Keywords: lattice model, brittle elements, mesh size, meso-structure, size effect.

We study the effect of the discretization density of lattice models. Two basic cases are examined: (i) homogeneous lattices, where all elements share the same strength and (ii) lattices in which the properties are assigned to the elements according to their correspondence to three phases of concrete, namely matrix, aggregates, and the interfacial transitional zone. These dependencies are studied with both, notched and unnotched beams loaded in three point bending. We report the results for regular discretization and irregular networks obtained using Voronoi tessellation. This is done for two types of models: with and without rotational springs (normal and shear springs are always present). All the springs are ideally brittle, i.e. after reaching the strength criterion; they are irreversibly removed from the structure. The dependence of strength is compared to various size effect formulas, and we show that in the case of homogeneous lattices, the fineness of discretization of the specimens of the same size can mimic variations in the size of lattice models with the same discretization density. In the case of heterogeneity (ii), we report how both the peak force and apparent fracture energy depend on the mesh resolution both for notched and unnotched structure.

The results are obtained for the rigid-body-spring network. The fracture criteria adopted is represented by the Mohr-Coulomb surface with tension cut-off. In this paper, we study two different mechanical models that differ in how internal forces (between rigid bodies) are transmitted at the connections of adjacent facets. In the first model type, only normal and shear springs act. In the other model type, also rotational springs transferring local bending moments are added. However, only stresses in normal and shear springs contribute to the fracture criteria in both model types.

The unnotched bent beams are analyzed and the known linear normal stress distribution along the beam depth is compared to the step-wise stress profile approximation of the homogeneous lattice model. Using this comparison, we were able to deliver closed-form formulas for the mesh size dependence of the ultimate strength.

The analysis of unnotched beams is more complicated due to the fact that the stress distribution is highly nonlinear. However, we were able to deliver some formulas for the strength prediction for the homogeneous lattice model.

In the case of heterogeneity, we report that even though the peak force dependence is influenced by the mesh resolution, and almost disappears in notched structures, a strong dependence on the apparent fracture energy remains. This is important for fracture studies with lattice models.

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