Issue 39

J. Eliáš, Frattura ed Integrità Strutturale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01 3     j j st s t j F c V 1    (3) where j runs over all nodes in contact with node i , F is a vector of contact force, c is the centroid of the contact facet and V is a volume of the i -th rigid body. The Mazar's equivalent stress serves as measure of the stress level, σ eq . Figure 1 : Adaptive refinement of discretization in steps; a) schematic explanation; b)-g) application to a 2D model. In probabilistic model, the contacts have random strength. Assuming that the random field does not change too much within one discrete body of the model ( l ρ < l min ), it is reasonable to estimate strength of hypothetical newly created contacts within the i -th coarse discrete body by strength at node i at coordinates i x . After every solution step, average stress tensors in all rigid bodies belonging to coarse discretization are evaluated and the Mazar‘s equivalent stresses are calculated. The refinement takes place whenever the equivalent stress exceeds chosen strength level γ   i f eq t    x (4) The node associated with the rigid body satisfying Eq. (4) serves as a center of the refinement sphere. The safe value of parameter γ was determined as 0.7, i.e. whenever equivalent stress reaches 70% of the tensile strength, the refinement takes place. The refinement is sketched in Fig. 1. All the nuclei inside the refinement sphere that does not belong to the fine discretization are removed. New nuclei are added into the refinement sphere according to the sequential algorithm described in the previous section. The parameter l min controlling the discretization density changes based on two additional length parameters, r f and r c . The linear transition from coarse ( l min =l c ) to fine ( l min =l f ) discretization is included within the circular ring of outer (inner) radius r c ( r f ) in order to minimize the shape distortion of the bodies. If the transitional regime is omitted, the sharp change in discretization density would produce significantly elongated body shapes inducing directional bias and anisotropy.