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Micromechanics of damage in brittle heterogeneous materials and composites requires analysis of a system of interacting cracks. The response path of a crack system typically exhibits bifurcations such that the states on each post-bifurcation branch can be stable yet only one branch can be reached in a stable manner. Recent results on thermodynamic criteria for stable states and stable response paths of inelastic structures are reviewed and formulated in terms of the incremental internal entropy of the system. The incremental entropy, which can be expressed in terms of second order work, is then calculated for various points on the response paths of some typical symmetric crack systems. It is shown that while the symmetric states may be stable, the path which leads to them is unstable and cannot occur in reality. Generally, nonsymmetry develops at the beginning of softening. The results show that it is insufficient to model distributed cracking only by means of crack systems and linear elastic fracture mechanics. Further aspects, such as material heterogeneity, residual stresses, and cohesive fracture zones for the microcracks, might have to be taken into account.

In recent research it has become clear that localization of damage due to strain softening as well as propagation of interactive cracks can result in bifurcations of the loading path characterized by localization of damage or fracture. Analysis of unnotched tension specimens by means of a particle model reveals a development of asymmetric response path after the peak load. A separate analysis based on inelastic stability theory is applied to an edge-notched tension specimen. It is found that the response path corresponding to symmetric propagation of both cracks is unstable and the propagation tends to localize into a single crack tip.

Mode III (antiplane) shear fracture energies of concrete and mortar were measured on the basis of the size-effect law. The specimens were cylinders with a circumferential notch at midlength, subjected to pure torsion at zero axial force. The specimens were geometrically similar and their sizes varied as 1:2:4. It was observed that the mortar specimens behave closer to linear elastic fracture mechanics than the concrete specimens. The ratio of Mode III to Mode I (tensile) fracture energies was found to be about 3 for concrete and about 8 for mortar. The results indirectly indicate that a volume expansion of the sheared fracture process zone may be a significant mechanism causing transverse tensile stresses across the ligament. The size effect measurements also yield an estimate of the size of the fracture process zone, which is found to be nearly the same as for Mode I fracture.

This paper presents a comparison of various models for strain‐softening due to damage such as cracking or void growth, as proposed recently in the literature. Continuum‐based models expressed in terms of softening stress—strain relations, and fracture‐type models expressed in terms of softening stress—displacement relations are distinguished. From one‐dimensional wave propagation calculations, it is shown that strain‐localization into regions of finite size cannot be achieved. The previously well‐documented spurious convergence is obtained with continuum models, while stress—displacement relations cannot model well smeared‐crack situations. Continuum models may, however, be used in general if a localization limiter is implemented. Gradient‐type localization limiters appear to be rather complicated; they require solving higher‐order differential equations of equilibrium with additional bourdary conditions. Non‐local localization limiters, especially the non‐local continuum with local strain, in which only the energy dissipating variables are non‐local, is found to be very effective, and also seems to be physically realistic. This formulation can correctly model the transition between homogeneous damage states and situations in which damage localizes into small regions that can be viewed as cracks. The size effect observed in the experimental and numerical response of specimens in tension or compression is shown to be a consequence of this progressive transition from continuum‐type to fracture‐type formulations.

This book is concerned with a leading-edge topic of great interest and importance, exemplifying the relationship between experimental research, material modeling, structural analysis and design. It focuses on the effect of structure size on structural strength and failure behaviour. Bazant's theory has found wide application to all quasibrittle materials, including rocks, ice, modern fiber composites and tough ceramics. The topic of energetic scaling, considered controversial until recently, is finally getting the attention it deserves, mainly as a result of Bazant's pioneering work. In this new edition an extra section of data and new appendices covering twelve new application developments are included. * The first book to show the 'size effect' theory of structure size on strength * Presents the principles and applications of Bazant's pioneering work on structural strength * Revised edition with new material on topics including asymptotic matching, flexural strength of fiber-composite laminates, polymeric foam fractures and the design of reinforced concrete beams

This book is concerned with a leading-edge topic of great interest and importance, exemplifying the relationship between experimental research, material modeling, structural analysis and design. It focuses on the effect of structure size on structural strength and failure behaviour. Bazant's theory has found wide application to all quasibrittle materials, including rocks, ice, modern fiber composites and tough ceramics. The topic of energetic scaling, considered controversial until recently, is finally getting the attention it deserves, mainly as a result of Bazant's pioneering work. In this new edition an extra section of data and new appendices covering twelve new application developments are included. * The first book to show the 'size effect' theory of structure size on strength * Presents the principles and applications of Bazant's pioneering work on structural strength * Revised edition with new material on topics including asymptotic matching, flexural strength of fiber-composite laminates, polymeric foam fractures and the design of reinforced concrete beams

This book is concerned with a leading-edge topic of great interest and importance, exemplifying the relationship between experimental research, material modeling, structural analysis and design. It focuses on the effect of structure size on structural strength and failure behaviour. Bazant's theory has found wide application to all quasibrittle materials, including rocks, ice, modern fiber composites and tough ceramics. The topic of energetic scaling, considered controversial until recently, is finally getting the attention it deserves, mainly as a result of Bazant's pioneering work. In this new edition an extra section of data and new appendices covering twelve new application developments are included. * The first book to show the 'size effect' theory of structure size on strength * Presents the principles and applications of Bazant's pioneering work on structural strength * Revised edition with new material on topics including asymptotic matching, flexural strength of fiber-composite laminates, polymeric foam fractures and the design of reinforced concrete beams

Motivated by the puzzle of sorption hysteresis in Portland cement concrete or cement paste, we develop in Part II of this study a general theory of vapor sorption and desorption from nanoporous solids, which attributes hysteresis to hindered molecular condensation with attractive lateral interactions. The classical mean-field theory of van der Waals is applied to predict the dependence of hysteresis on temperature and pore size, using the regular solution model and gradient energy of Cahn and Hilliard. A simple “hierarchical wetting” model for thin nanopores is developed to describe the case of strong wetting by the first monolayer, followed by condensation of nanodroplets and nanobubbles in the bulk. The model predicts a larger hysteresis critical temperature and enhanced hysteresis for molecular condensation across nanopores at high vapor pressure than within monolayers at low vapor pressure. For heterogeneous pores, the theory predicts sorption/desorption sequences similar to those seen in molecular dynamics simulations, where the interfacial energy (or gradient penalty) at nanopore junctions acts as a free energy barrier for snap-through instabilities. The model helps to quantitatively understand recent experimental data for concrete or cement paste wetting and drying cycles and suggests new experiments at different temperatures and humidity sweep rates., National Science Foundation (U.S.) (Grant DMS-0948071)

The sorption–desorption hysteresis observed in many nanoporous solids, at vapor pressures low enough for the liquid (capillary) phase of the adsorbate to be absent, has long been vaguely attributed to some sort of ‘pore collapse’. However, the pore collapse has never been documented experimentally and explained mathematically. The present work takes an analytical approach to account for discrete molecular forces in the nanopore fluid and proposes two related mechanisms that can explain the hysteresis at low vapor pressure without assuming any pore collapse nor partial damage to the nanopore structure. The first mechanism, presented in Part I, consists of a series of snap-through instabilities during the filling or emptying of non-uniform nanopores or nanoscale asperities. The instabilities are caused by non-uniqueness in the misfit disjoining pressures engendered by a difference between the nanopore width and an integer multiple of the thickness of a monomolecular adsorption layer. The wider the pore, the weaker the mechanism, and it ceases to operate for pores wider than about 3 nm. The second mechanism, presented in Part II, consists of molecular coalescence, or capillary condensation, within a partially filled surface, nanopore or nanopore network. This general thermodynamic instability is driven by attractive intermolecular forces within the adsorbate and forms the basis for developing a unified theory of both mechanisms. The ultimate goals of the theory are to predict the fluid transport in nanoporous solids from microscopic first principles, determine the pore size distribution and internal surface area from sorption tests, and provide a way to calculate the disjoining pressures in filled nanopores, which play an important role in the theory of creep and shrinkage., National Science Foundation (U.S.) (Grant DMS-0948071)

The two-parameter Weibull distribution has been widely adopted to model the lifetime statistics of dielectric breakdown under constant voltage, but recent lifetime testing for high-k gate dielectrics has revealed a systematic departure from Weibull statistics, evocative of lifetime statistics for small quasibrittle structures under constant stress. Here we identify a mathematical analogy between the dielectric breakdown in semiconductor electronic devices and the finite-size weakest-link model for mechanical strength of quasibrittle structures and adapt a recently developed probabilistic theory of structural failure to gate dielectrics. Although the theory is general and does not rely on any particular model of local breakdown events, we show how its key assumptions can be derived from the classical dielectric breakdown model, which predicts certain scaling exponents. The theory accurately fits the observed kinked shape of the histograms of lifetime plotted in Weibull scale, as well as the measured dependence of the median lifetime on the gate area (or size), including its deviation from a power law. The theory also predicts that the Weibull modulus for breakdown lifetime increases in proportion to the thickness of the oxide layer and suggests new ideas for more effective reliability testing., National Science Foundation (U.S.) (Grant no. CMS-0556323)

The two-parameter Weibull distribution has been widely adopted to model the lifetime statistics of dielectric breakdown under constant voltage, but recent lifetime testing for high-k gate dielectrics has revealed a systematic departure from Weibull statistics, evocative of lifetime statistics for small quasibrittle structures under constant stress. Here we identify a mathematical analogy between the dielectric breakdown in semiconductor electronic devices and the finite-size weakest-link model for mechanical strength of quasibrittle structures and adapt a recently developed probabilistic theory of structural failure to gate dielectrics. Although the theory is general and does not rely on any particular model of local breakdown events, we show how its key assumptions can be derived from the classical dielectric breakdown model, which predicts certain scaling exponents. The theory accurately fits the observed kinked shape of the histograms of lifetime plotted in Weibull scale, as well as the measured dependence of the median lifetime on the gate area (or size), including its deviation from a power law. The theory also predicts that the Weibull modulus for breakdown lifetime increases in proportion to the thickness of the oxide layer and suggests new ideas for more effective reliability testing., National Science Foundation (U.S.) (Grant no. CMS-0556323)

The failure probability of engineering structures such as aircraft, bridges, dams, nuclear structures, and ships, as well as microelectronic components and medical implants, must be kept extremely low, typically <10−6. The safety factors needed to ensure it have so far been assessed empirically. For perfectly ductile and perfectly brittle structures, the empirical approach is sufficient because the cumulative distribution function (cdf) of random material strength is known and fixed. However, such an approach is insufficient for structures consisting of quasibrittle materials, which are brittle materials with inhomogeneities that are not negligible compared with the structure size. The reason is that the strength cdf of quasibrittle structure varies from Gaussian to Weibullian as the structure size increases. In this article, a recently proposed theory for the strength cdf of quasibrittle structure is refined by deriving it from fracture mechanics of nanocracks propagating by small, activation-energy-controlled, random jumps through the atomic lattice. This refinement also provides a plausible physical justification of the power law for subcritical creep crack growth, hitherto considered empirical. The theory is further extended to predict the cdf of structural lifetime at constant load, which is shown to be size- and geometry-dependent. The size effects on structure strength and lifetime are shown to be related and the latter to be much stronger. The theory fits previously unexplained deviations of experimental strength and lifetime histograms from the Weibull distribution. Finally, a boundary layer method for numerical calculation of the cdf of structural strength and lifetime is outlined., Boeing, Inc. (Grant N007613), National Science Foundation (Grant CMS-0556323)

The failure probability of engineering structures such as aircraft, bridges, dams, nuclear structures, and ships, as well as microelectronic components and medical implants, must be kept extremely low, typically <10−6. The safety factors needed to ensure it have so far been assessed empirically. For perfectly ductile and perfectly brittle structures, the empirical approach is sufficient because the cumulative distribution function (cdf) of random material strength is known and fixed. However, such an approach is insufficient for structures consisting of quasibrittle materials, which are brittle materials with inhomogeneities that are not negligible compared with the structure size. The reason is that the strength cdf of quasibrittle structure varies from Gaussian to Weibullian as the structure size increases. In this article, a recently proposed theory for the strength cdf of quasibrittle structure is refined by deriving it from fracture mechanics of nanocracks propagating by small, activation-energy-controlled, random jumps through the atomic lattice. This refinement also provides a plausible physical justification of the power law for subcritical creep crack growth, hitherto considered empirical. The theory is further extended to predict the cdf of structural lifetime at constant load, which is shown to be size- and geometry-dependent. The size effects on structure strength and lifetime are shown to be related and the latter to be much stronger. The theory fits previously unexplained deviations of experimental strength and lifetime histograms from the Weibull distribution. Finally, a boundary layer method for numerical calculation of the cdf of structural strength and lifetime is outlined., Boeing, Inc. (Grant N007613), National Science Foundation (Grant CMS-0556323)

An extensive study of the micromechanics aspects and size effects associated with strain softening damage due to distributed cracking in brittle heterogeneous materials such as concrete, rocks and ceramics has been carried out and presented in a sequence of publications. The results may be grouped in five categories: (1) Size Effects, (2) Micromechanics Aspects, (3) Nonlocal Continuum Models, (4) Localization Instabilities, (5) Thermodynamic Analysis of Stable Path, and (6) Numerical Implementation of Localization Limiters. The law governing the size effect due to localization of strain softening into a finite size fracture process zone has been formulated, experimentally verified and calibrated. Knowledge of the size effect law has led to a method of determining nonlinear fracture properties on the basis of maximum loads of geometrically similar specimens of different sizes. The concept of a brittleness number characterizing the proximity of response of any structure to linear elastic fracture mechanics has been developed. Extensive experiments have been conducted on concrete, rock and ceramics. (JES)

Damage in concrete is characterized by microcracking and/or the development of large cracks. In this study, the physics underlying the relationships between microcracks and damage and the implications of these factors on the mathematical modeling of engineering structures have been studied. Particular issues which have been emphasized are: (1) the size effects on brittle failure, (2) shear fracture and (3) the mathematical and phenomenological consistency of strain-softening models with damage processes in concrete. The effect of the size of a concrete structure on the nominal stress at brittle failure is studied by dimension analysis and illustrative examples.

Contents: Variational Equation for the Eigenstates; Method of Solution on a Finite Element grid; and Numerical Solutions.

An extensive set of test data on the densification and stress-strain behavior of a dry sand at various relative densities and confining pressures and subjected to cyclic simple shear strains of different amplitudes is presented and used to verify a previously developed endochronic constitutive law and to determine its material parameters, including their dependence on relative density. In this law the accumulation of inelastic strain due to the irreversible rearrangement of grain configurations associated with deviatoric strains is characterized by a non-decreasing material variable, termed the rearrangement measure, which, in turn, forms the basis of an intrinsic time scale and related variables termed the densification measure and the distortion measure. The shear modulus is identified to be a function of the confining (volumetric) stress and the second invariant of the strain deviator. To fit the test data, step-by-step numerical integration of shear stresses is performed for a prescribed cyclic strain history. In contrast with current empirical methods, the material behavior is described herein by a constitutive law that satisfies all requirements of continuum mechanics; hence, this law should, in principle, be generally applicable, including the cases of non-sinusoidal loadings with varying amplitudes, general multiaxial stress states, and nonproportional stress component histories. In addition, the law automatically exhibits hysteretic damping and is fully continuous, i.e., it contains no inequalities, such as those used in plasticity to distinguish unloading. (Author)

The inelastic densification produced by shear straining saturated sands is opposed by the elasticity of the pore water and leads to a pore pressure increase, which causes a decrease in the intergranular frictional forces and consequent liquefaction of sand mass. This inelastic densification is accompanied by an inelastic strain of the fluid phase, and the magnitude of the developed pore water pressure is the product of the inelastic densification and the densification compliance, the latter being approximately equal to the drained compressibility of the sand. The tangent (incremental) elastic moduli are expressed in terms of the drained and undrained compressibilities of the two-phase medium and the compressibilities of water and the solid matter forming the grains. It is demonstrated that the volume change of the grains due to intergranular stresses has a negligible effect on the material parameters, even though it roughly equals the volume change of the grains due to the pore water pressure, which has an appreciable effect. Typical values are calculated for the material parameters. (Author)

A nonlinear constitutive law is developed to describe the densification and hysteresis of sand under cyclic loading. This law, together with the previously established model for an inelastic two-phase medium, gives a realistic prediction of the pore pressure buildup and associated liquefaction of sand due to cyclic shear. The law is of the endochronic type and consists of quasi-linear first-order differential equations expressed in terms of intrinsic time, which is an independent variable whose increments depend on the strain increments. This accounts for the accumulation of particle rearrangements, which are characterized by a parameter termed the rearrangement measure. The basic extensions of this work with regard to the application of endochronic theory to metals are due to the fact that sands densify upon shearing and are sensitive to confining stress; these two features are shared with the formulation of endochronic theory for concrete. Several typical examples are given to illustrate the ability of the developed theory to adequately characterize the densification, hysteresis, and liquefaction tendency of sands subjected to cyclic shear in laboratory tests. (Author)

The new theory presented in the preceding paper, which models the dynamic comminution of concrete due to very high shear strain rate, is now compared to recent test data on the penetration of projectiles through concrete walls of different thicknesses, ranging from 127 to 254 mm. These data are analyzed by an explicit finite element code using the new microplane constitutive model M7 for concrete, which was previously shown to provide the most realistic description of the quasi-static uni-, bi- and tri-axial test data with complex loading path and unloading. Model M7 incorporates the quasi-static strain rate effects due viscoelasticity and to the rate of cohesive crack debonding based on activation energy of bond ruptures, which are expected to extend to very high rates. Here model M7 is further enhanced by apparent viscosity capturing the energy dissipation due to the strain-rate effect of comminution. The maximum shear strain rates in the computations are of the order of 10(5) s(-1). The simulations document that, within the inevitable uncertainties, the measured exit velocities of the projectiles can be matched quite satisfactorily and the observed shapes of the entry and exit craters can be reproduced correctly. (C) 2014 Published by Elsevier Ltd., Postprint (published version)

SAFETEA-LU Project 2007¿2012, The overall goal of this project was to improve the safety and sustainability in the design of large, prestressed concrete bridges and other transportation structures. The safety of large concrete, structures, including bridges, has been insufficient. This is evidenced by the worldwide rate of failures, of very large concrete structures which has historically been about 1 in a thousand per lifetime, although 1 in a million is the maximum tolerable. Improvement necessitates taking into account the, size effect on quasibrittle failure loads, a phenomenon that has been mostly ignored by the ACI code, committee until recently, but now is considered seriously, largely as a result of this project.