Weakest-link theory

This theory was based on the assumption that when the network is stressed, the links between the microstructures are more likely to be stressed than the microstructures themselves or the structures within them. This is in fact reminiscent of the old adage the strength of a chain lies in its weakest link —the weakest links here are the links between the microstructures. This theory is simply, and appropriately, called the weak-link theory. Figure 7.14 shows a schematic of a fat network under extension when the weak-link theory is applicable. 

Madsen B (1978) In-grade testing problem analysis. Forest Products Journal, 28 4) 42-50 Madsen B and Buchanan AH (1986) Size effect in timber explained by a modified weakest link theory. Canadian Journal of Civil Engineering, 2 218-32 Maeglin RR and Boone RS (1983) Manufacture of quality yellow poplar studs using the saw-dry-rip (S-D-R) concept. Forest Products Journal, 55(3) 10-8 Maeglin RR and Boone RS (1988) Saw-dry-rip improves quality of random-length yellow-poplar 2 by 4 s. USDA, Forest Service, Forest Products Laboratory, Research Paper FPL RP-490... 

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 ... 

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). 

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] ... 

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]. 

Associated with the development of the Master Curve concept, which is discussed in Section 10.3.4, studies have concluded that it is possible to develop correlations describing a relationship between crack initiation and arrest toughness (Wallin, 2001). These studies have focused on clarifying which elements of the Master Curve approach should be modified for assessing crack arrest and finding possible correlations between initiation and arrest parameters. Due to different mechanisms and differences in factors controlling fracture initiation and arrest events (e.g. the local properties are crucial for crack initiation, but not so critical for crack arrest), the weakest link theory applied in the Master Curve approach is not directly suitable for crack arrest. 

There are a number of reasons for developing techniques for controlling network chain-length distributions one is to check the weakest-link theory for elastomer rupture, which states that the shortest chains are the culprits in causing rupture. Due to their limited extensibility, short chains supposedly break at relatively small deformations and then act as rupture nuclei. 

The theory of rubber elasticity (Section 9.7) assumes a monodisperse distribution of chain lengths. Earher, the weakest link theory of elastomer rupture postulated that a typical elastomeric network with a broad distribution of chain lengths would have the shortest chains break first, the cause of failure. This was attributed to the limited extensibility presumably associated with such chains, causing breakage at relatively small deformations. The flaw in the weakest link theory involves the implicit assumption that all parts of the network deform affinely (24), whereas chain deformation is markedly nonaffine see Section 9.10.6. Also, it is commonly observed that stress-strain experiments are nearly reversible right up to the point of rupture. 

However, to determine the field operating life, one needs to know the life of the parts in lab testing. The raw data obtained from the lab test could be fitted with a failure distribution to determine the mean life of the parts. Typical failure distributions include Weibull, Normal, Lognormal, and Exponential. For wear-out type of failures, the Weibull and Lognormal distributions are usually used, with Weibull being the most common. Weibull distributions are lowest value distributions derived from the weakest-link theory. Solder joint interconnects can be considered as connected in series. Usually, the failure of one joint at a critical location could cause the entire device to fail. The joints that fail early are usually located at the highest stress locations in the package. Devices with more resilient joints would not fail early. A Weibull distribution captures the minimum solder joint life, and the shape parameter captures the quality of the joints as a function of their construction and the applied stress. There are different types of Weibull distributions one-parameter, two-parameter, and three-parameter. The three-parameter Weibull Probability Distribution Function (PDF) is as shown in Eq.59.1. 

To predict component reliability for multiaxial stress states the Batdorf the-ory ° is used. Batdorf theory combines the weakest link theory with linear elastic fracture mechanics. It includes the calculation of the combined probability of the critical flaw being within a certain size range and being located and oriented so that it may cause fracture. 

S. B. Batdorf, and H. L. Heinisch Jr., Weakest Link Theory Reformulated for Arbitrary Fracture Criterion. Journal of the American Ceramic Society, Vol. 61, No. 7 8, 1978, pp. 355 358. 

The basic assumption of statistical fracture theory is that the reason for the variations in strength of nominally identical specimens is their varying content of randomly distributed (and generally invisible) flaws. The strength of a specimen thus becomes the strength of its weakest flaw, just as the strength of a chain is that of its weakest link. 

In the case of such noncrystalUzable, unfilled elastomers, the mechanism for network rupture has been elucidated to a great extent by studies of model networks similar to those already described. For example, values of the modulus of bimodal networks formed by end-linking mixtures of very short and relatively long chains as illustrated in Fig. 6.4 were used to test the weakest-link theory [7] in which rupture was thought to be initiated by the shortest chains (because of their very limited extensibility). It was observed that increasing the number of very short chains did not significantly decrease the ultimate properties. The reason [85] is the very nonaffine... 

Size eifect may be quantified through rather simple, but refined considerations. Metallurgical variability that results in the dependence of fatigue limit Of or Sf on specimen size is also responsible for the dependency of tensile strength specimen size. According to flie WeibuU theory of the weakest link (see Sect. 4.1.7) the ratio of the ultimate strength inversely proportional to the power of the ratio of the respective volumes... 

Harkegard, G. Fatigue assessment based on weakest-link Theory. Vortragim rahmendes kolloquiums filer technische wissenshaftenundess seminars in mechanik, 44, ETH Ziirich 18 (2007)... 

The probabilistic approach considers the flaw distribution and stress distribution in the material. This approach will be useful when high stresses and their complex distributions are present. In these situations, both empirical and deterministic designs have limitations. The flaw distribution can be characterized by the Weibull approach [1]. This approach is based on the weakest link theory. This theory says that a given volume of a ceramic under a uniform stress will fail at the most severe flaw. Thus, the probability of failure F is given by the following equation ... 

The theory of weakest link statistics was worked out by Weibull [49] in 1939, and rests on the sole assumption that a long sample is comprised of multiple smaller elements, each with a statistically independent yielding probability. For a sample to not fail or yield, each subelement must similarly not fail. If the... 

---