Bimodal Weibull Distribution Research Articles

Predictions of sheet molding compound composite structures using probabilistic finite element simulations with bimodal Weibull strength distributions

Due to the large scatters in Sheet Molding Compound (SMC) mechanical properties, the uncertainty in the prediction of mechanical responses of SMC composite structures is significantly high as compared to that for continuous fiber composite structures. Another challenge is the size effect resulting from the scatter in mechanical strengths, i.e., the measured material flexural strength is much higher than its tensile strength. As a result, classical approaches that rely on the mean tensile strength can substantially underpredict the flexural responses in Finite Element Analysis (FEA). This work investigates the use of a material model with bimodal Weibull distributions for its strength properties in Probabilistic FE (PFE) simulations. A Random Search-based optimization procedure was developed to obtain model parameters which are difficult to determine. The material model was verified by PFE simulations of standard experiments, and then examined in simulations of open-hole tension and disk bending tests.

Improved wind resource modeling using bimodal Weibull distribution

Despite the common norm of modeling wind regimes using two-parameter Weibull distribution (2WD), this probability density function (PDF) is not suitable for sites of calm wind regime. Furthermore, 2WD PDF lacks bimodality and exhibits a quasi-flat shape, which are phenomena seen occasionally in some sites. In this paper, the application of bimodal Weibull distribution (BWD) is proposed as a more comprehensive alternative to the conventional 2WD. A comparative analysis of BWD with 2WD, five-parameter Weibull and Weibull distribution, and three-parameter generalized extreme value distribution, across 32 sites spanning all five continents, reveals moderate to substantial improvements in root mean square error, χ2 statistic, and R2. In addition, the paper demonstrates and explores distinct attributes of BWD, such as bimodality, quasi-flat shapes, flat-start, and others.

Tensile strength prediction of unidirectional polyacrylonitrile (PAN)-based carbon fiber reinforced plastic composites considering stress distribution around fiber break points

Recent advancements in enhancing the mechanical characteristics of carbon fibers open up new application possibilities for carbon fibre-reinforced plastic (CFRP) composites. Particularly in unidirectional CFRPs, which form the basal structure of CFRP laminates, developing a micromechanics model capable of predicting the tensile strength of unidirectional CFRPs based on carbon fiber mechanical characteristics is a current aspiration. This study conducted a stress distribution analysis around the fiber fracture point to predict the tensile strengths of unidirectional CFRPs prepared with five types of polyacrylonitrile (PAN)-based carbon fibers, each with unique mechanical characteristics. Numerical simulation results obtained using a unidirectional CFRP model that considered the stress concentration, fiber axial stress, and bimodal Weibull distribution were reasonably consistent with the experimental results for the tensile strengths of unidirectional CFRP composites, regardless of the differences in the mechanical characteristics of the fiber. Our findings can provide guidance for designing further enhanced high-performance CFRP materials.

Hybrid model development and nonlinear model predictive control implementation for continuous dry granulation process

This study focuses on the development of a hybrid model to integrate roll compaction and ribbon milling operations to design and control a continuous dry granulation process. The proposed hybrid model has three features: (1) it compensates for underestimated roll gap measurements with knurled rolls, (2) it represents the bimodal size distribution of granules using five fitting parameters of the bimodal Weibull distribution instead of only using specific size percentiles, and (3) it considers the impact of rotor-screen gaps on granules. Furthermore, the hybrid model facilitates the implementation of nonlinear model predictive control in a roller compactor Alexanderwerk WP120. Compared to widely applied open-loop operations in the pharmaceutical industry, nonlinear model predictive control demonstrates better performance, indicated by lower integral absolute errors in controlling mass throughput and ribbon solid fraction, in which real-time measurements can be obtained using a near-infrared sensor and a novel spectra selection approach.

A statistical analysis of the second ‘pop-in’ behaviour of the spherical-tip nanoindentation of Zr-based bulk metallic glasses

Statistical analyses of the shear stresses, τy, at which the first ‘pop-ins’ occur during the spherical tip nanoindentation of a wide variety of materials is often conducted to understand the micromechanisms of incipient plasticity. In an earlier paper, we reported such a study on data generated on several different Zr-based bulk metallic glasses (BMGs) using a wide range of experimental variables, such as the tip radius, Ri, loading rate, P˙, and structural state of the glass. In the present work, we analyse the second pop-in stress, τ2, data employing the expectation maximization algorithm in conjunction with the Akaike Information Criterion to examine which of the single and mixed (bimodal) versions of the Gaussian, Lognormal and Weibull (both 2 and 3 parameter) statistical models best describes the stochasticity of τ2. Results show that the 3-parameter Weibull distribution also captures the stochasticity of τ2, just as it does for τy. For datasets of τ2 that are generated with larger Ri and P˙, the bimodal 3-parameter Weibull distribution is a better descriptor of the dispersion. While Weibull exponent, m, of τ2 datasets is marginally higher than that of τy, their kernel density estimates (KDEs) are similar. However, there is a relative shift of the KDEs of τ2 to smaller values compared to that of τy. From mechanistic arguments, τ2 is determined as the stress to nucleate a second shear band over a previously formed shear band and its stochasticity is attributed solely to the mechanical heterogeneity of the material within it. On this basis, the average shear strength of the shear band is estimated to be ∼8–11% and 15–17% lower than the strength of the undeformed BMG in the as cast condition and structurally relaxed conditions, respectively. This study provides an understanding of how plasticity develops in BMGs during nanoindentation.

Research on the Microstructure of Laser Remelted As-sprayed TiB2-TiC0.3N0.7 based Composite Coating

Ceramic coatings with excellent properties of wear resistance, corrosion resistance, oxidation resistance, stability in acid and alkaline environments, and good bond strength with a metallic substrate, are widely used in different industries. Air plasma spraying has high energy and efficiency for depositing ceramic materials with a high melting point. Due to intrinsic porosity and mechanical bond strength between coating and substrate, laser remelting technology can be applied to the coating surface for improving coating properties. As-sprayed TiB2-TiC0.3N0.7-based composite coating was surface remelted by laser. The surface roughness of as-sprayed and laser-remelted coatings was measured. The phase composition and the microstructure as well as the Vickers microhardness of coatings were studied by X-ray diffractometer (XRD), scanning electron microscopy (SEM), and Vickers indentation instrument. It is found that the phase composition does not change after laser remelting. However, the surface roughness of laser-remelted coating is smoother than that of as-sprayed coating. And the microhardness on the cross-section is increased after laser remelting. There is a micro-crack in the laser-remelted coating due to the fast heating and cooling of the laser. The microhardness of the laser-remelted coating is increased with bimodal Weibull distribution.

Detail fatigue rating method based on bimodal Weibull distribution for DED Ti-6.5Al-2Zr-1Mo-1V titanium alloy

Previous studies have shown that the fatigue life distribution of metal materials fabricated with Additive Manufacturing (AM) methods, such as Direct Energy Deposited (DED) Ti-6.5Al-2Zr-1Mo-1V alloys, exhibits two peaks. To promote the application of AM in aerospace and other engineering fields, developing a fatigue strength evaluation method suitable for AM materials based on their unique fatigue behaviours and fatigue life distributions is necessary. In this paper, a novel Detail Fatigue Rating (DFR) method was developed to evaluate the fatigue strength of DED Ti-6.5Al-2Zr-1Mo-1V based on a bimodal Weibull distribution and the excessive restriction on the allowable stress of potential was improved. Meanwhile, a Bimodal Weibull distribution model for fatigue life and its parameter estimation method were established based on a two-parameter Weibull distribution. The fatigue life at a specific reliability level and confidence level was calculated by using the bootstrap method. The calculation results showed that fatigue life estimated by using the bimodal Weibull distribution at the high reliability level and high confidence level is higher than that estimated by using the two-parameter Weibull distribution. Furthermore, The S-N curve at the specified confidence level and reliability level was fitted.

A bimodal Weibull distribution: properties and inference

ABSTRACT Modelling is challenging topic and using parametric models is important stage to reach flexible function for modelling. Weibull distribution has shape and scale parameters which play the main role for modelling. Bimodality parameter is added and so bimodal Weibull distribution can capture real data set with bimodality which can be actually combination of two populations. The properties of the proposed distribution and estimation method are examined extensively to show its usability in modelling accurately and safely for practitioners. After examination as first stage in modelling issue, it is appropriate to use bimodal Weibull for modelling bimodality in real data sets if it exists. Two estimation methods including objective functions are used to estimate the parameters of shape, scale and bimodality parameters of function. The second stage in modelling is overcome by using heuristic algorithms for optimization of function according to parameters due to the fact that converging to global point of objective function is performed by heuristic algorithms from stochastic optimization. Real data sets are provided to show the modelling competence of objective functions from bimodal forms of Weibull and Gamma distributions having well defined shape, scale and bimodality parameters and potentially less parameters when compared with the existing distributions.

Numerical analysis of the brittle–ductile transition of deeply buried marble using a discrete approach

In high geostress areas, rocks tend to experience plastic deformation processes and show ductility. Therefore, the study of the mechanical properties of rocks, especially the post-peak behavior, is crucial for underground geotechnical engineering. For this reason, the macroscopic mechanical property of marble is studied by using the PFC/PBM numerical method. An improved numerical marble sample with random properties is used by setting different contact strengths following a bimodal Weibull distribution. Thereafter, on the basis of an improved boundary servo mechanism, the brittle–ductile transition appearing in laboratory tests on marble is successfully realized by the discrete approach, and a servo velocity threshold function of axial loading speed and confining pressure is obtained. Furthermore, the marble model and servo method are applied in the excavation of typical underground caverns. The effects of constrained velocity and geostress level on the surrounding rock failure model of the cavern are described and analyzed. In addition, mechanism of the excavation failure under high geostress is discussed.

Statistical nature of the incipient plasticity in amorphous alloys

Statistical analyses of the first pop-in stress data, obtained by spherical tip-nanoindentation experiments on different metallic glasses (MGs) with tip radius, Ri, loading rate,P˙, and structural state of the glass as experimental variables, show that the 3-parameter bimodal Weibull distribution best captures the stochastic nature of the incipient plastic strengths. Significant bimodality in the strength distributions was observed only when a larger Ri and P˙ are employed. We hypothesize that the stress required for shear band nucleation has to exceed a critical value over a characteristic distance over which the stress gradients are minimum to rationalize the bimodality.

An Assessment of Onshore and Offshore Wind Energy Potential in India Using Moth Flame Optimization

Wind energy is one of the supremely renewable energy sources and has been widely established worldwide. Due to strong seasonal variations in the wind resource, accurate predictions of wind resource assessment and appropriate wind speed distribution models (for any location) are the significant facets for planning and commissioning wind farms. In this work, the wind characteristics and wind potential assessment of onshore, offshore, and nearshore locations of India—particularly Kayathar in Tamilnadu, the Gulf of Khambhat, and Jafrabad in Gujarat—are statistically analyzed with wind distribution methods. Further, the resource assessments are carried out using Weibull, Rayleigh, gamma, Nakagami, generalized extreme value (GEV), lognormal, inverse Gaussian, Rician, Birnbaum–Sandras, and Bimodal–Weibull distribution methods. Additionally, the advent of artificial intelligence and soft computing techniques with the moth flame optimization (MFO) method leads to superior results in solving complex problems and parameter estimations. The data analytics are carried out in the MATLAB platform, with in-house coding developed for MFO parameters estimated through optimization and other wind distribution parameters using the maximum likelihood method. The observed outcomes show that the MFO method performed well on parameter estimation. Correspondingly, wind power generation was shown to peak at the South West Monsoon periods from June to September, with mean wind speeds ranging from 9 to 12 m/s. Furthermore, the wind speed distribution method of mixed Weibull, Nakagami, and Rician methods performed well in calculating potential assessments for the targeted locations. Likewise, the Gulf of Khambhat (offshore) area has steady wind speeds ranging from 7 to 10 m/s with less turbulence intensity and the highest wind power density of 431 watts/m2. The proposed optimization method proves its potential for accurate assessment of Indian wind conditions in selected locations.

Characterizing Distributions of Tensile Strength and Crack Precursor Size to Evaluate Filler Dispersion Effects and Reliability of Rubber.

Undispersed filler agglomerates or other substantial inclusions/contaminants in rubber can act as large crack precursors that reduce the strength and fatigue lifetime of the material. To demonstrate this, we use tensile strength (stress at break, σb) data from 50 specimens to characterize the failure distribution behavior of carbon black (CB) reinforced styrene-butadiene rubber (SBR) compounds. Poor mixing was simulated by adding a portion of the CB late in the mixing process, and glass beads (microspheres) with 517 μm average diameter were introduced during milling to reproduce the effects of large inclusions. The σb distribution was well described with a simple unimodal Weibull distribution for the control compound, but the tensile strengths of the poor CB dispersion material and the compounds with the glass beads required bimodal Weibull distributions. For the material with the lowest level of glass beads—corresponding to less than one microsphere per test specimen—the bimodal failure distribution spanned a very large range of σb from 13.7 to 22.7 MPa in contrast to the relatively narrow σb distribution for the control from 18.4 to 23.8 MPa. Crack precursor size (c0) distributions were also inferred from the data, and the glass beads introduced c0 values in the 400 μm range compared to about 180 μm for the control. In contrast to σb, critical tearing energy (tear strength) was unaffected by the presence of the CB agglomerates and glass beads, because the strain energy focuses on the pre-cut macroscopic crack in the sample during tear testing rather than on the microscopic crack precursors within the rubber. The glass beads were not detected by conventional filler dispersion measurements using interferometric microscopy, indicating that tensile strength distribution characterization is an important complementary approach for identifying the presence of minor amounts of large inclusions in rubber.

Modelling energy and size distribution in cone crushers

The modelling of breakage in compression breakage has traditionally been done using population balance modelling, and the research has been developed over the last decades into advanced dynamic models. This paper presents a model for predicting particle size distribution and energy consumption. The particle size distribution model is derived using a first-order differential equation for how the coefficient of variance depends on the compression length. The coefficient of variance model is combined with a bimodal Weibull distribution to predict the cumulative size distribution. The power consumption is modelled in a similar way using Weibull analysis to determine the relationship between power consumption and the coefficient of variance.

Dynamic breakage of porous hexagonal boron nitride ceramics subjected to impact loading

Based on the Split Hopkinson Pressure Bar (SHPB) device, a series of experiments have been conducted to reveal the intrinsic mechanism of dynamic breakage for hexagonal boron nitride ceramics (HBNC) under impact loading at strain rates ranging from 400 s−1 to 1000 s−1. Firstly, high speed camera and the digital image correlation (DIC) method are employed to trace the strain field on specimen surface. Evolution of number density of strain amplitude is found to obey the bimodal (two-peak) Weibull distribution. The first peak (peak 1) and the second peak (peak 2) owe respectively to deformations of the matrix part and the failure part of specimen. Secondly, by using the 4-units HgCdTe infrared detector, the in-situ instantaneous temperatures elevating at three positions on specimen surface are measured. Contributions of two main mechanisms, e.g. the plastic work and the macro fracture energy, to the elevated temperature are distinguished when taking the temperature profiles of strain rate 400 s−1 as reference. Relations of peak 2 to the elevated temperatures induced by the macro fractures (TIF) are then established and analyzed in details. The TIFs show linear relationship to 1/R, where R denotes the characteristic fragment size. The results should be helpful for temperature controlling in powder grinding and medical pill refining.

A bimodal Weibull distribution - capacity factor for different heights at sulur

Due to developing environmental concern use of renewable energy source is very essential. The great demand for the energy supply coupled with inadequate energy sources creates an emergency to find a new solution for the energy shortage. The appropriate wind energy distribution is the fundamental requirement for the assessment of wind energy potential available at the particular site essential for the design of wind farms. Hence the proper specification of the wind speed distribution plays a vital role. In this paper the Bimodal Weibull distribution is used to estimate the Capacity factor at the proposed site. The shape and scale parameters estimated using Maximum likelihood method is used as the initial value for extrapolation. Application of this model will give an accurate result overwhelming the concept of overestimation or underestimation of Capacity factor.

TDDB Mechanism in a-Si/TiO2 Nonfilamentary RRAM Device

Mechanisms of time-dependent dielectric breakdown (TDDB) in nonfilamentary a-Si/TiO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> RRAM cell [a-vacancy modulated conductive oxide (VMCO)] have been examined in this paper, including defects generation in the grain boundary, defects clustering, and different defects generation rates in a-Si and TiO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> layers. The unique feature of a bimodal Weibull distribution at low-resistance state (LRS) and a single shallow slope distribution at high-resistance state (HRS) cannot be explained by the above mechanisms. By using a combination of constant voltage stress, time-to-breakdown Weibull distribution, and random telegraph noise (RTN)-based defect profiling in devices of various sizes, layer thicknesses, and processes, it is revealed that the defect profile is modulated when switching between HRS and LRS and the correlation of defect profile modulation with local defect generation rate can explain the difference in Weibull distributions at HRS and LRS. The transition from bimodal distribution at LRS to a single-steep slope with thinner a-Si layer, and the good area scaling of Weibull distribution at HRS but not at LRS, can also be explained. The critical layers affecting the TDDB in a-VMCO are identified, providing useful guidance for device performance improvement.

Порівняння шести функцій щільності розподілу для моделювання таксаційної будо¬ви за діаметром модальних ялицевих деревостанів Українських Карпат

Для дослідження таксаційної будови за діаметром ялицевого елемента лісу у семи підприємствах лісового господарства Українських Карпат та в НПП "Сколівські Бескиди" закладено 14 прямокутних тимчасових пробних площ та 61 кругову пробну площу (у 15 таксаційних виділах). Дані суцільного переліку дерев ялиці зі замірами їх діаметрів використано для визначення особливостей розподілу кількості дерев за діаметром. Моделювання фактичного розподілу кількості дерев здійснено за допомогою шести фунцій щільності розподілу: дво-, три-, та семипараметричного розподілу Вейбулла, гамма- та бета-розподілу, Sb розподілу Джонсона. Виявлено, що ялицеві деревостани характеризуються значною мінливістю діаметра. За результатами аналізу основних усереднених статистичних показників різних функцій розподілу за діаметром ялицевого елемента встановлено, що дво-, трипараметричний розподіл Вейбулла, гамма- та бета-розподіл завищують показники асиметрії та ексцесу порівняно із фактичним розподілом, Sb розподіл Джонсона занижує ці показники порівняно з фактичними, найближчі результати має бімодальний розподіл Вейбулла. Отримані результати непараметричної оцінки за допомогою χ² критерію Пірсона дають змогу зробити висновок, що найпридатнішою функцією для дослідження таксаційної будови модальних ялицевих деревостанів Українських Карпат є семипараметричний розподіл Вейбулла, а найменш точним – гамма-розподіл.

Heterogeneity in deformation of granite under dynamic combined compression/shear loading

By adding two blocks with symmetrical oblique surfaces, the dynamic behaviors of granite under combined compression/shear loading are investigated by the modified split Hopkins on pressure bar device (SHPB). High speed camera is employed for image acquisition of the specimens with manual speckle patterns. The transverse tensile strain field and maximum shear strain field are calculated and presented by the Digital image correlation (DIC) method. Evolution of number density of strain amplitude is found to obey the bimodal (two-peak) Weibull distribution. The first peak (peak 1) and second peak (peak 2) are dominated respectively by the evolution of local deformations in the matrix part and failure part of specimen. Amplitude of Peak 2 can also illustrate the distribution extent of activated microcracks under external loads, and is not obvious in quasi-static tests. A governing equation is proposed to describe the statistical evolution of strain fields. The evolution rate of quasi-elastic strain, the damage nucleation rate, and the contribution of heterogeneous deformation of the failure part to whole deformation in specimens are derived accordingly. The results show that the heterogeneity is loading-rate dependent and/or shear-component dependent.

Analysis of the concentration in rare metal ores during compression crushing

Given the increasing global demand for rare metals, there is a need for the development of fundamental predictive models to improve extraction processes. Comminution models commonly predict particle size reduction based on the compressive breakage behaviour; however, few of them include mineral concentration or mineral liberation at a coarse scale. This paper focuses on developing a model to predict the mineral concentration of rare metals as a function of the particle size distribution after a cycle of the compression crushing process. In this study, compressive breakage and geochemical analysis experiments were conducted on four different rare metal ores of tantalum and tungsten. The work is divided into two stages: the methodology of modelling particle size and modelling concentration by selecting a bimodal Weibull distribution for calibration. A novel model for simulating the concentration of rare metals as a function of the compression ratio is presented.

Dynamic breakage of glass sphere subjected to impact loading

The dynamic compression responses of four glass spheres with diameters 7.71 mm, 11.90 mm, 17.88 mm, and 24.87 mm are investigated by the modified split Hopkinson pressure bar device (SHPB). Based on the high speed images, the dynamic compression of the spheres can be divided into three stages, e.g. the quasi-elastic deformation stage, the non-linear deformation stage, and the brittle failure stage. The formation and growth of the shadow areas observed at the impact end and the support end in the spheres are the main causes of the catastrophic failure. The movement of the plane-like fronts of the shadow areas is activated and driven by local velocity gradients. Dynamic breakage of the spheres is then analyzed by the particle size distribution, the fractal properties, etc. The results show that there are two main failure mechanisms in the process of sphere breakage, e.g. the tensile failure mechanism and the shear failure mechanism. In order to investigate the transition of these two failure modes, a bimodal Weibull distribution model is preliminarily established on the Weibull statistical strength. The model shows both good simulations to the experimental data and obvious the transition process of two failure mechanisms. The scaling laws and the strain rate effects of breakage strength are then discussed. The results are helpful for the controlling of pulverization for brittle particles.