Continuum Microviscoelasticity Model for
Cementitious Materials: Upscaling Technique
and First Experimental Validation
S. Scheiner and C. Hellmich1
Abstract. We propose a micromechanics model for aging basic creep of earlyage concrete. Therefore, we formulate viscoelastic boundary value problems on
two representative volume elements (RVEs), one related to cement paste (composed of cement, water, hydrates, air), and one related to concrete (composed of
cement paste and aggregates). Homogenization of the non-aging elastic and viscoelastic properties of the material’s constituents involves the transformation of
the aforementioned viscoelastic boundary value problems to the Laplace-Carson
(LC) domain. There, formally elastic, classical self-consistent and Mori-Tanaka
solutions are employed, leading to pointwisely defined LC-transformed tensorial
creep and relaxation functions. Subsequently, the latter are backtransformed, by
means of the Gaver-Wynn-Rho algorithm, into the time domain. Temporal derivatives of corresponding homogenized creep tensors, evaluated for the current
maturation state of the material and for the current time period since loading of
the hydrating composite material, allow for micromechanical prediction of the aging basic creep properties of early-age concrete.
1 Fundamentals of Continuum Micromechanics
In continuum micromechanics [10, 16], a material is understood as a microheterogeneous body filling a macrohomogeneous representative volume element (RVE)
with characteristic length l II , l II >> d II , d II standing for the characteristic length
of inhomogeneities within the RVE, see Figure 1. These inhomogeneities are referred to as material phases, each exhibiting a homogeneous microstructure. The
homogenized mechanical behavior of the material on the observation scale of the
S. Scheiner and C. Hellmich
Institute for Mechanics of Materials and Structures, Vienna University of Technology,
Vienna, Austria
e-mail: [email protected],
[email protected]
www.imws.tuwien.ac.at
90
S. Scheiner and C. Hellmich
RVE, i.e. the relation between homogeneous deformations acting on the boundary
of the RVE and resulting macroscopic (average) stresses, can then be estimated
from the mechanical behavior of the material phases, their dosages within the
RVE, their characteristic shapes, and their interactions. If a single material phase
possesses a heterogeneous microstructure itself, its mechanical behavior can be estimated by introduction of RVEs within this phase, with characteristic lengths l I ,
l I ≤ d II , comprising again inhomogeneities with characteristic length d I << l I ,
and so on, see Fig. 1. Such an approach is referred to as multistep homogenization
and should, in the end, provide access to “universal” phase properties at sufficiently low observation scales.
Fig. 1 Separation of scales for multistep homogenization by means of continuum micromechanics, dI << lI ≤ dII << lII [8]
2 Continuum Microviscoelasticity Model for Concrete
Explanation of the mechanical behavior of concrete by means of continuum micromechanics requires resolution of the material microstructure down to the observation scale of at most 1 μm, where calcium silicate hydrates (CSH), reaction
products of cement grains and water, can be discerned [11]. Thereby, either distinction between high-density and low-density CSH [15] as well as portlandite and
aluminate is made [5], or, as implemented subsequently, hydration products are introduced as one material phase [9]. Consequently, at the observation scale of
cement paste, a typical RVE comprises the phases “hydration products”, “unhydrated cement grains”, “liquid (capillary) pores”, and “air pores (voids)” [5, 9].
Furthermore, at the observation scale of concrete, a typical RVE comprises the
phases “cement paste” and “aggregates” [5, 9]. Thus, a two-step homogenization
strategy is pursued. In the first homogenization step the mechanical behavior of
cement paste is determined under the assumption that, due to the disorder of the
hydrate phase, cement paste is reasonably represented as a polycrystal [5, 9], i.e.
no dominant material phase can be identified within the RVE of cement paste.
Thereby, CSH, unhydrated cement grains, water, and air pores are represented as
isotropic spherical inclusions. In the line of Laws and McLaughlin [13], the homogenized mechanical properties of cement paste are obtained by means of the
Continuum Microviscoelasticity Model for Cementitious Materials
91
viscoelastic correspondence principle. In detail, the transformation of the viscoelastic constitutive law describing the mechanical behavior of single constituents
from the time domain into the Laplace-Carson (LC) domain yields a mathematical
structure which is formally identical to the (corresponding) purely elastic case.
Thus, according to the correspondence principle, the LC-transformed material
properties of cement paste can be homogenized analogously to purely elastic
properties [5,9]. In order to finally obtain material properties in the time domain,
the LC-transformed homogenized material properties must be back-transformed.
This is carried out numerically by means of the Gaver-Wynn-Rho (GWR) algorithm [1]. In the second homogenization step, in turn, isotropic spherical aggregate
inclusions are embedded in the polycrystalline cement paste matrix. Again, the
LC-transformed material properties of concrete must be numerically backtransformed into the time domain. Furthermore, the microviscoelasticity theory (for details see [14]) considers the experimentally verified hypothesis that CSH is the
only viscoelastic constituent of concrete (the remaining constituents are purely
elastic) [2], described by the well-known Burgers model [6], whereas this phenomenon can be considered as purely deviatoric [4].
3 Model Validation
The microviscoelasticity model is experimentally validated on the basis of the
creep tests of Laplante [12] and of Atrushi [3], both carried out on sealed concrete
specimens. Laplante subjected concrete at the ages of 20 hours, 27 hours, 3 days, 7
days, and 28 days to uniaxial compressive loading. Thus, since at these ages
Fig. 2 Model-predicted aging creep functions compared to corresponding experimentally
2
obtained ones [12], r mean=91%
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S. Scheiner and C. Hellmich
Fig. 3 Model-predicted aging creep function rates compared to corresponding experimen2
tally obtained ones [3], r mean=96%
concrete still undergoes hydration, related aging of concrete, leading to a continuously changing microstructure, must be considered. Evaluating the microviscoelasticity model in terms of the aging creep function allows for comparison of
model predictions with corresponding experimental values, see Figure 2. The
satisfying agreement of model-predicted with experimentally obtained creep func2
tions at different age ( rmean
= 91%) allows for concluding that the presented microviscoelasticity model is capable to predict the aging viscoelastic behavior of
cementitious materials. This is further corroborated by comparison of the modelpredicted creep function rates with corresponding experimental results of Atrushi
[3], who subjected concrete to uniaxial compressive loading at the ages of 1 day, 2
2
days, 3 days, 4 days, 6 days, and 8 days ( rmean
= 96%), see Figure 3.
Acknowledgments. Financial support by “TUNCONSTRUCT – Technology Innovation
in Underground Construction” (IP011817-2), sponsored by the European Commission, is
gratefully acknowledged.
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