Two-parameter Weibull Statistics Research Articles

Vacancy-mediated inelasticity in two-dimensional vanadium-based dichalcogenides.

Amongst the two-dimensional (2D) transition metal dichalcogenide (TMDC) family, VX2 (X = S, and Se) are key members for next-generation electronic, spintronic, and energy storage applications. Structural defects may be introduced in these emerging atomically thin materials during synthesis, which can lead to both desirable and detrimental impacts on the physical and chemical properties. Owing to their brittle behaviour, defects may deteriorate the mechanical properties of 2D TMDCs drastically, causing reliability-related issues critical for long-term device-scale applications. In this context, more than 1600 classical molecular dynamics simulations were performed to obtain estimates of chirality-dependent failure stresses of 2D VX2 under technologically relevant conditions that are encountered in typical industrial and lab-scale experiments. The energy barriers associated with the fracture of defective samples were calculated using a thermal activation-based analytical model. Furthermore, the effect of various concentrations of randomly distributed defects on the mechanical reliability of VX2 samples was explored using two-parameter Weibull statistics. These monolayers were found to possess strong flaw tolerance and high Weibull modulus values, making them promising for flexible electronics and energy storage.

Tensile and flexural mechanical attributes of hybrid carbon/basalt fiber metal laminates under various hybridization and stacking sequences

In this work, experimental measurements and theoretical validation are adopted to examine the tensile and flexural behaviors of titanium-based carbon/basalt fiber metal laminates under various hybridization ratios and stacking sequences. Firstly, the mechanical response and damage patterns of fiber metal laminates (FMLs) subjected to different tensile and flexural loads have been explored. By observing the fracture surfaces of FMLs with various hybridization ratios and stacking sequences, scanning electron microscopy (SEM) has been used to identify the related microscopic damage patterns. The principal damage mechanism were attributed to fiber/matrix debonding, matrix microcrack, fiber pull-out, and delamination. Subsequently, to analyze the discreteness of experimental results and evaluate the theoretical flexural strength of FMLs under different conditions, the two-parameter Weibull statistics model for engineering application of FMLs was established. These results indicate that the tensile and flexural strength of FMLs can be improved by altering the hybridization ratios and stacking sequences. The thorough understanding of the mechanical behavior and failure mechanism of FMLs under various hybridization conditions can provide the basis for the design and utilization of FMLs structures.

Tensile mechanical behavior and failure mechanisms of fiber metal laminates under various temperature environments

This paper mainly investigates the tensile behavior of fiber metal laminates (FMLs) under various temperatures ranged from 25 ℃ to 175 ℃ combined experimental measurement, theoretical model and numerical technique. Firstly, based on numerous tensile tests under various temperatures, different tensile load–displacement response and damage modes for FMLs can be obtained and compared. And in virtue of the scanning electron microscopy (SEM), microscopic damage modes are identified through the fracture surface scanning of FMLs subjected to different temperatures. Subsequently, in order to analyze the discreteness of experimental results and evaluate the theoretical tensile strength of FMLs under different temperatures, the two-parameter Weibull statistics model is established for engineering application of FMLs. Finally, the effect of layer direction on damage modes and progressive damage evolution process of FMLs is characterized numerically. These results indicate that the tensile strength of FMLs represents a nonlinear downward trend with the increase of temperature. The complex failure mechanisms under different temperatures can be captured from SEM observation, including fiber/matrix debonding, fiber pull-out, matrix microcrack, delamination damage and so on. The deeper understanding of tensile behavior and failure mechanisms of FMLs under thermal conditions can be provided for the design and utilization of FML structures.

A PFA methodology to investigate UD composites in fatigue comprising a KTF-based model for matrix damage and stochastic fiber failure prediction

A physics-based, multiscale, progressive fatigue damage analysis methodology for continuous fiber-reinforced polymer composites was further developed and applied to the problem of fatigue damage prediction in open hole tension (OHT) coupons comprised of multidirectional laminates undergoing tension–tension loading. Based off lamina level calibration data, cumulative damage analysis of matrix and fiber constituents under cyclic loading condition was implemented with the objectives to: (i) predict the onset and the progression of subcritical matrix failures as well as the ultimate failure due to fiber fracture, and (ii) observe the effects of damage mode interactions on the progressive failure process. A kinetic theory of fracture (KTF) based durability prediction methodology was utilized to keep track of the matrix damage transpiring in the intra- and inter-laminar layers of the laminate. Tensile strengths were randomly assigned to the fibers using two-parameter Weibull statistics, while the maximum stress theory in tandem with the linear damage accumulation rule was applied to capture the tensile rupture of the reinforcements. Progressive fatigue damage prediction of OHT coupons consisting of the unidirectional composite IM7/977-3 and layup schedules of [0/45/90/−45]2S and [60/0/−60]3S were carried out and benchmarked against available experimental data. It was observed that the KTF-based fatigue model captured the damage state at individual layers, overall loss of mechanical performance, and stochastic nature of the ultimate failure well. The effects of interactions among the damage modes, namely matrix cracking, delamination, and fiber rupture were also investigated and it was recognized that such interactions are a critical feature of the progressive failure process in composite structures.

Statistical Analysis of Vickers Indentation Fracture Toughness of Y-TZP Ceramics

The purpose of this study was to investigate and analyse the fracture toughness of yttria stabilized tetragonal zirconia polycrystal (Y- TZP) dental ceramics by means of the Vickers indentation technique. Fracture toughness was analysed under the following loads: 29.42 N, 49.03 N, 196.13 N and 294.20 N. Cracks, developed from the vertices of a Vickers indentation, were measured and used for determination of fracture toughness using nine different models. It was found that the fracture toughness depends on the indentation load, morphology of indentation cracking and the applied model of fracture resistance. The set of fracture toughness data was analysed using the two-parameter Weibull statistics. The Weibull modulus and scale parameter were determined by the linear regression method. It was concluded that the Weibull distribution can describe fracture toughness of Y-TZP ceramics properly.

An approach to scaling size effect on strength of quasi-brittle biomedical materials

Two-parameter Weibull statistics is commonly used for characterizing and modeling strength distribution of biomedical materials and its size dependence. The calibrated scale parameter and shape factor are usually sensitive to specimen size. Since Weibull statistics is subject to the weakest link postulate, this work proposed to directly resort to the weakest-link formulation for the cumulative failure probability to characterize size effect on strength distribution of quasi-brittle biomedical materials. As a preliminary examination, the approach was assessed by two sets of published strength data. It shows that the resultant expression for the cumulative probability follows either Weibull distribution or other type of distributions. The calibrated model parameters are independent of specimen size, so they can be used to transfer strength distribution from one set of specimens to another set of specimens with geometrical similarity under same loading mode. These initial results motivate a more comprehensive validation of the proposed approach to proceed via a larger set of case studies covering different quasi-brittle biomedical materials over a wider range of size variation.

Tensile elongation of unidirectional or laminated composites combining a brittle reinforcement with a ductile strain and strain-rate hardening matrix

We use the long-wavelength model of Hutchinson and Neale (1977) and Ghosh (1977) to estimate the uniform tensile elongation of two-phase composites deforming quasistatically according to the equistrain rule of mixtures, in which one phase is ductile while the other fractures progressively according to two-parameter Weibull statistics. We use shear-lag models in the literature to quantify load transfer from the ductile phase to the fractured brittle phase, and to estimate the influence of matrix strain and strain-rate hardening, of brittle phase fracture characteristics, and of phase volume and strength ratios, on the composite strain to failure as dictated by the onset of unstable necking. Calculations show that strain and strain-rate hardening of the ductile phase do relatively little to increase the ductility of the composite. Two parameters play a dominant role, namely the brittle-phase Weibull modulus and a dimensionless parameter describing load transfer across the two phases. The main practical implication of this analysis is that, to produce reasonably ductile two-phase composites, the best strategy is to aim for small layer thicknesses.

Shear strength of pultruded composite pins with external confinement

Weight reduction using composites has gained increasing attention in recent times. In this study, pultruded composite pins (unconfined and confined) were manufactured and tested by using a custom double shear testing fixture. Different configurations were applied for confinement of the composite pins, including weft-knitted fabrics (plain, 1 × 1 rib, and Milano), woven fabrics and E-glass 130 tex fibers/adhesive cloth. They were externally wrapped and bonded to the unconfined composite pins. In each case, five identical specimens were tested, and shear strength data were analyzed by using two-parameter Weibull statistics. The results showed that the maximum shear strength took its highest value in the unconfined case for both average values of the test results and for 99% reliability under Weibull distribution. The confinement had a negative effect on the average shear strength of the unconfined pins. It was also seen that the 99% reliability values of shear strength were approximately equivalent to the 0.7 average value of the shear strength.

Power Output by Active Pitch-Regulated Wind Turbine in Presence of Short Duration Wind Variations

The inexact power output from a pitch angle-controlled (PAC) horizontal-axis turbine, as influenced by short duration wind variations like turbulence and gusts, has been the subject of well-documented experimental and empirical studies. In this paper, an analytical interpretation of the phenomena is presented under assumption of two-parameter Weibull statistics for short duration wind variations. The popular concept of turbulence intensity is used as a parametric measure for randomness of wind speed. The formulations culminate in analytical expressions for two distinct metrics of power distortion, namely: (a) the short duration output power from the turbine as a mean distinct from corresponding ideal zero-turbulence value; and (b) the output power variability as a quantification of randomness around the mean. Estimates of both metrics are presented for operating conditions in accordance with the well-known IEC 61400-1 standards. Trends indicated by computed estimates are found to be consistent with empirically observed data from field studies that are reported in the literature.

Performance prediction of active pitch-regulated wind turbine with short duration variations in source wind

Short duration wind variations affect real time performance of active pitch-regulated wind turbines in two ways as evident from reported experimental and empirical studies. First the mean output power, which may be referred to as the short duration output power, differs significantly from the corresponding zero-turbulence value obtained with ideal source wind streamlines. Second, random variation of output around the mean value appears with a significant standard deviation; the normalised value of which is referred to as the short duration variability. In this paper, analytical interpretation of both metrics is presented under assumption of two-parameter Weibull statistics for short duration wind variations. Statistical estimates for the metrics are presented for conditions described by the well known IEC 61400-1 Standards. Finally the statistical estimation procedure is applied to a Vestas V90 3MW zero-turbulence output curve as an illustrative application example.

A prediction scheme of the static fracture strength of MEMS structures based on the characterization of damage distribution on a processed surface

This paper presents a scheme for predicting the strength of MEMS structures patterned into arbitrary shapes by deep reactive ion etching of a silicon wafer. The scheme is based on the inhomogeneous defect distribution on the etched surfaces. Single-crystal silicon specimens with different shapes were subjected to four-point bending tests with monotonically increasing load. The distributions of the fracture strength were described using two-parameter Weibull statistics, where the two parameters are defined as functions of the etching depth representing the inhomogeneity of the damage on the etched surface in the direction perpendicular to the wafer. In order to estimate the distribution of the local strength determined by the local level of damage, the etched surfaces of specimens without a notch were tested in three different bending directions corresponding to three different stress distributions, and three different strength levels were given due to surface roughness conditions caused by the non-uniformity of the etching process. The estimated values of the parameters were used to estimate the fracture strength of four types of notched specimens with different notch tip radii. The results of comparison between the distributions of predicted strengths and experimental data showed that the fracture strength of arbitrarily shaped structures is predictable on the basis of the information obtained from specimens without notch, by taking into account the characteristics of etched surface, i.e. the inhomogeneous damage.

Impact of Short Duration Wind Variations on Output of a Pitch Angle Controlled Turbine

The distorted output of a pitch angle controlled (PAC) horizontal axis turbine, as influenced by short duration wind variations like turbulence and gusts, is analyzed under the assumption of two-parameter Weibull statistics. The well accepted concept of turbulence intensity is used as a parametric measure of short duration wind variations. Analytical expressions are formulated and used to estimate the short duration aggregate power output from a turbine, together with the fractional deviations from corresponding ideal zero-turbulence values. Operating conditions of constant turbulence intensity, and constant wind speed standard deviation, are considered separately for the studies. Sensitivity of turbine output power to cut-in and rated values of wind speed are evaluated and discussed.

Evaluation of bond strength of resin cements using different general-purpose statistical software packages for two-parameter Weibull statistics

ObjectivesThis study evaluated and compared Weibull parameters of resin bond strength values using six different general-purpose statistical software packages for two-parameter Weibull distribution. MethodsTwo-hundred human teeth were randomly divided into 4 groups (n=50), prepared and bonded on dentin according to the manufacturers’ instructions using the following resin cements: (i) Variolink (VAN, conventional resin cement), (ii) Panavia21 (PAN, conventional resin cement), (iii) RelyX Unicem (RXU, self-adhesive resin cement) and (iv) G-Cem (GCM, self-adhesive resin cement). Subsequently, all specimens were stored in water for 24h at 37°C. Shear bond strength was measured and the data were analyzed using Anderson–Darling goodness-of-fit (MINITAB 16) and two-parameter Weibull statistics with the following statistical software packages: Excel 2011, SPSS 19, MINITAB 16, R 2.12.1, SAS 9.1.3. and STATA 11.2 (p≤0.05). Additionally, the three-parameter Weibull was fitted using MNITAB 16. ResultsTwo-parameter Weibull calculated with MINITAB and STATA can be compared using an omnibus test and using 95% CI. In SAS only 95% CI were directly obtained from the output. R provided no estimates of 95% CI. In both SAS and R the global comparison of the characteristic bond strength among groups is provided by means of the Weibull regression. EXCEL and SPSS provided no default information about 95% CI and no significance test for the comparison of Weibull parameters among the groups. In summary, conventional resin cement VAN showed the highest Weibull modulus and characteristic bond strength. SignificanceThere are discrepancies in the Weibull statistics depending on the software package and the estimation method. The information content in the default output provided by the software packages differs to very high extent.

Microstructure and Performance of Four Casting Processes for Magnesium Alloy AZ91

The performance of four different casting processes for magnesium alloy AZ91 were evaluated through microstructure characterization, mechanical testing, and SEM analysis of fracture surfaces. Passenger car control arms were cast by indirect squeeze cast, low pressure permanent mold (LPPM), T-Mag, and ablation processes. Samples were cut from twelve locations of the control arms for microstructure characterization. The microstructure, grain size, porosity distribution, and defect analyses were performed using optical microscopy and an image analyzer. Five different defects including microporosity, oxide film, sponge shrinkage, gas pore, and a crack-like defect were identified. The mechanical behavior was characterized using four-point bending (FPB) and tensile tests. The four casting processes were evaluated in terms of reliability using two-parameter Weibull statistics of the ultimate bending strength (UBS) determined from the FPB test samples. Metallographic analyses were performed on these samples, revealing noticeable microstructural differences between them, with some showing possible evidence of oxide films.

Investigations of compressive strength on Cu-Hf-Al bulk metallic glasses: Compositional dependence of malleability and Weibull statistics

The study is focusing on how physical factors correlate with malleability for bulk metallic glasses (BMGs). Although great efforts have been done in these aspects, many problems still remain interesting. In this work, we investigated systematically the compositional dependence of the compressive malleability for three selected BMG samples of Cu52Hf39Al9 (C1), Cu49Hf42Al9 (C2) and Cu45Hf46Al9 (C3) in the high glass-forming Cu-Hf-Al system. Our results demonstrated that (i) with increasing of Hf content the malleability of BMGs becomes stronger, (ii) shear stress has an important impact in controlling the compressive deformation behaviors and (iii) the malleability increases linearly as the glass transition temperature and shear modulus decrease but it does not directly correlate with Poisson’s ratio and glass forming ability. Finally, through two-parameter Weibull statistics we further suggest that the C3 BMG indeed exhibits a more uniform distribution on the failing strength in a more reliable manner with respect to the other C1 and C2 cases.

Guidelines for Two-Parameter Weibull Analysis for Flaw-Containing Materials

Two-parameter Weibull statistics used in the analysis of mechanical data from flaw-containing materials, such as ceramics and castings, are reviewed. Guidelines to estimate Weibull parameters with the linear regression technique are provided. Moreover goodness-of-fit tests for Weibull fits and calculating confidence intervals for the estimated Weibull modulus are discussed. Methods of estimating lower bounds as well as a new hypothesis test for comparing two estimated Weibull moduli are explained. The use of these guidelines is demonstrated by using data from the literature.

A Statistical RKR Fracture Model for the Brittle Fracture of Functionally Graded Materials

A statistical Ritchie-Knott-Rice (RKR) [1 ] model for brittle fracture is considered for an FGM containing a slender notch. The FMG is modeled as linear elastic, with its strength described by two-parameter Weibull statistics. The Young's modulus is assumed to vary either linearly or sigmoidally. A compact tension (C(T)) fracture mechanics specimen is analyzed via the finite element method, considering the effect of modulus variation on the near-tip stress state. Results can be characterized by the stress intensity, K. For spatially constant Weibull parameters, the RKR model is used to predict the expected fracture toughness, K Φ , i.e., the K at which the first flaw failure occurs with probability Φ. For sufficiently high Weibull modulus, the failure occurs essentially at the notch tip. For sufficiently low Weibull modulus (m < 4), K Φ for an FGM is found to vary up to 25% from that of a homogeneous body.