Quasibrittle Structures Research Articles

Determination of strength distribution of quasibrittle structures from mean size effect analysis

This paper presents a new method to indirectly determine the probability distribution function of strength of quasibrittle structures. The method is derived within the framework of the finite weakest link model, which indicates that the size dependence of the mean structural strength can be calculated from the size effect on the probability distribution of the structural strength. By considering the inversion of this model, we show that the cumulative distribution function of structural strength can be explicitly determined from the parameters of the mean size effect curve. The proposed method is verified by a comprehensive set of experiments on asphalt mixture at a low temperature, which involves tests on both the strength histograms and the mean size effect. In the meantime, the agreement between the predicted and experimentally measured strength histograms of asphalt mixture specimens of different sizes further confirms the validity of the finite weakest link model.

Effect of Stress Singularity Magnitude on Scaling of Strength of Quasi-Brittle Structures

AbstractEngineering structures are often designed to have complex geometries, which could introduce stress singularities that are weaker than the conventional −1/2 crack-tip singularity. Extrapolating the results of small-scale laboratory tests to predict the response of a full-scale structure comprised of quasi-brittle materials requires an understanding of how the weak stress singularities modify the classical energetic and statistical scaling theories of quasi-brittle fracture. Through a theoretical and numerical study, a new scaling law for quasi-brittle fracture is derived, which explicitly relates the nominal structural strength to the structure size and the magnitude of the stress singularity. The theoretical analysis is based on a generalized weakest-link model that combines the energetic scaling of fracture with the finite weakest-link model. The model captures the transition from the energetic scaling to statistical scaling as the strength of the stress singularity diminishes. The new scaling la...

Size Effect Analysis in Laminated Composite Structure using General Bilinear Fit

Abstract A bilinear cohesive law analyzes the in-plane nominal strength of Quasi-brittle material containing a single hole in which damage around the open hole is represented by a line crack loaded on its faces. This analysis of quasi-brittle structures has been done by taking into account the geometry and length of fracture processing zone and the shape of the cohesive law. The nominal strength based on size effect is analyzed for similarly shape specimen geometry. Simple design tables for the bilinear cohesive law have been developed respect to the specimen types offering a very useful for engineering designers.

Indirect determination of size effect on strength of asphalt mixtures at low temperatures

Low temperature cracking of asphalt pavements is a major distress in cold regions. Accurate assessment of strength of asphalt mixtures at low temperatures is of great importance for ensuring the structural integrity of asphalt pavements. It has been shown that asphalt mixtures behave in a quasibrittle manner at low temperatures and consequently its nominal strength strongly depends on the structure size. The size effect on the strength of asphalt mixtures can be directly measured by testing geometrically similar specimens with a sufficiently large size range. Recent studies have shown in theory that for quasibrittle structures, which fail at the macrocrack initiation from one representative volume element, the mean size effect curve can also be derived from the scaling of strength statistics based on the finite weakest link model. This paper presents a comprehensive experimental investigation on the strength statistics as well as the size effect on the mean strength of asphalt mixtures at −24 °C. It is shown that the size effect on mean structural strength can be obtained by strength histogram testing on specimens of a single size. The present study also indicates that the three-parameter Weibull distribution is not applicable for asphalt mixtures.

Size Effect Law and Critical Distance Theories to Predict the Nominal Strength of Quasibrittle Structures

The design of structures with a nonuniform stress field is of great industrial interest. The ability of the size effect law and critical distance theories to predict the nominal strength of notched and open hole specimens is analyzed in the present paper. The results obtained with these methods are compared with the solution of the problem computed, taking into account the material cohesive law. A conclusion of this paper is that the role of the critical fracture energy in determining the structural strength is negligible, except in large cracked structures. For unnotched structures of any size and for small cracked structures, the key parameter is the initial part of the softening cohesive law. This allows us to define design charts that relate the structural strength to a specimen size normalized with respect to a material characteristic length.

Modeling of mode-I fatigue crack growth in quasibrittle structures under cyclic compression

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.

Scaling of Static Fracture of Quasi-Brittle Structures: Strength, Lifetime, and Fracture Kinetics

The paper reviews a recently developed finite chain model for the weakest-link statistics of strength, lifetime, and size effect of quasi-brittle structures, which are the structures in which the fracture process zone size is not negligible compared to the cross section size. The theory is based on the recognition that the failure probability is simple and clear only on the nanoscale since the probability and frequency of interatomic bond failures must be equal. The paper outlines how a small set of relatively plausible hypotheses about the failure probability tail at nanoscale and its transition from nano- to macroscale makes it possible to derive the distribution of structural strength, the static crack growth rate, and the lifetime distribution, including the size and geometry effects [while an extension to fatigue crack growth rate and lifetime, published elsewhere (Le and Bažant, 2011, “Unified Nano-Mechanics Based Probabilistic Theory of Quasibrittle and Brittle Structures: II. Fatigue Crack Growth, Lifetime and Scaling,” J. Mech. Phys. Solids, 1322–1337), is left aside]. A salient practical aspect of the theory is that for quasi-brittle structures the chain model underlying the weakest-link statistics must be considered to have a finite number of links, which implies a major deviation from the Weibull distribution. Several new extensions of the theory are presented: (1) A derivation of the dependence of static crack growth rate on the structure size and geometry, (2) an approximate closed-form solution of the structural strength distribution, and (3) an effective method to determine the cumulative distribution functions (cdf’s) of structural strength and lifetime based on the mean size effect curve. Finally, as an example, a probabilistic reassessment of the 1959 Malpasset Dam failure is demonstrated.

Size effect on strength and lifetime probability distributions of quasibrittle structures

Engineering structures such as aircraft, bridges, dams, nuclear containments and ships, as well as computer circuits, chips and MEMS, should be designed for failure probability < 10 − 6–10 − 7 per lifetime. The safety factors required to ensure it are still determined empirically, even though they represent much larger and much more uncertain corrections to deterministic calculations than do the typical errors of modern computer analysis of structures. The empirical approach is sufficient for perfectly brittle and perfectly ductile structures since the cumulative distribution function (cdf) of random strength is known, making it possible to extrapolate to the tail from the mean and variance. However, the empirical approach does not apply to structures consisting of quasibrittle materials, which are brittle materials with inhomogeneities that are not negligible compared to structure size. This paper presents a refined theory on the strength distribution of quasibrittle structures, which is based on the fracture mechanics of nanocracks propagating by activation energy controlled small jumps through the atomic lattice and an analytical model for the multi-scale transition of strength statistics. Based on the power law for creep crack growth rate and the cdf of material strength, the lifetime distribution of quasibrittle structures under constant load is derived. Both the strength and lifetime cdf’s are shown to be size- and geometry-dependent. The theory predicts intricate size effects on both the mean structural strength and lifetime, the latter being much stronger. The theory is shown to match the experimentally observed systematic deviations of strength and lifetime histograms of industrial ceramics from the Weibull distribution.

Computation of Probability Distribution of Strength of Quasibrittle Structures Failing at Macrocrack Initiation

AbstractEngineering structures must be designed for an extremely low failure probability, Pf<10-6. To determine the corresponding structural strength, a mechanics-based probability distribution model is required. Recent studies have shown that quasibrittle structures that fail at the macrocrack initiation from a single representative volume element (RVE) can be statistically modeled as a finite chain of RVEs. It has further been demonstrated that, based on atomistic fracture mechanics and a statistical multiscale transition model, the strength distribution of each RVE can be approximately described by a Gaussian distribution, onto which a Weibull tail is grafted at a point of the probability about 10-4 to 10-3. The model implies that the strength distribution of quasibrittle structures depends on the structure size, varying gradually from the Gaussian distribution modified by a far-left Weibull tail applicable for small-size structures, to the Weibull distribution applicable for large-size structures. Com...

General size effect on strength of bimaterial quasibrittle structures

This paper presents a general size effect equation for the strength of hybrid structures, which are made of two dissimilar quasibrittle materials with a thin and weak bimaterial interface. Depending on the material mismatch and structure geometry, a singular stress field could occur at the bimaterial corner. For structures with strong stress singularities, an energetic size effect is derived based on the equivalent linear elastic fracture mechanics and asymptotic matching. For structures without stress singularities, a finite weakest link model is adopted to derive the size effect. A general scaling equation that bridges the limits of strong and zero stress singularities is formulated by combining the energetic scaling of fracture of the bimaterial corner and the finite weakest link model.

Accurate simulation of mixed-mode cohesive crack propagation in quasi-brittle structures using exact asymptotic fields in XFEM: an overview

The extended finite element (XFEM) enriches the standard local FE approximations with known information about the problem, with the use of the partition of unity. This allows the use of meshes that do not conform to a discontinuity and avoids adaptive re-meshing as the discontinuity grows as required with the conventional FEM. When the crack tip asymptotic field is available and used as the enrichment function, XFEM is more accurate than FEM allowing the use of a much coarser mesh around the crack tip. Such asymptotic fields have been known for a long time for traction-free cracks (the Williams expansions) but have only recently been derived for cohesive cracks (Karihaloo–Xiao expansions). In this paper an overview of latter expansions is given for a range of cohesive laws and their usefulness in the simulation of cohesive crack propagation is demonstrated on two examples of concrete and fibre-reinforced concrete flexural members.

Unified nano-mechanics based probabilistic theory of quasibrittle and brittle structures: II. Fatigue crack growth, lifetime and scaling

This paper extends the theoretical framework presented in the preceding Part I to the lifetime distribution of quasibrittle structures failing at the fracture of one representative volume element under constant amplitude fatigue. The probability distribution of the critical stress amplitude is derived for a given number of cycles and a given minimum-to-maximum stress ratio. The physical mechanism underlying the Paris law for fatigue crack growth is explained under certain plausible assumptions about the damage accumulation in the cyclic fracture process zone at the tip of subcritical crack. This law is then used to relate the probability distribution of critical stress amplitude to the probability distribution of fatigue lifetime. The theory naturally yields a power-law relation for the stress-life curve (S-N curve), which agrees with Basquin's law. Furthermore, the theory indicates that, for quasibrittle structures, the S-N curve must be size dependent. Finally, physical explanation is provided to the experimentally observed systematic deviations of lifetime histograms of various ceramics and bones from the Weibull distribution, and their close fits by the present theory are demonstrated.

Unified nano-mechanics based probabilistic theory of quasibrittle and brittle structures: I. Strength, static crack growth, lifetime and scaling

Engineering structures must be designed for an extremely low failure probability such as 10 −6, which is beyond the means of direct verification by histogram testing. This is not a problem for brittle or ductile materials because the type of probability distribution of structural strength is fixed and known, making it possible to predict the tail probabilities from the mean and variance. It is a problem, though, for quasibrittle materials for which the type of strength distribution transitions from Gaussian to Weibullian as the structure size increases. These are heterogeneous materials with brittle constituents, characterized by material inhomogeneities that are not negligible compared to the structure size. Examples include concrete, fiber composites, coarse-grained or toughened ceramics, rocks, sea ice, rigid foams and bone, as well as many materials used in nano- and microscale devices. This study presents a unified theory of strength and lifetime for such materials, based on activation energy controlled random jumps of the nano-crack front, and on the nano-macro multiscale transition of tail probabilities. Part I of this study deals with the case of monotonic and sustained (or creep) loading, and Part II with fatigue (or cyclic) loading. On the scale of the representative volume element of material, the probability distribution of strength has a Gaussian core onto which a remote Weibull tail is grafted at failure probability of the order of 10 −3. With increasing structure size, the Weibull tail penetrates into the Gaussian core. The probability distribution of static (creep) lifetime is related to the strength distribution by the power law for the static crack growth rate, for which a physical justification is given. The present theory yields a simple relation between the exponent of this law and the Weibull moduli for strength and lifetime. The benefit is that the lifetime distribution can be predicted from short-time tests of the mean size effect on strength and tests of the power law for the crack growth rate. The theory is shown to match closely numerous test data on strength and static lifetime of ceramics and concrete, and explains why their histograms deviate systematically from the straight line in Weibull scale. Although the present unified theory is built on several previous advances, new contributions are here made to address: (i) a crack in a disordered nano-structure (such as that of hydrated Portland cement), (ii) tail probability of a fiber bundle (or parallel coupling) model with softening elements, (iii) convergence of this model to the Gaussian distribution, (iv) the stress-life curve under constant load, and (v) a detailed random walk analysis of crack front jumps in an atomic lattice. The nonlocal behavior is captured in the present theory through the finiteness of the number of links in the weakest-link model, which explains why the mean size effect coincides with that of the previously formulated nonlocal Weibull theory. Brittle structures correspond to the large-size limit of the present theory. An important practical conclusion is that the safety factors for strength and tolerable minimum lifetime for large quasibrittle structures (e.g., concrete structures and composite airframes or ship hulls, as well as various micro-devices) should be calculated as a function of structure size and geometry.

The problem of the critical angle for edge and center V-notched structures

In the present paper, the problem of detecting the critical notch angle, i.e. the angle providing the minimum failure load, for brittle or quasi-brittle structures containing either edge or center V-notches is investigated. The expression of the generalized fracture toughness is obtained according to Finite Fracture Mechanics. It is shown that a critical angle is always present: its value depends, through the brittleness number, on both material and geometric characteristics. Thus, the crack is not the most dangerous configuration. The result is supported by experimental results presented in the Literature.

Lifetime of high-k gate dielectrics and analogy with strength of quasibrittle structures

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.

Subcritical crack growth law and its consequences for lifetime statistics and size effect of quasibrittle structures

For brittle failures, the probability distribution of structural strength and lifetime are known to be Weibullian, in which case the knowledge of the mean and standard deviation suffices to determine the loading or time corresponding to a tolerable failure probability such as 10−6. Unfortunately, this is not so for quasibrittle structures, characterized by material inhomogeneities that are not negligible compared with the structure size (as is typical, e.g. for concrete, fibre composites, tough ceramics, rocks and sea ice). For such structures, the distribution of structural strength was shown to vary from almost Gaussian to Weibullian as a function of structure size (and also shape). Here we predict the size dependence of the distribution type for the lifetime of quasibrittle structures. To derive the lifetime statistics from the strength statistics, the subcritical crack growth law is requisite. This empirical law is shown to be justified by fracture mechanics of random crack jumps in the atomic lattice and the condition of equality of the energy dissipation rates calculated on the nano-scale and the macro-scale. The size effect on the lifetime is found to be much stronger than that on the structural strength. The theory is shown to match the experimentally observed systematic deviations of lifetime histograms from the Weibull distribution.

Scaling of strength and lifetime probability distributions of quasibrittle structures based on atomistic fracture mechanics

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.

Nano-mechanics based modeling of lifetime distribution of quasibrittle structures

The statistics of structural lifetime under constant load are related to the statistics of structural strength. The safety factors applied to structural strength must ensure failure probability no larger than 10 - 6 , which is beyond the means of direct verification by histogram testing. For perfectly brittle materials, extrapolation from the mean and variance to such a small tail probability is no problem because it is known that the Weibull distribution applies. Unfortunately, this is not possible for quasibrittle materials because the type of cumulative distribution function (cdf) has been shown to vary with structure size and shape. These are materials with inhomogeneities and fracture process zones (FPZ) that are not negligible compared to structural dimensions. A probabilistic theory of strength of quasibrittle structures failing at macro-crack initiation, which can be experimentally verified and calibrated indirectly, has recently been deduced from the rate of jumps of atomic lattice cracks governed by activation energy barriers. This paper extends this nano-mechanics based theory to the distribution of structural lifetime. Based on the cdf of strength and a power law for subcritical crack growth rate, the lifetime cdf of quasibrittle structures under constant loads is derived. The lifetime cdf is shown to depend strongly on the structure size as well as geometry. It is found that, for the creep rupture case, the mean structural lifetime exhibits a very strong size effect, much stronger than the size effect on the mean structure strength. The theory also implies temperature dependence of the lifetime cdf. For various quasibrittle materials, such as industrial ceramics and fiber composites, it is demonstrated that the proposed theory correctly predicts the experimentally observed deviations of lifetime histograms from the Weibull distribution.

Structural failure prediction of quasi-brittle structures: Modeling and simulation

Abstract In order to avoid the loss of well-posedness in the post-localization range, some continuum damage theories introduce higher order gradients of the damage variable in the constitutive model. This paper discuss the possibility of structural failure prediction of quasi-brittle materials through a special kind of gradient-enhanced damage theory in which the material is considered to possess a substructure or microstructure. In these theories the microscopic movements are accounted for by the damage variable and a reformulation of the kinematics (to include the possible “micro-motions’’) and of some basic governing principles of the classical continuum mechanics is necessary. The theory allows an adequate description of the strain-softening and localization behavior due to the material degradation. Within this framework, a numerically predicted macro-crack is the set of points in the structure where the damage variable has reached its critical value. A simple numerical technique, based on the finite element method, is proposed to approximate the solution of the resulting nonlinear problem. The main features of such kind of approach are discussed through examples concerning macro-crack initiation and propagation in plates under different loading conditions.

Random Lattice-Particle Simulation of Statistical Size Effect in Quasi-Brittle Structures Failing at Crack Initiation

The modeling of statistical size effect of concrete structures that fail at crack initiation is studied, with special attention to the interaction between the autocorrelation length and the size of the failure zone. The mechanical failure of concrete is modeled by a network of axial springs with degrading stiffness. The heterogeneity of concrete is idealized by spatial variation of the tensile strength and fracture energy. As an example, the direct tensile test in plane stress, with the size range of 1:20, is simulated. Furthermore, simulations of a four-point bending test with varying bending span are compared to the experimental results reported in the literature. The interaction of the autocorrelation length and the size of the failure zone is identified as a key parameter for the modeling of statistical size effect.