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The paper studies systems of rigid bodies with randomly generated geometry interconnected by normal and tangential bonds. The stiffness of these bonds determines themacroscopic elastic modulus while the macroscopic Poisson's ratio of the system is determined solely by the normal/tangential stiffness ratio. Discrete models with no directional bias have the same probability of element orientation for any direction and therefore the same mechanical properties in a statistical sense at any point and direction. However, the layers of elements in the vicinity of the boundary exhibit biased orientation, preferring elements parallel with the boundary. As a consequence, when strain occurs in this direction, the boundary layer becomes stiffer than the interior for the normal/tangential stiffness ratio larger than one, and vice versa. Nonlinear constitutive laws are typically such that the straining of an element in shear results in higher strength and ductility than straining in tension. Since the boundary layer tends, due to the bias in the elemental orientation, to involve more tension than shear at the contacts, it also becomes weaker and less ductile. The paper documents these observations and compares them to the results of theoretical analysis. [ABSTRACT FROM AUTHOR]

Static discrete models are advantageously used for the simulation of fracture in quasibrittle heterogeneous materials. In order to correctly capture strain localization during the fracture process, it is often necessary to represent material heterogeneity in the model directly via its discrete geometry. Depending on the specimen size and the size of the heterogeneities, these simulations are typically extremely computationally demanding. The contribution aims to reduce this computational cost via the implementation of adaptivity in the construction of the discrete model geometry. The simulation starts with coarse discretization, which provides correct elastic behavior and is then adaptively refined during the simulation in regions that suffer high stresses that induce cracking and strain localization. The technique is applied in deterministic and probabilistic simulations and demonstrated on several examples. [ABSTRACT FROM AUTHOR]

• Mesoscale discrete model of concrete enhanced by spatially random material parameters. • Notched and unnotched three point bending and uniaxial tension are analyzed. • Fracture process zone statistically evaluated for various random field parameters. • Nontrivial dependence of peak loads on correlation length and variance is reported. • Local averaging and weakest-link are identified as the underlying mechanisms. Numerical simulations of concrete fracture performed with a probabilistic mesoscale discrete model are presented. The model represents a substantial part of material randomness by assigning random locations to the largest aggregates. The remaining part of randomness is introduced by causing material parameters to fluctuate randomly via a homogeneous random field. An extensive numerical study performed with the model considers prisms loaded in uniaxial tension with both fixed and rotating platens, and also beams with and without a notch loaded in three point bending. The results show the nontrivial effect of (i) autocorrelation length and (ii) variance of the random field on the fracture behavior of the model. Statistics of the peak load are presented as well as the size and shape of the fracture process zone at the moment when the maximum load is attained. Local averaging within the fracture process zone and weakest-link are identified as underlying mechanisms explaining the reported results. The companion paper, Part II (Vořechovský and Eliáš, 2020), introduces an analytical model capable of predicting the distribution of the peak load obtained with the probabilistic discrete model via the simple estimation of extremes of a random field obtained as moving average of local strength. [ABSTRACT FROM AUTHOR]

• Strength prediction of quasibrittle material that removes problems of Weibull theory. • Statistics of peak load are obtained analytically based on extremes of local averages. • Local averaging of Gaussian random field provides effective strength of material. • The weakest-link mechanics is incorporated via extremes of effective strength field. • Four model parameters are used to represent a wide range of random field parameters. The paper delivers an analytical model for prediction of the peak force in concrete specimens loaded in bending (both notched and unnotched). The model is capable of predicting peak force statistics by computing the extreme values of sliding averages of random strength fields. The local strength of the specimen is modeled by a stationary isotropic random field with Gaussian distribution and a given autocorrelation function. The averaging operation represents the progressive loss in material integrity and the associated stress redistribution that takes place prior to reaching the peak load. Once the (linear) averaging process is performed analytically, the resulting random field of averaged strength is assumed to represent a series of representative volume elements (RVEs) and the global strength is found by solving for the minimum of such an effective strength field. All these operations can be written analytically and there are only four model parameters: the three dimensions of the averaging volume (RVE) and the length of the final weakest-link chain. The model is verified using detailed numerical computations of notched and unnotched concrete beams simulated by mesoscale discrete simulations of concrete fracture performed with probabilistic distributions of model parameters. These are presented in the companion paper Part I (Eliáš and Vořechovský, 2020). The numerical model used for verification represents material randomness both by assigning random locations to the largest aggregates and by simulating random fluctuations of material parameters via a homogeneous random field. [ABSTRACT FROM AUTHOR]