Fracture strength distribution

The validity of these fracture strength distributions (3.18a) and (3.186) has not been checked yet extensively in experiments, as the above two forms differ very little numerically and require very accurate data for the failure strength distribution F a) for the analysis (see however the next subsection). Various accurate numerical simulation experiments have however been performed. 

Sahimi and Arbabi (1993) also studied the fracture strength distribution of a two-dimensional (triangular lattice) randomly diluted network with both central and bond-bending forces (with Hamiltonian given by (1.11) in Section 1.2.1 (f)). The results showed that, although the Weibull distribution fits the data initially for small disorder (p near unity), the data fits the Gumbel distribution considerably better and much more accurately as disorder increases (p Pc)- Iii fact, one can define a quantity A as... 

For the fracture strength distribution of superelastic networks (see Section 1.2.1(f)), containing bonds of finite strength, which break beyond a fixed amount of stretching, and infinitely rigid bonds, which do not break, Sahimi and Arbabi (1993) however observed the Weibull distribution to fit better than the Gumbel distribution. 

Fig. 3.13. Experimental data for the fracture strength distribution of porous silica extrudates. (a) Cumulative failure distribution F a), normalised to a constant volume, (b) Weibull distribution fit A versus In...

Whereas in the second approach of the size effects it is also assumed that fracture is controlled by defects, the strength is now considered a statistically distributed parameter rather than a physical property characterised by a single value. The statistical distribution of fibre strength is usually described by the Weibull model [22,23]. In this weakest-link model the strength distribution of a series arrangement of units of length L0 is given by... 

Several additional, non-microstructural, inputs are required for the fracture model (i) Particle critical stress intensity factor, KIc. Here, the value determined in a previous study (Klc = 0.285 MPa in )[3] was adopted for all four graphites studied. This value is significantly less than the bulk Klc of graphites (typically -0.8-1.2 MPa rn). However, as discussed in the previous section, when considering fracture occurring in volumes commensurate in size with the process zone a reduced value of Klc is appropriate (ii) the specimen volume, taken to be the stressed volume of the ASTM tensile test specimens specimen used to determine the tensile strength distributions and (iii) the specimen breadth, b, of a square section specimen. For cylindrical specimens, such as those used here, an equivalent breadth is calculated such that the specimen cross sectional area is identical, i.e.,... 

There is a significant reduction in strength when microstructures consisting of a broad grain diameter distribution are generated. When a fine equiaxed microstructure is generated, both the fracture strength and the fracture resistance are reduced [33, 350]. 

Average grain size Type of distribution Fracture strength (MPa)... 

Fig. 1.21 In situ strength distributions measured for Nicalon fibers on three CMCs, using the fracture mirror approach.

We will discuss these fracture properties of disordered solids, modelled by the random percolation models, and concentrate on their statistics, given by the cumulative failure strength distribution F a) under stress a, and the most probable fracture strength erf of such samples. We will discuss separately the cases for weak disorder p 1) and strong disorder p Pc)-The scaling properties of <7f near p pc and the nature of the competition between the percolation and extreme statistics here, will be discussed in detail. 

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. 

Fig. 3.12. Fit to Gumbel distribution for the computer simulation results of fracture strength for triangular network of springs with bond bending force l3 = 0.1), with the linear size L of the network fixed (L = 60). Plot of A versus with = 1. (a) For p = 0.9 and (b) p = 0.5...

For the distribution of fracture strength in such systems, the scaling fit was better obtained for the Weibull distribution (power law with sample size variations) than with the double exponential Gumbel distribution (Sahimi and Arbabi 1993). 

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