Numerical and analytical effective elastic properties of degraded cement pastes B. Bary a, ⁎, M. Ben Haha a , E. Adam b , P. Montarnal c a CEA, DEN, DPC, SCCME, Laboratoire d'Etude du Comportement des Bétons et des Argiles, F-91191 Gif-sur-Yvette, France b CEA, DEN, DM2S, SFME, Laboratoire de Génie Logiciel et de Simulation, F-91191 Gif-sur-Yvette, France c CEA, DEN, DM2S, SFME, Laboratoire de Simulation des Ecoulements et du Transport, F-91191 Gif-sur-Yvette, France abstract article info Article history: Received 23 January 2009 Accepted 11 June 2009 Keywords: Elastic moduli Numerical simulations Cement paste Homogenization Degradation Cement pastes are heterogeneous materials composed of hydrates, anhydrous products and pores of various shapes. They are generally characterized by a high particle concentration and phase contrasts, in particular in the case of degraded materials which exhibit important porosity. This paper compares the performance of several classical effective medium approximation schemes (Mori–Tanaka, Zheng-Du, self-consistent) through their ability to estimate the mechanical parameters of cement paste samples obtained numerically. For this purpose, finite element simulations are performed on 3D structures to compute for each sample accurate values of these mechanical properties. For these simulations, the cement paste is considered as a matrix of C–S–H in which are embedded inclusions of anhydrous, hydration products, and pores. In order to evaluate the importance of the particle shape, two types of samples are generated, one with only spherical inclusions and the other containing both spherical and prismatic particles. Simulations with three perpendicular loading directions and both uniform and mixed boundary conditions are performed in order to verify that the dispersion in the computed elastic moduli is low, or equivalently that the generated structures are close to representative volume elements (RVEs). It is shown that the considered effective medium approximation schemes, except the self-consistent one, provide relatively good estimations of the overall mechanical parameters when compared to simulation results, including when both particle volume fraction and phase contrast are high. The analytical methods taking into account the particle shapes also give estimates close to the corresponding numerical simulations, the latter confirming the influence of the particle form. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction The macroscopic properties of the composite materials depend mainly on the properties of the constitutive phases, their volume fraction, microgeometry and relative arrangement. The effects of the concentration and the shapes of inclusions on the physical properties of various compounds attracted the attention of researchers in many fields, in particular in materials science and mechanics. Analytical evaluations of the effective elastic properties of the composite can be obtained by several well-known methods. After the earliest work of Eshelby [1] on the stress field of an isolated ellipsoidal inclusion in an elastic infinite matrix, various schemes were developed to calculate the elastic properties of the composites of the matrix-inclusion type, as in the method of Mori–Tanaka (MT) [2,3] and self-consistent (SC) scheme [4–6]. They were based on the approximation of the mean- field, which supposes that the fields of stresses and strains in the matrix and in the inclusions are adequately represented by their volume fraction average, and differed from the way that they consider the elastic interaction between inclusions. Owing to the linearity condition, these average stresses or strains are directly related via fourth-order localization tensors to the macroscopic homogeneous stresses or strains. Obviously, these schemes converge when the volume fraction of particles or the contrast between the properties of the two phases is small, but the differences among them (and between upper and lower Hashin-Shtrikman bounds, [7]) may be very large otherwise. Their precision and extent of applicability may be established through analytical solutions (which do not exist in general for real materials due to their complex microgeometry) or by solving numerically the boundary value problem on a sample of the microstructure corresponding to a representative volume element of the composite. The characterization and definition of the RVE is essential in these numerical techniques. Indeed, the RVE has to contain a sufficiently important number of particles and heterogene- ities to resemble cementitious materials, but must remain small enough to be considered as a volume element of continuum mechanics. Generally, the size of the RVE depends on the sought physical property, the contrast of properties between the different components, the volume fraction of the phases, and the number of microstructure realizations and the precision of the wanted estima- tion of macroscopic properties [8,9]. Another alternative method for obtaining numerical estimations of these elastic properties consists to Cement and Concrete Research 39 (2009) 902–912 ⁎ Corresponding author. E-mail address: benoit.bary@cea.fr (B. Bary). 0008-8846/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cemconres.2009.06.012 Contents lists available at ScienceDirect Cement and Concrete Research journal homepage: http://ees.elsevier.com/CEMCON/default.asp