Three-parameter Weibull Analysis Research Articles

Increasing Bending Strength of Polycarbonate Reinforced by Carbon Fiber Irradiated by Electron Beam.

In an interlayered carbon fiber-reinforced polycarbonate polymer (CFRPC) composite composed of three sized of CF plies, alternating between four PC sheets, designated [PC]4[CF]3, and a new process of activating CF cross-weave cloth plies directly on both sides with homogeneous low-energy electron beam irradiation (HLEBI) before lamination assembly and hot pressing at 6.0 MPa and 537 K for 8 min was produced. Experimental results show that a dose of 215 kGy of HLEBI raised the bending strength, σb, at each experimental accumulative probability, Pa, with the σb at a median Pa of 0.50, increasing by 25% over that of the untreated sample. Three-parameter Weibull analysis showed that when quality can be controlled, a dose of 215 kGy of HLEBI can raise the statistically lowest bending strength, σs, at Pa = 0 (94.3 Mpa), with a high correlation coefficient. This is because, although it had a higher bending strength than that in the other experimental conditions, the weakest sample of the 215 kGy data set had a much lower σb value than that of the others. Electron spin resonance (ESR) of the CF showed that naturally occurring dangling bonds in CF were increased at 215 kGy. Charge transfer to the PC occurs, apparently generating stronger bonds, which are possibly covalent, resulting in enhanced adhesion at the CF-PC interface.

An improved crack probability model for silicon oxide layers using three-parameter Weibull analysis

For semiconductor dielectric layer crack detection in real-time we developed an efficient and very accurate method, which is based on acoustic emission (AE) testing and simulates the wafer probing process. The crack probability is determined using clustered and filtered AE data, that is collected by an advanced sensor-indenter system. The derived distribution function is based on two-parameter Weibull analysis, which is commonly applied for stress-induced failures of brittle materials like silicon oxide (SiO2). During indenter experiments, we observed, that the probability model for a higher number of data points is becoming inaccurate, especially at low fracture forces. Therefore, we introduce a more accurate statistical model for probing-induced SiO2 layer cracks, which is, according to a three-parameter Weibull distribution, considering a location parameter. Experimental results have shown, that the improved distribution function fits much better for predicting low-ppm failure rates of back-end-of-line (BEOL) stacks of CMOS technologies. This enables more reliable pad stack design rules and increases the reliability of complex chip designs regarding mechanical stress during test and assembly processes.

Parameters estimation and fatigue life prediction of sisal fibre reinforced foam concrete

Based on the Glivenko-Cantelli theorem and Weibull distribution functions, the fatigue life of foam concrete with different sisal fibre volume fraction (0–0.3% vol.) was performed. When the two-parameter Weibull function was selected, a linear regression analysis showed that the fatigue life had high correlation between the independent and dependent variables (correlation coefficients above 0.75). But the three-parameter Weibull analysis gave more accurate results (correlation coefficients above 0.90). Then, the corresponding Weibull probability density function distributions of the fatigue lives were compared. Based on the parameters obtained, the fatigue lives of foam concrete with different survival probabilities were analyzed and compared with results taken from the literature. Finally, for specific survival probabilities, the fatigue limit and fatigue life of foam concrete under specified stress level were predicted.

Tensile Strength Reliability Analysis of Cu48Zr48Al4 Amorphous Microwires

Uniform Cu48Zr48Al4 amorphous microwires with a high surface quality are fabricated by a melt extraction technique. The mechanical property of microwires is evaluated via tensile tests. To estimate the strength scattering, statistical analysis of fracture strengths is conducted using logarithmic normal distribution, and two- and three-parameter Weibull analysis, severally. The results show that the tensile strengths of Cu48Zr48Al4 amorphous microwires range from 1724 to 1937 MPa with the arithmetical average value of 1836 MPa, and the arithmetical standard deviation of 56.4 MPa. The geometric mean of fracture strength is 1840 MPa using logarithmic normal distribution statistical analysis. Using two- and three-parameter Weibull analysis, the Weibull modulus and fracture threshold value are respectively calculated for 34.8 and 1483 MPa, which shows the excellent tensile mechanical properties with a high predictability of Cu48Zr48Al4 amorphous microwires and further indicates the great potential of application.

Tensile properties and fracture reliability of a glass-coated Co-based amorphous microwire

Co68.15Fe4.35Si12.25B15.25 (at%) amorphous microwires with a smooth surface and a circular cross-section were fabricated by the glass-coated melt spinning method. Their mechanical properties were evaluated through tensile tests of the glass-coated amorphous microwires, and their fracture reliability was estimated using two- and three-parameter Weibull analysis. X-ray diffraction and transmission electron microscopy results showed that these glass-coated Co-based microwires were mostly amorphous. The coated Co-based microwires exhibit a tensile strength of 1145 to 2457 MPa, with a mean value of 1727 MPa and a variance of 445 MPa. Weibull statistical analysis showed that the tensile two-parameter Weibull modulus of the amorphous microwires is 4.16 and the three-parameter Weibull modulus is 1.61 with a threshold value as high as 942 MPa. These results indicate that the fabricated microwires exhibit good tensile properties and fracture reliability, and thus appear to be good candidates for electronics reliability engineering applications.

Weibull statistical analysis of the effect of melt conditioning on the mechanical properties of AM60 alloy

The mechanical properties of the standard tensile samples of AM60 Mg-alloy obtained by conventional high pressure die casting (HPDC) and by melt conditioned high pressure die casting (MC-HPDC) have been assessed with the Weibull statistical model in association with the microstructural features, porosity levels and fracture morphology. It is found that three-parameter Weibull analysis is capable of evaluating the mechanical properties of AM60 magnesium alloy and assessing the improvement in the mechanical properties with melt conditioning in die casting process. The increased values of Weibull characteristic parameters show improvement in the strength and ductility of the AM60 alloy with melt conditioning. The reduced porosity levels and the improved microstructural uniformity in the die castings are found to be responsible for the improvement of the mechanical properties and the variation of the characteristic parameters in Weibull statistical distribution.

Cast defects induced sample-size dependency on compressive strength and fracture toughness of Mg–Cu–Ag–Gd bulk metallic glass

In this work, Mg59.5Cu22.9Ag6.6Gd11 bulk metallic glass (BMG) is selected to investigate the specimen-size dependency of the compressive fracture strength and notch toughness. As the diameter of cylindrical sample increases, apparent fracture strength is reduced by 25% from ∼950MPa at 1-mm down to ∼710MPa at 10-mm sample, which is associated with the population and size of as-cast porosity in the BMG rods as indicated by X-ray computed tomography (CT). Meanwhile, it is inappropriate to describe the size dependence of fracture strength using Weibull or Gumbel statistics owing to the inhomogeneous distribution of cast defects in different sized rods. Similar to the compressive fracture strength, cast defects in Mg59.5Cu22.9Ag6.6Gd11 BMG also yield specimen-size dependency of the measured notch toughness, when the plate-specimen thickness varies within the range of 1–8mm. With three-parameter Weibull analysis, threshold toughness Kμ of as-cast Mg59.5Cu22.9Ag6.6Gd11 BMG can be estimated, which represents a minimum toughness value below which the cracks arrest. Using finite element analysis, it is confirmed that the specimen-thickness dependency of notch toughness does not result from the transition from plane-stress to plane-strain in the specimen, but from the extrinsic flaws.

Variability in the yield strength of a metallic glass at micron and submicron length scales

This work presents an in-depth investigation of the dispersion in the compressive yield strength of a Zr-based bulk metallic glass as the specimen dimensions decrease from the micron to the submicron length scale. While the investigated alloy belongs to one of the so-called “ductile” glass families, the degree of variation in strength exhibited in the submicron size range approaches that of conventional brittle ceramics. Using a three-parameter Weibull analysis, however, we find that the engineering reliability of the glass is independent of specimen size. The glass exhibits a size-independent failure-free stress of 1825MPa, below which yielding is not expected. The three-parameter Weibull modulus is also size independent. The dispersion in yield strengths above the failure-free stress is due to the change in the sampling volume for specimens 1μm in diameter or larger, while interactions between defects and the surface actually limit the breadth of the strength distribution observed in specimens 200–330nm in diameter relative to what may be expected for that volume. Using an instability criterion for a penny-shaped crack under shear with interfacial friction, a relationship between yield strength and the corresponding defect size is established, and the defect concentration is estimated.

Weibull analysis of fracture strength for Zr55Ti2Co28Al15bulk metallic glass: Tension–compression asymmetry and porosity effect

Abstract

Mechanical properties distributions of PM manganese steels analysed by Gaussian and Weibull statistics

Powder metallurgy (PM) parts acceptance is determined by compositional and processing parameters and their controls. Statistical procedures are used for assessment; normal distribution and 'six sigma' appear to predominate. For fatigue of metallic materials and strengths of ceramics, fibres and composites, however, Weibull probability of survival analyses are widely used. The original analysis considers a threshold stress at which the probability of failure is zero. This stress is frequently taken to be zero, simplifying the analysis to two parameters. The yield stress has been suggested for the threshold stress, probably not sufficiently conservative for less ductile PM materials. A new three-parameter Weibull analysis, in which it is taken to be the fracture strength minus six standard deviations, is presented. Powder metallurgy manganese steels are under commercial consideration and this approach is applied to 12 variants of such laboratory processed specimens. It is compared with the two-parameter Weibull and Gaussian, the least conservative, analyses.

The Effect of Fe-Rich Intermetallics on the Weibull Distribution of Tensile Properties in a Cast Al-5 Pct Si-3 Pct Cu-1 Pct Fe-0.3 Pct Mg Alloy

A comparison was made between the tensile properties of an Al-5 pct Si-3 pct Cu-1 pct Fe-0.3 pct Mg (319-type) alloy in the as-cast condition. The alloy was either unmodified or modified with Mn or Sr. The additions of 0.6 wt pct Mn, or 0.9 wt pct Mn, or 300 ppm Sr were made. All modifying additions increased the average and decreased the scatter in tensile strength, and increased elongation, with the Sr addition being most effective among the three. Fractographic examinations showed that the β phase, the polyhedral α phase, and porosity appear to be the main defects responsible for failure. The distribution of tensile strength has been quantitatively assessed using both two- and three-parameter Weibull distribution functions. In the two-parameter Weibull analysis, the high Weibull modulus (46) achieved by Sr modification is due to its binary effect on eutectic silicon and β platelets. Reservations about potentially misleading implications of the two-parameter approach are explored. The three-parameter Weibull analysis provides new information. In particular, minimum values of strength below which the material is extremely unlikely to fail are found. These values again show the clear benefit of Sr addition. The results are explained by the effects of modifying agents on the microstructure.

Rapid Temperature Cycling (RTC) Methodology for Reliability Assessment of Solder Interconnection in Tape Ball Grid Array (TBGA) Assembly

In this study, a thermoelectric cooler-based rapid temperature cycling (RTC) testing method was established and applied to assess the long term reliability of solder joints in tape ball grid array (TBGA) assembly. This RTC testing methodology can significantly reduce the time required to determine the reliability of electronic packaging components. A three-parameter Weibull analysis characterized with a parameter of failure free time was used for assembly reliability assessment. It was found that the RTC not only speedily assesses the long-term reliability of solder joints within days, but also has the similar failure location and failure mode observed in accelerated temperature cycling (ATC) test. Based on the RTC and ATC reliability experiments and the modified Coffin-Manson equation, the solder joint fatigue predictive life can be obtained. The simulation results were found to be in good agreement with the test results from the RTC. As a result, a new reliability assessment methodology was established as an alternative to ATC for the evaluation of long-term reliability of electronic packages.

Life analysis of coated tools using statistical methods

The lifetime improvements achieved using physical vapour-deposited (PVD) TiN on hexagonal washer punch tools used in the fastener industry have been investigated. Tools were coated with PVD TiN initially in an unmodified form and later with surface preparation changes prior to coating. An original analysis, relying mainly on the visual appraisal of scanning electron micrographs and a rudimentary study of the lifetime data, found that the failure modes of the tools were altered by modifying the tool manufacturing process as well as by applying a surface coating. Sample mean lifetime of the best tools was over six times that of the mean sample lifetime of the unmodified, uncoated tools.By retrospectively applying experimental design techniques to the data, the influence on the lifetime from the tool modifications and the coating application have been decoupled. This shows that the tool modifications produce a larger lifetime response than the presence or not of a PVD coating. However, the interaction of the PVD coating and the tool modification produces a stronger positive response. Simple visualisation techniques have been used to effectively present this information in the form of ANOM (analysis of means), response charts, box plots and scatter diagrams.A three-parameter Weibull analysis also revealed evidence to suggest that, as well as different failure modes between treatments, some treatments underwent a number of early life failures by different failure modes. This is revealed purely from the distribution of the data and as such greatly enhances the visual analysis and interpretation of the previous analysis. The characteristic life for the improved tooling in this particular case is 465200 operations as opposed to 72000 for the original tooling. This study highlights the benefits of a full statistical analysis in assessing the benefits of coatings on production tooling, especially where the use of a coating is coupled with other changes in tool design and preparation.

Investigation of Weibull Statistics in Fracture Analysis of Cast Aluminum

The fracture strengths of two large batches of A357-T6 cast aluminum coupon specimens were compared by using two-parameter Weibull analysis. The minimum number of these specimens necessary to find the fracture strength of the material was determined. The applicability of three-parameter Weibull analysis was also investigated. A design methodology based on the combination of elementary stress analysis and Weibull statistical analysis is advanced and applied to the design of a spherical pressure vessel shell. The results from this design methodology are compared with results from the applicable ASME pressure vessel code.

Weibull analysis of strengths of various reinforcing filaments

The strengths of various reinforcing filaments were analysed using a three-parameter Weibull analysis computer program. The filaments studied included three kinds of tungsten-substrate filament and six kinds of non-substrate filament. The results showed that the log-log plot of coefficient of variation versus average strength of all the filaments is linear with the negative slope. The Weibull shape parameter and log of scale parameter are linearly related to the converted strength, the average strength minus location parameter, with the (0,0) intercept. The location parameter is also linearly related to the converted strength, but with the negative slope, and fairly large negative values are obtained for several filaments. In all four plots tungsten-substrate and non-substrate filament data are completely mixed. This suggests that the same mechanism prevails in the fracture of both types of filament.