Weibull theory

In this paper the Weibull theory is applied to very small specimens. The analysis follows the ideas presented in [13]. The relationships between flaw population, size of the fracture initiating flaw and strength are discussed. It is shown that a limit for the applicability of the classical fracture statistics (i.e. Weibull statistics based on the weakest link hypothesis) exists for very small specimens (components). 

Fig.S Characteristic strength versus effective volume in a double logarithmic plot (for details see [23]). The size effect on strength predicted by Weibull theory (based on bending strength data, 4PB) is indicated by the straight line (m=l6). It is obvious that the strength data determined by the notched ball test (set A, B and C) do not follow the predicted trend. This is caused by some surface damage, which is different in each of the investigated data sets (A, B and C).

Notably, these prerequisites are not valid for ductile and fiber-reinforced materials. In the first case, stresses are accommodated by plastic deformation, whilst in the latter case the load can be transferred from the matrix into the fibers. Furthermore, a negligible interaction between flaws is only possible if the flaw density is low. Thus, Weibull theory does not apply to porous materials. 

It has also been shown that Weibull theory does not apply to very small specimens [62], in which the fracture-causing flaws are also very small. For a Weibull material (i.e., for a material with relative flaw density corresponding to g oc a P), the density of the flaws becomes so high for small flaws that an interaction between the flaws will undoubtedly occur. 

The knowledge of the stress held is a hrst step towards the analysis of the risks of failure of ceramic materials subjected to multiaxial stress, such as provided by the Weibull theory [122]. A postprocessing procedure based on the the principle of independent action (PIA) is straightforward to implement. In the case future studies highlight the limitations of this simplihcation, more rehned theories exist, such as the Batdorf theory [57]. A complete analysis tool, CARES [94] is available. 

The parameter is a crack propagation velocity and n(e) is a crack activation law driven by the bulk tensile strain e and specified by the Weibull fracture theory... 

Weibull statistics are widely used in analysing the lifetime of components ( lifing ). Based on probability theory, they apply the formula ... 

Goda, K., Park, J. M. and Netravali, A. N., A new theory to obtain weibull fibre strength parameters from a single-fibre composite test, J. Mater. Sci., 30, 2722 (1995). 

Weibull (1995) Evolutionary Game Theory, MIT Press, Cambridge, Mass. 

Drug release from controlled release matrix tablets has been described by many kinetic theories [20,21]. Fig. 4 illustrates the release profiles of the validated dissolution (experiments 12, 12b and 12c) and the release profiles obtained from fits to the Weibull, Higuchi and Hixson-Crowell models. Figs 5 and 6 show the curves after linear transformation. 

The molecular mobility is usually discussed by T2 on the basis of BPP theory.25 However, parameters fci-fc4 cannot be directly compared any longer. The mobility should be discussed by using the width of the broad-line spectrum. The obtained crystalline components are Fourier-transformed into broad-line spectra for simultaneous evaluation of changes in the component ratio and molecular motion. The integral peak width on a frequency scale was calculated from these broadline spectra. Each component ratio of crystalline (Weibull/sine), intermediate... 

Weibull, W. (1939) A statistical theory of the strength of materials. Proc. Ing. Vetensk. Akad. (Stockholm), 151, 5—45. 

A Statistical Theory of the Streugth of Materials, Weibull aud Waloddi, lugeuiors Veteuskops Akadamieu-Hawdliugar, Stockholm 1939, No. 151, p. 1-45. 

Weibull developed his statistical theory of brittle fracture on the basis of the weakest link hypothesis, i.e. the specimen fails if its weakest element fails [6, 7], In its simplest form and for an uniaxial homogenous and tensile stress state, ct, and for specimens of the volume, F, the so called Weibull distribution of the probability of failure, F, is given by ... 

Bazant formulated a statistical theory of fracture for quasibrittle materials [5, 23, 24]. He assumed that there exist several hierarchical orders which each can be described by parallel and serial linking of so-called representative volume elements (RVEs). For large specimens (and low probability of failures) the fracture statistics is equal to the Weibull statistics, i.e. if the specimens size is larger than 500 to 1000 times of the size of one RVE. In the actual case this is similar to the diameter of the critical flaw. For smaller specimens the volume effect disappears and the fracture... 

W. Weibull, A Statistical Theory of the Strength of Materials, Ingenidrsvetenskapsakademiens, HandlingarNr 151, Generalstabens Litografiska Anstalts Fdrlag, Stockholm, 1 -45, (1939). 

The strength variability of ceramic materials can be evaluated using Weibull stahshc, which is based on the weakest-link theory, where the more severe flaw results in fracture propagation and determine the strength [69]. The Weibull two-parameter distribution is given by [14] ... 

G. J. DeSalvo, Theory and Structural Design Applications of Weibull Statistics, Report WANL-TME-2688, Westinghouse Electric Corporation, 1970. 

Klein, J.P. Basu, A.R1981. Weibull accelerated life test when there are competing causes of failure. Communications in Statistics Theory and Methods, 10(20) 2073-2100. 

When discrepancies between the strength predicted by the conventional Weibull model and the experimentally obtained data occur, then multimodal Weibull distributions have been used [18, 37]. Similarly, as mentioned earlier, the Weibull/ weakest-link model can be utilized to predict the scale effect by combining the classical Weibull distribution with the weakest link theory. All these take into account the shape and scale of the distribution. Such models are reported to have shown improved accuracy of prediction including those for wool fibers [38]. 

The Weibull approach, although useful when comparing fibres, does not take account of the failure mechanism. On the other hand, the deterministic theory of LEFM overcomes some of the limitations of the Weibull method but is applicable only when the initial crack geometry is known. There are intermediate cases where the crack geometry is known but not its precise location, as, for example, inner circular cracks located randomly. In these circumstances, LEFM can provide answers to questions such as what is the maximum rupture stress for a postulated maximum defect size of unknown location ... 

W. Weibull A Statistical Theory of the Strenght of Materials, Generalstabens Litografiska Anstalts Foerlag, Stockholm, 1939... 

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