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 ???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.     

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.     

Probability distribution of energetic-statistical size effect in quasibrittle fracture

The physical sources of randomness in quasibrittle fracture described by the cohesive crack model are discussed and theoretical arguments for the basic form of the probability distribution are presented. The probability distribution of the size effect on the nominal strength of structures made of heterogeneous quasibrittle materials is derived, under certain simplifying assumptions, from the nonlocal generalization of Weibull theory. Attention is limited to structures of positive geometry failing at the initiation of macroscopic crack growth from a zone of distributed cracking. It is shown that, for small structures, which do not dwarf the fracture process zone (FPZ), the mean size effect is deterministic, agreeing with the energetic size effect theory, which describes the size effect due to stress redistribution and the associated energy release caused by finite size of the FPZ formed before failure. Material randomness governs the statistical distribution of the nominal strength of structure and, for very large structure sizes, also the mean. The large-size and small-size asymptotic properties of size effect are determined, and the reasons for the existence of intermediate asymptotics are pointed out. Asymptotic matching is then used to obtain an approximate closed-form analytical expression for the probability distribution of failure load for any structure size. For large sizes, the probability distribution converges to the Weibull distribution for the weakest link model, and for small sizes, it converges to the Gaussian distribution justified by Daniels' fiber bundle model. Comparisons with experimental data on the size-dependence of the modulus of rupture of concrete and laminates are shown. Monte Carlo simulations with finite elements are the subject of ongoing studies by Pang at Northwestern University to be reported later.     

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.     

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.     

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.     

The asymptotic stochastic strength of bundles of elements exhibiting general stress–strain laws

The fiber bundle model is widely used in probabilistic modeling of various phenomena across different engineering fields, from network analysis to earthquake statistics. In structural strength analysis, this model is an essential part of extreme value statistics that governs the left tail of the cumulative probability density function of strength. Based on previous nano-mechanical arguments, the cumulative probability distribution function of strength of each fiber constituting the bundle is assumed to exhibit a power-law left tail. Each fiber (or element) of the bundle is supposed to be subjected to the same relative displacement (parallel coupling). The constitutive equations describing various fibers are assumed to be related by a radial affinity while no restrictions are placed on their particular form. It is demonstrated that, even under these most general assumptions, the power-law left tail is preserved in the bundle and the tail exponent of the bundle is the sum of the exponents of the power-law tails of all the fibers. The results have significant implications for the statistical modeling of strength of quasibrittle structures.     

Incorporation of statistical length scale into Weibull strength theory for composites

In this paper an extension of Weibull theory by the introduction of a statistical length scale is presented. The classical Weibull strength theory is self-similar; a feature that can be illustrated by the fact that the strength dependence on structural size is a power law (a straight line on a double logarithmic graph). Therefore, the theory predicts unlimited strength for extremely small structures. In the paper, it is shown that such a behavior is a direct implication of the assumption that structural elements have independent random strengths. By the introduction of statistical dependence in the form of spatial autocorrelation, the size dependent strength becomes bounded at the small size extreme. The local random strength is phenomenologically modeled as a random field with a certain autocorrelation function. In such a model, the autocorrelation length plays the role of a statistical length scale. The focus is on small failure probabilities and the related probabilistic distributions of the strength of composites. The theoretical part is followed by applications in fiber bundle models, chains of fiber bundle models and the stochastic finite element method in the context of quasibrittle failure.     

An efficient sampling scheme: Updated Latin Hypercube Sampling

An efficient sampling scheme called Updated Latin Hypercube Sampling is presented. The proposed method is an improved variant of Latin Hypercube Sampling. It uses specially modified tables of independent random permutations of rank numbers which form the strategy of generating input samples for a simulation procedure. The method is presented in order to obtain these specially modified tables. The aim of this paper is to compare estimates of certain widely used statistical parameters obtained by Updated Latin Hypercube Sampling, Latin Hypercube Sampling and Simple Random Sampling. It is shown that Updated Latin Hypercube Sampling usually results in a substantial decrease of the variance in the estimates of commonly used statistical parameters and that the bias is quite small for a moderate number of simulations. This sampling technique seems to be generally very useful, efficient and superior to other methods especially in the case of statistical, sensitivity and probability analyses of complex analytical models with random input variables.     

Latin hypercube sampling used in the calculation of the fracture probability

The paper proposes Latin hypercube sampling combined with the stratified sampling of variance reduction technique to calculate accurate fracture probability. In the compound sampling, the number of simulations is relatively small and the calculation error is satisfactory.     

Probabilistic analysis of concrete cracking using neural networks and random fields

In this paper an algorithm for the probabilistic analysis of concrete structures is proposed which considers material uncertainties and failure due to cracking. The fluctuations of the material parameters are modeled by means of random fields and the cracking process is represented by a discrete approach using a coupled meshless and finite element discretization. In order to analyze the complex behavior of these nonlinear systems with low numerical costs a neural network approximation of the performance functions is realized. As neural network input parameters the important random variables of the random field in the uncorrelated Gaussian space are used and the output values are the interesting response quantities such as deformation and load capacities. The neural network approximation is based on a stochastic training which uses wide spanned Latin hypercube sampling to generate the training samples. This ensures a high quality approximation over the whole domain investigated, even in regions with very small probability.     

Computational modeling of size effects in concrete specimens under uniaxial tension

The paper presents a follow-up study of numerical modeling of complicated interplay of size effects in concrete structures. The major motivation is to identify and study interplay of several scaling lengths stemming from the material, boundary conditions and geometry. Methods of stochastic nonlinear fracture mechanics are used to model the well published results of direct tensile tests of dog-bone specimens with rotating boundary conditions. Firstly, the specimens are modeled using microplane material and also fracture-plastic material laws to show that a portion of the dependence of nominal strength on structural size can be explained deterministically. However, it is clear that more sources of size effect play a part, and we consider two of them. Namely, we model local material strength using an autocorrelated random field attempting to capture a statistical part of the combined size effect, scatter inclusive. In addition, the strength drop noticeable with small specimens which was obtained in the experiments could be explained either by the presence of a weak surface layer of constant thickness (caused e.g. by drying, surface damage, aggregate size limitation at the boundary, or other irregularities) or three dimensional effects incorporated by out-of-plane flexure of specimens. The latter effect is examined by comparison of 2D and 3D models with the same material laws. All three named sources (deterministic-energetic, statistical size effects and the weak layer effect) are believed to be the sources most contributing to the observed strength size effect; the model combining all of them is capable of reproducing the measured data. The computational approach represents a marriage of advanced computational nonlinear fracture mechanics with simulation techniques for random fields representing spatially varying material properties. Using a numerical example, we document how different sources of size effects detrimental to strength can interact and result in relatively complicated quasibrittle failure processes. The presented study documents the well known fact that the experimental determination of material parameters (needed for the rational and safe design of structures) is very complicated for quasibrittle materials such as concrete.     

Postgrouped sampling method of estimation

A postgrouped sampling is considered for estimating the (finite or finite), population mean. Double sampling and an empirical-weighted estimator is used. Unbiasedness, variance and efficiency are considered. Its properties are discussed allowing the simple random sampling with replacement (SRSWR) design in the first phase, and in each stratum for the second phase. It is shown that for a fixed sample size in each postgroup, the variance of the proposed estimator with less prior information is asymptotically equivalent to the usual stratified estimator for fixed allocation. Some examples are provided for natural populations., The method is also extended to simple random sampling without replacement (SRSWOR) design in the first phase, and in each stratum for the second phase. Unbiased variance estimation is provided for both types of sampling designs.     

Simulation of local material properties based on moving-window GMC

When analyzing the behavior of composite materials under various loading conditions, the assumption is generally made that the behavior due to randomness in the material can be represented by a homogenized, or effective, set of material properties. This assumption may be valid when considering displacement, average strain, or even average stress of structures much larger than the inclusion size. The approach is less valid, however, when considering either behavior of structures of size at the scale of the inclusions or local stress of structures in general. In this paper, Monte Carlo simulation is used to assess the effects of microstructural randomness on the local stress response of composite materials. In order to achieve these stochastic simulations, the mean, variance and spectral density functions describing the randomly varying elastic properties are required as input. These are obtained here by using a technique known as moving-window generalized method of cells (moving-window GMC). This method characterizes a digitized composite material microstructure by developing fields of local effective material properties. Once these fields are generated, it is straightforward to obtain estimates of the associated probabilistic parameters required for simulation. Based on the simulated property fields, a series of local stress fields, associated with the random material sample under uniaxial tension, is calculated using finite element analysis. An estimation of the variability in the local stress response for the given random composite is obtained from consideration of these simulations.     

坯料随机局部弯曲对滚弯成形结果影响的蒙特卡洛分析

采用蒙特卡洛有限元模拟方法研究了坯料局部的随机弯曲对滚弯成形结果的影响。为了提高滚弯模拟的效率,提出了基于欧拉网格和经典梁单元的滚弯模拟方案,并与传统有限元模型和理论解对比验证了该方案的正确性。在此基础上模拟了具有零均值正态分布的局部曲率的超长坯料的滚弯过程,并统计产品曲率半径的分布规律。结果表明:输出曲率半径分布近似满足正态分布,且随着坯料曲率标准差的增大,均值减小,方差增大,宏观上表现为产品半径减小。产品的目标半径越大,代表性单元长度越长,受初始弯曲的影响就越大;对于给定的目标形状,辊轮位置参数对实际输出半径的分布没有影响。     

Dynamic crack propagation with cohesive elements: a methodology to address mesh dependency

In this paper, two brittle fracture problems are numerically simulated: the failure of a ceramic ring under centrifugal loading and crack branching in a PMMA strip. A three‐dimensional finite element package in which cohesive elements are dynamically inserted has been developed. The cohesive elements' strength is chosen to follow a modified weakest link Weibull distribution. The probability of introducing a weak cohesive element is set to increase with the cohesive element size. This reflects the physically based effect according to which larger elements are more likely to contain defects. The calculations illustrate how the area dependence of the Weibull model can be used to effectively address mesh dependency. On the other hand, regular Weibull distributions have failed to reduce mesh dependency for the examples shown in this paper. The ceramic ring calculations revealed that two distinct phenomena appear depending on the magnitude of the Weibull modulus. For low Weibull modulus, the fragmentation of the ring is dominated by heterogeneities. Whereas many cracks were generated, few of them could propagate to the outer surface. Monte Carlo simulations revealed that for highly heterogeneous rings, the number of small fragments was large and that few large fragments were generated. For high Weibull modulus, signifying that the ring is close to being homogeneous, the fragmentation process was very different. Monte Carlo simulations highlighted that a larger number of large fragments are generated due to crack branching. Copyright © 2003 John Wiley & Sons, Ltd.     

基于区间分析的不确定性结构动力学模型修正方法

提出了一种基于区间分析的不确定性有限元模型修正方法。在区间参数结构特征值分析理论和确定性有限元模型修正方法基础上,假设不确定性与初始有限元模型误差均较小,采用灵敏度方法推导了待修正参数区间中点值和不确定区间的迭代格式。以三自由度弹簧-质量系统和复合材料板为例,采用拉丁超立方抽样构造仿真试验模态参数样本,开展仿真研究。结果表明,当仿真试验样本能准确反映结构模态参数的区间特性时,方法的收敛精度和效率均较高;修正后计算模态参数能准确反映试验数据的区间特性。所提出方法适用于解决试验样本较少,仅能得到试验模态参数区间的有限元模型修正问题。     

Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems

The following techniques for uncertainty and sensitivity analysis are briefly summarized: Monte Carlo analysis, differential analysis, response surface methodology, Fourier amplitude sensitivity test, Sobol' variance decomposition, and fast probability integration. Desirable features of Monte Carlo analysis in conjunction with Latin hypercube sampling are described in discussions of the following topics: (i) properties of random, stratified and Latin hypercube sampling, (ii) comparisons of random and Latin hypercube sampling, (iii) operations involving Latin hypercube sampling (i.e. correlation control, reweighting of samples to incorporate changed distributions, replicated sampling to test reproducibility of results), (iv) uncertainty analysis (i.e. cumulative distribution functions, complementary cumulative distribution functions, box plots), (v) sensitivity analysis (i.e. scatterplots, regression analysis, correlation analysis, rank transformations, searches for nonrandom patterns), and (vi) analyses involving stochastic (i.e. aleatory) and subjective (i.e. epistemic) uncertainty.     

Direct simulations of fiber network deformation and failure

A finite element model for 3D random fiber networks was constructed to simulate deformation and failure behavior of networks with dynamic bonding/debonding properties. Such fiber networks are ubiquitous among many living systems, soft matters, bio-materials, and engineering materials (papers and non-woven). A key feature of this new network model is the fiber–fiber interaction model that is based on AFM measurements from our earlier study. A series of simulations have been performed to investigate strain localization behavior, strength statistics, in particular, the variations of strength, strain-to-failure and elastic modulus, and their size dependence. Other variables investigated are fiber geometries. The result showed that, in spite of its disordered structure, strength and elastic modulus of a fiber network varied very little statistically, as long as the average number of fibers in the simulated specimen and the degree of fiber orientation are kept constant. However, strain-to-failure showed very significant statistical variations, and thus more sensitivity to the disordered structures.     

Probabilistic analysis of multi-site damage in aircraft fuselages

 Most aircraft fleets nowadays are operating under the concept of damage tolerance, which requires an aircraft to have sufficient residual strength in the presence of damage in one of its principal structural elements (PSE) during the interval of service inspections. The residual strength however is significantly reduced due to multi site damage (MSD). In the present paper, a probabilistic framework for the computation of the failure probability is developed. The MSD problem of a PSE is considered, where the uncertainties in crack initiation and crack growth as well as yield stress and fracture toughness are described by random variables. For the crack growth calculations the finite element alternating method [1], which avoids a remeshing of the finite element problem, is used. After specifying link up and failure criteria, importance sampling is employed to obtain the probability of failure of the PSE due to MSD. Received: 29 July 2002 / Accepted: 18 December 2002