Keywords: FFT, homogenization, cement paste, concrete, elastic properties, hydration.

This paper shows how the simulation of elastic properties of hydrating concrete may be accomplished via the combination of a hydration model and various homogenization techniques. Under certain assumptions of length separation, the framework of homogenization allows independent analysis of the composite levels found in the concrete. Therefore, it is convenient to disassemble concrete into simpler components, where independent analysis on mutually different scales may be performed:

• C-S-H level typically spans the characteristic length between 10 nm - 1 μm,
• cement paste level is found at the scale of 1 μm - 100 μm. Clinker minerals, gypsum, portlandite, homogenized C-S-H and some capillary porosity is present,
• mortar level is considered at the scale between 1 mm and 1 cm. It contains homogenized cement paste, fine aggregates such as sand and associated interfacial transition zone (ITZ). Air voids may be found too,
• concrete level spans the characteristic length of 1 cm - 1 dm. Mortar, coarse aggregates such as gravel and associated ITZ are typically found.

The most important level is that of cement paste, where the hydration and accompanying reactions change the disconnected, heterogeneous material to a solid one. This situation is complicated for an analysis since the material changes its porosity, connectedness and chemical phases during hydration.

The microstructure of the cement paste will be reconstructed in the periodic discrete form, using the hydration model CEMHYD3D [1]. The resolution of the model is 1 μm and the basic chemical reactions of Portland cement are implemented. The reasonable size of representative volume element (RVE) is 50x50x50 μm for the simulation of elastic properties.

Moulinec and Suquet [2] introduced a new method of homogenization, based on a fast Fourier transformation (FFT). The strain field is decomposed into the behaviour of the reference medium and the polarization strain within a heterogeneous composite. The system of equations leads to the convolution that might be treated efficiently via FFT. The time of one iteration grows as , where is the number of elements. Another advantage of this method is no improvement of results under sampling refinement, as opposed to finite element mesh refinement.

To validate the multiscale approach, cement paste is homogenized via a FFT at discrete time points. When air is considered as entrained, i.e. remains on the level of cement paste, the Mori-Tanaka scheme [3] homogenizes the cement paste with the air, followed by a Hervé-Zaoui scheme [4] on the mortar level. This level contains fine aggregates, associated ITZ and cement paste with entrained air. The concrete level is homogenized via the same scheme and contains coarse aggregates, associated ITZ, and homogenized mortar level. The elastic properties of ITZ on fine or coarse aggregates were derived from the cement paste by a reduction of E-modulus by 50 %. Poisson's ratio remained unchanged.

Validation on the concrete with the water-to-cement ratios of 0.27 and 0.5 shows good agreement with experimental results on a macroscale. Moreover, elastic properties at early ages of concrete may be treated as well. The multiscale approach brings a new insight into behaviour of characteristic levels found in ordinary concrete, with a minimum of unknown variables needed for homogenization.

1

D.P. Bentz, "CEMHYD3D, A Three-Dimensional Cement Hydration and Microstructure Development Modeling Package. Version 3.0", Building and Fire Research Laboratory Gaithersburg, Maryland 20899, 2005.

2

H. Moulinec and P. Suquet, "A fast numerical method for computing the linear and nonlinear mechanical properties of composites", Comptes Rendus de l'Academie des Sciences Serie II, 318(11), 1417-1423, 1994.

3

T. Mori, K. Tanaka, "Average stress in matrix and average elastic energy of materials with misfitting inclusions", Acta Metallurgica, 21(5), 1605-1609, 1973. doi:10.1016/0001-6160(73)90064-3

4

E. Hervé, A. Zaoui, "N-layered Inclusion-based Micromechanical Modelling", Int. J. Engng Sci., 31, 1-10, 1993. doi:10.1016/0020-7225(93)90059-4

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