Bimodal Weibull Distribution Function Research Articles

Effect of the inter-fiber friction on fiber damage propagation and ballistic limit of 2-D woven fabrics under a fully confined boundary condition

This paper investigates the effect of the inter-fiber friction coefficient on fabric ballistic performance. Firstly, dynamic stress responses within a fabric due to first fiber failure are analyzed. Relations between inter-fiber friction and progressive fiber failure are assessed. Then, ballistic perforation processes of Kevlar KM2 fabrics are simulated using a fiber level micro-approach, the digital element approach (DEA). In this model, each yarn is discretized into many fibers and each fiber is divided into many rod elements. A Monte Carlo process is utilized to assign a unique strength to each element following a bimodal Weibull distribution function. DEA is used to simulate the ballistic perforation processes. A comparative investigation using a broad range of inter-fiber friction coefficients are conducted. Relations of ballistic limits and inter-fiber friction coefficients are presented. Simulation results show that ballistic limit improves as the inter-fiber friction coefficient increases up to a critical value. Beyond that point, ballistic strength decreases slightly as the inter-fiber friction coefficient increases. The inter-fiber friction also changes the ballistic perforation mechanisms.

Weibull Distribution to Describe Grading Evolution of Materials with Crushable Grains

This work presents a study on the evolution of the Grain Size Distribution (GSD) of an artificial granular material with crushable grains and its relation with the observed mechanical behaviour. The main aim of the work is to find a relationship between the initial GSD, its evolution and the mechanical behaviour. The artificial material used is a Light Expanded Clay Aggregate (LECA) subject to one-dimensional compression at various stress levels with eight initial GSDs characterised by four coefficients of uniformity and two mean diameters. The evolution of the GSD is characterised by a double mechanism depending on the initial GSD and on the applied stress level. A link between the evolution of the GSD, breakage and the compressibility curves is observed. A bimodal Weibull distribution function is proposed to describe the GSD before and after testing. The long term objective of the work is to link the evolution of breakage, from moderate to high stress, with the mechanical behaviour and to formulate a constitutive model able to describe the observed behaviour.

Fabrication and characterization of carbon fiber reinforced clay/epoxy composite

In this study, dynamic mechanical analysis (DMA), thermogravimetric analysis (TGA), and flexural tests were performed on unfilled, 1, 2, 3, and 4 wt% clay filled SC-15 epoxy to identify the effect of clay weight fraction on thermal and mechanical properties of the epoxy matrix. The flexural results indicate that 2.0 wt% clay filled epoxy showed the highest improvement in flexural strength. DMA studies also revealed that 2.0 wt% system exhibit the highest storage modulus and Tg as compared to neat and other weight fraction. However, TGA results show that thermal stability of composite is insensitive to the clay content. Based on these results, the nanophased epoxy with 2 wt% clay was then utilized in a vacuum assisted resin transfer molding set up with carbon fabric to fabricate laminated composites. The effectiveness of clay addition on thermal and mechanical properties of composites has been evaluated by TGA, DMA, tensile, flexural, and fatigue test. 5 °C increase in glass transition temperature was found in nanocomposite, and the tensile and flexural strengths improved by 5.7 and 13.5 %, respectively as compared to the neat composite. The fatigue strength was also improved significantly. Based on the experimental result, a linear damage model combined with the Weibull distribution function has been established to describe static failure processing of neat and nanophased carbon/epoxy. The simulated stress–strain curves from the model are in good agreement with the test data. Simulated results show that damage processing of neat and nanophased carbon/epoxy described by bimodal Weibull distribution function.

Simulation on the relation of distribution of overall critical transport current to that of local one in bent-damaged Bi2223 superconducting composite tape

The superconducting Bi2223/Ag/Ag alloy composite tape specimens with a length 6 cm, composed of a series circuit of six local elements with a length 1 cm, were bent by 0.37% and 1.0% for measurement of distributions of the overall (6 cm) and local (1 cm) critical transport current and n-value. When the damage amount was small (0.37% bending strain), the distributions of the measured local and overall critical currents were described by the three parameter Weibull distribution function, while when the damage amount was large (1.0% bending strain), those were described by the bimodal Weibull distribution function. The distributions of n-value in the diagram of n-value versus critical current for both local elements and overall specimens were expressed by the regression curve of n-value as a function of critical current and the normal distribution function for the deviation of the measured n-values from the regression curve. The experimentally measured critical current distribution and the diagram of the n-value versus critical current of the overall specimens were reproduced successfully in the computer by inputting the formulated distribution functions for the critical current and n-value of the local elements into the proposed simulation method, which used a Monte Carlo simulation method and a one-dimensional series circuit model for the generated voltage near the transition of superconducting to normal state.

曲げ変形により損傷を受けたBi2223超伝導複合テープにおける局所臨界電流の分布と試料全体の臨界電流との関係

The damage extent and therefore the critical current are different from position to position in a long sample especially when the sample is deformed severely. In the present work, a 48 cm long Bi2223 composite tape bent by 0.7%, being composed of 48 elements with a length of 1 cm and of 8 parts with a length of 6 cm, was used as the test sample. The V-I curve, critical current and n-value were measured for the 48 elements, 8 parts and overall sample, and the relation of the distribution of the critical current of the elements to the critical current of the parts and overall sample and the sample length dependence of critical current were discussed. It was found that the critical current and n-value of parts and overall sample are described well by the voltage summation model which regards the parts and overall sample to be composed of a series circuit of elements. The critical current distribution of the elements was described by the bi-modal Weibull distribution function. Applying the simulation method, in which the estimated Weibull distribution function for the critical current of elements, the empirical relation between the critical current and n-value, the series circuit model for voltage generation and the Monte Carlo method were combined, the relation of average critical current to sample length could be described well. From these results and the features of the V-I curve of the damaged part, it was shown that the critical current and n-value decrease with increasing sample length and the extent of the decrease in critical current with sample length is dependent on the criterion for the definition of critical current.

An experimental study on the tensile behavior of a unidirectional carbon fiber reinforced aluminum composite at different strain rates

In this paper, dynamic and quasi-static tensile behavior of a unidirectional carbon fiber reinforced aluminum composite (Cf/Al) have been studied. The complete stress–strain curves of the composite in strain rate range from 0.001 to 1000 s −1 were obtained. The experimental results show that the tensile strength and critical strain of the Cf/Al composite are strain rate-sensitive. Micro-graphs show the presence of surface cracks as well as internal cracks in the fibers. A coated fiber bundles model has been combined with Weibull strength distribution function to establish a one-dimensional damage constitutive equation for the Cf/Al composite. This study shows that the strength of the Cf/Al composite can be described by a bimodal Weibull distribution function. The scale parameter σ 0 and shape parameter β of the Weibull distribution and their relationship with strain rate are obtained from the experimental results. The simulated stress–strain curves from the model are in good agreement with the test data.

Tensile mechanical behavior of T300 and M40J fiber bundles at different strain rate

Tensile mechanical behavior of T300 fiber bundles and M40J fiber bundles have been studied in the strain rate range from 0.001 1/s to 1300 1/s and complete stress strain curves were obtained. Results show that both ultimate strength and failure strain of two materials are strain rate insensitive, and T300 fiber and M40J fiber can be regarded as strain rate insensitive materials. On basis of the fiber bundles model and the statistic theory of fiber strength, single Weibull distribution model and bimodal Weibull distribution model have been developed to describe mechanical behavior of fiber bundles. And a method for determine the statistic parameters of fibers by tensile tests of fiber bundles is established, too. The simulated stress strain curves from the model are in good agreement with the test data. Simulated results show that the strength of T300 fiber can be described by single Weibull distribution function, and the strength of M40J fiber should be described by bimodal Weibull distribution function.

Experimental and theoretical study on the strain rate and temperature dependence of mechanical behaviour of Kevlar fibre

Abstract The tensile impact experiments on Kevlar 49 fibre bundles were carried out at strain rates 140, 440, 1350 s −1 and at temperatures −60, −20, 15, 50 and 90°C. It was found that the tensile mechanical properties of Kevlar 49 fibre bundles depend both upon the strain rate and the temperature. A bimodal Weibull distribution statistical model of the strain rate and temperature dependence of fibres, and a test method of determining mechanical properties and Weibull parameters of fibres from the fibre bundle test were adopted to characterize the combined effects of strain rate and temperature on Kevlar 49 fibre strength distribution. The simulated results are in good agreement with the experimental data, which proves that the bimodal Weibull distribution function is suitable to represent the statistical strength distribution of Kevlar 49 fibre which has peculiar ‘skin–core’ physical structure, and that the test method is valid and reliable.

Experimental evaluation of the strength distribution of E-glass fibres at high strain rates

A bimodal Weibull distribution function was applied to analyse the strength distribution of glass fibre bundles under tensile impact. The simulation was performed using a one-dimensional damage constitutive model. The results show that there were two concurrent flaw populations in the fracture process. The regression analysis using the bimodal Weibull distribution function was in good agreement with experiment.

Application of Reliability and Fiber Probabilistic Strength Distribution Concepts to Composite Vessel Burst Strength Design

With the expanding use of composite vessels, including space and tactical applications, accurate prediction of performance becomes increasingly important. A criti cal parameter in performance prediction is the burst strength as measured by the delivered fiber ultimate strain-to-failure. This paper discusses a composite pressure vessel design approach that is based on reliability and probabilistic fiber strength distribution concepts.1 The method is based on the fiber strain-strength interference reliability theory. A substan tial part of this paper examines how fiber strength distribution parameters generated by tow tensile tests of impregnated carbon fiber strands can be used in pressure vessel strength design. The fiber statistical strength distribution is analyzed using the Weibull distribution function. It is shown that the bimodal Weibull distribution function fits the impregnated fiber strand strength distribution data much better than the single mode Weibull function. This is particularly important because the bimodal function gave a much more accurate representation of the lower tail of this distribution. According to the strain- strength interference theory, the lower tail of the fiber strength distribution is the most sig nificant in pressure vessel strength reliability calculations. The impregnated fiber strand strength distribution agrees well with the strength distribu tion of 4-in. biaxial composite tubes manufactured from the same carbon fiber lot. Good agreement was also shown with the strength distribution of pooled fiber lot tow tensile data and hydroburst strength distribution of full-scale composite pressure vessel manufactured from these lots. In this comparison, the A-basis fiber strength allowable calculated from the strand data was slightly lower (3%) than those calculated from the full-scale pressure vessel strength data, whereas, the B-basis allowables calculated from the strand and the full-scale pressure vessel strength data were the same. However, because A-basis and/or B-basis allowables require many data points from many material lots, the use of fiber lot qualification data is advantageous because it meets these requirements.

The evaluation of the strength distribution of silicon carbide and alumina fibres by a multi-modal Weibull distribution

The strengh distributions of silicon carbide and alumina fibres have been evaluated by a multimodal Weibull distribution function. This treatment is based on the concept that the fracture of the fibre is determined by competition among the strength distributions of several kinds of the defect sub-population. Since those fibres were observed to have two types of fracture mode, the evaluation of a bi-modal Weibull distribution was performed in comparison with the single Weibull distribution usually employed. The accuracy of the fit for these two distributions was judged from maximum logarithm likelihoods and cumulative distribution curves. The result showed that the logarithm likelihood calculated using the bi-modal Weibull distribution function gave a larger value, as compared with those using the single Weibull distribution function. The curve predicted from the former function was also in good agreement with the data points. In addition, the strength distribution and the average value at a different gauge length were extrapolated from the Weibull parameters estimated at the original gauge length. In this case, also, the bi-modal Weibull distribution gave a more accurate prediction of the data points.

多重モード・ワイブル分布のパラメータ推定における多段相関係数法と多段最ゆう法の比較

Sakai et al. established the “correlation factor method” for estimating 3-parameters in Weibull distribution function. However, this method can only be applied to mono-modal fracture models.In this work, for analyzing multi-modal Weibull distribution function we suggested a new estimating method, namely the multi-correlation factor method, by employing Kaplan-Meier method which can separate the data into individual causes of fracture. By Monte Carlo simulation for bimodal Weibull distribution function (6-parameters) we compared the multi-correlation factor method with the multi-maximum likelihood method. It was concluded that the multi-maximum likelihood method was more accurate than the multi-correlation factor method. Further, we estimated six Weibull parameters for 403 data of Si3N4 obtained by 3-point bending tests.

セラミックスの曲げ試験における強度分布モデルの選択とモンテカルロシミュレーション

Generalized stepwise structural models of multi-modal Weibull distribution functions have been established for the bending strength of ceramics. The maximum likelihood estimators of the Weibull parameters for each model were obtained by Nelder-Mead method. By using Akaike's information criteria (AIC), a mono-modal model was selected as the most suitable one for two kinds of bending test data on steatite ceramics. However, it was pointed out that some additional data such as diagnostics data and/or additional strength data were needed.The direct and the two-step maximum likelihood methods have been applied to Monte Carlo simulation data of the bi-modal Weibull distribution function for small sample size. The standard deviation and the mean squared error of the maximum likelihood estimators by the two-step method were two or three figures smaller than those of the direct method.