13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- continuum cell, e.g. [9, 10]. In both cases explicit relations between local and continuum parameters can be established for regular lattices [11], but the only isotropic material that can be represented in 3D is a material with zero Poisson’s ratio. A bi-regular lattice that can represent all materials of practical interest has been proposed recently [12]. This lattice, currently formed by beams clamped at sites, is used in the current work together with microstructure data for concrete obtained with X-ray computed tomography. Failure models based on microstructure data and the new lattice have been previously used for modelling tensile and compressive behaviour of cement [13] and the compressive behaviour of concrete under various complex loading conditions [14]. This work makes a step into developing our understanding of the micro-crack population and its relation to macroscopic damage. Most of the work relating micro-crack populations to elastic moduli follows the fundamental paper [15], where analytical statistical derivation of the relation was provided. We follow the interpretation given in [16], in which the damage is measured as a relative change of the elastic modulus and related to micro-crack population via ( ) ( ) ∑ ≥ = − = 1 3 0 1 T c c N c E N E D β , (1) where c is some measure of micro-crack size, N(c) is the number of micro-cracks of size c, NT is the total number of sites capable of nucleating micro-cracks, and β is a scaling parameter reported as 0.47π for cracks in a 2D medium. Eq. (1) is our point of comparison for the simulations performed with the lattice model for various tensile loading cases. In the current work we are interested in testing the range of applicability of Eq. (1) and understanding the reasons for deviation from this rule, should such occur, by explicitly analysing the micro-crack population growth. 2. Model and method 2.1. The site-bond model The lattice model used in this work is illustrated in Fig. 1. The unit cell, shown in Fig. 1(a) is a truncated octahedron – a solid with six square and eight regular hexagonal boundaries. The 3D space can be compactly tessellated using such cells, with each cell representing a material meso-scale feature, e.g. grain, in an average sense. This representation is supported by physical and statistical arguments [12]. A discrete lattice is formed by placing sites at the centres of the cells and connecting each site to its 14 nearest neighbours; example is shown in Fig. 4(b). The lattice contains two types of bonds. Bonds denoted by B1 are normal to square boundaries and form orthogonal set. For convenience this set is coincident with the global coordinate system and B1 are referred to as principal bonds. Bonds denoted by B2 are normal to hexagonal boundaries. The hexagons lie on the octahedral planes with respect to the selected system, hence B2 are referred to as octahedral bonds. Figure 1. Lattice illustration: (a) Unit cell showing the site with 14 coordinating bonds: six principal, B1, and eight octahedral, B2; (b) Discrete lattice of beam elements.
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