COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Finer-Scale Characterization and Scale Transition of Viscoelastic
Properties within the Multiscale Framework of Material Description
Roman Lackner *, Andreas Jäger, Christian Pichler
Institute for Mechanics of Materials and Structures, Vienna University of Technology (TU Wien), Karlsplatz 13/202,
1040 Vienna, Austria
Email: {Roman.Lackner,Andreas.Jaeger,Christian.Pichler}@tuwien.ac.at
Abstract Recent progress in both finer-scale experimentation (atomic force microscopy, nanoindentation, ...) and
theoretical and numerical upscaling schemes provides the basis for the development of so-called multiscale models,
taking finer scales of observation into account. Hereby, chemical, physical, and mechanical processes taking place at
finer scales can be considered and their effect on the macroscopic material performance is obtained via appropriate
upscaling schemes. The success of multiscale models is strongly linked to the proper identification of material
properties at finer scales, serving as input for the upscaling schemes. In this paper, extraction of material parameters by
means of the nanoindentation technique is presented. Whereas the extraction of elastic parameters dates back to the
1990s, the focus of this paper is on the extraction of viscoelastic properties. Hereby, the experimental data are
compared with the respective analytical solution for the mathematical problem of a rigid indenter penetrating a
viscoelastic medium, giving access to the viscoelastic material properties at finer scales. Finally, continuum
micromechanics is employed for upscaling of the identified viscoelastic parameters. The proposed identification and
upscaling scheme is applied to cement-based and bituminous mixtures, both characterized by a matrix-inclusion
morphology, with cement and bitumen, respectively, serving as binder material.
Key words: multiscale, upscaling, nanoindentation, bitumen, cement, concrete, asphalt, viscoelastic
NANO-TO-MACRO MECHANICS OF BUILDING MATERIALS
Most (man-made) building materials are composed of several constituents in order to optimize both their performance
as regards mechanical properties and durability and their cost of production. The different constituents are
characterized by different properties and different spatial distribution (e.g., matrix or inclusion). The effect of the
constituents on the overall performance of building materials are properly represented by so-called multiscale models.
At the Institute for Mechanics of Materials and Structures at TU Wien, multiscale models for the most commonly used
building materials, i.e., concrete [1] and asphalt [2], were developed. These models comprise several observation
scales ranging from the hydrate- and bitumen-scale, respectively, towards the macroscale (see Fig. 1). Both materials
exhibit a time-dependent, viscoelastic material response. As indicated in Fig. 1, the macroscopically observable
viscoelastic response results from the viscoelastic behavior of bitumen, on the one hand, the CSH (reaction products of
cement hydration), on the other hand. Aggregates and, in case of concrete, unhydrated cement do not exhibit any time
dependent behavior. Accordingly, in order to develop a multiscale model for the prediction of the macroscopic
viscoelastic material response, the behavior of bitumen and CSH must be identified. In this paper, the nanoindentation
technique, which is explained in more detail in the following section, is employed for this purpose.
IDENTIFICATION OF VISCOELASTIC PROPERTIES BY MEANS OF NANOINDENTATION (NI)
The main goal of NI is the identification of mechanical properties of the indented material. During NI measurements,
a tip with defined shape penetrates the specimen surface with the indentation load P [N] and the penetration h [m]
recorded as a function of time. Commonly, each indent consists of a loading, holding, and unloading phase (see Fig. 2).
The hardness of the material, defined as H = Pmax/Ac [Pa], is obtained from the loading phase of the NI test.
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Figure 1: Observation scales for asphalt and cement-based material
Figure 2: Illustration of (a) load history and (b) load-penetration curve of NI tests
Hereby, Ac [m2] is the horizontal projection of the contact area and Pmax [N] denotes the applied maximum load.
According to [3, 4], the Young’s modulus E of materials exhibiting elastic or elastoplastic behavior is obtained from
and the indentation modulus
the relation between the measured initial slope of the unloading curve
, reading
(1)
where ν is the Poisson’s ratio.
Parameter identification of materials exhibiting, in addition to elastic and plastic material response, time-dependent
behavior (e.g., polymers, bitumen, etc.) requires back calculation of the parameters from the holding phase of the
measured indentation data h(t). Recently, analytical solutions for the indentation of axisymmetric, rigid tips into a
linear viscoelastic halfspace were reported in [5] for spherical tips and in [6] for perfect conical tips. Whereas both [5]
and [6] considered indenter tips characterized by exact geometric properties, the shape of real indenter tips varies in
consequence of the production process and in the course of testing due to attrition. By means of calibration, NI-testing
equipment give access to the real tip geometry [4]. In order to consider the so-obtained geometrical properties of the tip
for back-calculation of material parameters, analytical solutions for the indentation of a tip into a viscoelastic material,
taking the real tip geometry into account, are considered. For this purpose, the geometrical representation of the
for perfect conical tips) is extended to
indenter shape (with
(2)
where Atip [m2] is the area of the cross section and ρ [m] is the corresponding radius of the axisymmetric tip (see Fig. 3).
C0 [-] and C1 [m] are constants describing the tip shape, which are generally provided during calibration of the
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Figure 3: Contact between a rigid axisymmetric tip of shape f (ρ) and an infinite halfspace (P is the applied load,
h is the indentation depth, a is the contact radius, and Ac is the projected area of contact)
NI-testing equipment. In a first step, the elastic indentation problem for the indenter shape given in Eq. (2) is solved.
According to [7], the viscoelastic solution is obtained by replacing the operators of the elastic solution by the Laplace
transforms of the associated viscoelastic operators. Back transformation gives access to the solution for viscoelastic
indentation in the time domain. Finally, the viscoelastic solutions are employed for the identification of viscoelastic
properties of bitumen from NI-test data.
1. Elastic indentation problem For the solution of the elastic indentation problem, i.e., a rigid indenter penetrating
the elastic halfspace, the so-called Sneddon solution [3] is employed. According to [3], the relation between the
penetration h and the corresponding load P is given for an axisymmetric indenter of shape f (ρ) (see Fig. 3) by
(3)
Hereby, a [m] is the radius of the projected contact area Ac, ρ [m] is the radius of the axisymmetric tip, f (ρ) [m] is a
For the case of conical indenters, f (ρ) = ρ/tanα, where
smooth function describing the tip shape, and
α is the semi-apex angle. Accordingly, for the commonly used Berkovich indenter, which may be represented by a
cone of α = 70.32◦, f (ρ) becomes linear in ρ. In general, however, because of inaccuracies during the tip-production
process and attrition, the aforementioned linear relation is nonlinear. During calibration of the NI-testing equipment,
this nonlinearity is specified, following the procedure outlined in [4]:
(1) perform indents in a material with given elastic properties (e.g., fused quartz) in the depth range of the indentation
experiments;
(2) compute the projected contact area
, where S [N/m] is the initial unloading slope of the
loadpenetration curve and M is the indentation modulus, with
, where E = 72 GPa and ν = 0.17 for
fused quartz;
(3) plot Ac as a function of the contact depth hc, with
, and approximate the so-obtained function by
(4)
where C0 [-] and C1 [m] are constants describing the tip shape.
Replacing Ac and hc in Eq. (4) by ρ2π and f (ρ), respectively, f (ρ) is obtained as
(5)
For the case of a conical indenter with a semi-apex angle α, where C0 = π tan2 α and C1 = 0, Equ. (5) gives f (ρ) = ρ/tan
α. Fig. 4 shows the tip-shape function f (ρ) and the area function Ac (hc) for a perfect Berkovich tip (C0 = 24.5 and C1 =
0) and a Berkovich tip with a value of C1 deviating from zero.
Inserting Eq. (5) into Eq. (3) gives the penetration and the applied load as a function of the contact radius a,
(6)
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Figure 4: (a) Tip-shape function f (ρ), and (b) area function Ac(hc) of a perfect Berkovich tip (C0 = 24.5 and C1 = 0)
and a real Berkovich tip with C0 = 24.5 and C1 = 2314 nm
(7)
where 2F1(a; b; c; z) denotes a hypergeometric function (see, e.g., [8]).
2. Viscoelastic indentation problem – Application to trapezoidal load history In the following, the elastic
indentation problem outlined in the previous subsection is extended to linear viscoelasticity. For this purpose, the
method of functional equations [7] is employed. Hereby, the elastic constants in the solution of the equivalent elastic
boundary value problem are replaced by the Laplace transforms of the associated viscoelastic operators. Rewriting the
result for the solution of the indentation in an elastic halfspace (Eq. (7)) in the form
(8)
allows us to split solution (7) into the material dependent indentation modulus M, with M = E/(1−ν2), and the function
F(a) depending only on geometric properties, such as the tip shape (represented by the constant parameters C0 and C1)
and the unknown contact radius a, reading
(9)
Following the method of functional equations, the viscoelastic solution for the indentation problem is obtained by
, giving
replacing the elastic operators P, M, and F(a) in Eq. (9) by their Laplace transforms
(10)
Re-arrangement yields an expression for the Laplace transform of the function F(a(s)) as
(11)
where
was replaced by the Laplace transform of
, in the following referred to as indentation
compliance function. Considering that (1) a multiplication by s in the Laplace domain is equivalent to a derivation in
the time domain and (2) a multiplication of two Laplace-transformed functions is equivalent to the convolution product
of the two functions in the time domain, F(a(t)) is obtained from Eq. (11) as
(12)
Finally, combining Eqs. (9) and (12) allows determination of the unknown contact radius a(t). With a(t) at hand, Eq. (6)
provides access to the history of the penetration h(t).
The method of functional equations is restricted to an increasing contact area, and may be applied only to
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monotonically increasing and constant load histories [7]. Since indentation tests are commonly conducted under load
control, Eq. (12) is specified to the trapezoidal load history depicted in Fig. 2(a), reading
(13)
where τL, τH, and τU are the loading, holding, and unloading durations, respectively. Considering the load history P(t)
given in Eq. (13) in Eq. (12), the function F(a(t)) becomes for the loading and holding regime
(14)
(15)
Based on FL and FH in Eqs. (14) and (15), the history of the penetration, h(t), for the loading and holding time is
determined in three steps:
1) Determination of the indentation compliance function
appearing in Equations (14) and (15) for the considered
viscoelastic model: The indentation compliance function
is determined for two deviatoric creep models, i.e., the
three-parameter (3P) and the Power-Law (PL) model (see Fig. 5). In the case of elastic material response, the
indentation modulus M can be expressed by the bulk modulus K and the shear modulus μ0, reading
(16)
Figure 5: Considered viscoelastic deviatoric creep models:
(a) the three-parameter (3P) model and (b) the Power-Law (PL) model
By the way of application of the method of functional equations, the elastic constants K and μ0 in Eq. (16) are replaced
by the associated Laplace-transformed operators
and
, reading
(17)
where
for the case of deviatoric creep only. The shear relaxation modulus, on the other hand, is time
dependent, and is given by (see, e.g., [9]):
(18)
(19)
where
denotes the Laplace transformation of
and applying the inverse Laplace transformation to
. Considering
,
of the 3P model given in Eq. (18) in Eq. (17)
is obtained as
(20)
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For the PL model, the respective expression for the case of incompressible materials (K → ∞) reads
(21)
2) Determination of FL(a(t)) and FH(a(t)) using Eqs. (14) and (15): Considering the indentation compliance functions
for the two viscoelastic models given in Equations (20) and (21) in Equations (14) and (15), the function F(a(t)) is
obtained for the loading and holding regime, FL(a(t)) and FH(a(t)), as
(22)
(23)
(24)
(25)
3) Determination of the contact radius a(t) and, hence, via Eq. (6), the penetration h(t) by combining the expressions
for FL(a(t)) and FH(a(t)) with Eq. (9): The history of the contact radius, a(t), is obtained from combining the expressions
for FL(a(t)) and FH(a(t)) given in Eqs. (22) to (27) with Eq. (9). The so-obtained (nonlinear) expression for a(t) is solved
numerically, employing a Newton-iteration scheme. With a(t) at hand, the history of the penetration, h(t), is given by
Eq. (6) for a given load history P(t) and the material model describing the behavior of the viscoelastic half space.
From the two viscoelastic models considered so far, simpler models may be obtained by eliminating a spring and/or
dashpot. E.g., a nonlinear dashpot (nlDP) is obtained from the PL model by setting J0 equal to zero, giving FL−nlDP(a(t))
and FH−nlDP(a(t)) in the form
(26)
(27)
3. Extraction of parameters from NI-test data In order to determine material parameters from NI-test data, the error
between the experimentally-obtained function Fexp (determined from the penetration history using Eqs. (6) and (9)) for
the holding period and the analytical result for the considered viscoelastic model (see previous subsection) is
minimized by adapting the unknown model parameters (see Fig. 6 for the case of nlDP model). For the case of the
nlDP model, the mentioned error is defined by
(28)
with
(29)
The error given in Eq. (28) is minimized by adapting Ja and k, using a simplex iteration [10]. With the model
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Figure 6: Illustration of the definition of the error used for parameter identification from the holding phase
of NI-test data for the nlDP model: (a) penetration history h(t) and (b) function FH
parameters at hand, the history of the contact radius, anlDP(ti), and subsequently hnlDP(ti) are computed from Eqs. (9) and
(6), respectively.
APPLICATION TO BITUMEN
The presented parameter-identification scheme is applied to B50/70, a bitumen commonly used for the construction of
flexible pavements. Fig. 7 shows the frequency plots for Ja, k, and the error RnlDP for 10 × 10 indentations at −1◦C. The
obtained frequency plots are approximated by a Gaussian distribution, giving mean values and standard deviations for
the model parameters. The computed mean value of the error RnlDP is about 1%, confirming the proper choice of the
nlDP model for fitting the viscoelastic response of bitumen. For the considered type of bitumen (B50/70), the mean
values of the model parameters corresponding to T = −1◦C are: Ja = 8.4 GPa−1 and k = 0.7. The outlined mode of
parameter identification is employed to study (1) the influence of the loading rate and the maximum load on the
determined parameters, (2) the temperature dependence of the model parameters and, finally, (3) the microstructure
and the mechanical properties of the different bitumen phases.
Figure 7: Identification of model parameters for B50/70 tested at −1◦C: frequency plots of (a) Ja [GPa−1], (b) k [-],
and (c) error RnlDP [%] (NI-test conditions: Pmax =20 μN,
, τH =5 s)
Fig. 8 shows the mean values of the nlDP-model parameters for different loading rates and different values for the
maximum load (10 × 10 indentations each) on bitumen B50/70 at −1◦C (with = dP/dt =20, 40, 80, 160 μN/s and
Pmax =10, 20, 50, 120, 240 μN). Whereas the influence of the loading rate on the parameters is quite small for all
considered load levels, the maximum load itself has a higher impact on the obtained model parameters, indicating the
development of plastic deformations during the loading phase. Accordingly, assigning all (viscoelastic and plastic)
deformations to the viscoelastic behavior leads to higher values for Ja as the maximum load increases (see Fig. 8(a)).
The effect of the temperature on the identified model parameters is illustrated in Figs. 9(a) and (b), showing the mean
values of the nlDP-model parameters for bitumen B50/70 tested at T = −4.5, −1, 2, 5.5, 9◦C. The temperature
dependence of Ja depicted in Fig. 9(a) and (c) is well-described by an Arrhenius-type law, with
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Figure 8: Influence of loading rate and maximum load on nlDP-model parameters: (a) mean value of Ja [GPa−1] and
(b) mean value of k [-] (bitumen B50/70 tested at T = −1◦C; Pmax =10, 20, 50, 120, 240 μN;
= dP/dt =20, 40, 80, 160 μN/s; τH =5 s (for Pmax = 10, 20 μN) and 10 s (for Pmax = 50, 120, 240 μN))
Figure 9: Temperature dependence of nlDP-model parameters for bitumen B50/70: (a) Ja [GPa−1],
(b) k [-], and (c) Identification of Arrhenius-type law describing the temperature dependence of Ja
( = 273 K) (NI-test conditions: Pmax =10, 20 μN, = dP/dt =20, 40 μN/s, τH =10 s)
(30)
where
energy.
[K] is the reference temperature, R = 8.31 [J/MOL/K] is the gas constant, and Ea [J/MOL] is the activation
Finally, the obtained values for Ja and k for different testing temperatures were compared with the results obtained
from standard bitumen tests, such as the bending beam rheometer (BBR) and the dynamic shear rheometer (DSR). Fig.
10 shows a good agreement between the obtained parameters from NI and the respective standard-test data. The
obtained NI data fits well into the larger temperature range covered by the standard test methods (−24 to 40◦C).
Figure 10: Comparison of material parameters obtained from NI testing with DSR and BBR data: (a) Identification of
Arrhenius-type law describing the temperature dependence of Ja ( = 273 K), (b) temperature dependence of k [-]
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1. Bitumen treated as a multiphase composite Bitumen is the remaining part during crude-oil distillation and, hence,
its chemical composition strongly depends on the origin of the crude oil. Its main constituents are hydrocarbons with
different amount of polarity, and a molecular mass ranging from 300 to 100000 g/mol [11]. This complex chemical
composition results in the development of a certain microstructure of bitumen, characterized by a string-like structure
embedded in a matrix substance (see Fig. 11(a)) [12, 13]. Interestingly, the string-like structures change their
properties with aging [12] and they align under external loading [13].
With the characteristic dimension of the bitumen microstructure of 10 μm (diameter of strings according to [12]), the
distance between adjacent points of the 10×10 indentation grid was set to 5 μm. Using a Berkovich tip for the
indentation experiments, the distance between adjacent indents of 5 μm gives a maximum penetration of 400 nm,
ensuring no interaction between two adjacent indents.
Figs. 11(b) and (c) show histograms and the corresponding grid plots for the parameters of the nlDP model obtained at
−1◦C. The grid plots confirm the microstructure already observed by environmental scanning electron microscopy
(ESEM) [13, 12], showing a string like microstructure with typical dimensions of about 10 μm embedded into a matrix
material. The histograms emphasize the presence of two bitumen phases exhibiting different mechanical behavior,
with the strings characterized by lower values of Ja and k. The approximation of the histograms by two Gaussian
distributions gives access to mean value and standard deviation of the viscoelastic model parameters for the two
bitumen phases.
Figure 11: (a) Environmental scanning electron microscopy (ESEM) image of unaged bitumen;
Frequency plot and corresponding grid plot for parameters of nlDP-model: (b) Ja [GPa−1] and
(c) k [-] (bitumen B50/70 tested at T = −1◦C, Pmax=10 μN, τL =0.25 s, τH =5 s)
APPLICATION TO CSH
Having encountered a significant influence of plastic deformation on the identified model parameters in the previous
section, viscoelastic material properties of CSH are backcalculated from the viscous part of the penetration history hv
= htot −he during the holding phase (see Figs. 12 and 13), taking the increased penetration in consequence of
elastoplastic loading into account. For this purpose, a logarithmic-type deviatoric compliance is employed:
(31)
with 1/μ denoting the elastic part of the deviatoric compliance, and Jv and τv as material parameters related to
magnitude and duration, respectively, of the creep response. So far, the volumetric deformations are assumed to be
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Figure 12: NI test on OPC paste: typical test result
Figure 13: NI test on OPC paste: backcalculation of viscous material properties from the dwelling phase
purely elastic. The respective indentation compliance function
case of incompressible materials
(see Eq. (21) for
for bitumen) reads for the
(32)
In this section, a perfect indenter geometry was assumed for backcalculation of viscoelastic properties. Moreover, the
loading phase was described by a step load, which is justified by the rather high loading rate employed in the NI tests
on ordinary Portland cement (OPC) paste characterized by a water/cement-ratio of 0.5 (see Fig. 12). With Young’s
modulus E determined from the initial slope of the unloading curve (see Eq. (1)), the loading phase of the NI test can be
(see Fig. 14). The experimental data indicate a
plotted in terms of the dimensionless quantity
progressive switch from spherical to conical indentation. The initial contact of the indenter tip is characterized by
spherical indentation due to the small (but ever present) bluntness of the indenter tip. The initially large values for
reflect this type of loading situation. With increasing penetration depth, however, the experiment is characterized by
reaching a constant value. The observed horizontal asymptote indicates that, for the
conical indentation, with
employed loading rate of =100 μN/s (see Fig. 12), the loading phase of the NI test is characterized by mainly
, on the other hand, are related to a combined
elastoplastic material response. Monotonic decreasing values for
viscoelastic-plastic material response during the loading phase of the NI test [14]. When the value for
representing
the horizontal asymptote is scaled by the elastic solution”,
, an elastoplastic scaling relation obtained from
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
”
The elastic solution of the cone penetration problem is given as
The correction factor β in Equation (33) takes the radial displacement of points along the surface of contact under the
indenter into account, which was assumed to be zero in the derivation of the elastic solution of the cone penetration
problem [15, 16, 3]. For θ = 70.32◦ and ν = 0.24 [17], β is given as 1.081 [14].
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Figure 14: NI test on OPC paste, extraction of plastic material properties
numerical analyses of the cohesive indentation problem [14] gives access to the cohesion/stiffness-ratio c/E of the
tested material (see Fig. 14) [14]. This scaling relation reads
(33)
where the three parameters were identified as a0 =20.62, b0 =0.6852, and c0 =1.360. For the NI test shown in Fig. 12,
the obtained material parameters can be summarized as E =24.8 GPa, c =149 MPa, Jv = 8.9×10−3 GPa−1, and τv=0.47 s,
showing an excellent agreement between NI-test data and model response (see Fig. 13).
UPSCALING OF VISCOELASTIC PROPERTIES
According to the correspondence principle [18, 19], the Laplace-Carson transform” of the effective shear compliance
, is related to and the same way as Je f f is related to Jm and Ji (subindices “m” and “i” refer to matrix
and inclusion, respectively) in the elastic case (see, e.g., [20] for an application of the latter to the selfconsistent
scheme). As for homogenization of asphalt or concrete, viscoelastic behavior is assigned to the matrix material
(binders, i.e., bitumen and cement paste, respectively), while the inclusions exhibit elastic deformations only. E.g., the
Laplace-Carson transform of the deviatoric creep compliance of the matrix material for the logarithmic-type creep
model identified in the NI experiments on OPC paste reads
(34)
with Γ denoting the incomplete gamma-function””. As regards the elastic material behavior, the effective (elastic)
shear compliance of a matrix-inclusion type composite is related to the elastic properties of the material phases
employing continuum micromechanics, reading [21]
(35)
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
”
The Laplace-Carson transformation of f (t) is defined as
with p as the complex variable. The inverse Laplace-Carson transformation is defined in the complex plane as
where ∧ is a parallel to the imaginary axis having all poles of f _(p) to the left.
””
The incomplete gamma function Γ[a, z] satisfies
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where fm and Jm = 1/μm denote the volume fraction and (elastic) shear compliance of the matrix m, fr and Jr are the
volume fraction and the shear compliance of the r-th inclusion phase, β = 6(Km+2/Jm)/[5(3Km +4/Jm)], where Km is the
bulk modulus of the matrix material. Applying the correspondence principle to Eq. (35), the Laplace-Carson transform
of the effective (viscoelastic) creep compliance is obtained as
(36)
, where
and
for the case of elastic material response.
with
Inserting Eq. (34) into Eq. (36) and performing the inverse Laplace-Carson transformation gives access to the effective
creep compliance,
.
Upscaling of CSH creep A previously developed multiscale model for upscaling of autogenous-shrinkage
deformation [22] is extended towards upscaling of viscoelastic properties. This multiscale model considers different
material phases at four length scales, which are identified as follows (see Fig. 1):
(I) Scale I comprises the four clinker phases, high-density CSH (CSH −HD) and low-density CSH (CSH − LD), and
the water and air phase. As regards upscaling of creep properties, the four clinker phases, which do not exhibit
time-dependent behavior, are condensed into one material phase (Scale Ia). The creep-active constituents, are
combined at Scale Ib-1, where CSH −HD is located in the space confined by the previously formed CSH −LD. At the
porous CSH scale (Scale Ib-2), water and air are considered as inclusions in a matrix constituted by the homogenized
material of Scale Ib-1.
(II) At the cement-paste scale, anhydrous cement (homogenized material of Scale Ia), gypsum
, portlandite CH,
and reaction products from C3A and C4AF hydration form inclusions in a matrix constituted by the homogenized
material of Scale Ib-2.
(III) At the mortar or concrete scale, aggregates are represented as inclusions in the cement paste (homogenized
material of Scale II).
(IV) Finally, at the macroscale, concrete is treated as a continuum.
Volume fractions at Scales I and II were determined using a extended version of the hydration kinetic laws for the
individual Portland cement clinker phases given in [23]. Hereby, three stages for hydration of tricalciumaluminate in
the presence of gypsum were considered [22]. With volume fractions of the different phases at the respective length
scales of observation at hand, classical homogenization schemes can be employed for upscaling of elastic and viscous
properties. For homogenization of Scale Ia, the self-consistent scheme, suitable for a polycrystalline microstructure, is
employed [24, 25]. During homogenization of Scales Ib to III, the matrix – inclusion type morphology is taken into
account, rendering the Mori-Tanaka scheme [26] as an appropriate homogenization method. Hereby, the inverse
(see Eq. (36)) was performed in a pointwise manner (for discrete
transformation of the effective creep compliance
values of t > 0) by applying the Gaver-Stehfest algorithm [27]. Implying an affine form of the effective creep
compliance Jeff (t) with respect to the creep compliance at the μm-scale [Eq. (31)], the discrete points from inverse
transformation were approximated by
(37)
given in Eq. (35). This non-aging creep compliance, i.e., related to a frozen
with the effective elastic compliance
microstructure or constant values of fm and fr, is determined for different time instants of loading t0, with the
corresponding degree of hydration ξ0 at t0, giving
(38)
with
being related to the microstructure encountered at the time instant t0. The aging creep compliance taking
the changing microstructure (in terms of changing volume fractions of the different phases) for t >t0 into account, is
obtained by
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(39)
The described upscaling procedure is also applied at Scales II and III, where effective properties determined at the
next-finer scale of observation serve as input. Fig. 15 shows first results from upscaling in terms of the effective shear
, for different time instants of loading t0.
compliance rate,
Figure 15: Result from upscaling procedure: rate of aging creep compliance for different time instants of loading t0
CONCLUDING REMARKS
Departing from [6], the theoretical basis for the identification of viscoelastic properties by means of nanoindentation
(NI), taking the real tip geometry into account, was presented in this paper. The presented mode of identification was
formulated for the three-parameter model, the power-law model, and the logarithmic model, and applied to NI-test data
of bitumen and cement paste. From the performed experimental studies and the identified model parameters, the
following conclusions can be drawn:
1) For the NI tests on bitumen, ...
(1) the increase of the maximum load resulted in a significant variation of the identified model parameters (this effect
was explained by plastic deformations during the loading phase of the NI test),
(2) the loading rate showed no influence on the parameters, indicating that the time-dependent deformation during the
loading phase are well captured by the presented parameter-identification tool,
(3) the temperature dependence of the model parameter Ja (a nonlinear dashpot was used for the description of bitumen
behavior) follows an Arrhenius-type law,
(4) the model parameters obtained from NI testing fit well into results from standard test methods, and
(5) the results from grid indentation confirm the string-like microstructure of bitumen already observed by
environmental scanning electron microscopy [12].
2) For the NI tests on cement paste, ...
(1) a logarithmic-type creep law was identified for CSH, what is also reflected by macroscopic creep tests for mortar or
cement paste,
(2) the loading phase in the NI tests is dominated by elastoplastic material response for the chosen loading rate of 100
μN/s.
The theoretical formulation of the parameter identification was based on the assumption of purely-viscoelastic
material response. As highlighted in an experimental study, considering different maximum loads in the NI tests,
significant plastic deformations resulted in an increase of the identified viscous compliance parameter with increasing
NI loading. As a first approach, only the viscous part of the intentation compliance function was used to approximate
the penetration increase during the holding phase of NI tests on cement paste. The extraction of the plastic material
from NI-test data, however, still represents one main challenge in NI research and is a topic of ongoing work.
For the upscaling of viscoelastic properties within the multiscale framework, the Laplace-Carson transform of the
classical Mori-Tanaka homogenization scheme [26] (continuum micromechanics) was employed, exploiting the
viscoelastic correspondence principle. With the NI technique used as “identification experiment” for the identification
of viscous properties of the material phases of cement paste and bitumen, so-called “verification experiments”
performed on the composite (binder and aggregate) are currently in progress. For an overview over the experimental
techniques as regards (a) identification of properties of the material phases and (b) verification of homogenization
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schemes, the reader is referred to [2]. Respective experimental results on binder material and binder-aggregate
composites will be presented at the congress.
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