Modeling of Nanoindentation by a Visco-elastic
Porous Model with Application to Cement Paste
D. Davydov and M. Jirásek
*
Abstract. In order to derive a meaningful constitutive relationship for the Calcium-Silicate-Hydrate (CSH) phase in hardened cement paste, a micromechanical
approach is used. It is well known that CSH gel can be considered as a porous medium composed of certain particles, but the precise shape of these particles has not
been well established yet, neither experimentally nor theoretically. Different authors consider spheres, platelets or fibers. In the present study we propose a viscoelastic constitutive relationship for CSH spherical particles and upscale it to the
level at which nanoindentation takes place. The resulting contact problem can be
solved semi-analytically using the Laplace-Carson transform. Material properties
can be identified by solving an inverse problem based on the nanoindentation experiment. An example of application to white cement paste is presented.
1 Introduction
Nanoindentation is a widely used technique for measuring properties of materials
at the micron and submicron levels. It consists of establishing contact between a
substrate (sample) and an indenter with known properties and geometry. The force
P acting on the indenter is applied as the control variable and the corresponding
penetration depth h is recorded. Commonly, a trapezoidal loading program with
loading, holding and unloading periods is used (Fig. 1).
Elastic properties of the substrate can be evaluated from the unloading part of
the load-penetration curve by the standard Oliver-Pharr procedure [5], which is
based on the analytical solution of the contact problem for an axisymmetric
indenter and a linear elastic isotropic homogeneous infinite half-space [6].
Although this procedure is based on the assumption of material homogeneity, it
has been applied to heterogeneous materials as well [3].
D. Davydov
Department of Mechanics, Faculty of Civil Engineering, Czech Technical
University in Prague
e-mail: denisdavydov@fsv.cvut.cz
M. Jirásek
Department of Mechanics, Faculty of Civil Engineering, Czech Technical
University in Prague
e-mail: milan.jirasek@fsv.cvut.cz
188
D. Davydov and M. Jirásek
Another problem with the interpretation of indentation results is caused by
deviations from an ideal shape of the indenter. The geometry of the actual
indentation tip varies due to the production process and wear out during
experiments. The asssociated error can be reduced by calibration based on series
of indentations into a reference material with known properties [5]. The results
can be used to determine an approximation of the tip shape for which the OliverPharr procedure gives the correct elastic modulus. Usually, the cross-sectional
area
Atip = ρ 2π is approximated by several terms of the series
Atip = C0 f 2 + C1 f + C2 f 1/2 + C3 f 1/4 ...
ρ
(1)
f are respectively the radius of the section and its distance from the
apex of the axisymmetric tip (Fig.1), and Ci are constants describing the tip shape.
where
and
Fig. 1 a) Illustration of the load-penetration P-h curve; b) Trapezoidal loading program
used in a nanoindentation experiment; c) Indentation of material by a rigid axysimmetric
indenter
Vandamme [8] has developed a viscoelastic solution of the contact problem for
a conical indenter using several simple creep models. In view of the current
understanding of the structure of CSH – the main creeping phase in the cement
paste – as a porous medium [3,4], it is necessary to extend this procedure to viscoelastic two-phased porous materials.
2 Viscoelastic Solution of Contact Problem
Analysis of the indentation problem with a rigid indenter of an arbitrary
axisymmetric shape can be based on the so-called Galin-Sneddon solution [6]. The
relation between the penetration depth h and the corresponding load P is
parametrically described by
a
f ′( ρ )dρ
0
a −ρ
h = a∫
2
2
, P=2
E
1 −ν 2
a
ρ 2 f ′( ρ )dρ
0
a −ρ
∫
2
(2)
2
a is the radius of the projected contact area Ac (Fig. 1), ρ is the tip
radius, and f ( ρ ) is a smooth function describing the tip shape and implicitly
where
defined by Eq. (1). An ideal conical indenter is described by
Modeling of Nanoindentation by a Visco-elastic Porous Model with Application
f ( ρ ) = ρ / tan(α )
189
(3)
α is the semi-apex angle. In Eq. (1), this case would correspond to
C0 = π tan 2 α and Ci = 0, i = 1,2,... . For f ( ρ ) given by Eq. (3), the
where
improper integrals in Eq. (2) can be calculated analytically, and the following
formula can be derived for evaluation of the indentation modulus from the initial
slope of the unloading part of the indentation curve:
π
E
1 dP
=
2
1 −ν
2 dh
(4)
Ac
The foregoing elastic solution can be extended to the viscoelastic case by the
method of functional equations [2]. It consists of replacing elastic constants in the
contact problem solution by Laplace-Carson transform (LCT) of bulk and shear
relaxation functions. The second part of Eq. (2) can be rewritten in the form
P(t ) = M F (a(t )), F (a(t )) = 2 ∫
a
ρ 2 f ′( ρ )dρ
Here, function
(5)
a2 − ρ 2
0
F (a (t )) depends only on the tip geometry, whereas the material
properties are reflected by the indentation modulus M ≡ E/(1 −ν
elastic solution of the contact problem can then be written as
2
) . The visco-
Pˆ ( s ) = Mˆ ( s ) Fˆ (a ( s ))
(6)
•ˆ ( s ) denotes the LCT of • (t ) . The LCT of the indentation modulus can
be expressed in terms of the LCTs Kˆ ( s ) and Gˆ ( s ) of the bulk and shear
where
relaxation functions as
Mˆ ( s ) = 4Gˆ ( s )(3Kˆ ( s ) + Gˆ ( s )) /(3Kˆ ( s ) + 4Gˆ ( s ))
(7)
Using homogenization procedures in the Laplace-Carson domain, it is possible to
upscale the visco-elastic properties from the microstructure level to the level of the
indentation experiment. For the self-consistent homogenization scheme, the
effective moduli K hom and Ghom are obtained by solving the system of two
nonlinear equations
K hom = ∑ f r K r [1 +
α hom
Ghom = ∑ f r Gr [1 +
β hom
r
r
K hom
Ghom
( K r − K hom )]−1 / ∑ fω [1 +
ω
(Gr − Ghom )]−1 / ∑ f ω [1 +
ω
α hom
K hom
β hom
Ghom
( K ω − K hom )]−1
(Gω − Ghom )]−1
(8)
190
D. Davydov and M. Jirásek
in which f r are the volume fractions of individual phases,
bulk and shear moduli, and
α hom =
K r and Gr are their
3K hom
6 K hom + 2Ghom
, β hom =
3K hom + 4Ghom
5 3K hom + 4Ghom
A rearrangement of Eq. (6) yields an expression for the LCT of
Fˆ (a ( s )) = Pˆ ( s ) / Mˆ ( s ) ≡ Pˆ ( s )Yˆ ( s )
F (a (t )) ,
(9)
ˆ ( s ) = the LCT of the indentation
where we have denoted Yˆ ( s ) = 1/ M
compliance.
Exploiting the properties of the LCT, Eq.(9) can be written in the time domain
as
t
F ( a(t )) = ∫ Y (t − τ ) P& (τ )dτ
0
(10)
where the dot over P denotes the time derivative. To calculate the improper
integrals in Eq. (2), f ′( ρ ) is needed. Thus, Eq. (1) has to be solved for f ( ρ ) . For
a simple shape of the indenter (two constants) these integrals can be calculated
analytically [1], but for the case when the contact area of the tip is approximated
using three or more terms, a closed-form solution is complicated or even not
available. Numerical integration can then be used. Eq. (1) is solved for f ( ρ ) at
ρi ∈ [0, ρ max ] , i = 1,2,... N . Afterwards, f ( ρi ) is interpolated by piecewise
cubic Hermite polynomials. The Monte-Carlo integration method is used to
compute the integrals. The obtained values hi = h( ai ) and Fi = F ( ai ) is then
interpolated for later usage.
For very simple visco-elastic models, such as the Kelvin-Maxwell-Voight
model, the exact form of Y (t ) can be derived analytically [8]. However, for a
logarithmic creep model, even in its simplest form, numerical inversion has to be
done. Since the basic creeping phase CSH is believed to be porous, we consider a
two-phase polycrystal composite with one phase being viscoelastic and another
corresponding to the pores. From given viscoelastic model parameters and volume
fraction of pores, Kˆ hom ( s ) and Gˆ hom ( s ) for the homogenized medium can be
calculated by solving (8) in the Laplace-Carson domain. Substituting the result
ˆ ( s ) and thus Yˆ ( s ) = 1/ Mˆ ( s ) can be
into Eq. (7), the indentation modulus M
calculated. The Stehfest algorithm [7] is used for the inverse transform and
calculation of Y (t ) .
Indentation experiments are usually conducted under trapezoidal load control
(Fig. 1), described by
Modeling of Nanoindentation by a Visco-elastic Porous Model with Application
PL (t ) = t/τ L Pmax
⎧
⎪
P(t ) = ⎨
PH (t ) = Pmax
⎪ P (t ) = (τ + τ + τ − t ) P / τ
L
H
U
max
U
⎩ U
191
0 ≤ t ≤τL
(11)
τL ≤ t ≤τL +τH
τ L + τ H ≤ t ≤ τ L + τ H + τU
where τ L , τ H and τ U are the loading, holding and unloading durations,
respectively. Considering this load history in Eq. (10) leads to
F ( a (t )) =
Pmax
F ( a (t )) =
Pmax
τL
τL
t
∫ Y (t − τ )dτ
0
∫
0 ≤ t ≤τL
(12)
τL
0
Y (t − τ )dτ τ L ≤ t ≤ τ L + τ H
Knowing Y (t ) , the integrals in Eq. (12) are numerically calculated using an
adaptive Simpson quadrature. With F (t ) at hand, the previously found
interpolation of the values F ( ai ) and h( ai ) allows obtaining h(t ) . Thus, an
optimization procedure to fit the experimentally observed penetration depth
history can be invoked.
3 Example of Application and Conclusions
In order to check the proposed method, one and a half year old cement paste CEM
I 42.5 prepared at w/c = 0.4 has been tested. Indentation has been performed by
τ L = 10s ,
= 1000 μ N .
a Hysitron nanoindenter using a trapezoidal loading program with
τ H = 100s
and τ U = 10s and the maximum applied force Pmax
One of the obtained indentation curves has been considered. The indenter
geometry parameters have been determined as C0 = 24.5, C1 = 4793,
C2 = −82009, C3 = 30934, C4 = −236130 . The Poisson ratio has been set
to 0.24. From the Oliver-Pharr procedure, elastic modulus of 31.64 GPa has been
obtained. The porosity of CSH has been taken as 0.26 [3,4]. The deviatoric creep
has been described by the logarithmic compliance function
J (t − t0 ) = 1/ G0 + J log(1 + (t − t0 ) / τ )
(13)
with J = 0.0082 GPa and τ = 0.0287s .
Fig. 2 shows the best fit of function F (t ) and the corresponding fit of the
experimental penetration depth curve.
The considered example shows the applicability of the proposed method to
extraction of viscoelastic properties of porous media from the nanoindentation
experiment. This method could be useful in further studies of the CSH gel in
cement paste as well as of other porous media.
−1
192
D. Davydov and M. Jirásek
Fig. 2 a) Experimental F(t) curve and its fit in dimensionless form; b) Experimental
penetration depth h(t) and the corresponding fit by a porous logarithmic creep model
Acknowledgments. The authors are grateful to the European Community for the full
support of Denis Davydov under the Marie Curie Research Training Network MRTN-CT2005-019283 “Fundamental understanding of cementitious materials for improved
chemical, physical and aesthetic performance” and to the Ministry of Education, Youth and
Sports of the Czech Republic for the full support of Milan Jirásek under the Research Plan
MSM6840770003.
References
1. Jager, A., et al.: Identification of viscoelastic properties by means of nanoindentation
taking the real tip geometry into account. Meccanica 42, 293–306 (2007)
2. Lee, E., et al.: The contact problem for viscoelastic bodies. J. Appl. Mech. 82, 438–444
(1960)
3. Constantinides, G.: The nanogranular nature of C-S-H. J. Mech. Phys. Solids. 55, 64–90
(2007)
4. Jennings, H., et al.: A multi-technique investigation of the nanoporosity of cement
paste. Cem. Concr. Res. 37, 329–336 (2007)
5. Oliver, W., et al.: An improved technique for determining hardness and elastic modulus
using load and displacement sensing indentation experiments. J. Mater. Res. 7, 1564–
1582 (1992)
6. Sneddon, I.: The relation between load and penetration in the axisymmetric Boussinesq
problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3, 47–57 (1965)
7. Stehfest, H.: Numerical inversion of Laplace transforms. CACM 13(1), 438–444 (1970)
8. Vandamme, M., et al.: Viscoelastic solutions for conical indentation. Int. J. Sol.
Struct. 43, 3142–3165 (2006)