Issue 39

J. Eliáš, Frattura ed Integrità Strutturale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01 2 In some applications of fracture simulations, it might be important to consider additional material randomness (besides the one covered by the random location of nodes in the discrete model) usually represented by a random field [15-18]. An extension of the discrete model by fluctuation of material parameters according to a random field was developed in [18,19]. In this contribution, the adaptive concept is extended for such probabilistic discrete fracture models. P ROBABILISTIC DISCRETE MODEL he model uses random geometry to avoid directional bias that occurs in any regular structure. Domain of the modeled body is filled with nuclei with randomly generated positions. These nuclei are added sequentially with restricted minimal distance l min . The parameter l min controls size of the discrete bodies and therefore it should correspond to the size of heterogeneities in the material. In concrete, this is typically a size of the mineral aggregates. Each of the nuclei will serve as one model node with associated six degrees of freedom, three translational and three rotational. The connectivity of the nodes is given by Delaunay triangulation. Dual diagram of Delaunay triangulation called Voronoi tessellation then creates geometry of the rigid bodies. Rigid bodies have common contact facets, which are perpendicular to their connections because of the Voronoi tessellation properties. There is a complex damage-mechanics based constitutive law used at the facets. Its deterministic version has been adapted from [9], where it is also described in detail. The main material parameters for fracture behavior are tensile strength, f t , and tensile fracture energy, G F . The probabilistic extension of the model is elucidated in [19,20]. Here, only brief description of the probabilistic part is given. Both the tensile strength and fracture energy in tension are assumed to be governed by single random field H with mean value 1 and probabilistic distribution with Gaussian core and Weibull left tail. The correlation structure of the random field is given by square exponential function with single parameter, l ρ , called the correlation length. The strength and fracture energy of every model contact with centroid c are given by     f f H t t  c c (1)     G G H 2 F F      c c with X being the mean value of the material parameter X . The square in the equation for fracture energy is added to preserve constant material characteristic length [20]. In the adaptive model, new contacts are created after every refinement. Therefore, the random field values at the new contact centers must be generated after every refinement. This is effectively done using kriging. Initially, standard Gaussian random field realizations ( H ˆ ) are generated on points arranged in a regular grid with spacing l ρ /4. Random field value at point c is then estimated using the optimal linear estimation method [21]   K T k k cg k k H 1 ˆ      c ψ C (2) and finally standard Gaussian field is transformed onto the Weibull-Gauss field ( H H ˆ  ) using isoprobabilistic transformation. Vector ξ collects realizations of K independent standard Gaussian variables, λ and ψ are K eigenvalues and eigenvectors of the grid covariance matrix and cg C is the covariance vector between the grid points and point c . A DAPTIVITY nly brief description of the adaptive concept in deterministic model is given. Deep elucidation is provided in [5]. The refinement criterion is intuitive. It is based on an average stress in the rigid bodies calculated using the fabric stress tensor. For rigid body associated with node i, the average stress components st  are T O