Applicability of Weibull analysis for brittle materials

Weibull analysis for the interpretation of strengths of brittle materials was previously justified for a particular flaw size distribution. The present results show that the Weibull distribution provides a close approximation to the distribution of failure stress for all the flaw size distributions considered. However certain reservations are noted in the interpretation of the Weibull modulus.     

On estimating the Weibull modulus for a brittle material

Common methods of estimating the Weibull modulus are surveyed. Computer simulation is used to obtain the statistical properties of different estimators. Most estimators are shown to be biased and their respective adjustment factors, for a range of experimentally feasible sample sizes, are given.     

Probabilistic Weibull behavior and mechanical properties of MEMS brittle materials

The objective of this work is to present a brief overview of a probabilistic design methodology for brittle structures, review the literature for evidence of probabilistic behavior in the mechanical properties of MEMS (especially strength), and to investigate whether evidence exists that a probabilistic Weibull effect exists at the structural microscale. Since many MEMS devices are fabricated from brittle materials, that raises the question whether these miniature structures behave similar to bulk ceramics. For bulk ceramics, the term Weibull effect is used to indicate that significant scatter in fracture strength exists, hence requiring probabilistic rather than deterministic treatment. In addition, the material's strength behavior can be described in terms of the Weakest Link Theory (WLT) leading to strength dependence on the component's size (average strength decreases as size increases), and geometry/loading configuration (stress distribution). Test methods used to assess the mechanical properties of MEMS, especially strength, are reviewed. Four materials commonly used to fabricate MEMS devices are reviewed in this report. These materials are polysilicon, single crystal silicon (SCS), silicon nitride, and silicon carbide.     

On the estimation of Weibull's parameters in brittle materials

The parameters in Weibull's specific risk function are estimated using several methods, both in compression and in bending. To this end, several formulae are employed for different values of the limiting stress _L. The parameters are estimated by minimizing ² in order to compare the capacity of the different formulae for approximating these parameters. In the case of bending, the error committed when obtaining the parameters from a uniform-stress-field Weibull function is of the order of 100%.     

Investigation on the Weibull parameters identification for local approach application in the ductile to brittle transition regime

Fracture of BCC metals is a temperature dependent process characterized by a temperature transition regime where material failure is the result of the competing action of two different mechanisms of rupture, namely, cleavage and ductile fracture. In this paper, two micromechanics based methodologies, the local approach and a non-linear CDM model, are used to model fracture processes in the temperature transition regime for a low alloy steel. The study revealed that approaching the nihil ductility temperature (NDT), the identification procedure for the material parameters needed in the local approach becomes no longer stable resulting in a non unique set of σ_W and m values. This occurrence can be used to discriminate in the experimental data set those data failed in a non-complete brittle manner. Using m exponent equal to 4, a linear dependency, between σ_W and K_J, also in upper shelf regime, is demonstrated.     

Methods of estimating hydraulic and transport parameters for the unsaturated zone

Methods of estimating hydraulic and transport parameters for the unsaturated zone are presented. Two approaches to parameter estimation are discussed. The first approach is based on inversion of the governing initial-boundary value problem for unsaturated flow and/or transport using data from transient experiments. The second approach utilizes particle size distribution data to obtain estimates of soil hydraulic properties. Example applications of the two different approaches are presented and their advantages and limitations are discussed. Procedures for identifying and quantifying sources of uncertainty in parameter estimates are examined. Ongoing research on the scale-up of parameters for applications in large-scale numerical simulations is reviewed. Financial support for this study was provided by the Electrical Power Research Institute (Contract RP2485-06), and by the American Petroleum Institute (Contract WM-5-324-7).     

Effect of material parameters on the erosion resistance of brittle materials

Erosion data are compared with two theories that have been suggested to explain the erosive behaviour of solids. A dimensional analysis is applied to the variables that are important to erosion, and a multivariate, linear regression analysis is used to fit the data to the dimensional analysis. The results of the linear regression analyses are compared with the two theories in order to evaluate the applicability of these theories to erosion. Although semi-quantitative agreement of the data with the theories is obtained, some discrepancies are apparent. In particular, the dependence of erosion rate on hardness and critical stress intensity factor is greater than predicted by either of the two theories. These discrepancies are attributed primarily to microstructural aspects of erosion that are not modelled by either of the theories.     

Failure criteria for brittle elastic materials

The validity of several known failure initiation criteria at reentrant corners in brittle elastic materials is examined and a simple one is proposed. Their predictions, under mode I stress field, are compared to experimental observations carried out on PMMA (polymer) and Alumina-7%Zirconia (ceramic) V-notched specimens. Because all realistic V-notched reentrant corners are blunt, a detailed experimental procedure has been followed, focusing on specimens with different notch tip radii. It is demonstrated that by assuming a sharp V-notch, some failure criteria predict reasonably well the experimental findings, and that corrections are needed in order for these to take into consideration the realistic radius at the notch tip.     

On the validity of the Weibull failure model for brittle particles

The Weibull theory of material strength and fracture assumes that the Weibull modulus m is a material parameter, which does not depend on shape and size of the loaded object. Based on large data sets from single-particle fracture experiments with brittle materials (glass, clinker cement, limestone), the authors show that the Weibull modulus of nearly spherical particles seems to decrease with increasing particle diameter. A possible explanation is that the inner structure of the particles depends on their size so that small particles are much stronger than large ones. Received: 9 December 1999     

A statistical strength criterion for brittle materials

Based on a probabilistic model of brittle microfracture (microdamage) leading to macrofracture of materials, we propose a statistical interpretation of strength criteria, which relate start of macrofracture (manifested by macrocrack formation in tension or internal structure stability loss in compression) with attainment of a certain (critical) value of density of microdefects in the material under study. The criterion is reduced to comparative analysis of microdefect density induced by the particular loading type and its critical value, which is intricnsic to this material and is invariant to the stressed state type.     

The single-filament-composite test: a new statistical theory for estimating the interfacial shear strength and Weibull parameters for fiber strength

For over two decades the single-filament-composite (SFC) test has been an important tool in the study of the failure of fibrous composites. The SFC test itself involves a single brittle fiber embedded along the center-line of a matrix specimen of both large cross-sectional area and strain to failure. With increasing strain, the fiber fractures progressively, breaking into an increasing number of shorter and shorter fragments. Surrounding each break a shielded or exclusion zone develops within which no further breaks typically occur. At some strain level ‘saturation’ occurs abruptly as the shielded zones finally occupy the whole fiber, thus leaving a final distribution of fiber fragments end-to-end. Two uses for the SFC test have emerged: one has been to estimate the interfacial shear stress, τ, in the exclusion zone, sometimes called the interfacial shear strength and usually idealized as a constant over this zone. The other has been to estimate the fiber strength distribution and in particular the Weibull shape and scale parameters, ρ and σ_l, for fiber strength appropriate to some characteristic ‘gage’ length, l, such as the mean fragmentation length. In the past, theoretical bases for these estimates have handled the statistics of shielding in ways that have led to quite large biases. The purpose of the present paper is to use some recent theoretical advances to develop more sophisticated estimation procedures for τ and the Weibull fiber strength parameters ‘ in situ’, and thus to eliminate various errors in previous methods. Straightforward computer programs (written in release 3 of Maple), which calculate the various quantities in the paper, will be provided by the first or second author on request.