13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- If the spacing between sites in the principal directions is denoted by L, bonds B1 have length L1 = L, and bonds B2 have length L2 = √3 L / 2. Presently, the bonds are represented by structural beam elements of circular cross sections, with R1 and R2 denoting the radii of beams B1 and B2, respectively. The beams are clamped at the lattice sites. The two types of beams have identical modulus of elasticity, Eb, and Poisson’s ratio, νb. With this setup, it has been previously shown that by calibrating four parameters: R1 / L, R2 / L, Eb, and νb, the lattice can produce a large class of isotropic elastic materials with Poisson’s ratios of practical interest [12]. The reference material in this work is a concrete with E = 46 GPa and ν = 0.27, for which the calibration, assuming isotropic elasticity, yields R1 / L = 0.2; R2 / L = 0.32; Eb = 90 GPa; and νb = 0.4 [14]. The commercial software Abaqus [17] with Euler-Bernoulli beam formulation has been used for the calibration and the analyses reported in this work. The behaviour of the beams is linear elastic. 2.2. Pore distribution and failure criterion Microstructure data for the reference material was obtained using X-ray Computed Tomography as reported in [14]. The pore size distribution was obtained by segmentation of reconstructed 3D images. The studied regions of interest had dimensions of 1700 x 1200 x 1200 voxels with a voxel size of ca. 15 μm, allowing for a minimum detectable pore radius of ca. 15 μm. The number of pores measured experimentally was n ≈ 41500. The measured pore radii, ci, were used to construct a cumulative probability distribution (CPD) with standard median ranking, where for pore radii ordered as c1 ≤ c2 ≤…≤ cn, the cumulative probability for pores with radii less than ci is given by F(c < ci) = (i - 0.3) / (n + 0.4). The CPD for the reference material is shown in Fig. 2(a), where the minimum and maximum pore radii are also depicted. The CPD is used to assign pore sizes to the lattice bonds. For each bond a uniformly distributed random number 0 ≤ r < 1 is generated and the assigned pore radius is calculated from c = F-1(r). This ensures that the distribution of pore sizes in the model comes from the same population as in the experiment. A fragment of the model with distributed pores is given in Fig. 2(b). The cell size, L, is calculated such that the volume of all distributed pores divided by the volume of the cellular structure equals the material porosity, which is ca. 5% for the reference material. The pore sizes shown are to scale with the sketched cellular structure. With respect to the cellular structure pores reside at cell boundaries, i.e. interfaces between grains. The lattice bonds are also depicted (diameters not to scale) in order to show that pores reside at bond centres. Figure 2. Pore distribution: (a) Cumulative probability of pore radii in the concrete; (b) Segment of model illustrating pores distributed to cell boundaries and corresponding. Pore sizes are to scale with the cell size. Damage in the lattice model is introduced by removal of bonds. Propensity for bond failure is measured by the parameter f f f f M M T T S S N N Π= + + + , (2)
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