Two-parameter Weibull Distribution Function Research Articles

Probabilistic Stress-Life (P-S-N) Study on Bearing Steel Using Alternating Torsion Life Test

Probabilistic stress-life (P-S-N) studies using alternating torsion life test were conducted with the bearing steel JIS SUJ2/AISI 52100 having a hardness range of Rockwell C scale of 58-62 HRC to determine the existence of a fatigue limit for this through-hardened steel. One hundred and fifty (150) alternating torsion tests were performed at six torsion maximum shear stress amplitudes of 0.5, 0.63, 0.76, 0.80, 0.95, and 1.0 GPa. The number of specimens that were run at each stress amplitude varied from 19 to 33. There is no observable fatigue limit for JIS SUJ2/AISI 52100 bearing steel. On log-log plotted data, life is linear and is inversely proportional to shearing stress to a 10.34 exponent. The failure mode of the experiment is due to the conjugate tensile and compressive alternating stresses in accordance with alternating shearing stresses. The three-parameter Weibull life distribution function with a constant value of the Weibull slope m = 3/2 was found to have the best conformity followed by the log-normal distribution and the two-parameter Weibull distribution function. The shear stress-life exponent (A = 10.34) for JIS SUJ2/AISI 52100 bearing steel was independent of the Weibull slope.

Is Normal Distribution the Most Appropriate Statistical Distribution for Volumetric Properties in Asphalt Mixtures?

Abstract Volumetric properties are essential for a better understanding of how to design the mixtures. Due to the irregularity of aggregate shape, these properties vary randomly with the same gradation. In this paper, the volumetric properties [bulk specific gravity (γf), air voids (VV), voids in mineral aggregate (VMA), and voids filled with asphalt (VFA)] were determined on two mixtures types, namely, A and B. These mixtures were statistically analyzed using different distribution methods. Seventy-two samples of Mixture A and 52 of Mixture B were fabricated using the Marshall design method. Results of the study showed that the Weibull distribution could reflect the probability distribution law of the volumetric properties of the mixtures. The two-parameter Weibull distribution function appears to be more appropriate or use with γf and VFA, whereas the three-parameter Weibull distribution function was most appropriate for VV and VMA. The analysis of test results also revealed that there are significant differences in the volumetric properties of the two mixtures tested at the same confidence level. The confidence interval decreases with a decrease in reliability.

The reduction of carbon nanotube (CNT) length during the manufacture of CNT/polymer composites and a method to simultaneously determine the resulting CNT and interfacial strengths

Carbon nanotube/epoxy composites with an excellent dispersion of carbon nanotubes (CNTs) were prepared using a three-roll calandering technique. CNT length after processing of composites is measured and then characterized using a two-parameter Weibull distribution function. Significant reduction of the CNT length is observed as a result of the processing and it is thus suggested that great attention should be paid to the retention of CNT length after processing in order to obtain good mechanical properties. Because of the difficulties in manipulating nanometer sized CNTs during measurement of CNT strength and CNT–polymer interfacial strength, CNT strength and CNT–polymer interfacial strength have previously been determined using complicated methods with expensive or specially designed equipments. In this work a simple methodology based on the modified rule of mixtures is proposed to simultaneously determine the CNT strength and CNT–polymer interfacial strength.

Large scale testing and statistical analysis of dynamic fracture toughness of ductile cast iron

The present paper deals with the experimental determination and statistical analysis of dynamic fracture toughness values of ductile cast iron. K Id data from 140 mm thick single edge bend specimens of two dynamic fracture toughness test series on ductile cast iron from heavy-walled castings were analysed. At first, the statistical analysis of data at −40 °C was done based on ASME Code Case N-670 using a two-parameter Weibull distribution function. Weibull analyses of three samples covering different pearlite contents (⩽4%, ⩽9%, ⩽20%) were performed and characteristics of the distribution functions as well as two-sided confidence intervals were calculated. The calculated characteristics show that K Id of ductile cast iron decreases with increasing pearlite content. In a second step, the applicability of the Master curve procedure according to ASTM E 1921 to ductile cast iron materials was investigated and it was formally used for statistical analysis of ductile cast iron dynamic fracture toughness data. Although the Master curve method was originally introduced for static fracture toughness data of ferritic steels, the successful individual analyses performed here support the engineering way taken to apply the method to ductile cast iron materials too. The results of both methods, the Master curve procedure and the ASME Code Case N-670, show acceptable congruity. At the same time, it is concluded from the present study that further investigations and experiments are required to improve precision and for verification before the results could be applied within component safety analyses.

Determination of Turboprop Reduction Gearbox System Fatigue Life and Reliability

Two computational models to determine the fatigue life and reliability of a commercial turboprop gearbox are compared with each other and with field data. These models are (1) Monte Carlo simulation of randomly selected lives of individual bearings and gears comprising the system and (2) two-parameter Weibull distribution function for bearings and gears comprising the system using strict-series system reliability to combine the calculated individual component lives in the gearbox. The Monte Carlo simulation included the virtual testing of 744,450 gearboxes. Two sets of field data were obtained from 64 gearboxes that were first-run to removal for cause, were refurbished and placed back in service, and then were second-run until removal for cause. A series of equations were empirically developed from the Monte Carlo simulation to determine the statistical variation in predicted life and Weibull slope as a function of the number of gearboxes failed. The resultant L 10 life from the field data was 5,627 h. From strict-series system reliability, the predicted L 10 life was 774 h. From the Monte Carlo simulation, the median value for the L 10 gearbox lives equaled 757 h. Half of the gearbox L 10 lives will be less than this value and the other half more. The resultant L 10 life of the second-run (refurbished) gearboxes was 1,334 h. The apparent load-life exponent p for the roller bearings is 5.2. Were the bearing lives to be recalculated with a load-life exponent p equal to 5.2, the predicted L 10 life of the gearbox would be equal to the actual life obtained in the field. The component failure distribution of the gearbox from the Monte Carlo simulation was nearly identical to that using the strict-series system reliability analysis, proving the compatibility of these methods.

Statistical analysis of bending fatigue life data using Weibull distribution in glass-fiber reinforced polyester composites

The bending fatigue behaviors were investigated in glass fiber-reinforced polyester composite plates, made from woven-roving with four different weights, 800, 500, 300, and 200 g/m 2, random distributed glass-mat with two different weights 225, and 450 g/m 2 and polyester resin. The plates which have fiber volume ratio V f ≅ 44% and obtained by using resin transfer moulding (RTM) method were cut down in directions of [0/90°] and [±45°]. Thus, eight different fiber-glass structures were obtained. These samples were tested in a computer aided fatigue apparatus which have fixed stress control and fatigue stress ratio [ R = −1]. Two-parameter Weibull distribution function was used to analysis statistically the fatigue life results of composite samples. Weibull graphics were plotted for each sample using fatigue data. Then, S– N curves were drawn for different reliability levels ( R = 0.99, R = 0.50, R = 0.368, R = 0.10) using these data. These S– N curves were introduced for the identification of the first failure time as reliability and safety limits for the benefit of designers. The probabilities of survival graphics were obtained for several stress and fatigue life levels. Besides, it was occurred that RTM conditions like fiber direction, resin permeability and full infiltration of fibers are very important when composites (GFRP) have been used for along time under dynamic loads by looking at test results in this study.

Fatigue reliability analysis of composites based on residual strength

The fatigue reliability of Gr/PEEK composites has been investigated in the present study. In the experimental investigation, constant amplitude loading, multiple-level loading and random loading have been considered step by step during the fatigue tests. In particular, a residual strength degradation model was presented for predicting the fatigue life of composites subjected to constant amplitude fatigue loading. The accumulation of fatigue damage was determined, using the residual strength degradation model. The residual strength distribution after an arbitrary fatigue cycle can be represented by a two-parameter Weibull distribution function. A fatigue reliability analysis method for arbitrary load history was thus developed through the Weibull distribution function. Good correlation between theory and experiment was obtained for constant amplitude loading, two-level loading, three-level loading and random loading. The result of the present study can be used as a basis for the reliability design of mechanical components made of Gr/PEEK composites.

Effects of the entrained surface film on the reliability of castings

The tilt pouring and gravity top pouring of an Al-4.5 pct Cu alloy have been studied. A computercontrolled rollover casting wheel was used to perform the tilt pouring. Filling sequences with tranquil or turbulent flow patterns have been visualized using real-time X-ray video radiography and modeled using a computational fluid dynamics (CFD) code. The area of the free surface film entrained into the bulk of liquid metal in different filling conditions has been calculated using a filling sequence free from surface turbulence as a baseline. The tensile properties of the castings have been quantitatively assessed for reliability using a two-parameter Weibull distribution function. The study reveals that the liquid metal flow in the mold filling process can be accurately simulated using a CFD code. In addition, the computed total surface area of the entrained surface film can be used as a criterion to judge the deterioration of reliability. The high Weibull modulus achieved by filling a mold without surface turbulence was reduced by a factor of 2.5 of its original value by entrained surface films. Entrainment of bubbles required surface turbulence, but folded films could be entrained simply by contraction of the free surface, creating excess surface film that necessarily folds inward.

Estimation of the minimum number of test specimens for fatigue testing of acrylic bone cement

In the literature on fatigue testing of acrylic bone cements, data sets of various sizes have been used in different test series for the same cement formulation. There are two important consequences of this situation. First, it means that some test series last much longer than others, with all the implications for the cost of testing. Second, it makes drawing conclusions about the fatigue performance of a cement, based on the results of different literature series, a problematic issue. Clearly then, a recommendation as to what should be the minimum number of test specimens to use that would allow for confidence in the results of the statistical treatment of the test results ( G min) would be desirable. In the present work, a method that could be used to culminate in such a recommendation is described. This method involves (i) obtaining experimental fatigue test results and (ii) analyzing those results using the Weibull probability distribution function and other statistical methods. This methodology is illustrated using fatigue life results obtained from uniaxial tension–compression fatigue tests on specimens fabricated from the polymerizing dough of one commercially available acrylic bone cement. For a tolerable error of 5%, we estimated G min to be either 7 (if the fatigue life results are treated using the two-parameter Weibull distribution function) or 11 (if the fatigue life results are treated using the three-parameter Weibull distribution function). To be on the conservative side, we therefore recommend that G min be 11. Three key limitations of the methodology presented here are discussed.

A Numerical Analysis of Ceramics Strength Affected by Material Microstructure

To clarify microstructural influences on the ceramic strength, a numerical analysis of the elastic stress distribution around the crack tip for several crack lengths was conducted by using a finite element method (FEM), in which each element was regarded as one grain in a ceramic polycrystal. In the analysis, the anisotropy expected in actual grains was dealt with by randomly assigning different values to the rigidity and the size of each element (i.e., each grain). The FEM results showed that the crack tip stress was dependent on the grain rigidity but was hardly affected by the grain size. A microstructural modification factor f M for the stress intensity factor was newly introduced to reflect microstructural influences on ceramic strength. The factor f M was defined as the ratio of the crack tip stress obtained by the FEM analysis to the K-based stress. Statistical aspects of f M were investigated by generating virtual materials with different combinations of grain rigidity and size. When the f M distribution was fitted to the two-parameter Weibull distribution function, it was clarified that the distribution shifted toward a lower side with increasing the crack length. The R-curve expressed by an exponential form was applied so that the grain bridging effect in ceramics could be included in the analysis of the strength depending on crack length. It was revealed that the estimated scatter band in the relation of the strength versus the crack length represented the central trend of dispersed experimental results.

Fatigue and Reliability Analysis of Unidirectional GFRP Composites under Rotating Bending Loads

Rotating bending fatigue tests have been conducted on unidirectional glass fiber reinforced polyester (GFRP) composites. Standard test specimens were manufactured in form of circular rods with various fiber volume fraction ( Vf) ratios. Failure modes of the composite rods have been examined using scanning electron microscope. The two-parameter Weibull distribution function was used to investigate the statistical analysis of the experimental fatigue life results. Safe design life based on time to first failure (TTFF) concept was calculated at high confidence level γ=0.99 and at two values of reliability, R=0.368 and 0.99. S–N diagrams of mean life, 50% survival life, lower bound life and safe design fatigue life have been constructed for GFRP rods with various values of Vf ratios. These diagrams are of considerable value to the designer specially when the structure contains a critical component where any failure is catastrophic. No rigorous fatigue limit was observed within 107 cycles in these S–N diagrams. The fiber volume fractions have insignificant effect on the slope of the power function that fits the mean fatigue life. At the same number of cycles the stress amplitude required to fracture the specimens with Vf=44.7% was increased by an order equal to 1.58 and1.25 than specimens with Vf=15.8 and 31.8% respectively. The widest scatter was observed at the life range of 105 and 106 cycles for GFRP rode with different Vf ratios. This tendency in the dispersion of fatigue life at varying stress levels is extremely important and deserves much attention for the design and application of GFRP composites.

Deformation and fracture behavior of notched and unnotched unidirectional C/C-Mg composite with Young's modulus 520 GPa and tensile strength 1 GPa

Experimental study on tensile fracture behavior of the newly developed C/C-Mg composite, prepared by infiltration of Mg into the pores in the C/C composite heat-treated at 3000°C, was carried out. The volume fraction of the filled Mg was 9–10%. The composite had a specific density 2.1, Young's modulus 520 GPa and Poisson's ratio 0.26. The average tensile strength measured for the specimen with a nominal width 8 mm, gage length 40 mm and thickness 1 mm was 1 GPa. The Young's modulus was improved from 450 to 520 GPa and the strength from 0.9 to 1.0 GPa by Mg-infiltration. The specific Young's modulus and specific strength based on the average measured values were 2.5 × 107 m and 5 × 104 m, respectively, showing high potential as light-weight, stiff and strong structural material. The strength distribution of the composite was described by the two-parameter Weibull distribution function with a shape parameter 7.6 and scale parameter 1060 MPa. Prior to the overall fracture of the composite, the longitudinal cracking arose at the notch tip, due to which the notch tip was blunted and the ligament portion behaved like an unnotched specimen. As a result, the notched strength could be described by the net stress criterion. The apparent critical energy release rate at formation of the longitudinal crack was around 70–90 J/m2.

Statistical analysis of the failure stresses of ceramic fibres: Dependence of the Weibull parameters on the gauge length, diameter variation and fluctuation of defect density

The strength distribution of ceramic fibres is commonly described using a two-parameter Weibull distribution function. This study shows that the determination of these parameters from around 30 tensile tests obtained at one single gauge length (L0) does not allow the strength distribution at other gauge lengths to be correctly predicted. The reliability in the Weibull parameter determination is lowered by variations in fibre diameter (D) and the insufficient number of fibres tested. An effective failure stress σE = σ · (π D · L0)1/m is first introduced to take into account fibre diameter variations and to extract the two Weibull parameters from the 180 tests obtained at 6 gauge lengths. It is then shown that the linear size effect, which is expected from the standard Weibull model, is not appropriate to fit correctly this experimental strength distribution. The length dependence follows a power law (L0β) leading to an effective failure stress σE = σ · (π D · L0β)1/m. Diameter variations along the gauge length cannot be responsible for this non linear variation with the length, which is attributed to a large scale fluctuation of the density of defects. The value of β can bring valuable information about fluctuations in the fibre processing conditions.

Indentation toughness of ceramics: a statistical analysis

Direct indentation method was employed to measure the toughnesses for soda-lime glass and a TiC particulate reinforced Al 2O 3 composite. The statistical aspect of the resultant indentation toughness was then analyzed. It was found that the indentation toughness measured at a given indentation load for a given material exhibits a large scatter and can be described with a two-parameter Weibull distribution function. Analyses based on the observations of the crack/microstructure interactions showed that the statistical variability in the measured indentation toughness may be attributed to the effect of microstructural inhomogeneity on the local crack resistance.

A Numerical Investigation on Variation of Shape Parameter in Two-Parameter Weibull Distribution Function Fitted to Distribution of Ceramic Strength

In this study, the dependency of the variation in the shape parameter m, which was determined in the two-parameter Weibull distribution function fitted to a strength distribution, on the number of samples, n, was investigated by using data in numerically simulated distributions. By random sampling, five groups, i. e. (a) forty sets of n=5, (b) twenty sets of n=10, (c) eight sets of n=25, (d) five sets of n=40, and (e) two sets of n=100, were prepared for the present analysis.It was clarified that the variation of m-value was enlarged with decreasing the number of samples, and its variation range as an absolute value became larger as the shape parameter determined for the original population is increased. Values of the shape parameter, which were obtained for respective numbers of samples, were normalized by the shape parameter for the original population. By such normalization, it was revealed that a relative m-variation with respect to the number of samples was hardly dependent on the shape parameter for the original population. The simulated behavior of m-variation was also found to coincide with that observed in four-point bending test using a pressureless sintered alpha silicon carbide.Finally, it should be noted that the number of samples to be used in a strength test, which is recommended in JIS, is not always enough for a better understanding of the statistical strength characteristics in ceramics.

Statistical Properties of Fatigue Life Simulated for Notched Components under Combined Loading.

Failure lives of notched components in multiaxial fatigue were statistically estimated by Monte Carlo simulations for materials with different microstructures using a proposed crack growth model. An analytical algorithm for crack growth in a microstructure, which was modeled as an aggregate of hexagons, was based on competition between two possible growth modes. In simulations, various microstructure configurations, which resulted in distinct failure lives, were randomly generated for a material under consideration. Ranges of simulated life almost coincided with experimental results observed in four kinds of materials. The coefficient of variation and the shape parameter in the fitted two-parameter Weibull distribution function for the simulated life distribution were investigated to clarify statistical properties of the failure life in multiaxial fatigue. The life dispersion was found to be larger under a loading mode with a higher proportion of shear stress component and/or in a material with coarser grains.

Size and heterogeneity effects on the strength of fibrous composites

Probabilistic fiber composite strength distributions and size scalings depend heavily on both the stress redistribution mechanism around broken fibers and properties of the fiber strength distribution. In this study we perform large scale Monte Carlo simulations to study the fracture process in a fiber composite material in which fibers are arranged in parallel in a hexagonal array and their strengths are given by a two-parameter Weibull distribution function. To calculate the stress redistribution due to several broken fibers, a realistic 3D shear-lag theory is applied to rhombus-shaped domains with periodic boundary conditions. Empirical composite strength distributions are generated from several hundred Monte Carlo replications, particularly for much lower values of fiber Weibull modulus γ, and larger composite sizes than studied previously. Despite the localized stress enhancements due to fiber failures, predicted by the shear-lag model, composite response displays a transition to equal load sharing like behavior for approximately γ≤1. Accordingly, the results reveal distinct alterations in size effect, failure mode, and weak-link scaling behavior, associated with a transition from stress-driven to fiber strength-driven breakdown.

Powder particle size relationship in microwave synthesised ceramic powders

In this investigation, titania and zirconia ceramic powders were synthesised from titanium isopropoxide and zirconyl nitrate in methanol respectively by exposure to microwave radiation. The powder size distribution data determined using laser particle sizer followed two-parameter Weibull distribution function. The mean particle size determined from the distribution function showed a linear relationship with increase in applied microwave power.

Relation of strength distribution of Nb 3Al filaments to strength of multifilamentary superconducting composite wire

The distribution of tensile strength of Nb 3Al filaments and its relation to tensile strength of Nb 3Al multifilamentary superconducting composite wire were studied. The main results can be summarized-as follows: (1) The tensile strength of the extracted filaments with an average diameter of 24 μm was estimated based on the two-parameter Weibull distribution function. The shape and scale parameters were 7.0 and 530 MPa (for a standard length of 1 m), respectively. (2) Although the scatter of the strength of the Nb 3Al filaments was large, that of the multifilamentary composite was very small. This means that, when a large number of filaments are embedded in a composite, the stress leading to overall fracture of the composite is not very different from sample to sample, even though the strengths of the embedded filaments are significantly different. This feature was confirmed by means of a computer-aided Monte Carlo simulation. (3) It was shown by experiment and simulation that the strength of the Nb 3Al composite wire has a very slight dependence on length, although the strength of the Nb 3Al filament decreases markedly with increasing length. This result indicates that, even if the length of the composite wire is extended from a short, laboratory scale sample to an industrial scale, the reduction in strength will be very small.

Relation of the strength distribution of Nb3Sn to the critical current of a pre-stressed multifilamentary composite superconductor

From an analysis of the change in the superconducting critical current at 4.2 K caused by pre-stressing treatment at room temperature, the distribution of tensile strength of Nb3Sn in multifilamentary bronze-processed composite wires heat-treated at 973 K for 8.6 (sample A), 43 (B) and 260 ks (C) is estimated using the two-parameter Weibull distribution function. The shape parameters of the Weibull distribution are 7.2, 12 and 14, and the average strengths of the Nb3Sn are 1.3, 1.0 and 0.79 GPa for samples A, B and C respectively for the present specimen length of 25 mm. This means that the scatter of strengths and also the average strength for specimens of length 25 mm decrease with increasing heat-treatment time. An attempt to predict the length dependence of the critical current of pre-stressed specimens from the data for short specimens is presented. It seems that the permissible pre-stress below which there is no breakage of Nb3Sn (for all Nb3Sn transport currents) is reduced by 20-30% when the specimen is lengthened from 25 mm to 300 m, while the average strength of Nb3Sn is reduced by 50-70%.