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This paper presents a stochastic analysis of the size effect on the nominal strength of quasibrittle structures with a large preexisting notch. This type of scaling behavior has been extensively studied within a deterministic framework. Little attention is paid to stochastic analysis, which is essential for reliability-based analysis and design of engineering structures. Stochastic finite element simulations are performed to study the failure of geometrically similar beams of different sizes. The numerical analysis uses a continuum damage model, in which the tensile strength and fracture energy are modeled by homogeneous random fields. Different correlation lengths are considered as a parametric study. The analysis yields the size effects on the mean and coefficient of variation (CoV) of the nominal structural strength. It is shown that the size effect on the mean strength agrees well with the Bazant size effect model. The simulation predicts a strong size effect on the CoV of the nominal strength. Small-, intermediate-, and large-size asymptotes are derived analytically for the scaling behavior of the strength CoV. Based on these asymptotes, an approximate scaling equation is proposed for the CoV of nominal strength. The effect of the correlation length on the simulated failure behavior is discussed. [ABSTRACT FROM AUTHOR]

• Elastic behavior of isotropic discrete models with an arbitrary geometry is analyzed. • An analytical estimation of the macroscopic elastic characteristics is derived. • It is shown that the widest limits of the Poisson's ratio are obtained for Voronoi and Power tessellation. • The Poisson's ratio limits cannot be further extended by manipulations with the model geometry. The use of discrete material representation in numerical models is advantageous due to the straightforward way it takes into account material heterogeneity and randomness, and the discrete and orientated nature of cracks. Unfortunately, it also restricts the macroscopic Poisson's ratio and therefore narrows its applicability. The paper studies the Poisson's ratio of a discrete model analytically. It derives theoretical limits for cases where the geometry of the model is completely arbitrary, but isotropic in the statistical sense. It is shown that the widest limits are obtained for models where normal directions of contacts between discrete units are parallel with the vectors connecting these units. Any deviation from parallelism causes the Poisson's ratio limits to shrink. A comparison of the derived equations to the results of the actual numerical model is presented. It shows relatively large deviations from the theory because the fundamental assumptions behind the theoretical derivations are largely violated in systems with complex geometry. The real shrinking of the Poisson's ratio limit is less severe compared to that which is theoretically derived. [ABSTRACT FROM AUTHOR]

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]

The Audze–Eglājs (AE) criterion was developed to achieve a uniform distribution of experimental points in a hypercube. However, the paper shows that the AE criterion provides strongly nonuniform designs due to the effect of the boundaries of the hypercube. We propose a simple remedy that lies in the assumption of periodic boundary conditions. The biased behavior of the original AE criterion and excellent performance of the modified criterion are demonstrated using simple numerical examples focused on (i) the uniformity of sampling density over the design space and, (ii) statistical sampling efficiency measured through the ability to correctly estimate the statistical parameters of functions of random variables. An engineering example of reliability calculation is presented, too. [ABSTRACT FROM AUTHOR]

The paper presents a discrete meso-scale model for fracture of concrete taking into account random spatial variability of material parameters. Beams of various sizes, with notches of various depths, are simulated numerically to study the combination of energetic and statistical size effects. A substantial part of material randomness is shown to be caused by random locations of the largest aggregates. Further randomness, due to random fluctuations of material parameters, is considered and an effect of introducing a spatially auto-correlated random field is analyzed. The results of the simulations are compared with recently published experimental data on concrete beams in three-point bending. The differences in the role of randomness in beams of various sizes, with different notch depths, are demonstrated, and differences in energy dissipation are discussed. [ABSTRACT FROM AUTHOR]

Discrete models of solids have been motivated, in large part, by the discontinuous and heterogeneous nature of material structure and its breakdown under loading. The capabilities of discrete models have evolved over the past several decades, offering novel means for investigating material structure–property relationships. However, lack of understanding of both the utilities and disadvantages of discrete models limits their further development and applications. This paper reviews relevant features of discrete approaches applied to modeling the mechanical behavior of geomaterials, concrete materials in particular. The discrete models are classified according to their form and abilities to represent elastic and fracture behaviors in the presence of large-scale material heterogeneity. Discretization of the material domain plays a large role in this respect. Emphasis is placed on particle-based lattice models. The relative merits of various strategies for introducing reinforcing components, which are essential for many applications, are outlined. Recent advances are highlighted, including the use of discrete models for coupled, multi-field analysis. The merits of discrete approaches are summarized in the conclusions. • Discrete methods for simulating concrete fracture are critiqued. • Model capabilities depend on the choice of physical or non-physical forms of discretization. • The topics of stress oscillation, Poisson's ratio and volumetric effects are examined. • Dual-lattice models are suited to modeling hygro-mechanical couplings in cracked media. • Application examples demonstrate the utilities and expanding use of discrete methods. [ABSTRACT FROM AUTHOR]

• numerical integration with designs optimized via miniMax (mM) criterion is developed. • mM designs with standard metric are nonuniform which may cause biased integration. • the proposed periodic metric ensures point uniformity and unbiased robust integration. • efficient construction algorithm for mM designs (standard or periodic) is developed. The selection of points for numerical integration of the Monte Carlo type, largely used in analysis of engineering problems, is developed. It is achieved by modification of the metric in the minimax optimality criterion. The standard minimax criterion ensures the design exhibits good space-filling property and therefore reduces the variance of the estimator of the integral. We, however, show that the points are not selected with the same probability over the space of sampling probabilities: some regions are over- or under-sampled when designs are generated repetitively. This violation of statistical uniformity may lead to systematically biased integral estimators. We propose that periodic metric be considered for calculation of the minimax distance. Such periodic minimax criterion provides statistically uniform designs leading to unbiased integration results and also low estimator variance due to retained space-filling property. These conclusions are demonstrated by examples integrating analytical functions. The designs are constructed by two different algorithms: (i) a new time-stepping algorithm resembling a damped system of attracted particles developed here, and (ii) the heuristic swapping of coordinates. The designs constructed by the time-stepping algorithm are attached to the paper as a supplementary material. The computer code for construction of the designs is attached too. [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]

Anchored by the theory of extreme value statistics, Weibull distribution is the most widely used mathematical model for strength distribution of brittle structures. In a series of recent studies, a finite weakest-link model was developed for strength distribution of quasi-brittle structures, and the classical Weibull distribution was shown to represent the large-size asymptote of the model. By employing a length scale, the finite weakest-link model is capable of capturing correctly the size effects on both the probability distribution and the mean value of structural strength. However, the connection of this length scale with the basic material properties is still missing. This study investigates the relationship between the length scale of the finite weakest-link model and the material length scales by analyzing the size effect on the mean structural strength. The mathematical form of this relationship is derived through dimensional analysis. To validate the model, a set of mean size effect curves is obtained through stochastic simulations, which use a nonlinear constitutive model involving both the Irwin characteristic length and the crack band width. The internal length scale of the weakest-link model is determined by optimum fitting of the benchmark size effect curves in the small-size range. Furthermore, the effect of stress field on this internal length scale is studied by considering three different loading configurations. The present analysis reveals the importance of the mean size effect analysis for the calibration of finite weakest-link model. [ABSTRACT FROM AUTHOR]

• Uniform distribution of points in a unit hypercube. • Optimal Design of Experiments for computationally intensive models. • The analogy to a dynamical system of repulsive particles in periodic space is exploited. A method is proposed for the construction of uniformly distributed point sets within a design domain using an analogy to a dynamical system of interacting particles. The possibility of viewing various distance-based optimality criteria as formulas representing the potential energy of a system of charged particles is discussed. The potential energy is employed in deriving the equations of motion of the particles. The particles are either attracted or repelled and dissipative dynamical systems can be simulated to achieve optimal and near-optimal arrangements of points. The design domain is set up as an N var -dimensional unit hypercube, with N var being the number of variables (factors). The number of points is equal to the number of simulations (levels). The periodicity assumption of the design domain is shown to be an elegant way to obtain statistically uniform coverage of the design domain. The ϕ p criterion, which is a generalization of the Maximin criterion, is selected in order to demonstrate its analogy with an N-body system. This criterion guarantees that the points are spread uniformly within the design domain. The solution to such an N-body system is presented. The obtained designs are shown to outperform the existing optimal designs in various types of applications: multidimensional numerical integration, statistical exploration of computer models, reliability analyses of engineering systems, and screenings or exploratory designs for the global optimization/minimization of functions. [ABSTRACT FROM AUTHOR]