that the NDT value is the
independent variable (along the X axis) and the compressive strength is
the dependent variable (along the Y axis). Regression analysis is
performed as a conventional least-squares analysis on the data pairs to obtain
the best-fit estimate for the strength relationship. The technical literature
calls the attention that the boundary conditions of the conventional
least-squares analysis are violated in the case of rebound index vs. strength
relationships [54]. The conventional least-squares analysis is not recommended
because the uncertainty in the strength relationship could be underestimated.
Statistical analysis of the surface hardness
vs. compressive strength relationships usually also indicates heteroscedastic
behaviour; i.e. increasing standard deviation in strength (Y variable)
for increasing rebound index (X variable). Even the manufacturer of the
original design rebound hammers suggests increasing standard deviations to be
taken into account for increasing rebound indices [43]. Examples for the heteroscedastic
behaviour are indicated in Fig. 2 [18]
and Fig. 3 [55].
Fig. 2. Heteroscedastic behaviour of the rebound surface hardness vs.
compressive strength relationship (based on [18]). Note: 1 psi =
6,894×10-3 N/mm2.
Fig. 3. Heteroscedastic behaviour of the rebound surface hardness vs.
compressive strength relationship (based on [55]). Note: 1 psi =
6,894×10-3 N/mm2.
It should be highlighted that researchers
usually do not separate the experimental data of the corresponding rebound
index vs. compressive strength results by different influencing parameters in
the graphical representations – and the situation has not changed during the
last 60 years. Therefore, exclusively the one-parameter regression curves are
available in the technical literature that was introduced in Section 2. Surface
hardness and compressive strength of concrete, however, are depending on
several parameters (e.g. type of cement, amount of cement, type of aggregate,
amount of aggregate, compaction of structural concrete, type of formwork, method
of curing, quality of concrete surface, age of concrete, carbonation depth in
the concrete, moisture content of concrete, mass of the structural element,
temperature and state of stress) of which influences may be represented when a
two- or more-parameter regression analysis is carried out.
The most significant influencing parameters for
the compressive strength of normal weight concretes are the water-cement ratio, the type of cement and the age of the concrete. The amount of
cement, the amount of aggregate, the storing method and further concrete
technology parameters have only secondary influences. The type and amount of
aggregate can have significant influence in the case of lightweight aggregate concretes.
It is shown here as an example that
non-separation of experimental data can lead to completely misleading trends of
the analysis and the separation of experimental data can clearly uncover the
real material behaviour and, therefore, gives the only way to understand the mechanisms
of the rebound surface hardness testing of concrete. Two from the earliest
publications are referred as examples, i.e. papers by Schmidt and Herzig
[17,56]. Both papers are based on detailed laboratory tests carried out at EMPA
Laboratories, Switzerland
Fig. 4. Heteroscedastic behaviour of the rebound surface hardness vs.
compressive strength relationship (based on [17]).
Fig. 5. Influences of data separation on the rebound surface hardness vs.
compressive strength relationship (based on [56]).
a) selected data separated by cement amount and storage of specimens
b) all data points in non-separated representation
4.
GENERATING A PHENOMENOLOGICAL CONSTITUTIVE MODEL
The primary factor that governs the
characteristics of cementitious materials is porosity. It was found
experimentally that the development of the porosity in concrete can be
described reasonably well by the gel-to-space ratio [57]. It is necessary to
know the degree of hydration in the hardened cement paste to work with
gel-to-space ratio, therefore, the water-cement ratio (w/c) is a much more
practical measure for the porosity of concrete [58]. For practical purposes it
can be accepted that the water-cement ratio (w/c) determines the capillary
porosity of a properly compacted concrete at any degree of hydration [59]. As a
consequence, strength and related properties of concrete can be accepted to
depend primarily on the w/c ratio as it was realized more than 100 years ago [60,61].
Surface hardness of concrete is also considerably influenced by the w/c ratio
in addition to the modulus of elasticity of the aggregate particles (which is
usually considered to be constant in time). The interaction volume (i.e. the
volume of hardened cement paste adjacent to the testing device contact) plays
the most important role in hardness testing; being several mm in depth for
rebound hardness testing. Hydration of clinker minerals in the hardened cement
paste makes the per se heterogeneous concrete to be a material with time
dependent properties. Transport phenomena (mostly of carbon-dioxide) in the
near surface zone of concrete structures have also considerable influence on
the rebound hardness characteristics through the chemical reaction referred as carbonation, whenever the hydrated
lime content of the hardened cement paste forms limestone due to the chemical
reaction with carbon-dioxide [53,62]. Based on the above general behavioural
scheme, a phenomenological constitutive model (SBZ-model; the abbreviation is indicating the names of the authors)
can be formulated for the surface hardness of concrete as a time dependent
material property.
The generation scheme of the SBZ-model as well
as the symbolic shapes of the individual functions given by Eq. (1) to Eq. (5)
can be studied in Fig. 6.
Fig. 6. The generation scheme of the SBZ-model.
The formulation of the model includes the
following experimental findings:
A) The compressive strength of concrete at the
age of 28 days can be described by an exponential function of the w/c ratio (Eq. 1).
Eq. (1)
with
a1
> 1
a2
< 0
0
< a3 < 1
B) The development of the compressive strength of concrete with time can
be followed by an exponential function of time (Eq. 2).
Eq. (2)
with
0
< a4 < 1
0
< a5 < 1
and both parameters a4 and
a5 are a function of w/c
C) An empirical relationship of a power function can be assumed between
the strength of concrete and the rebound index at the age of 28 days (Eq. 3).
Eq.
(3)
with
a6 > 0
a7 ≥ 1
D) The development of the carbonation depth (xc)
in concrete with time can be described by models based on Fick’s law of diffusion (Eq. 4).
Eq. (4)
with
0
< a8 < 1
0
< a9 < 1
0
< a10 < 1
E) Carbonation of concrete results an increase in the surface hardness
that can be assumed to be modelled by a power function of the carbonation depth
(Eq. 5).
Eq. (5)
with
a11
< 0
a12
> 0
The SBZ-model can provide corresponding
compressive strength fc(t) and rebound index R(t)
values for any w/c ratio at any age of concrete (t).
A typical fc(t) vs. R(t)
relationship is represented in Fig. 7: the output of the model is a set
of curves corresponding to different w/c ratios at different ages of the
concrete. It should be noted that the shape and curvature of the individual
curves are depending on the actual values of the twelve empirical constants a1
to a12 covered in Eqs. (1) to (5) and Fig. 7 indicates
a possible general case.
It can be realized that the SBZ-model provides
a reasonable depiction of the rebound surface hardness of concrete as a time
dependent material property. It should be also noted that the SBZ-model gives a
clear explanation for the experimental findings about the apparent heteroscedastic behaviour of the rebound index vs.
compressive strength data pairs. The SBZ-model calls the attention that the graphical
representation of the experimental results should not be carried out by the
simplifying assumption that concretes of different w/c ratios and different
ages provide data being in the same population. It can be clearly seen that the
simplification could result misleading representation and the influencing
parameters should be separated in the graphical visualization as it is
suggested by the SBZ-model.
Fig. 7. Typical schematic fc(t) vs. R(t) response as an
output of the SBZ-model:
a set of curves corresponding to different w/c ratios at different ages of the
concrete.
5.
EXPERIMENTAL VERIFICATION OF THE SBZ-MODEL
An extensive experimental study was carried out
at the Budapest University of Technology and Economics (BME), Department of
Construction Materials and Engineering Geology for the verification of the developed
SBZ-model. The tested concrete mixes in the experiments were prepared in
accordance with present concrete construction needs, i.e. slightly
over-saturated mixes with different admixtures were designed. Danube sand and gravel was used as
aggregate. Consistency of the tested concrete mixes was constant: 500±20 mm flow. Design air
content of the compacted fresh concrete for the tested concrete mixes was 1.0 V%.
The specimens were cast into steel formworks and the compaction of concrete was
carried out by a vibrating table. The specimens were stored in water tank for 7
days as curing. After 7 days the specimens were stored at laboratory condition
(i.e. 20±3°C
temperature and 65±5% relative humidity). Test parameters were:
0.38 – 0.41 – 0.43 – 0.45 – 0.47 –
0.50 – 0.51 – 0.55 – 0.60
Cement
type:
Cement
content (kg/m3):
300 – 350 – 400
Mixing
water content (kg/m3):
150 – 165 – 180
Cement
paste content (litres/m3):
247 – 263 – 278 – 293 – 309
Aggregate-cement
ratio:
4.5 – 4.6 – 4.7 – 5.3 – 5.4 – 5.5 –
6.3 – 6.5 – 6.6
Admixture
type:
accelerator admixtures (3 types)
Age of
concrete at testing (days):
7 – 14 – 28 – 56 – 90 – 180
The experimental programme made possible a
detailed verification study to be carried out on a wide range of compressive
strengths and ages of concrete at testing. Total number of 864 specimens (150 mm cubes) were tested at
six different ages for present verification study. Typical results are
indicated in Fig. 8 corresponding to concrete specimens prepared with CEM
I 42.5 N cement. Fig. 8 represents test results for only 104 specimens.
Fig. 8. Experimental verification of the
SBZ-model on concrete cube specimens prepared with CEM I 42.5 N cement.
a) data in non-separated representation
b) data separated by the applied water-cement ratios
c) data represented together with the fitted SBZ-model
The following observations can be emphasized:
1) An apparently coherent population of data is resulted if one does not
differentiate water-cement ratios and ages of concrete in the graphical
representation of test data (Fig. 8a). A completely misleading trend of
results is realized and an apparent power function or exponential function
relationship can be the output of a regression analysis (usually with
considerably good correlation coefficients which further ratifies the
misleading direction of the analysis). In Fig. 8a 52 data points are
indicated as the pair-averages of the 104 specimens (covering 9 different
water-cement ratios and 6 different ages of concrete at testing). Regression
curve of an exponential function is also indicated. The correlation coefficient
was found to be r2 = 0.84 for this false relationship.
2) An apparent heteroscedastic behaviour of the rebound index vs.
compressive strength data pairs is realized if one does not differentiate
water-cement ratios and ages of concrete in the graphical representation of
test data (Fig. 8a). It can be studied in Fig. 8a that the
distance between the lower and upper limit curves corresponding to the
increasing rebound index values is increasing that can result the apparent
heteroscedasticity (i.e. increasing standard deviation in strength for
increasing rebound index).
3) The real performance appears only if one separates the rebound index
vs. compressive strength data pairs by the water-cement ratio (Fig. 8b).
For the sake of better visualisation only 5 curves are represented in Fig. 8b
from the 9 different water-cement ratios studied. It can be realized that the
apparently coherent population of data comes loose to separate monotonic curves
for the different water-cement ratios.
4) It can be seen in the real performance that rebound index vs.
compressive strength relationships are sensitive (but not uniformly) to the
water-cement ratio applied (Fig. 8b). The gradients and directions of
the responses clearly indicate the influence of the capillary pores of
different water-cement ratios on the strength development and carbonation depth
development differences. It can be postulated that the water-cement ratio
dependent strength development and carbonation depth development behaviour of
concretes gives the complete explanation of the observed results. Results of
the verification study confirmed that the most significant influencing
parameters are the water-cement ratio,
the type of cement and the age of the concrete. The cement
content, the aggregate content, the cement paste content and further parameters
have much less pronounced influences; as it was presumed.
5) The application of the SBZ-model is reasonable for the rebound index
vs. compressive strength data (Fig. 8c). A suitable fit of the empirical
parameters of the SBZ-model can result an acceptable numerical reproduction of
any experimental data. The detailed verification study demonstrated the
applicability of the SBZ-model for CEM I 42.5 N, CEM II/A-V 42.5 N and CEM
III/B 32.5 N cements on a wide range of water-cement ratios and ages of
concrete at testing (further details of the verification is to be presented
separately due to the space limitation of present paper). The SBZ-model also provides
a clear understanding of the rebound surface hardness of concrete as a time
dependent material property. Based on its composition the SBZ-model is deemed
to be a unique constitutive model available now for the rebound surface
hardness of concrete.
6. FUTURE
WORK
The theoretical considerations covered in the
development of the SBZ-model were confirmed by the extensive experimental
verification introduced in Section 5. However, further studies are
needed for the ratification of the SBZ-model for practical applications. The
SBZ-model provides a clear and transparent explanation to the rebound surface
hardness of concrete in its introduced form. The observations predict that the
general scheme of the SBZ-model allows an extension of the model also for
concretes older than 180 days. It was found that the predictions made by the
SBZ-model are far more accurate than that was available earlier by simple
regression analyses. On the other hand, the number of the empirical constants
included in the SBZ-model may result a challenging parameter fitting work
before any practical application. Further types of concretes should be studied in
the future to be able to find simplification possibilities. Typical form of
generating functions should be clarified and the limits of the practical
application should be determined. It is to be highlighted, however, that the
main purpose of the development of the SBZ-model was to provide a better
understanding of the rebound surface hardness of concrete and to explain the
experimental findings. The direct practical application of the model is not
started yet. Authors are working on further developments and hope that the
SBZ-model can be successfully used in practice in the future.
7. CONCLUSIONS
Rebound surface hardness testing of concrete is
one of the most widespread NDT methods for in-situ strength estimation of
concrete structures. Rebound surface hardness methods are available in the
civil engineering testing practice for more than 60 years. However,
understanding and modelling of the rebound surface hardness of concrete as a
time dependent material property is not available in the technical literature.
Present paper introduces the SBZ-model developed by the authors of
the paper which is a phenomenological constitutive model for the rebound
surface hardness of concrete as a time dependent material property. Origination
of the SBZ-model is based on the time dependent development of the capillary
pore system of the hardened cement paste in concretes that is characterised by
the water-cement ratio as a practical simplification. The model covers the
following empirical material laws: relationship between the water-cement ratio
and the compressive strength of concrete at the age of 28 days; development of
the concrete compressive strength in time; relationship between the compressive
strength of concrete and the rebound index at the age of 28 days; the development
of carbonation depth of concrete in time; the influence of carbonation depth of
concrete on the rebound index.
An extensive experimental verification of the
SBZ-model clearly demonstrated its reasonable application possibilities for CEM
I 42.5 N, CEM II/A-V 42.5 N and CEM III/B 32.5 N cements on a wide range of
water-cement ratios (w/c = 0.38 to 0.60) and ages of concrete at testing (7 to
180 days). The transparency of the SBZ-model offers further promising
development, however, provides also in its present form the long time missing
fill to the gap of knowledge appeared in the last 60 years.
8.
ACKNOWLEDGEMENTS
The authors gratefully acknowledge the support
of the Bolyai János research scholarship by the Hungarian Academy
9.
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[62] The People’s Republic of China China
In memoriam of
our colleague, mentor and friend,
István Zsigovics PhD, who has left us so
early.
1949-2015
Dear István, thank you for everything.
Dear István, thank you for everything.
May your soul rest in peace.
K. Szilágyi, A. Borosnyói
