Weibull Distribution-Volume Effect

In order to verify the volume effect it has been used a metal with a very low Weibull exponent like ferritic ductile cast iron (DCI). To that purpose, live sets of 15 specimens each have been prepared. Each set has the same volume that varies from set to set from a minimum of 500 mm to a maximum of 400,000 mm. Each set of specimens has been subjected to a monotonic traction test and the yield strength analysed with the Weibull approach to infer the relative m exponent and the scale factor ag. Results are shown in Fig. 4.28 and listed in the following table. 

The minimum m value is 9, rather poor and typical of low quality DCI, the maximum is 40 typical of a very good quality material. The equation of the curve of Fig. 4.28 interpolating the five experimental results is 

---

It should be noted that on the basis of a small sample size, e.g. only 30 specimens, it is not possible to differentiate between a Weibull, a Gaussian, or any other similar distribution functions, as shown by Lu et al. [14] using statistical measures or by Danzer et al. [12] using Monte Carlo simulations. This is caused by the inherent scatter of the data and the difference between sample and true population. The ultimate test for the existence of a Weibull distribution is to test a material on different levels of (effective) volumes. 

For the data presented below, determine the Weibull modulus in and the charaeteristie strength cr for each set of data. Are the two sets of data consistent with a single flaw distribution The effective volumes are LpF=30 mm for the four-point flexure specimens and LpF=0.1 m for the internally pressurized cylindrical tubes. 

In reality, fibre properties are statistically distributed. This is true for their geometry (length and diameter), but also, especially in the case of ceramic fibres, for their mechanical properties that are distributed according to Weibull statistics (see section 7.3). Non-ceramic fibres are also usually not identical since they may contain surface defects, for instance. Because of this statistical distribution of their properties, not all fibres fail simultaneously even in a homogeneously loaded composite. Instead, the weakest fibre will fail first. Due to the volume effect (see section 7.3.1), the failure probability of a long fibre is greater than that of a short one. 

A Weibull type strength distribution also may arise for inhomogeneous stress and non uniaxial stress states (then the volume has to be replaced by an effective volume, [1 - 3]). If failure is caused by surface flaws, the volume has to be replaced by the surface [1,3, 12]. 

---