Weibull material

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. 

In cases where the relative frequency of flaw sizes has a shape different to that of a Weibull material, the Weibull modulus becomes stress-dependent. The same happens for bimodal and multimodal flaw distributions [70, 71]. 

Table 1. Calculation of Weibull Material Constants with Suspended Tests...

Weibull - Fatigue enduranee strength of metals and strength of eeramie materials. 

It has been shown that the ultimate tensile strength, Su, for brittle materials depends upon the size of the speeimen and will deerease with inereasing dimensions, sinee the probability of having weak spots is inereased. This is termed the size effeet. This size effeet was investigated by Weibull (1951) who suggested a statistieal fune-tion, the Weibull distribution, deseribing the number and distribution of these flaws. The relationship below models the size effeet for deterministie values of Su (Timoshenko, 1966). 

As ean be seen from the above equation, for brittle materials like glass and eeramies, we ean seale the strength for a proposed design from a test speeimen analysis. In a more useful form for the 2-parameter Weibull distribution, the probability of failure is a funetion of the applied stress, L. 

A high shape faetor in the 2-parameter model suggests less strength variability. The Weibull model ean also be used to model duetile materials at low temperatures whieh exhibit brittle failure (Faires, 1965). (See Waterman and Ashby (1991) for a detailed diseussion on modelling brittle material strength.)... 

Note that /3 and /4 are stress components in the plane of isotropy and, therefore, have the same Weibull parameters. The parameters i and /3i would be obtained from uniaxial tensile experiments along the material orientation direction, dt. The parameters a2 and /Efe would be obtained from torsional experiments of thin-walled tubular specimens where the shear stress is applied across the material orientation direction. The final two parameters, a3 and /33, would be obtained from uniaxial tensile experiments transverse to the material orientation direction. 

This technique was employed in calculating the reliability contours depicted in Fig. 11.4. The reliability contours represent a homogeneously stressed material element, and for dimensionless , the Weibull parameter /3 has units of stress (volume)1/a. Here a, = 5, j3, = 0.2, ac = 35, /3C = 2, abc — 35, and f bc = 2.32. The three surfaces correspond to 0tj = 0.95, 0.5, and 0.05. Note that the reliability contours retain the general behavior of the deterministic failure surface from which they were generated. In general, as the a values increase, the spacing between contours diminishes. Eventually the contours would not be distinct and they would effectively map out a... 

Following Huttinger (1990), we can correlate the modulus and strength of carbon fiber to its diameter. We make use of Weibull statistics to describe the mechanical properties of brittle materials (see Chapter 10). Brittle materials show a size effect, i.e. the experimental strength decreases with increasing sample size. This is demonstrated in Fig.8.10 which shows a log-log plot of Young s modulus as a function of carbon fiber diameter for three different commercially available carbon fibers. The curves in Fig. 8.10 are based on the following expression ... 

We can regard a fiber as consisting of a chain of links. We assume that fiber failure occurs when the weakest link fails. This is called the weakest-link assumption. It turns out that such a weak-link material is well described by the statistical distribution known as the Weibull distribution (Weibull, 1939,1951). We first describe the general Weibull treatment for brittle materials and then describe its application for fibers. 

The Weibull distribution is called a parametric distribution, i.e. it is an empirical distribution and does not concern itself with the origin of the defects. The Weibull distribution for the strength (cr) of a brittle material takes the following form... 

Table 10.1 Typical Weibull modulus (P) values for bulk materials.

This distribution appears whenever g a) is given by a power law in (j, coming from the power law variation of the density of linear cracks g l) with their length 1. In the random percolation model considered here, this does not normally occur (except at the percolation threshold p = Pc)- However, for various correlated disorder models, applicable to realistic disorders in rocks, composite materials, etc., one can have such power law distribution for clusters, which may give rise to a Weibull distribution for their fracture strength. We will discuss such cases later, and concentrate on the random percolation model in this section. 

Figure 5.9 is a schematic comparison between the Normal (Gaussian) and Weibull distributions for plane strain fracture toughness Kic of a material in terms of the respective probability density distributions, The minimum and char-... 

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