Validated Prediction of the Strength Size Effect in Polycrystalline Silicon Using the Three‐Parameter Weibull Function

It has recently been shown that a three‐parameter Weibull function with a large threshold strength of ≈2 GPa is needed to accurately describe the failure strength statistics of micromachined polycrystalline silicon samples. Here, we explore how to apply this function to predict strength size effects over a size range of 100. A two‐parameter function is unsatisfactory in predicting the size effect. If a three‐parameter Weibull fit to only the largest specimen is used, the prediction also does not satisfactorily agree with strength data in smaller specimens. The prediction is greatly improved if the two largest specimens, a factor of 10 different in size, are used for fitting. It is further demonstrated that the threshold strength depends on geometry in notched samples due to their large stress gradients.     

Standardized Weibull statistics of ceramic strength

In order to estimate Weibull parameters in the Weibull statistical fracture theory as truly material properties independent of specimen geometry and loading mode, first the Weibull statistical fracture theory is transformed into the ordinary Weibull distribution function under certain approximation. Then the standardized format of ordinary Weibull distribution is introduced to enable Weibull modulus as the single parameter for estimation via the maximum likelihood method. The method of using standardized Weibull distribution for strength data synchronization and Weibull modulus estimation is validated by analyzing extensive strength data sets measured from uniaxial flexure, biaxial flexure and their combination, and from smooth and notched specimens. The technical path to estimate the scale parameter and threshold strength as material properties in the Weibull statistical fracture theory and effect of sample size on the estimation accuracy are also discussed.     

Fracture of Alumina Tubes under Combined Tension/Torsion

A fracture criterion for Al₂O₃ tubes is investigated for various loading paths under combined tension and torsion. Experimental data are compared with the Weibull statistical fracture theory. As the stress state approaches a shear dominant state, a tensile principal stress at fracture from the Weibull theory underestimates strengthening effects. By including the effect of the minimum principal stress in the tensile principal stress criterion, an empirical fracture criterion is proposed. Agreement between the proposed criterion and experimental data for Al₂O₃ tubes under combined tension/torsion and balanced biaxial tension is very good.     

Fracture of Al2O3 in Combined Tension/Torsion: II, Weibull Theory

The Weibull statistical fracture theory for multiaxial stresses has been extended to conditions of combined tension/torsion loading. At fracture, a tensile principal stress ratio σ₁ (tension/torsion) σ₁ (uniaxial tension) greater than one is predicted which is dependent on stress state, Weibull modulus, and fracture location. Comparison to experimental tension/torsion data for Al₂O₃ shows that the Weibull theory, although predicting correct trends, generally underestimates strengthening effects of the compressive principal stress, thus providing a conservative failure prediction. This discrepancy may be related to influences of stress state on crack-tip "process zone" behavior.     

Weibull Crack Density Coefficient for Polydimensional Stress States

A structural ceramic analysis and reliability evaluation code has recently been developed encompassing volume and surface flaw induced fracture, modeled by the two-parameter Weibull probability density function. A segment of the software involves computing the Weibull polydimensional stress state crack density coefficient from uniaxial stress experimental fracture data. The relationship of the polydimensional stress coefficient is derived herein for a shear-insensitive material with a random surface flaw population. In addition, a critique is made on the derivation of this relationship by Ikeda and Igaki, which appeared in a recent article in the Journal of the American Ceramic Society.     

Potential of SiC as a heat exchanger material in combined cycle plant

The short and long term mechanical properties of a sintered silicon carbide intended as a heat exchanger material have been investigated. The short term strength shows an acceptable scatter characterised by a Weibull modulus of seven from room temperature up to 1400°C. In the time-dependent regime failure occurs by subcritical crack growth from surface located inherent defects at high stresses. Below a threshold stress oxidation blunting of these surface defects occurs and causes a transition from subritical crack growth to diffusion creep as life-limiting mechanism. Unlike other ceramics, the threshold stress for subcritical crack growth falls within the low probability range of fast fracture. Failure mechanism maps presenting the life-limiting mechanisms of the investigated sintered silicon carbide over a range of stresses and temperatures are presented.     

Ceramics Reliability: Statistical Analysis of Multiaxial Failure Using the Weibull Approach and the Multiaxial Elemental Strength Model

Methodology for designing reliable ceramic components requires a precise evaluation and correlation of strengths in different stress states. The present paper compares the merits of the Weibull approach and the multiaxial elemental strength model on an experimental case involving mixed-mode failure in the presence of bimodal flaw populations (surface and volume flaws). The experimental data were obtained using flexure specimens of Si₃N₄ tested at various spans, with the purpose of enhancing shearing effects. The analysis of data was refined by developing an advanced postprocessor program to finite-element codes for failure probability determination based upon the Barnett-Freudenthal approximation of the Weibull approach and the multiaxial elemental strength model. In a second step, the strengths of the specimens exhibiting failures from the two concurrent populations of flaws (intermediate span) were predicted using both approaches from data obtained with different span lengths (long and short spans). Comparison with experimental data showed that the multiaxial elemental strength model is an improvement over the Weibull approach. It also allowed the short-span bending test to be assessed. Finally, important implications for structural design with ceramics are discussed.     

Large-scale Weibull analysis of H-451 nuclear-grade graphite rupture strength

A Weibull analysis was performed of the strength distribution and size effects for 2000 specimens of H-451 nuclear-grade graphite. The data, generated elsewhere, measured the tensile and four-point-flexure room-temperature rupture strength of specimens cut from a single extruded graphite log. Strength variation versus specimen location, size, and orientation relative to the parent body were compared. In our study, data were progressively and extensively pooled into larger data sets to discriminate overall trends from local variations and investigate the strength distribution. Issues regarding size effect, Weibull parameter consistency, and nonlinear stress–strain response were investigated using the Ceramics Analysis and Reliability Evaluation of Structures Life Prediction Program (CARES/Life) and WeibPar codes. Overall, the Weibull distribution described the behavior of the pooled data very well. The Weibull modulus was shown to be clearly consistent between different tensile specimen sizes and orientations. However, the issue regarding the smaller-than-expected size effect remained. This exercise illustrated that a conservative approach using a two-parameter Weibull distribution is best for designing graphite components with low probability of failure for the in-core structures in the proposed Generation IV high-temperature gas-cooled nuclear reactors. This exercise also demonstrated the continuing need to better understand the mechanisms driving stochastic strength response.     

Controlling and Testing the Fracture Strength of Silicon on the Mesoscale

Strength characterizations and supporting analysis of mesoscale biaxial flexure and radiused hub flexure single-crystal silicon specimens are presented. The Weibull reference strengths of planar biaxial flexure specimens were found to lie in the range 1.2 to 4.6 GPa. The local strength at stress concentrations was obtained by testing radiused hub flexure specimens. For the case of deep reactive ion-etched specimens the strength at fillet radii was found to be significantly lower than that measured on planar specimens. This result prompted the introduction of an additional isotropic etch after the deep reactive ion etch step to recover the strength in such regions. The mechanical test results reported herein have important implications for the development of highly stressed microfabricated structures. The sensitivity of the mechanical strength to etching technique must be accounted for in the structural design cycle, particularly with regard to the selection of fabrication processes. The scatter of data measured in the mechanical tests clearly illustrated the need to use a probabilistic design approach. Weibull statistics may be the appropriate means to describe the data, although a simple two-parameter Weibull model only provides a moderately good fit to the experimental data reported in this study.     

Maximum likelihood estimation for the three-parameter Weibull cdf of strength in presence of concurrent flaw populations

For a correct strength characterization of brittle materials, not only the maximum stress at fracture, but also the geometry of the specimens has to be considered thus taking into account the variable stress state and the size effect. Additionally, fracture may occur due to different fracture modes, as for example surface or edge defects. The authors propose a maximum likelihood estimator to obtain the cumulative distribution functions of strength for surface and edge flaw populations separately, both being three-parameter Weibull cdfs referred to an elemental surface area or elemental edge length, respectively. The method has been applied to simulated 3-point bending test data. The estimated Weibull parameters have been used to compute the cdfs of strength for specimens with different size, providing also the confidence bounds calculated by means of the bootstrap method. Finally, fracture data of 4-point bending tests on silicon carbide have been evaluated with the proposed method.     

Loading (L) Factors for the Biaxial Flexure Test

The dependence of the loading ( L ) factor on the Weibull modulus ( m ) was derived for the biaxial flexure test assuming a 2-parameter Weibull stress function and noninteracting orthogonal stresses. The specimen and test fiture dimensions influence the dependence significantly, whereas little difference is found between a 3-point or ring specimen support. The L factors are more strongly influenced by the Weibull modulus than in a 3-point bend test.     

Weibull analysis of ceramics under high stress gradients

The Weibull parameter m of the strength distribution of ceramics under high stress gradients differs from that for moderate stress gradients. This is shown for contact loading. Bars were loaded by oppositely concentrated forcers via rollers. For most investigated materials, measured contact strengths showed strongly reduced Weibull exponents compared with those from 4-point bending tests. This was the reason for a study, in which the effective volumes and surfaces for the two tests were compared and the influence of the strong stress gradients was considered. Under the assumption of the Weibull theory being valid, the effective surfaces and volumes were computed for the normal stress and the energy release rate criteria. In the second part, it will be shown that the strongly non-homogeneous stresses lead to a reduced Weibull exponent.     

Evaluation of the polymer-nonpolymer adhesion in particle-filled polymers with the acoustic emission method

The acoustic emission (AE) test was employed to study the debonding of glass beads from a polystyrene (PS) matrix for uniaxially loaded specimens of composites with a different adhesion level between the glass surface and PS. The adhesion was varied by tailoring the interface with endfunctionalized PS or polystyrene-block-poly-(2-vinylpyridine) copolymers adsorbed onto the glass surface. The number of beads was about 30 000 per specimen, which allowed good statistics to be obtained in the experimental data. The number of AE events, the debonding stress, the AE amplitude (AEA) for every signal, and the elongation of the specimen were recorded in the test. The experimental distribution of the number of AE signals per stress unit was fitted with a Weibull function and the maximum of the function was associated with the average debonding stress (ADS). The distribution of AE signals via AEA was also fitted with a Weibull function and the amplitude that corresponded to the maximum of the function was used as a parameter to characterize the AE energy released. All the parameters were used for the analysis of the failure mechanism of the composites. The ADS increases as the interface strength increases. The AEA measurement data usually should be fitted by two Weibull functions with two AEAs of relatively small and large energy (I and II, respectively). Both AEA-I and AEA-II decrease as the adhesion increases. These two maxima are assumed to characterize two different microdefects at the interface. AEA-I is caused by the propagation of the microdefects that were formed at the interface during the material preparation. The dewetting of the glass beads at larger stress affects the AE signals with larger AEA-II. It is suggested that the decrease in AEA with adhesion is caused by the propagation of microdefects towards the matrix. Simultaneous consideration of ADS and AEA allows the interfacial strength and location of the microcracks to be evaluated. The polydispersity of the PS matrix, the duration of sample preparation (melting under pressure), the rate of specimen deformation in the test, and the volume fraction of the filler within the range 2-15 vol.% have very slight effects on the results.     

Distribution distortion in the statistical analysis of strength data when using multiple specimen batches

The strength distribution of Zerodur® (Schott AG) prepared by grinding and etching is determined through statistical analysis to be best described as normal. Although a 3-parameter Weibull appearance is exhibited when multiple specimen batches are combined and analyzed together, this appearance is the result of mixing of components of variation. As a result of the particular type of flaw population, little effect of scale is exhibited. Caution is advised when assuming a 3-parameter Weibull distribution and associated threshold solely based on distribution shape. Fractography to understand the flaw distribution should be employed.     

不同拉伸条件下基于Weibull模型的粘胶基碳纤维强度分布研究

从粘胶基碳纤维的拉伸实验得到其 S-S曲线和强度、模量、断裂伸长等力学性能数据 ,表明该材料是典型的脆性断裂 ,且断裂分散性较大。采用 VB编程软件设计了 Weibull程序 ,该模型能计算出碳纤维的平均强度、Weibull模数、尺度参数 ,并能模拟碳纤维强度的累积概率分布和概率密度曲线。不同氧化拉伸条件下强度的实验数据基本上落在程序模拟出的累积概率分布直线上 ,证明了该数学模型适用于分析碳纤维强度分布。在氧化完全松弛的条件下 ,粘胶基碳纤维的平均强度较高 ,但 Weibull模型分析的结果表明氧化拉伸比为 -5 %时 ,Weibull模数最大 ,不匀率最小 ,而氧化拉伸对粘胶基碳纤维模量没有显著影响。     

Mechanical Test for Anisotropy; Failure of Aluminoborosilicate Glass Fibers Under Combined Loadings of Tension and Torsion

The failure behavior of aluminoborosilicate glass fibers in pure tension, pure torsion, and under combined loadings of tension and torsion was investigated. It was established that the glass fibers failed in accordance with the predictions of the Weibull statistical theory of failure and the maximum principal stress theory of failure. Therefore, it could be concluded that the fibers were isotropic. A general procedure was developed for detecting structural anisotropy by experimentally evaluating the average torsional and tensile strengths of a material and comparing their ratio with the ratio predicted when the experimental data are analyzed in terms of the Weibull theory.     

Large-Deformation Strength Analysis of Chemically Strengthened Glass Disks

A large-deformation stress analysis of a ring-on-ring bending fracture test, using the finite-element method, was used to derive the strength of chemically strengthened and nonstrengthened glass disks from experimental test data. Agreement between measured and predicted disk deflections was very good. The predicted strength data were analyzed using Weibull statistics.