The basis of catalyst solid mechanics

A large number of test results indicate that supported metal catalysts and metal-oxides have properties similar to that of brittle materials. That is to say, basically plastic deformation does not occm before the materials fracture under external stress, and the fracture is instantaneous without any obvious forewarning. Solid catalysts are highly porous with crystal boundaries and a lot of defects, such as dislocation or discontinuities arising from the presence of promoters. They are similar to cracks, as defined in fracture mechanics, where fracture stresses are focused, eventually leading to breakup of the catalyst particles.  

The well-known Griffith s theory constitutes the principle of fracture mechanics. In fracture mechanics, the fracture of a brittle material is caused by the increase of tension-stress at the edges of micro-cracks (disfigurements) in the material. The breaking strength of a catalyst follows the Griffith equation. 

In 1951, Weibull proposed the weakest link model , and gave corresponding distribution functions on the basis of the model. The model is formulated using a chain link under symmetrical tension-stress as shown in Fig. 8.35. The chain has n links, and the whole chain would disconnect if any one link is broken under the action of the a stress. The broken strength of each link follows a uniform distribution. The critical defect or crack is the smallest crack or flaw with the least stress. In a brittle material, the critical defect or crack for a given stress distribution is the weakest link. A brittle material can be regarded as a material with n defects or cracks which are independent of each other, with the cracks under a uniform distribution of damage stress. Under the effect of an external force, the invalidation of the material 

Two necessary conditions are required by the Weibull statistical distribution all samples should have the same size and distribution of tensile stress all samples contain a unified and uniform distribution of defects. The derivation of Weibull s distribution is as follows. 

Assuming that the number of cracks is proportional to the size of the space under consideration, the cracks in this space are randomly distributed, and the tensile stress is uniform, and combining with the weakest link assumption, the probability of structural fracture may be expressed as  

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