Finite Weakest Link Model Research Articles

Statistical Scaling in Localization-Induced Failures

Abstract The investigation of statistical scaling in localization-induced failures dates back to da Vinci's speculation on the length effect on rope strength in 1500 s. The early mathematical description of statistical scaling emerged with the birth of the extreme value statistics. The most commonly known mathematical model for statistical scaling is the Weibull size effect, which is a direct consequence of the infinite weakest-link model. However, abundant experimental observations on various localization-induced failures have shown that the Weibull size effect is inadequate. Over the last two decades, two mathematical models were developed to describe the statistical size effect in localization-induced failures. One is the finite weakest-link model, in which the random structural resistance is expressed as the minimum of a set of independent discrete random variables. The other is the level excursion model, a continuum description of the finite weakest-link model, in which the structural failure probability is calculated as the probability of the upcrossing of a random field over a barrier. This paper reviews the mathematical formulation of these two models and their applications to various engineering problems including the strength distributions of quasi-brittle structures, failure statistics of micro-electromechanical systems (MEMS) devices, breakdown statistics of high– k gate dielectrics, and probability distribution of buckling pressure of spherical shells containing random geometric imperfections. In addition, the implications of statistical scaling for the stochastic finite element simulations and the reliability-based structural design are discussed. In particular, the recent development of the size-dependent safety factors is reviewed.

A Simplified Probabilistic Model for Nanocrack Propagation and Its Implications for Tail Distribution of Structural Strength

This paper presents a simplified probabilistic model for thermally activated nanocrack propagation. In the continuum limit, the probabilistic motion of the nanocrack tip is mathematically described by the Fokker-Planck equation. In the model, the drift velocity is explicitly related to the energy release rate at the crack tip through the transition rate theory. The model is applied to analyze the propagation of an edge crack in a nanoscale element. The element is considered to reach failure when the nanocrack propagates to a critical length. The solution of the Fokker-Planck equation indicates that both the strength and lifetime distributions of the nanoscale element exhibit a power-law tail behavior but with different exponents. Meanwhile, the model also yields a mean stress-life curve of the nanoscale element. When the applied stress is sufficiently large, the mean stress-life curve resembles the nasquin law for fatigue failure. nased on a recently developed finite weakest-link model as well as level excursion analysis of the failure statistics of quasi-brittle structures, it is argued that the simulated power-law tail of strength distribution of the nanoscale element has important implications for the tail behavior of the strength distribution of macroscopic structures. It provides a physical justification for the two-parameter Weibull distribution for strength statistics of large-scale quasi-brittle structures.

Rate-Dependent Scaling of Dynamic Tensile Strength of Quasibrittle Structures

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.

A Probabilistic Crack Band Model for Quasibrittle Fracture

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.

Size effect on reliability indices and safety factors of quasibrittle structures

Many modern engineering structures are made of brittle heterogenous (quasibrittle) materials, such as concrete, composites, and tough ceramics. Recent studies have shown that the strength distribution of quasibrittle structures varies from a Gaussian distribution modified by a power-law tail to a Weibull distribution as the structure size increases, and such size dependence can be well captured by a finite weakest link model. This paper investigates the implication of the size-dependent strength distribution on the reliability-based analysis and design of quasibrittle structures by considering a linear failure limit state. The size effect on the statistical properties of structural strength directly leads to a size-dependent Cornell index, which could give a reasonable estimation of the failure risk in some cases. To improve the prediction of the structural failure risk, the size dependence of the strength distribution is further incorporated into the computation of the Hasofer–Lind index, where it is shown that the transformation method for the non-Gaussian random structural strength is dependent on the structure size and geometry. By assuming that the applied load follows a Gaussian distribution, approximate size effect equations are developed for the central and nominal safety factors through asymptotic matching. The proposed formulation of the size effect on reliability indices and safety factors is applied to analyze the failure of the Malpasset arch dam.

Finite weakest-link model of lifetime distribution of quasibrittle structures under fatigue loading

The design of various engineering structures, such as buildings, infrastructure, aircraft, ships, as well as microelectronic components and medical implants, must ensure an extremely low probability of failure during their service lifetime. Since such a low probability is beyond the means of histogram testing, we must rely on some physically based probabilistic model for the statistics of structural lifetime. Attention is focused on structures consisting of quasibrittle materials. These are brittle materials with inhomogeneities that are not negligible compared with the structure size, as exemplified by concrete, fiber composites, tough ceramics, rocks, sea ice, bone, wood, and many more at the micro- or nano-scale. This paper presents a finite weakest-link model of the fatigue lifetime of quasibrittle structures that fail at the fracture of one representative volume element (RVE). In this model, the probability distribution of critical stress amplitude is first derived by assuming a prescribed number of loading cycles and a fixed stress ratio. The probability distribution of fatigue lifetime is then deduced from the probability distribution of critical stress amplitude through the Paris law for fatigue crack growth. It is shown that the present theory matches well with the experimentally measured lifetime histograms of various engineering and dental ceramics, which systematically deviate from the two-parameter Weibull distribution. The theory indicates that the mean fatigue lifetime of quasibrittle structures must strongly depend on the structure size and geometry. Finally, the present model indicates that the probability distribution of fatigue lifetime can be determined from the mean size effect analysis.

Energetic-statistical size effect in fracture of bimaterial hybrid structures

This paper investigates the size effect on the strength of bimaterial hybrid structures, which consist of a weak bimaterial interface. A general scaling relation was recently derived by combining the energetic scaling of fracture of the bimaterial notch and the finite weakest link model. This scaling relation is now studied through a stochastic cohesive crack model, which is able to capture the fracture process at the bimaterial notch tip as well as the random fracture properties of the bimaterial interface. The numerical example includes a series of metal-composite hybrid beams with a centered V-notch under three-point bending. A wide range of notch angles is considered, which represents various orders of stress singularities. The simulation shows that, as the stress singularities get weaker, there exists a transition from the energetic scaling to the statistical scaling. Within this transition range, the size effect on the structural strength can be explained by a combined energetic-statistical mechanism.

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...

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.

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.

A finite weakest-link model of lifetime distribution of high- k gate dielectrics under unipolar AC voltage stress

This paper presents a new probability distribution function for the breakdown lifetime of high- k gate dielectrics under unipolar AC voltage stress. This function is derived from a finite weakest-link model, where the gate oxide layer is considered to consist of many potential breakdown cells. Each potential breakdown cell is modeled as a series coupling of several subcells, which is analogous to the fiber-bundle model for the strength statistics of structures. The present model indicates that the type of lifetime distribution varies with the gate area and the dependence of the mean lifetime on the gate area deviates from the classical Weibull scaling law. It is shown that the model agrees well with the observed lifetime histograms of HfO 2 based gate dielectrics under unipolar AC voltage stress.

Strength distribution of dental restorative ceramics: Finite weakest link model with zero threshold

Ensuring a small enough failure probability is important for the design and selection of restorative dental ceramics. For this purpose, the two-parameter Weibull distribution, which is based on the weakest link model with infinitely many links, is usually adopted to model the strength distribution of dental ceramics. This distribution has been thoroughly validated for perfectly brittle materials. However, dental ceramics are generally quasibrittle because the inhomogeneity size is not negligible compared to the size of the ceramic part. For such materials, the experimental histograms of many quasibrittle materials have been shown to exhibit strong deviations from the two-parameter Weibull distribution. As a remedy, the three-parameter Weibull distribution, which has a nonzero threshold, has been proposed. However, the improvement of the fits of histograms of quasibrittle materials has been only partial. Instead of making the threshold non-zero, the correct remedy is to consider the weakest link model to have a finite number of links, each of them representing one finite-size representative volume element of material. This model has recently been justified on the basis of the probability of random jumps of atomic lattice cracks over the activation energy barriers on the free energy potential of the lattice. It is shown that, in similarity to other quasibrittle materials, this new model allows excellent fits of the experimental strength histograms of various types of dental ceramics.

Statistics of strength of ceramics: finite weakest-link model and necessity of zero threshold

It is argued that, in probabilistic estimates of quasibrittle structure strength, the strength threshold should be considered to be zero and the distribution to be transitional between Gaussian and Weibullian. The strength histograms recently measured on tough ceramics and other quasibrittle materials, which have been thought to imply a Weibull distribution with nonzero threshold, are shown to be fitted equally well or better by a new weakest-link model with a zero strength threshold and with a finite, rather than infinite, number of links in the chain, each link corresponding to one representative volume element (RVE) of a non-negligible size. The new model agrees with the measured mean size effect curves. It is justified by energy release rate dependence of the activation energy barriers for random crack length jumps through the atomic lattice, which shows that the tail of the failure probability distribution should be a power law with zero threshold. The scales from nano to macro are bridged by a hierarchical model with parallel and series couplings. This scale bridging indicates that the power-law tail with zero threshold is indestructible while its exponent gets increased on each passage to a higher scales. On the structural scale, the strength distribution except for its far left power-law tail, varies from Gaussian to Weibullian as the structure size increases. For the mean structural strength, the theory predicts a size effect which approaches the Weibull power law asymptotically for large sizes but deviates from it at small sizes. This deviation is the easiest way to calibrate the theory experimentally. The structure size is measured in terms of the number of RVEs. This number must be convoluted by an integral over the dimensionless stress field, which depends on structure geometry. The theory applies to the broad class of structure geometries for which failure occurs at macro-crack initiation from one RVE, but not to structure geometries for which stability is lost only after large macro-crack growth. Based on tolerable structural failure probability of <10−6, the change from nonzero to zero threshold may often require a major correction in safety factors.