Identification of Logarithmic-Type Creep
of Calcium-Silicate-Hydrates by Means
of Nanoindentation
Ch. Pichler* and R. Lackner†
*Institute for Mechanics of Materials and Structures, Vienna University of Technology, Vienna, Austria
†
Material-Technology Unit, University of Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria
In order to relate the complex macroscopic behaviour of concrete with finer-scale
properties of cement paste, aggregates, porous space, etc. multiscale models are developed. Here,
in addition to homogenisation schemes establishing the relation between the properties of the
constituents, e.g. matrix and inclusions, and the properties of the composite material, experimental
methods for identification of finer-scale properties are required. Recently, the elastic properties of
the main clinker phases in ordinary Portland cement (OPC) and calcium-silicate-hydrates (CSH)
were identified by nanoindentation (NI). In this paper, the NI technique is extended towards
identification of the viscous, time-dependent behaviour of CSH, i.e. the main finer-scale binding
phase in OPC-based material systems. Hereby, a logarithmic-type creep behaviour is identified,
corresponding perfectly to the creep behaviour encountered for mortars and concrete.
ABSTRACT:
KEY WORDS: logarithmic creep, nanoindentation, Sneddon solution, Tresca criterion, viscoelastic–
cohesive indentation
Introduction
Objectives of this study
Experimental methods for determination of
mechanical properties (elasticity, time-dependent
behaviour, strength) may be classified according to
load history (static or cyclic tests) and the stress distribution induced by loading of the specimen (uniform or non-uniform). Among the experiments for
the identification of mechanical properties of concrete characterised by a uniform stress distribution
(apart from the domain of load application) are uniaxial, biaxial and triaxial tests. Experiments with
non-uniform stress distribution include the Brazilian
test, pull-out tests and four-point bending experiments. Indentation tests rank among identification
experiments inducing a non-uniform stress distribution. They have been employed in material science
for determination of material ‘hardness’ for over a
century (Johan A. Brinell presented his ‘Spherical
Indentation Test’ at the World Fair in Paris in 1900).
As the only prerequisite for indentation tests is a
sufficiently plane sample surface, they may be conducted at small length scales of observation. This has
led to the development of the so-called nanoinden-
tation (NI) technique in the 1980s. Hereby, the
applied load ranges in the micro-Newton range and
the penetration depth in the nanometre level. An NI
test is characterised by driving a tough (usually a
diamond-) tip into the ground and polished sample
surface, with load history P(t) (N) prescribed and
penetration history h(t) (m) being monitored. Loadcontrolled NI tests usually include a loading, a
dwelling and an unloading phase (see Figure 1).
Based on the so-obtained penetration history data,
mechanical properties of the indented material can
be computed. In this paper, the NI technique is
extended from extraction of elastic and strength
(hardness) properties of calcium silicate hydrates
(CSH) to the identification of the viscous, timedependent behaviour of CSH. With this information
at hand, recently developed multiscale models can be
used to predict and optimise the creep and relaxation
behaviour of concrete and concrete structures [1–4].
Classical Indentation Analysis – Literature
Review
Assuming purely elastic unloading, the indentation
(plane stress) modulus M ¼ E/(1)m2), with m as Poisson’s ratio and E as Young’s modulus, is determined
2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd
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Identification of Logarithmic-Type Creep of CSH : Ch. Pichler and R. Lackner
(A)
(C)
(B)
Figure 1: Typical NI-test (single indent) result for OPC paste: (A) load history, (B) penetration history, and (C) load-penetration
curve
from the initial slope of the load-penetration curve
during the unloading phase S ¼ dP/dh|h¼hmax (see
Figure 1C) according to the so-called BASh equation
(Bulychev–Alekhin–Shoroshorov equation) [5, 6]:
pffiffiffiffiffiffi
2
S ¼ pffiffiffi M Ac ;
p
(1)
where Ac is the projected area of contact of the
indenter. The projected area of contact for conical
tips is given as
Ac ¼ ph2c tan2 h;
(2)
where h is the indenter half angle and hc denotes the
contact depth, i.e. the height of the (conical)
indenter tip in contact with the sample. Pyramidshaped indenter tips might be represented by a cone
such that the cross-sectional area of the pyramidshaped tip and the cone are equal [6]. Accordingly,
the three-sided Berkovich tip may be represented by a
cone with a half angle of h ¼ 70.32. In order to avoid
measuring the projected area of contact, a procedure
to determine the contact depth based on the solution
of the elastic contact problem was proposed in Ref.
[5] as
2 Pmax
hc ¼ hmax 2 1 ;
S
p
(3)
with Pmax denoting the maximum load acting at the
beginning of the unloading phase. As pointed out in
Ref. [7], Equation (3) is widely used to estimate the
contact depth for elastic–plastic solids. However,
Equation (3) may only be applied for (2 · cohesion)/
18
stiffness ratios of 2c/E > 0.05 in case of a Tresca-type
material, when no ‘piling-up’ occurs. Equation (1)
was derived using the analytical solution (Sneddon
solution) for the contact problem of a rigid indenter
penetrating an elastic halfspace, given as [8–10]:
P
2 tan h
¼
Eh2 p 1 m2
(4)
and
hc 2
¼ :
p
h
(5)
Hereby, the radial displacement of points along the
surface of contact under the indenter was constrained for the derivation of Equations (4) and (5).
Multiplication of the right-hand side of Equation (4)
with the correction term b [6] gives access to the
so-called ‘modified Sneddon solution’:
P
2 tan h
¼b
:
Eh2
p 1 m2
(6)
According to Ref. [6], b depends on Poisson’s ratio m
and the indenter half angle h. Results of a parameter
study focusing on b as a function of indenter half
angle and Poisson’s ratio [4, 11, 12] (see Figure 2) are
in good agreement with the numerical results given
in Ref. [6]. For a Berkovich indenter characterised by
a half angle of h ¼ 70.32 and m ¼ 0.24, b was found
to be 1.081. (In Ref. [19], a value of m ¼ 0.24 was
suggested for CSH.) Differentiation of Equation (6)
with respect to h, specialised for h ¼ hmax, and considering S ¼ dP/dh|h¼hmax and Equations (5) and (2)
2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd
j Strain (2009) 45, 17–25
Ch. Pichler and R. Lackner : Identification of Logarithmic-Type Creep of CSH
Figure 2: Correction term b as a function of indenter half angle h and Poisson’s ratio m [4, 11, 12]
gives access to the corrected form of the BASh equation as
2
E pffiffiffiffiffiffi
Ac
S ¼ b pffiffiffi
p 1 m2
(7)
which is used for determination of the elastic properties of the constituents of cement paste throughout
this paper.
Utilisation of the NI Technique Towards
Identification of Creep and Strength
Properties of CSH
Experimental programme
The test series presented in this paper focus on the
extraction of viscoelastic and strength properties of
CSH, i.e. the main finer-scale binding phase in
hydrated ordinary Portland cement (OPC) paste. The
NI tests were performed in grids, with the so-called
‘grid-indentation technique’ [13], giving access to: (i)
the spatial distribution and (ii) material properties of
the different material phases at the micrometre scale
of observation of cement paste. Table 1 summarises
all relevant information on the experimental programme presented in this paper. Whereas the first
test series (see Table 1) focused on the extraction of
Table 1: Experimental parameters for NI-testing of ordinary
elastic properties of CSH, the remaining test series
were characterised by a vast dwelling phase, hence,
aimed at the extraction of viscous material behaviour. Hereby, the influence of the Blaine (grinding)
fineness was investigated. All tests presented in this
paper were characterised by
• the use of well-hydrated samples with an age of
approximately 1 year;
• a distance between neighbouring grid points of
5 lm for grid indentation; and
• the use of a Hysitron TriboIndenter and a Berkovich diamond tip (Hysitron Inc., Minneapolis,
MN, USA).
Figure 3 shows: (i) the spatial and (ii) the frequency
distribution of Young’s modulus E determined using
Equation (7) for OPC paste characterised by a Blaine
fineness of 4000 cm2 g)1 and a water/cement ratio
w/c of 0.5 (see Table 1). Hereby, m in Equation (7) was
set to 0.24 [14].
Extraction of strength properties
While elastic material properties are determined from
the unloading branch of NI tests, strength properties
may be deduced from the loading branch, characterised by plastic deformations in the vicinity of the
indenter tip. The hardness H (Pa) of the material is
determined at the end of the loading phase with the
maximum load Pmax and its corresponding projected
area of contact as
Portland cement paste
H¼
No. indents ())
Blaine
fineness
w/c
Load history*
(cm2 g)1) ratio ()) (s–s–s/lN)
Results
given in:
25 · 25 ¼ 525
4000
0.5
5–5–5/500
Figure 3
20 · 20 ¼ 400
3890
0.4
1–100–1/1000
Figures
6B and 7
20 · 20 ¼ 400
3000
0.4
1–100–1/1000
[4]
20 · 20 ¼ 400
4850
0.4
1–100–1/1000
[4]
1 (single indent) 4000
0.5
10–350–10/1000 Figures
1 and 5
*Duration of loading–dwelling–unloading/maximum load.
Employed indenter: Hysitron TriboIndenter with a Berkovich tip; age of specimens:
1 year; distance between neighbouring grid points: 5 lm.
Pmax
:
Ac
(8)
The hardness represents the material response corresponding to the stress state under the indenter tip,
characterised by predominant hydrostatic loading.
As the tip shape influences the latter, the hardness
depends on the indenter tip employed (Berkovich
hardness, Vickers hardness, etc.). Hence, H is not
a fundamental material property. In order to determine material parameters describing plastic material
behaviour, experimental data have to be analysed in
combination with analytical/numerical methods.
2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd
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Identification of Logarithmic-Type Creep of CSH : Ch. Pichler and R. Lackner
Figure 3: Young’s modulus E for material phases encountered in cement paste determined by the corrected form of the BASh
equation (Equation 7) (grid-indentation with 25 · 25 indents and distance between adjacent grid points of 5 lm on OPC paste
characterised by w/c ¼ 0.5 and a Blaine fineness of 4000 cm2 g)1)
1 In case the material strength is a linear function of
the applied hydrostatic pressure, i.e. the material
behaviour can be described by, e.g. the Mohr–
Coulomb failure criterion, the so-called ‘dualindentation technique’ for identification of
cohesion and angle of internal friction was
developed in Ref. [15]. This technique is based on
the use of two different, but geometrically similar
indenter tips, e.g. two conical tips with different
half angles. A Berkovich tip and a ‘cube-corner’
tip, characterised by h ¼ 70.32 and h ¼ 42.28,
respectively, were used in Ref. [15]. Numerically
obtained (employing a limit analysis-based
approach) relations for the hardness/cohesion
ratio as a function of the angle of internal friction
yield two constraints for the determination of the
unknown values for the cohesion and the angle of
internal friction.
2 For elastoplastic materials obeying the von Mises
yield criterion, Tabor [16] postulated a relation
between the yield strength fy and the (Brinell)
hardness H as: H 2.6/3fy. Recently, this relation
was expanded towards conical indentation of
strain-hardening materials [7]. In Ref. [7],
numerical results are given in the form (H/fy) as a
function of (fy/E) for different values of the strainhardening exponent n.
3 In Refs [4, 11, 12], scaling relations for conical
viscoelastic–cohesive indentation are derived
employing the finite element method. For an
elastic–ideally plastic Tresca-type material, given
by Young’s modulus E and cohesion c, the scaling
function G0(c/E) [4, 11, 12] is given by
n
h
ioc0
;
G0 ðc=EÞ ¼ 1 exp a0 ðc=EÞb0
(9)
with a0 ¼ 20.62, b0 ¼ 0.6852 and c0 ¼ 1.360. G0
accounts for the effect of plastic material response
on the respective (constant) value of the dimensionless quantity Pp, with Pp ¼ P(t)/[Eh(t)2].
20
Figure 4: Scaling function G0(c/E) obtained from numerical
simulation of conical indentation [4, 11, 12], with Pp ¼
P(t)/[Eh(t)2] and Pep ¼ 2b=p tan h=ð1 m2 Þ (see Equation 6)
Thus, G0 ¼ 1 refers to the purely elastic loading
situation (see Figure 4). With Young’s modulus at
hand [determined according to Equation (7)], the
experimental data of the loading phase of the NI
test can be plotted in terms of the dimensionless
quantity Pp ¼ P(t)/[Eh(t)2] (see Figure 5). These
data indicate a progressive switch from spherical
to conical indentation. The initial contact of the
indenter tip is characterised by spherical indentation because of the small (but ever present)
bluntness of the indenter tip. The initially large
values for Pp reflect this type of loading situation.
With increasing penetration depth, however, the
experiment is characterised by conical indentation, with Pp reaching a constant value. The
observed horizontal asymptote indicates that, for
the employed loading rate of 100 lm s)1 (see
Figure 5), the loading phase of the NI test is
characterised by mainly elastoplastic material
response. Viscoelastic–plastic material response,
on the other hand, would have resulted in a
monotonically decreasing value for Pp during the
loading phase of the NI test [4, 11, 12]. When the
value for Pp representing the horizontal asymptote is scaled by the respective value corresponding to the elastic solution, Pp =Pep , with
Pep ¼ 2b=p tan h=ð1 m2 Þ [see Equation (6)], the
elastoplastic scaling relation G0(c/E) [see Equation
2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd
j Strain (2009) 45, 17–25
Ch. Pichler and R. Lackner : Identification of Logarithmic-Type Creep of CSH
Figure 5: Extraction of plastic material properties from loading phase of NI-test data for OPC
(9)] gives access to the cohesion/stiffness-ratio c/E
of the tested material (see Figure 5) and, thus, to
the cohesion c. Based on NI tests on OPC presented in the next section, c/E at the investigated
observation scale is in the range 1/100 to 1/200.
Remark 1
The cohesion of the Tresca criterion extracted from
NI-test data characterises plastic material failure
because of predominant hydrostatic loading under
the indenter tip. Hence, the so-obtained failure
surface may not be used to determine material
properties describing material failure under mainly
deviatoric loading, such as, e.g. the uniaxial tensile or
compressive strength.
Identification of logarithmic-type creep
properties
Whereas the loading branch is characterised by
mainly elastoplastic material response and unloading
is predominated by elastic behaviour, hence, are used
for determination of strength and elastic properties,
the viscous material response during the dwelling
phase of NI tests may be employed to quantify the
creep behaviour of cement-based materials at the
micrometre scale. The NI data are characterised by a
pronounced increase of the penetration depth during
the dwelling phase (see Figure 1C), particularly during the first minute after load application. Recently,
analytical solutions for conical indentation of the
viscoelastic halfspace characterised by the Maxwell
model, the Kelvin–Voigt model and the threeparameter model were derived in Ref. [14]. In Refs [4,
11, 12], this approach is extended towards:
1
a logarithmic-type creep model [17, 18]: according to Ref. [18], the two mechanisms leading to
logarithmic creep in crystalline solids are either (i)
2
work hardening: dislocations move forward under
the applied stress by overcoming potential barriers, while successively raising the height of the
potential barriers or (ii) exhaustion: while
neglecting work hardening the barriers to dislocation motion do not have equal activation
energies; those with relatively small activation
energies are overcome faster than those with relatively large activation energies; if each barrier
which is overcome contributes an equal increment of strain, the total strain increases linearly
with the increasing activation energy, where the
latter is a logarithmic function of time; and
the incorporation of plastic deformations modelled by the Tresca criterion, using, similar to the
previous section, scaling relations for the viscoelastic–plastic material response.
The mentioned logarithmic-type model is characterised by a creep compliance proportional to
ln (1 + t/sv), where sv denotes the characteristic time
of the creep process. In fact, this model shows the
best agreement with the obtained experimental data.
Whereas elastic material behaviour is considered for
hydrostatic loading, the viscoelastic deviatoric creep
compliance function
J dev ðtÞ ¼
1
t
þ J v;dev ln 1 þ v;dev
l
s
(10)
is employed [4, 12]. Hereby, l ¼ E/2/(1 + m) (Pa)
denotes the shear modulus given by the unloading
branch of the NI data, and Jv,dev(Pa)1) and sv,dev (s) are
the creep compliance parameters to be identified. In
case of a step load P(t) ¼ H(t)Pmax applied during the
loading phase onto a conical indenter, where H(t)
denotes the Heaviside step function, the deviatoric
creep law (Equation 10) gives the penetration history
for the dwelling phase as [4, 12]
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Identification of Logarithmic-Type Creep of CSH : Ch. Pichler and R. Lackner
HðtÞPmax
p
1 m2
h2 ðtÞ ¼
2 tan h E
b
1
t þ sv;dev
v;dev
:
1þb
ln v;dev
lJ
2ð1 mÞ
s
An experimental study was conducted in order to
identify the influence of the Blaine fineness / on the
finer-scale creep properties (see Table 1). Hereby, the
three test series were characterised by / ¼ 3000, 3890
and 4850 cm2 g)1, respectively, w/c ¼ 0.4, and a relative humidity in the test chamber of hexp ¼ 50%.
The result for / ¼ 3890 cm2 g)1 is shown in Figure 7.
The grid indentation, with 20 · 20 ¼ 400 indents for
each grid, gives access to the spatial distribution of
mechanical properties. Hereby, the specimen with /
¼ 3000 cm2 g)1 is characterised by a coarser microstructure with some anhydrous clinker grains, the
specimens with / ¼ 3890 and 4850 cm2 g)1 by a
finer and more homogeneous microstructure.
Whereas the frequency distributions for Young’s
modulus E show two (or three) maxima in the range
15–35 GPa, corresponding to different types of CSH
[19] [low-density (LD) and high-density (HD) CSH]
and portlandite, the frequency distributions of
v;dev
1=JCSH
are characterised by one pronounced peak.
Hereby, this peak value amounts to 22.5 GPa for / ¼
3000 cm2 g)1, 27.5 GPa for / ¼ 3890 cm2 g)1, and
32.5 GPa for / ¼ 4850 cm2 g)1 [4]. The corresv;dev
are 0.044, 0.036 and
ponding values for JCSH
)1
0.031 GPa [4].
(11)
In order to account for the plastic deformations
during the loading phase, the analytical solution
(Equation 11) is scaled by [G0 + glog(1)G0)])1, i.e.
HðtÞPmax
p
1 m2
h2 ðtÞ ¼
2 tan h E
b
1
t þ sv;dev
1þb
lJ v;dev ln v;dev
2ð1 mÞ
s
1
G0 þ glog ð1 G0 Þ ;
(12)
where G0(c/E) denotes the elastoplastic scaling relation already used in the previous section (see Figure 5) and glog(m,c/E, Jv,dev, sv,dev) is an interpolation
function, which accomplishes the transition between
elastoplastic (short-term) response and viscoelastic
(long-term) response (see Figure 6A; [4, 12]). The
unknown deviatoric creep parameters Jv,dev and sv,dev
are determined employing a least-square fitting
method. Hereby, the error between the measured
penetration history in the dwelling phase, h2exp ðtÞ,
and h2(t,Jv,dev,sv,dev) for a step load according to
Equation (12) is minimised:
X 2
½hexp ðtn Þ h2 ðtn ; J v;dev ; sv;dev Þ2 ;
R J v;dev ; sv;dev ¼
Remark 2
The reasonable duration of the dwelling phase of an
NI test is limited by the so-called machine drift of
the NI-testing rig, h_ rig . This drift is mainly caused by
creep of the three-axis piezo-scanner [20], which is
part of the measuring head holding the indenter tip
[whereas a capacitive transducer is employed for
load application, a three-axis piezo-scanner is
adopted for 3D imaging of the sample surface, as
well as for the (final) approach of the indenter tip
to the sample surface before NI testing]. It is
determined prior to the actual NI test by putting
the indenter tip on the sample surface, applying a
small force, e.g. 2 lN and recording the deformation rate – ideally until a steady state is reached.
n
(13)
where n denotes the number of data points of the
dwelling phase considered in the minimisation
scheme. Figure 6B shows the distribution of R
obtained for one indent on OPC as a function of the
unknown model parameters Jv,dev and sv,dev, indicating a clear (global) minimum. Thus, the proposed
parameter-identification scheme leads to a unique
solution for Jv,dev and sv,dev for the respective indent.
The steps necessary for the identification of the creepcompliance parameters are summarised in Box 1.
(A)
(B)
Figure 6: (A) Interpolation function glog(m,c/E,Jv,dev,sv,dev) for different values of Pl ¼ lJv,dev, v ¼ 0.24, and Pc ¼ c/E ¼ 1/100
[4, 12]; (B) distribution of R as a function of Jv,dev and sv,dev (see Equation 13) showing uniqueness of the solution obtained by
the proposed parameter-identification scheme
22
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j Strain (2009) 45, 17–25
Ch. Pichler and R. Lackner : Identification of Logarithmic-Type Creep of CSH
Box 1: Identification of creep-compliance parameters by nanoindentation
1. Determination of Young’s modulus E from the initial slope of the load-penetration curve during the unloading phase according to Equation (7)
2. Calculation of cohesion c based on the scaling function G0(c/E) (see Equation 9) and the measured dimensionless quantity Pp ¼ P(t)/Eh(t)2] at the
end of the loading phase
3. Identification of creep parameters by backcalculation of the penetration history during the dwelling phase of the indentation test (see Equation 12)
(A)
(B)
(C)
Figure 7: (A) Young’s modulus E according to Equation (7); creep parameters (B) Jv,dev and (C) sv,dev obtained from backcalculation
of the dwelling phase (grid indentation with 20 · 20 indents and distance between adjacent grid points of 5 lm on OPC paste
characterised by w/c ¼ 0.4 and a Blaine fineness of 3890 cm2 g)1)
However, for materials with low viscosity (i.e. high
compliance), the compliance of the indented specimen may spuriously be interpreted as machine
drift when employing the mentioned procedure.
During the NI tests presented in this paper, jh_ rig j
ranged from 0.01 to 0.1 nm s)1. Once jh_ rig j is in the
range of the penetration rate, the test data lose
their relevance. For the tested OPC pastes and the
employed NI-testing equipment, a reasonable duration of the dwelling phase was found to be around
2–3 min. On the other hand, the average value of
the penetration rate at the beginning of the dwelling phase was approximately 30 nm s)1 for the NI
tests conducted, which is significantly larger than
the jh_ rig j encountered.
Another source of measuring errors is the so-called
thermal drift. It is associated with thermal dilation of
test rig and specimen. The thermal-drift error can be
minimised by: (i) setting up the NI test rig in a
temperature-controlled environment and (ii) storing
2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd
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Identification of Logarithmic-Type Creep of CSH : Ch. Pichler and R. Lackner
the specimen in the test rig (which is hooded) prior
to testing.
Remark 3
Viscous material response influences the unloading
slope and, thus the determination of elastic properties [7]. This effect can even cause ‘outbulging’ in the
load-penetration diagram, characterised by increasing penetration depth for decreasing loading of the
indenter tip. Two measures minimise the described
deviation from purely elastic unloading: (i) a sufficiently long dwelling phase: when the creep response
subsides, i.e. the creep potential is consumed to a
great part, unloading is not influenced either (ii) fast
unloading: for the theoretical situation of infinitely
fast unloading, only elastic material response occurs.
The two measures are also visible when deriving a
criterion for a suitable duration of the unloading
time, su, in order to exclude viscoelastic behaviour
during the unloading branch [14]. For a finite
unloading period following a step load, this criterion
reads for the logarithmic-type model [4]
su 2ð1 mÞ
ðtu þ sv;dev Þ;
blJ v;dev
(14)
with tu denoting the time instant at the beginning of
the unloading phase. Hence, for the load history
considered in the NI tests performed, characterised
by a duration of the dwelling phase of 100 s, and l ¼
E/[2(1 + m)] ¼ 20/[2(1 + 0.24)] ¼ 8 GPa, su ¼ 1 s>2
(1)0.24)(101 + 0.1)/(1.081Æ8Æ0.037) ¼ 480 s,
thus
satisfying condition (14).
Concluding Remarks
In this paper, elastic, strength, and, finally, viscous
properties of cement paste at the micrometre scale
were determined by means of NI. The material
parameters obtained serve as input for the recently
developed multiscale model for upscaling of viscoelastic properties [3, 4], autogenous shrinkage deformations [2, 4], and strength properties [21, 22] of
early-age cement-based materials. Whereas the elastic
properties of the constituents of OPC paste at finer
scales of observation were similar to recently published values in the open literature [13, 19, 23, 24], a
logarithmic-type creep behaviour was identified
during NI creep tests. Interestingly, this type of creep
behaviour is also observed at the macroscale, i.e. is
immanent to the scale of observation. Recently, the
origin of these viscous deformations was explained
by the rearrangement (dislocation) of CSH globules
24
under shear stresses [25]. [In Ref. [25] CSH is
described as an aggregation of precipitated, colloidsized particles with the ‘basic building blocks’ (radius
of 1.1 nm) aggregating into ‘spherical globules’
(radius of 2.8 nm). The latter aggregate further into:
(i) LD CSH or (ii) HD CSH, depending on w/c ratio,
age and environmental conditions]. According to
Ref. [18], dislocation-based mechanisms lead to
logarithmic-type creep behaviour. In order to analyse
the obtained NI data, analytical solutions for
conical indentation with viscoelastic–plastic material
response, constructed in dimensionless form by
means of the finite element (FE) method, were
employed [4, 11, 12]. The scaling relations derived in
Ref. [4, 12] provide the mean to identify logarithmictype creep behaviour and to backcalculate the
respective model parameters.
ACKNOWLEDGEMENTS
The authors thank Franz-Josef Ulm, Georgios Constantinides and Matthieu Vandamme for fruitful discussions and helpful comments during and after their
stay at the CEE department at MIT. The authors further
thank Erwin Schön, Rupert Friedle and Gernot Tritthart
(LAFARGE CTEC, Mannersdorf) for providing the
cements employed in the experimental study, and
Andreas Jäger for conducting the nanoindentation tests
included in this paper. Financial support granted by the
Austrian Science Fund (FWF) via project P15912-N07 is
appreciated.
REFERENCES
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multiscale micromechanics–hydration model for the early-age elastic properties of cement-based materials. Cement
Concrete Res. 33, 1293–1309.
2. Pichler, Ch., Lackner, R. and Mang, H. A. (2007) A
multiscale micromechanics model for the autogenousshrinkage deformation of early-age cement-based materials. Eng. Fracture Mech. 74, 34–58.
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