Miner累积损伤理论的试验验证和统计分析

本文通过对LY12-CZ铝合金板材光滑试件和中心孔试件的常幅加载和变幅加载疲劳试验,对Miner累积损伤理论进行了试验验证,并对累积周比的分散性作了统计分析,发现累积周比的分散性服从Weibull分布,因此本文提出以最小累积周比值或以与一定存活率相应的累积周比值进行安全寿命估计的观点。

加卸载响应比与损伤变量关系研究

首先介绍加卸载响应比的2个基本参量,并从损伤力学的基本理论出发,引入损伤变量,结合加卸载响应比方法的思想,推导用损伤和应变定义的加卸载响应比,并以Weibull分布作为随机分布函数,建立且分析损伤变量与加卸载响应比之间的联系,并进一步考察Weibull指数对两者关系的影响.然后利用岩石破坏声发射实验的数据,用声发射数密度描述岩石试件的损伤演化过程,并由此根据损伤和应变定义的加卸载响应比,结合实验过程中的损伤和应变,计算花岗岩试件的加卸载响应比曲线,并与用Benioff应变作响应量计算的加卸载响应比曲线进行比较,发现两者具有相当程度的一致性,两者的加卸载响应比值都经历了出现异常→升高到最大值→急剧回落→岩石试件破裂或失稳,这一规律与实际地震震例中加卸载响应比的演化趋势是一致的,同时也说明在实际地震预测中用Benioff应变作为响应量来计算加卸载响应比是合理的.最后介绍意大利那不勒斯大学完成的二层楼房的加卸载实验,且对实验结果进行分析.建立的加卸载响应比与损伤变量的关系,不仅为加卸载响应比用来定量研究脆性介质的损伤程度提供更为坚实的基础,而且也可能为评估大型建筑的损伤和预测工程事故开辟新道路.

一种岩石概率型本构及其应用

在岩石材料非均匀特性的基础上,建立了弹塑性概率材料本构.认为材料单元的切线模量与破坏强度都是服从二参数Weibull分布的随机量,通过用户材料子程序将该本构导入到LS-DYNA3D中,推导了程序流程,分析了导入过程中的关键技术.算例验证表明,应用该本构模拟得出的现象与实验观测完全一致,这是确定性本构无法实现的.

Effects of Microstructural Heterogeneity on The Spallation Behavior of Materials

It is of utmost importance to understand the spallation behaviour of heterogeneous materials. In this paper, a driven nonlinear threshold model with stress fluctuation is presented to study the effects of microstructural heterogeneity on continuum damage evolution. The spallation behavior of heterogeneity material is analyzed with this model. The heterogeniety of mesoscopic units is characterized in terms of Weibull modulus m of strength distibution and stress fluctuation parameter k. At high stress, the maximum damage increases with m; while at low stress, the maximum damage decreases. In addition, for low stress, severe stress fluctuation causes higher damage; while for high stress, causes lower damage.

Catastrophic Rupture Induced Damage Coalescence in Heterogeneous Brittle Media

In heterogeneous brittle media, the evolution of damage is strongly influenced by the multiscale coupling effect. To better understand this effect, we perform a detailed investigation of the damage evolution, with particular attention focused on the catastrophe transition. We use an adaptive multiscale finite-element model (MFEM) to simulate the damage evolution and the catastrophic failure of heterogeneous brittle media. Both plane stress and plane strain cases are investigated for a heterogeneous medium whose initial shear strength follows the Weibull distribution. Damage is induced through the application of the Coulomb failure criterion to each element, and the element mesh is refined where the failure criterion is met. We found that as damage accumulates, there is a stronger and stronger nonlinear increase in stress and the stress redistribution distance. The coupling of the dynamic stress redistribution and the heterogeneity at different scales result in an inverse cascade of damage cluster size, which represents rapid coalescence of damage at the catastrophe transition.

平台用高强度钢多次补焊及TIG熔修的疲劳分析

海洋石油平台在服役期内，由于受到各种环境载荷的交变作用，其受力的焊接构件不可避免地会出现疲劳裂纹。对出现的疲劳裂纹进行补焊修复，可以延长平台的使用期，带来巨大的经济和社会效益。本文通过实验研究，考察了多次补焊对海洋平台用高强度钢焊接接头疲劳性能的影响，以及是否可以通过TIG熔修工艺的采用提高补焊构件的疲劳寿命，达到延长补焊后平台的使用寿命，减少补焊次数的目的。本文涉及的主要研究内容如下：1、考察了多次补焊对焊接接头疲劳寿命的影响。通过对平台用高强度钢焊接接头初焊试样，以及重复三次补焊试样的疲劳实验，得到了各次补焊试样在对数正态分布和威布尔分布下的寿命估算式和考虑了存活率的R-S-N曲线。对实验结果进行了比较和分析，从而考察了多次补焊对焊接接头疲劳性能的影响。2、在每次补焊后，由于焊趾外移造成了焊趾角的变化，使得焊趾处的应力集中系数发生了变化。本文采用有限元方法计算了试样的应力场分布，得到了焊趾处的应力集中系数与焊趾角的关系。考虑到应力集中系数的变化后，对多次补焊试样疲劳实验的实验数据进行了重新处理。3、考察了多次补焊熔修对焊接接头疲劳寿命的影响。本文通过对平台用高强度钢焊接接头补焊及补焊熔修试样的疲劳实验，得到了多次补焊和补焊熔修试样对数正态分布和威布尔分布下的寿命估算式和考虑了存活率的R-S-N曲线。对实验结果进行了比较和分析，从而考察了补焊后采用TIG熔修工艺对焊接接头疲劳性能的影响。4、一次补焊后进行熔修处理可以极大的改善试样的疲劳性能，提高试样的疲劳寿命。但这一结果是在平焊位条件下得出了。为了将这一结果用于实际，还要考察实际操作中大量采用的立焊位条件下熔修的效果。本文完成了立焊位条件下补焊及补焊熔修试样的疲劳实验，得到了相应的S-N曲线。5、疲劳实验数据的处理中，大量采用威布尔分布作为数据统计的分布类型。但由于威布尔分布具有三个参数，给求解带来一定的困难。本文提出一种以在威布尔坐标纸上，数据分布的线形相关系数作为判据的求解威布尔参数的方法，并通过程序加以实现。为了对疲劳实验数据的处理更加方便和直观，编制了“疲劳实验数据处理程序“，将对疲劳实验数据的编辑、分布参数的求解、不同存活率下的疲劳寿命估算和各种分布类型图以及R-S-N曲线的绘制工作包括在该程序中。

非均匀脆性介质破坏的非平衡非线性统计演化

在白以龙小组已有工作的基础上，利用他们提出的简化的耦合斑图模型，几种可能的应力重分配模型在该论文中得到了进一步的讨论。通过不同的应力重分配模型，我们发现了该类演化过程中的三个一般规律，这些规律对于类似的动力过程的预报（例如，非均匀介质的破坏）提供了线索。首先，我们采用系综统计的方法，对相同宏观参量的大量样本的强度分布作了考察，结果表明：宏观强度的统计结果可以非常好的拟合为Weibull分布，且其Weibull模数与系统的大小、应力重分配的方式以及细观单元的强度分布相关。其次，在模拟过程中，对演化过程中的能量释放、损伤事件的统计发现，它们存在标度行为，而且这一标度主要归因于灾变点附近的损伤事件。这一现象表明这一转变具有某种临界特征。最后，对于我们模型中的动力学过程，我们发现了灾变预报的线索。从样本演化过程 中的能量释放规律来看，我们发现有两件事是有意义的：一是辨别出主破坏的发生点（在这一时修，系统中的大部分能量得到释放）；另外，给出转变点（整体稳定转化为演化诱致灾变）的预报。对前一个问题，我们通过考察系统在GS和EIC段的应力损伤涨落特征可以给出回答，通常，在EIC段的最大应力涨落（通常出现在主破坏过程中）比在GS过程中的最大应力涨落高一个数量级，根据这一差异，可以设立一个应力涨落的警戒值来判断系统所处的演化状态。对于后者，受到地震预报中采用的加卸载响应比（LURR）的启发，我们通过对系统中的外回转应力或损伤单元施加一个微增扰动，然后，根据系统在扰动前后释放的能量和相应的扰动，就可以得到临界敏感系数，临界敏感系数在灾变点附近迅速增加，在灾变点之迅速下降到1附近-我们称这一特征为临界敏感性。不同的应力重分配模型下得到了类似的现象，由此看来，对于 类似的动力学过程，临界敏感性是一个一般的特征。这一特征可能为我们对非均匀脆性介质的破坏提供线索。

Fumar antecipa a menopausa ? Evidências do Estudo Pró-Saúde

A menopausa é definida pelo momento em que a menstruação cessa permanentemente como conseqüência da falência ovariana. Uma vez estabelecida, há um aumento no risco de doença coronariana, doença de Alzheimer, osteoporose e fraturas e sua antecipação está relacionada a maiores índices de mortalidade. Com o envelhecimento geral da população mundial, as mulheres passaram a viver de um terço a metade de suas vidas no período pós-menopáusico e, conseqüentemente, a pesquisa de condições associadas à menopausa ganham importância. A idade da menopausa e os fatores que a influenciam variam entre os diversos estudos e poucos estudos brasileiros abordam o tema. O fumo tem sido associado à Antecipação da idade da menopausa. O objetivo deste estudo é analisar as diversas dimensões de associação entre o fumo e a idade da menopausa, levando em consideração possíveis relações de dose-resposta. Com base em dados do Estudo Pró-Saúde, foi realizado um estudo seccional utilizando-se para a análise o modelo de sobrevida paramétrico de riscos proporcionais com distribuição de Weibull. Os resultados apontam para uma redução em 32% do risco de menopausa entre fumantes ativas, que a alcançam 2,5 anos mais tarde que nunca fumantes, ajustando-se para escolaridade e paridade. Entre fumantes ativas, no entanto, foram sugeridos aumentos de risco de 123% e 192% para fumantes de 10 a 20 cigarros por dia e de mais de 20 maços-ano respectivamente, quando comparadas às fumantes de menos de 10 cigarros por dia e menos de 10 maços-ano. Nesses grupos, a menopausa foi antecipada em 3,3 anos e 4,4 anos, respectivamente. Em relação ao tempo decorrido entre cessar o tabagismo e a idade menopausa, duração do fumo ou idade de início, não foram encontradas associações. Neste estudo, em possíveis efeitos de dose-resposta, o fumo apresenta-se como fator associado à antecipação da idade da menopausa em alguns aspectos, embora não em outros.

Comportamento mecânico de cerâmica produzida a partir de reciclagem de para-brisa de automóvel

É crescente o interesse nos materiais cerâmicos, devido as suas características como baixa massa específica e maior resistência a ambientes agressivos do que a maioria das ligas metálicas. Este trabalho tem o objetivo de produzir a partir de material descartado, cerâmicas em diferentes temperaturas de sinterização e avaliar a sua tensão de ruptura em ensaio de flexão de três pontos e a confiança desta medida. Devido ao custo de produzir pó de vidro, tanto pelo alto gasto de energia para fundir a matéria-prima como pelo consumo de minerais industriais, foi proposto utilizar vidro de para-brisa obsoleto reduzindo despesas operacionais e definindo um destino econômico e ambiental viável para estes rejeitos. A metodologia consistiu-se na obtenção do pó de vidro com características adequadas para ser conformado e sinterizado. Foram usadas duas composições e quatro tratamentos térmicos para obter oito materiais. Uma composição com apenas o pó oriundo da moagem de para-brisa e outra com este pó mais 4% de óxido de nióbio. A resistência à flexão dos produtos obtidos foi avaliada. Utilizou-se a estatística de Weibull para caracterizar estes resultados. Os resultados obtidos indicam que o material de composição pó de vidro e temperatura final de sinterização de 650C obteve a maior resistência mecânica entre os materiais sintetizados. A adição do óxido de nióbio provoca um decréscimo na resistência mecânica se comparada com o material sem a adição deste óxido. Entretanto, comparando as duas composições na mesma temperatura final de sinterização, a adição de óxido de nióbio provocou um aumento no módulo de Weibull, excetuando-se dois de oito materiais obtidos. As diferentes composições e temperaturas de sinterização afetaram as propriedades mecânicas dos materiais obtidos.

An evaluation of back-calculation methodology using simulated otolith data

I simulated somatic growth and accompanying otolith growth using an individual-based bioenergetics model in order to examine the performance of several back-calculation methods. Four shapes of otolith radius-total length relations (OR-TL) were simulated. Ten different back-calculation equations, two different regression models of radius length, and two schemes of annulus selection were examined for a total of 20 different methods to estimate size at age from simulated data sets of length and annulus measurements. The accuracy of each of the twenty methods was evaluated by comparing the back-calculated length-at-age and the true length-at-age. The best back-calculation technique was directly related to how well the OR-TL model fitted. When the OR-TL was sigmoid shaped and all annuli were used, employing a least squares linear regression coupled with a log-transformed Lee back-calculation equation (y-intercept corrected) resulted in the least error; when only the last annulus was used, employing a direct proportionality back-calculation equation resulted in the least error. When the OR-TL was linear, employing a functional regression coupled with the Lee back-calculation equation resulted in the least error when all annuli were used, and also when only the last annulus was used. If the OR-TL was exponentially shaped, direct substitution into the fitted quadratic equation resulted in the least error when all annuli were used, and when only the last annulus was used. Finally, an asymptotically shaped OR-TL was best modeled by the individually corrected Weibull cumulative distribution function when all annuli were used, and when only the last annulus was used.

北京地区植被景观的空间分布特征

本文利用地理信息系统(GIS)技术、景观生态学理论和方法、分形理论以及统计分析方法对北京地区植被景观的空间分布特征进行了分析，并对景观格局和景观多样性的分析方法进行了探讨，结果表明： (1)对几乎所有的斑块类型，其斑块大小的分布都不是对称的，而是右偏的。4种概率分布（Г—分布、对数正态分布、Weibull分布和（负）指数分布）都只能刻划部分斑块类型，并且服从对数正态分布的斑块类型最多，服从（负）指数分布的斑块类型最少。 (2)随着斑块面积的增加，边界效应越来越小，而斑块形状越来越不紧凑。 (3)分形分析识别出本地区植被景观中的两个尺度域：一个是斑块面积小于(大约)2.7km2，另一个是斑块面积大于(大约)2.7km2。两个域中的斑块复杂程度有很大差异，后一个域中的斑块明显比前一个域中的斑块复杂，并且随着斑块面积的增加，斑块形状越来越复杂。 (4)用斑块数作为多度指标时，该景观的斑块类型一多度分布服从（截断）对数正态分布和（截断）负二项分布，不服从对数级数分布和几何分布。用斑块面积作为多度指标时，该景观的斑块类型一多度分布服从对数正态分布、Weibull分布和Г—争布，不服从正态分布。从而该景观的斑块类型一多度分布不是对称的，也是右偏的。在4个优势度／多样性模型中，“生态位优先占领”模型和Zipf-Mandelbrot模型可以较好地刻划该景观的斑块类型一多度关系。 (5)样本大小对多样性测度有直接的影响。如果这种影响比较小，就说明测度指标比较稳定。三个丰富度指数中，Ri比R2和R3更稳定;五个多样性性指数中，D和Di最稳定，OD最不稳定，因此，OD是用于景观多样性监测的理想指标;五个均匀度指数中，Jgi最稳定。根据设计的3种计算临界样方数量（即多样性测度指标达到稳定时的样方数量）方法的计算结果，上述几个最稳定的测度指标在通常情况下只需要几个样方(即总抽样面积为数百km2)就达到稳定状态。 (6)斑块类型数目随面积的增加而增加。根据四个评价指标的评价结果，认为双曲线对该景观的斑块类型一面积关系的拟合效果最好。 (7)样本较大（对于一阶刀切估计，大于30个样方;对于二阶刀切估计，大于60个样方）时，刀切法能够给出斑块类型数目(NPT)较好的估计;样本较小（小于30个样方）时，Mingoti和Meeden提出的经验贝叶斯方法能够对NPT给出比刀切法和自助法更好的估计。斑块类型一面积曲线外推虽然也能给出NPT较好的估计，但这种方法需要慎重使用，不能外推得很远。 (8)列联表分析表明，该植被景观中的斑块类型与土壤类型、岩石类型、海拔高度和坡向各因子之间均存在显著的相关性。植被景观多样性与岩石类型多样性和地形多样性之间也均呈显著的正相关关系，即植被景观多样性随岩石类型多样性和地形多样性的增加而增加。但植被景观多样性与土壤类型多样性之间不存在显著的线性相关或秩相关关系，这可能是由于二者的分类体系不吻合。植被景观多样性与总的道路密度和第二类道路密度之间均呈显著的负相关关系，而与第一类和第三类道路密度之间的关系都不显著。这反映出景观样本单元(10kmxlOkm)的尺度对应于第二类道路的影响尺度。而道路密度在一定程度上反映了人类活动的强度，因此，在10kmxlOkm这个尺度上，人类活动愈剧烈，景观多样性就愈小。

内蒙古锡林河流域湿地植被研究及道路对该区草地退化的影响

通过群落生态学和景观生态学方法，结合GIS、RS技术对锡林河流域湿地植被进行了研究。结果表明：流域湿地面积为301.62km2，占流域面积的3%左右。尽管面积相对较小，但是物种丰富，群落结构多样。植被调查数据显示基本确定的植被型4个，植被亚型6个，群系组16个，群系68个，区系成分以泛北极种为主，占69%，相对简单;按照水分生态型划分，中生物种占最多，为44.32％;按生活型分以多年生草本为主占50％以上;科属分布相对复杂，隶属39个科，其中禾本科和菊科是最大的两个科，所占比例仅有17.30％和12.43％，其他科没有明显的优势性，充分说明湿地优越的生境可以满足多种植物共同生长。 多度分布是研究物种多样性分布的重要组分，同时反映了群落结构的特性。以常用的Lognormal、Logseries和Weibull、Exp、Power模型来拟和6个典型草甸群落和踏头草甸群落的物种多度分布，分log-相对多度-物种级数和物种-游程两种形式进行比较;同时，对于典型草甸群落和踏头群落区分常见种、偶然种等进行细化，深入分析群落多度的变化。结果表明，5个模型对于log-相对多度-物种级数在整个群落水平上均不能很好的拟和，50%以上的点都落在95%置信区间以外;但是对常见种和偶然种的拟和情况要好，Weibull、Power和Logseris模型分别对典型草甸群落常见种、偶然种和中间种能很好的拟和，而Logseries和Power模型对于踏头群落的常见种和偶然种拟和较好。5个模型都能较好的拟和物种-游程分布，其中K—S检验结果表明：Lognormal模型对于无脉苔草、针苔草和荸荠这类相对湿润环境下的典型草甸群落拟和较好，对于长叶火绒草和密花风毛菊群落Weibull拟和最好，Power 模型拟和箭叶橐吾最好，踏头草甸拟和最好的是Logseries模型，踏头间拟合最好的是Exp模型。不同的拟和模型应用于不同的群落类型，可以看出湿地群落的复杂性和生境的多样性。区分常见种和偶然种的拟合结果表明典型群落和踏头群落表现一致，即Lognormal模型对所有种拟和是最好的，而Power模型对偶然种的拟和是最好的，同时，Lognormal对典型草甸群落的中间种拟和也是最好。从中可以看出典型草甸群落和踏头群落尽管在表现形式上不同，但是群落的内部仍存在相似的联系，可能跟相似物种的作用有关。 根据湿地表观类型、植被水分状态和航片判别能力，结合实地调查，采用监督分类的方法将锡林河流域的湿地划分为低湿地草甸、盐化草甸和沼泽三种类型。自1984年以来20多年的时间中，锡林河流域的湿地发生了巨大的变化。尽管总的面积没有太大变化，但是湿地类型发生转化。中上游的低湿地草甸面积减少8.94％，沼泽面积减少30.82％，同时，盐化草甸的面积增加了15.98％。增加的盐化草甸主要是另外两种湿地类型转化而成的，中游水库截流，加速中下游草甸的盐化是锡林河流域湿地变化的主要原因。利用GIS技术依据探讨不同湿地的空间变化，分析沙化对湿地变化的影响，结果表明：沙化只对少数湿地有影响，发育良好的湿地即使处在相对强烈的沙化环境下，仍能保持不变。接着，分析了人类直接干扰对湿地变化的影响，缓冲区居民点分析结果表明：近20年来，位于湿地周边的居民点分布格局发生显著的变化。1980年代，居民点分布在盐化草甸周边的最多，到2004年，居民点在沼泽草甸分布数量为最多，该类湿地水、草和资源最为丰富，人类直接的干扰最大，进而转化成另外两类，减少面积最大。低湿地草甸是物种丰富，结构复杂的一种湿地，抗干扰能力强，恢复能力也强，因此相对的变化面积较小。以锡林浩特市水库上下游的湿地植被物种和群落结构的变化，证明了水量减少是湿地数量、结构改变的直接影响因子。

Fracture of brittle lattice materials: A review

The mechanics of failure for elastic-brittle lattice materials is reviewed. Closed-form expressions are summarized for fracture toughness as a function of relative density for a wide range of periodic lattices. A variety of theoretical and numerical approaches has been developed in the literature and in the main the predictions coincide for any given topology. However, there are discrepancies and the underlying reasons for these are highlighted. The role of imperfections at the cell wall level can be accounted for by Weibull analysis. Nevertheless, defects can also arise on the meso-scale in the form of misplaced joints, wavy cell walls and randomly distributed missing cell walls. These degrade the macroscopic fracture toughness of the lattice. © 2010 Springer Science+Business Media B.V.

Fracture Properties of LPCVD Silicon Nitride and Thermally Grown Silicon Oxide Thin Films From the Load-Deflection of Long Si3N4 and SiO2/Si3N4 Diaphragms

The bulge test is successfully extended to the determination of the fracture properties of silicon nitride and oxide thin films. This is achieved by using long diaphragms made of silicon nitride single layers and oxide/nitride bilayers, and applying comprehensive mechanical model that describes the mechanical response of the diaphragms under uniform differential pressure. The model is valid for thin films with arbitrary z-dependent plane-strain modulus and prestress, where z denotes the coordinate perpendicular to the diaphragm. It takes into account the bending rigidity and stretching stiffness of the layered materials and the compliance of the supporting edges. This enables the accurate computation of the load-deflection response and stress distribution throughout the composite diaphragm as a function of the load, in particular at the critical pressure leading to the fracture of the diaphragms. The method is applied to diaphragms made of single layers of 300-nm-thick silicon nitride deposited by low-pressure chemical vapor deposition and composite diaphragms of silicon nitride grown on top of thermal silicon oxide films produced by wet thermal oxidation at 950 degrees C and 1050 degrees C with target thicknesses of 500, 750, and 1000 mn. All films characterized have an amorphous structure. Plane-strain moduli E-ps and prestress levels sigma(0) of 304.8 +/- 12.2 GPa and 1132.3 +/- 34.4 MPa, respectively, are extracted for Si3N4, whereas E-ps = 49.1 +/- 7.4 GPa and sigma(0) = -258.6 +/- 23.1 MPa are obtained for SiO2 films. The fracture data are analyzed using the standardized form of the Weibull distribution. The Si3N4 films present relatively high values of maximum stress at fracture and Weibull moduli, i.e., sigma(max) = 7.89 +/- 0.23 GPa and m = 50.0 +/- 3.6, respectively, when compared to the thermal oxides (sigma(max) = 0.89 +/- 0.07 GPa and m = 12.1 +/- 0.5 for 507-nm-thick 950 degrees C layers). A marginal decrease of sigma(max) with thickness is observed for SiO2, with no significant differences between the films grown at 950 degrees C and 1050 degrees C. Weibull moduli of oxide thin films are found to lie between 4.5 +/- 1.2 and 19.8 +/- 4.2, depending on the oxidation temperature and film thickness.

Fracture properties of silicon carbide thin films by bulge test of long rectangular membrane

This paper reports the mechanical properties and fracture behavior of silicon carbide (3C-SiC) thin films grown on silicon substrates. Using bulge testing combined with a refined load-deflection model of long rectangular membranes, which takes into account the bending stiffness and prestress of the membrane material, the Young's modulus, prestress, and fracture strength for the 3C-SiC thin films with thicknesses of 0.40 and 1.42 mu m were extracted. The stress distribution in the membranes under a load was calculated analytically. The prestresses for the two films were 322 +/- 47 and 201 +/- 34 MPa, respectively. The thinner 3C-SiC film with a strong (111) orientation has a plane-gstrain moduli of 415 +/- 61 GPa, whereas the thicker film with a mixture of both (111) and (110) orientations exhibited a plane-strain moduli of 329 +/- 49 GPa. The corresponding fracture strengths for the two kinds of SiC films were 6.49 +/- 0.88 and 3.16 +/- 0.38 GPa, respectively. The reference stresses were computed by integrating the local stress of the membrane at the fracture over edge, surface, and volume of the specimens and were fitted with Weibull distribution function. For the 0.40-mu m-thick membranes, the surface integration has a better agreement between the data and the model, implying that the surface flaws are the dominant fracture origin. For the 1.42-mu m-thick membranes, the surface integration presented only a slightly better fitting quality than the other two, and therefore, it is difficult to rule out unambiguously the effects of the volume and edge flaws.