Weibull Theory Research Articles

Fracture of Alumina Tubes under Combined Tension/Torsion

A fracture criterion for Al2O3 tubes is investigated for various loading paths under combined tension and torsion. Experimental data are compared with the Weibull statistical fracture theory. As the stress state approaches a shear dominant state, a tensile principal stress at fracture from the Weibull theory underestimates strengthening effects. By including the effect of the minimum principal stress in the tensile principal stress criterion, an empirical fracture criterion is proposed. Agreement between the proposed criterion and experimental data for Al2O3 tubes under combined tension/torsion and balanced biaxial tension is very good.

A general strength distribution function for brittle materials

A new strength distribution function for brittle materials is developed, which applies to materials with an inhomogeneous distribution of flaws. The probability of failure is F=1− exp [−〈N c,s 〉] where 〈N c , s 〉 is the mean number of critical defects in the specimen of size S. The well-known Weibull statistics are a special case of the new statistics for a special flaw size distribution. Several aspects of the relationships between the Weibull statistics and material structure are analysed in the light of the new formalism. Examples are materials with several different flaw distributions or rising crack resistance. The conditions necessary to get a Weibull distribution as well as the reasons why Weibull distributions are observed so often in the daily material testing practice are discussed. Finally, the minimum number of test specimens necessary to guarantee a reliable prediction of the component's reliability using Weibull's theory is given. This number depends on the necessary reliability as well as on the loaded (effective) volumes of the test specimens and components, respectively. Es wurde eine neue Festigkeitsverteilungsfunktion für spröde Werkstoffe entwickelt, die Werkstoffe mit einer inhomogenen Fehlerverteilung beschreibt. Die Bruchwahrscheinlichkeit ergibt sich aus F=1− exp [−〈N c,s 〉] Hierbei ist 〈N c , s 〉 die mittlere Anzahl kritischer Defekte in einer Probe der Größe S. Die bekannte Weibullstatistik ist ein Sonderfall der neuen Statistik und zwar für eine besondere Fehlergrößenverteilung. Verschiedene Aspekte der Beziehungen zwischen der Weibullstatistik und der Materialstruktur werden aus der Sicht des neuen Formalismus analysiert. Beispiele hierzu bilden Werkstoffe mit mehreren verschiedenen Fehlerverteilungen oder mit zunehmendem Rißwiderstand. Die erforderlichen Bedingungen, die zu einer Weibull-Verteilung führen, werden diskutiert. Außerdem werden die Ursachen, weshalb die Weibull-Verteilung so häufig in der täglichen Praxis der Werkstoffprüfung beobachtet wird, besprochen. Schließlich wird die minimale Probenanzahl angegeben, die für eine Gewährleistung einer sicheren Vorhersage der Zuverlässigkeit der Komponente nach der Weibull-Theorie erforderlich ist. Diese Anzahl hängt sowohl von der erforderlichen Zuverlässigkeit als auch von dem effektiv belasteten (effektiven) Volumen der Probe bzw, der Komponente ab. Une nouvelle fonction de distribution de la résistance mécanique pour des matériaux cassant a été développée, qui s'applique aux matériaux avec une distribution inhomogène des défauts. La probabilité de défauts est F=1− exp [−〈N c,s 〉] où 〈N c , s 〉 est le nombre de défauts critiques moyen dans l'échantillon de taille S. La statistique bien connue de Weibull est un cas particulier de la nouvelle statistique pour une distribution particulière de défauts. Plusiers aspects de la relation entre la statistique Weibull et la structure du matériau sont analysés à la lumière de ce nouveau formalisme. Des exemples sont des matériaux avec plusieurs distributions de défauts ou une résistance croissante à la fissure. Les conditions nécessaires pour avoir une distribution Weibull, aussi bien que les raisons pour lesquelles des distributions Weibull sont observées si souvent dans la pratique courante des tests sur les matériaux, sont discutées. Finalement on a donné le nombre minimum d'échantillons d'essais nécessaires pour garantir une bonne mesure de la fiabilité du composant en utilisant la théorie de Weibull. Ce nombre dépend de la fiabilité nécessaire ainsi que des volumes en charge (effective) des échantillons d'essais et des composants.

Statistical Size Effect in Quasi‐Brittle Structures: II. Nonlocal Theory

The failure probability of structures must be calculated from the stress field that exists just before failure, rather than the initial elastic field. Accordingly, fracture‐mechanics stress solutions are utilized to obtain the failure probabilities. This leads to an amalgamated theory that combines the size effect due to fracture energy release with the effect of random variability of strength having Weibull distribution. For the singular stress field of linear elastic fracture mechanics, the failure‐probability integral diverges. Convergent solution, however, can be obtained with the nonlocal‐continuum concept. This leads to nonlocal statistical theory of size effect. According to this theory, the asymptotic size‐effect law for very small structure sizes agrees with the classical‐power law based on Weibull theory. For very large structures, the asymptotic size‐effect law coincides with that of linear elastic fracture mechanics of bodies with similar cracks, and the failure probability is dominated by the...

Transmission overhaul and component replacement predictions using Weibull and renewal theory

Two methods of estimating the time between transmission overhauls and the number of replacement components needed were presented. The first method assumes replacement of all components during an overhaul of a failed transmission (full replacement method). The second method assumes replacement of failed components only (partial replacement method). Both methods assume the transmission components follow a two-parameter Weibull failure distribution. Renewal theory was presented to estimate the number of component replacements in a transmission for both methods. For the partial replacement method, renewal theory was used with the individual component life predictions to estimate the number of component replacements needed and the transmission time between overhauls. For the full replacement method, renewal theory was used with a transmission system life model to estimate the number of replacement transmissions needed and the transmission time between overhauls. Confidence statistics were applied to both methods to improve the statistical estimate of sample behavior. A transmission example was presented to illustrate use of both methods.

Statistical Size Effect in Quasi‐Brittle Structures: I. Is Weibull Theory Applicable?

The classical applications of Weibull statistical theory of size effect in quasi-brittle structures such as reinforced concrete structures, rock masses, ice plates, or tough ceramic parts are being reexamined in light of recent results. After a brief review of the statistical weakest-link model, distinctions between structures that fail by initiation of macroscopic crack growth (metal structures) and structures that exhibit large macroscopic crack growth prior to failure (quasi-brittle structures) are pointed out. It is shown that the classical Weibull-type approach ignores the stress redistributions and energy release due to stable large fracture growth prior to failure, which causes a strong deterministic size effect. Further, it is shown that, according to this classical theory, every structure is equivalent to a uniaxially loaded bar of variable cross section, which means that the mechanics of the failure process are ignored. Discrepancies with certain recent test data on the size effect are also pointed out. Modification of the Weibull approach that can eliminate these shortcomings is left for a subsequent paper.

Numerical simulation and field testing of flame-jet thermal spallation drilling—1. Model development

Certain polycrystalline rocks will fracture into thin, disk-like fragments when exposed to rapid surface heating. Some current hard-rock drilling methods using supersonic flame-jets as heat sources exploit this behavior for efficient granite quarrying or blasthole formation. Recent extensions of Weibull's theory of rock failure to quantitatively analyze rock spallation are employed to estimate spall size distributions and rock surface temperatures at the onset of spallation. A numerical drilling simulation code incorporating rock failure criteria and turbulent flow effects is developed to predict spallation rates and hole radii under field-drilling conditions. Because an accurate description of heat transfer through a turbulent wall jet is required, the wall-function method is generalized in a crude way to account for compressible flow past roughened, non-adiabatic rock surfaces.

Relationship between strength variability and size effect in unidirectional carbon fibre/epoxy

Abstract Unidirectional carbon fibre/epoxy exhibits a size effect which is not necessarily consistent with the variability of specimen strengths predicted on the basis of classical Weibull theory. In some cases the variability is higher than expected and this may be due to other sources of variability in the tests apart from the material. However, in bending tests consistently lower variability than expected has been measured. This is explained qualitatively in terms of the splitting that occurs during failure which means that the composite behaves between the extremes of a brittle solid and a loose fibre bundle. This also implies that the size effect should be more strongly dependent on specimen length than indicated by Weibull theory. Three-point bending tests on specimens of different lengths but the same cross-section have been carried out which support this.

Size effect on the strength of a concrete member

The size effect of concrete strength has been analysed by statistical theories of brittle materials in this paper. On the basis of available theories and experiments, normal distribution was used in preference to any other. The use of Weibull's theory in a non-uniformly tensile stress state has been discussed in this paper, then the authors proposed a new method to count the volume of tensile stress for beams and eccentric loading columns. Finally, we got a good coincidence between the tested and theoretical calculated results for cubic, prismatic compressive specimens, splitting specimens and fracture specimens.

Rock failure mechanisms of flame-jet thermal spallation drilling—theory and experimental testing

Certain polycrystalline rocks will fracture into thin, disk-like fragments when exposed to rapid surface heating. Several current hard-rock drilling methods using supersonic flame-jets as heat sources exploit this behaviour for efficient granite quarrying or blasthole formation. The recent application of Weibull's theory of rock failure to quantitatively analyze rock spallation has been extended to allow prediction of chip size distributions and rock surface temperatures at spallation under any intense heat source. Therefore, with measured values of the Weibull distribution parameters, surface spallation temperatures can be estimated theoretically. However, finite-element analysis for intergranular stresses developed during transient heating prior to spall formation indicates that additional microcrack initiation will be favoured thereby altering Weibull parameters measured at room temperature. Nonetheless, experimentally observed spallation characteristics of Barre and Westerly granites are consistent with theory when mechanically determined Weibull parameters are substituted. Spallation temperatures induced by flame-jet heating were estimated to be below 520°C. In all cases, quartz or other mineral solid phase transitions are not expected at these conditions. Laser-induced spallation experiments were less successful, partially because extremely localized heating delivered by the beam greatly complicated the subsequent thermal stress analysis.

Application of the Weibull statistics to the characterization of metallic glass ribbons

Metallic glasses are of commercial interest because of their magnetic and mechanical properties. Apart from their embrittlement under given testing conditions which has not yet been fully understood, the ribbons exhibit some inhomogeneities that lead to high failure probabilities for unnotched specimens. Two main types of quenched-in defects can be observed on the rupture surfaces. They are responsible for the specimens failure. For the first time, Weibull theory is successfully applied to the defects distribution in metallic glasses. Three distinct Weibull families of results are distinguished according to Weibull statistics. Two examples of commercial alloys are presented which show the interest of Weibull plots to control the ribbons quality and to follow the evolution of their properties under different conditions of use.

The application of Weibull theory to primary fragmentation of a coal during devolatilization

Primary fragmentation occurring during the devolatilization of a South African coal in a laboratory fluidized-bed combustor has been investigated. Fracture orientation in respect to geological bedding planes has been examined and the tensile strength of the coal determined by means of dia- metrical compression tests of spheres and bending tests of bars at ambient temperature and at temperatures up to 600 °C. A mechanical model of primary fragmentation accounting for anisotropy due to bedding planes is proposed for spherical particles. The starting point for prediction of the occurrence of fragmentation is a simple monoaxial stress field, consisting of internal forces due to volatile pressure opposite to resistance forces of the material. Experimental fragmentation probabilities, particle multiplication and fragment size distributions have been compared with model predictions when both the critical stress theory and, taking into account the non homogeneities of the material, the Weibull theory, are used to characterize coal and char strength.

Note on Composite Size Effect

The purpose of this note is to propose a relation somewhat similar to the size-strength relationship of brittle materials failing in tension in accordance with the Weibull theory. According to the failure theory of composites, a plot of the natural log of failure stress (ln σf) versus the natural log of volume (ln V) for a uniaxially reinforced composite is a curve that is concave upward. A simple analytical relation representing this concave failure curve could be very useful, since current methods of constructing the line are quite complex and necessitate the adoption of debatable assumptions about the magnitudes of a number of the many parameters that appear in the theory. The relation proposed in this note gives the theoretical failure stress of uniaxial composites to a surprising degree of accuracy.

Primary fragmentation of a coal in fluidized bed combustion

Primary fragmentation is related to the build up of pressure in the pore network of a coal particle during devolatilization. Experiments of primary fragmentation of a bituminous, non-swelling South African coal have been carried out by injecting irregular particles, spheres and cubes of size ranging between 1 and 15.5 mm in a basket-equipped fluidized bed combustor of 40 mm ID. Depending on the size of coal, fragmentation occurs with a different degree of probability and with the generation of fragments characterized by a variety of shapes. This may include up to eight different types of fragments for the coarser coal particles tested. Material shows an elastic-brittle behavior with fractures parallel to bedding, planes of coal. Separate experiments have been carried out in order to determine average tensile strength of coal at different temperatures and its dispersion around the average due to non-homogeneities of material. A model of primary fragmentation has been developed. It is made of a serial structure of four submodels which account for the heatup of coal particles in the bed, the kinetics of devolatilization, the volatile convective transport and build up of pressure in the particle pore network and, according to the statistical theory of Weibull, the fragmentation of particles when internal pressure force overcomes the resistance force of the material. The basic model assumes spherical symmetry of coal particles. A modified version considers that non-spherical fragments still rich in volatiles may experience further fragmentation as if they were fresh spheres of equivalent diameter. The study shows that effects of primary fragmentation may noticeably influence fluidized combustion behavior of coal tested particularly in the case of coarse feed. Probabilities of particle fragmentation, multiplication factors of the number of particles, average sizes of fragmented material predicted by the modified model are in good agreement with experimental results.

Decrease of apparent tensile and bending strength with specimen size: Two different explanations based on fracture mechanics

The influence of the shape and size distribution of defects on material strength is investigated. Constant reference is made to Weibull Theory and Fracture Mechanics, and a family of regular polygonal voids is considered with the two limit-cases of Griffith cracks and circular pores. A defect size distribution of proportionality is defined, for which the maximum defect size turns out to be proportional to the linear size of the body. In this way, a very general analytical expression for the tensile strength decrease with size is obtained and then confirmed by experimental evidence.Specimen size is also shown to have a fundamental influence on global structural behaviour, which can range from ductile to brittle when strain softening and strain localization are taken into account. The brittle behaviour coincides with a snap-back instability in the load-deflection path, which shows a positive slope in the softening branch. Such a virtual branch may be revealed only if the loading process is controlled by a monotonically increasing function of time (e.g. the crack opening displacement). Otherwise, the loading capacity will present a discontinuity with a negative jump.A general explanation of the well-known decrease in bending strength by increasing the specimen sizes is given in terms of dimensional analysis. Due to the different physical dimensions of strength [FL−2] and toughness [FL−1 ], the true value of such material property may be found exactly only with comparatively large specimens.

Weibull statistical fracture theory for the fracture of ceramics

The Weibull statistical fracture theory is widely applied to the fracture of ceramic materials. The foundations of the Weibull theory for brittle fracture are reviewed. This theory predicts that brittle fracture strength is a function of size, stress distribution, and stress state. Experimental multiaxial loading results for A12O3 tubes are compared to the stress state predictions of the Weibull theory. For the most part, the Weibull theory yields reasonable predictions, although there may be some difficulties in dealing with shear stress effects on fracture.

A New Procedure for Determination of Point Load Strength in Site Investigation

Abstract As a result of developments over the last twenty years, the point load test has become a cheap and practical method for field and laboratory determination of rock strength. Since point load strength is size dependent, the test results are generally corrected to a standard thickness of 50 mm by using empirical correction charts and the median I (50) value is given as the point load strength. This may be used to estimate other rock properties. As the failure of specimens is in the tensile mode and strength is size dependent, the point load test results are well accounted for by the Weibull theory. When mean point load strength values are plotted against corresponding specimen sizes on a log-log scale, a linear relation with a negative slope is obtained. The slope of the best fit line gives the reciprocal of the Weibull modulus ( m ). I (50) is found from the intercept with the 50 mm thickness line. Similarly I ( s ) values for any thickness can be determined. If rock cores of single size are tested, the “ m ” modulus is estimated first from the Weibull distribution plot of the point load strengths. A strength-size relation is thus established from which I (50) can be determined. These methods are simple to apply and do not require a correction chart.

Strength of Resin Mortar Tunnel

Although the strength of resin mortar tunnel rings is less than 75% of the bending strength of rectangular specimens, the reason for this has not been clarified. In this paper, the resin mortar strength is analyzed statistically using Weibull's theory. Taking size effects and stress distribution into account, the relation between the ring strength and the bending strength of rectangular specimens is determined. Based on this relation, the resin mortar tunnel reliability can then be estimated.

Probabilistic approach and size effect in brittle materials under multiaxial stress states

The problem of evaluating the strength of brittle materials demands a statitistical approach owing to the considerable dispersion shown by these materials, particularly if subject to stress states in which the tensile component is predominant. For this purpose, the Weibull theory can be conveniently applied, which is based on the concept of the weakest link in a chain allowing the determination of both the failure probability of a model under certain loading conditions and the corresponding mean strength.

Uncertainty on the mean strength and Weibull's modulus of an alumina batch as a function of the number of samples

Sintering of powders is the main process used to produce ceramic materials. It may induce various defects, such as pores, voids or inclusions. Moreover, other defects are induced when machining pieces (cutting, polishing, etc.) the most common of them being streaks and the pull-out of grains. The defects control the initiation of brittle fracture and their efficiency as crack initiators is dependent on their size and shape. Besides, this efficiency is greater when the critical defect lies in the vicinity of a high stress tensile zone. The random distribution of defects leads to a random distribution of the measured values of strength: so, these values can be understood only in the frame of a statistical analysis. Among the studies dealing with this question, Weibull's theory of the "weakest link" [1] is the most commonly used, but more refined analyses can be developed [2, 3]. From the experimental point of view, an important question concerns the number of samples which is required for an accurate determination of Weibull's parameters. The present note is devoted to this point. Weibull's analysis links the survival probability of a piece, Ps, to the applied stress, a, by means of the relation:

Multiaxial Loading Fracture of Al2O3 Tubes: II, Weibull Theory and Analysis

The Weibull statistical fracture theory for multiaxial loading was developed for thick‐ and thin‐walled tube geometries subjected to multiaxial loading in tension‐internal pressure, compression‐internal pressure, and pure torsion. As compared to uniaxial tension, lower strengths are predicted for tension‐tension stress states and higher strengths for tension‐compression stress states. Comparison to experimental results for A1203 tubes indicates a reasonable agreement with Weibull theory predictions for tension‐internal pressure and compression‐internal pressure conditions, but an underestimation of stress state effects in pure torsion. Results indicate a weakening effect of in‐flaw‐plane tensile stresses, with no observed influence of in‐plane compressive stresses.