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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]

Diffusion behaviors of heterogeneous materials are of paramount importance in many engineering problems. Numerical models that take into account the internal structure of such materials are robust but computationally very expensive. This burden can be partially decreased by using discrete models, however even then the practical application is limited to relatively small material volumes. This paper formulates a homogenization scheme for discrete diffusion models. Asymptotic expansion homogenization is applied to distinguish between (i) the continuous macroscale description approximated by the standard finite element method and (ii) the fully resolved discrete mesoscale description in a local representative volume element (RVE) of material. Both transient and steady‐state variants with nonlinear constitutive relations are discussed. In all the cases, the resulting discrete RVE problem becomes a simple linear steady‐state problem that can be easily pre‐computed. The scale separation provides a significant reduction of computational time allowing the solution of practical problems with a negligible error introduced mainly by the finite element discretization at the macroscale. [ABSTRACT FROM AUTHOR]

Quasibrittle materials are featured by a strain-softening constitutive behavior under many loading scenarios, which could eventually lead to localization instability. It has long been known that strain localization would result in spurious mesh sensitivity in finite element (FE) simulations. Previous studies have shown that, for the case of fully localized damage, the mesh sensitivity can be mitigated through energy regularization of the material constitutive law. However, depending on the loading configuration and structural geometry, quasibrittle structures could exhibit a complex damage process, which involves both localized and diffused damage patterns at different stages of loading. This study presents a generalized energy regularization method that considers the spatial and temporal evolution of damage pattern. The method introduces a localization parameter, which describes the local damage pattern. The localization parameter governs the energy regularization of the constitutive model, which captures the transition from diffused to localized damage during the failure process. The method is cast into an isotropic damage model, and is further extended to rate-dependent behavior. The energy regularization scheme is directly incorporated into the kinetics of damage growth. The model is applied to simulate static and dynamic failures of ceramic specimens. It is shown that the present model is able to effectively mitigate the spurious mesh sensitivity in FE simulations of both types of failure. The present analysis demonstrates the essential role of mechanism-based energy regularization of constitutive relation in FE simulations of quasibrittle fracture. [ABSTRACT FROM AUTHOR]

This article proposes a sampling technique that delivers robust designs, that is, point sets selected from a design domain in the shape of a unit hypercube. The designs are guaranteed to provide a statistically uniform point distribution, meaning that every location has the same probability of being selected. Moreover, the designs are sample uniform, meaning that each individual design has its points spread evenly throughout the domain. The sample uniformity (often measured via a discrepancy criterion) is achieved using distance-based criteria ( ϕ p or Maximin), that is, criteria normally used in space-filling designs. We show that the standard intersite metrics employed in distance-based criteria (Maximin and ϕ p (phi)) do not deliver statistically uniform designs. Similarly, designs optimized via centered L2 discrepancy or support points are also not statistically uniform. When these designs (after optimization based on intersite distances) are used for Monte Carlo type of integration, their statistical nonuniformity is a serious problem as it may lead to a systematic bias. This article proposes using a periodic metric to guarantee the statistical uniformity of the family of distance-based designs. The presented designs used as benchmarks in the article are only taken from the class of Latin hypercube designs, which forces univariate projections to be uniform and improves accuracy in Monte Carlo integration of some functions. for this article are available online. [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]

This paper investigates the effect of strain rate on the scaling behavior of dynamic tensile strength of quasibrittle structures. The theoretical framework is anchored by a rate-dependent finite weakest link model. The model involves a rate-dependent length scale, which captures the transition from localized damage to diffused damage with an increasing strain rate. As a result, the model predicts a rate- and size-dependent probability distribution function of the nominal tensile strength. The transitional behavior of the strength distribution directly leads to the rate and size effects on the mean and standard deviation of the tensile strength. The model is verified by a series of stochastic discrete element simulations of dynamic fracture of aluminum nitride specimens. The simulations involve a set of geometrically similar specimens of various sizes subjected to a number of different strain rates. Both random microstructure geometry and fracture properties are considered in these simulations. The simulated damage pattern indicates that an increase in the strain rate results in a more diffusive cracking pattern, which supports the theoretical formulation. The simulated rate and size effects on the mean and standard deviation of the nominal tensile strength agree well with the predictions by the rate-dependent finite weakest link model. [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]

This paper presents a new crack band model (CBM) for probabilistic analysis of quasibrittle fracture. The model is anchored by a probabilistic treatment of damage initiation, localization, and propagation. This model regularizes the energy dissipation of a single material element for the transition between damage initiation and localization. Meanwhile, the model also takes into account the probabilistic onset of damage localization inside the finite element (FE)for the case where the element size is larger than the crack band width. The random location of the localization band is related to the random material strength, whose statistics is described by a finite weakest link model. The present model is applied to simulate the probability distributions of the nominal strength of different quasibrittle structures. It is shown that for quasibrittle structures direct application of the conventional CBM for stochastic FE simulations would lead to mesh-sensitive results. To mitigate such mesh dependence, it is essential to incorporate the strain localization mechanism into the formulation of the sampling distribution functions of material constitutive parameters. [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]

Abstract: Shape optimization in connection with numerical modelling is used to reduce bending and associated flexural stresses in buried concrete arches. Modelling of the arch is carried out via a nonlinear finite element model that accounts for soil constitutive relations, soil–structure interactions, sequential construction stages and soil compaction. Centre line of the arch is parameterized by Bézier curve with three degrees of freedom that are subjected to optimization by genetic and Levenberg–Marquardt algorithm. The paper presents a parametric study which aims to determine the optimal shapes for buried arches of various span/rise ratios, backfill depths and foundation soil types. In the second part of the paper it shows a theoretical reduction in tensile stresses obtained by shape optimization of concrete arch culvert with a 9.4m span tested at the University of Massachusetts at Amherst. [Copyright &y& Elsevier]

Abstract: A cyclic cohesive zone model is proposed to simulate the mode-I crack growth in quasibrittle structures under compressive fatigue. The constitutive behavior of the cohesive elements is formulated for both tension and compression regimes. A strain-softening cohesive law is adopted for the tension regime whereas a plastic-type cohesive behavior is considered for the compression regime. It is shown that the proposed model is able to capture some essential fracture behaviors of quasibrittle structures under compressive fatigue, which include the onset of fatigue crack growth, the gradual decrease in crack growth rate, and the exhaustion of residual tensile stress over the cycles. Based on a fracture process zone (FPZ)-equivalence principle, it is further shown that the existing kinetics equation for tensile fatigue crack can be extended to the crack growth under cyclic compression. [Copyright &y& Elsevier]

In this paper, we study a mixed-mode fracture process using a conventional two dimensional lattice model with incorporated meso-level internal material structure. Simple elasto-brittle elements of the network are divided into three phases according to a projected grain layout. The stiffness of any element that fulfils a failure criterion is removed. As a new feature of the otherwise standard lattice approach, we added the recovery of normal stiffness when a severed element enters the compressive regime. This enhancement enables capture of the shear resistance of an existing crack caused by crack roughness, i.e. what is termed aggregate interlocking. We demonstrate this enhancement via the simulation of mixed-mode experiments on concrete performed at a laboratory at the Technical University of Denmark. Double notched concrete specimens were initially pre-cracked in tension. Then, various combinations of tensile and shear load (normal and tangential to the crack plane) were applied. Simulated crack patterns and load-displacement curves are compared to the experimental observations. [ABSTRACT FROM AUTHOR]

Abstract: The article proposes an improvement over the widely used sequentially linear solution procedure often utilized for fracture simulations. In the classical secant version of this method, a partial solution of a step is scaled to reach a stress limit in exactly one element and the mechanical properties of the critical element are reduced. Non-proportional loading is generally unfeasible due to avalanches of ruptures caused by stress redistribution. Because only one loading vector can be scaled at a time, all others have to remain constant during the step. However, the constant load vectors do not allow proper determination of the critical element. A modified procedure based on redistribution of released stresses is developed here. It preserves the linearity of each step. After rupture of the critical element, a sequentially linear redistribution process of stress release takes place until a static equilibrium state is reached. During the redistribution, other elements may break. The proposed enhanced sequential procedure is also compared with another recently published “event-by-event” linear method for non-proportional loading. It is shown here, with the help of simple examples, that the proposed redistribution method yields correct results for non-proportional loading, unlike the other methods under comparison. [Copyright &y& Elsevier]

• 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]

An adaptive discretization refinement strategy for steady state discrete mesoscale models of coupled mechanics and mass transport in concrete is presented. Coupling is provided by two phenomena: the Biot's theory of poromechanics and an effect of cracks on material permeability coefficient. The model kinematics is derived from rigid body motion of Voronoi cells obtained by tessellation of the domain. Starting with a coarse discretization, the density of Voronoi generator points is adaptively increased on the fly in regions where the maximum principal stress exceeds a chosen threshold. Purely elastic behavior is assumed in the coarse discretization, therefore no transfer of history/state variables is needed. Examples showing (i) computational time savings achieved via the adaptive technique and (ii) an agreement of the outputs from the fine and adaptive models during simulations of hydraulic fracturing and three-point bending combined with a fluid pressure loading are presented. • Adaptive discretization refinement for multi-physical discrete models is developed. • Verification is provided in both 2D and 3D by simulation concrete fracturing. • Quality of the solution (displacements, stresses, pressure and fluxes) is preserved. • Speed-up factors between 8 and 20 are achieved in the verification examples. [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]

Biogenesis of F1Fo ATP synthase, the key enzyme of mitochondrial energy provision, depends on transmembrane protein 70 (TMEM70), localized in the inner mitochondrial membrane of higher eukaryotes. TMEM70 absence causes severe ATP-synthase deficiency and leads to a neonatal mitochondrial encephalocardiomyopathy in humans. However, the exact biochemical function of TMEM70 remains unknown. Using TMEM70 conditional knockout in mice, we show that absence of TMEM70 impairs the early stage of enzyme biogenesis by preventing incorporation of hydrophobic subunit c into rotor structure of the enzyme. This results in the formation of an incomplete, pathologic enzyme complex consisting of F1 domain and peripheral stalk but lacking Fo proton channel composed of subunits c and a. We demonstrated direct interaction between TMEM70 and subunit c and showed that overexpression of subunit c in TMEM70-/- cells partially rescued TMEM70 defect. Accordingly, TMEM70 knockdown prevented subunit c accumulation otherwise observed in F1-deficient cells. Altogether, we identified TMEM70 as specific ancillary factor for subunit c. The biologic role of TMEM70 is to increase the low efficacy of spontaneous assembly of subunit c oligomer, the key and rate-limiting step of ATP-synthase biogenesis, and thus to reach an adequately high physiologic level of ATP synthase in mammalian tissues. [ABSTRACT FROM AUTHOR]