The Mathematical-physical Formulation of the Size Effect of Compression Crushing Forces

The sizing of crushers depends on the expected crushing forces. However, the crushing forces develop differently with every individual rock while it is compressed by the working surfaces of the crusher. For sizing, this compression process has to be characterized by the maximum crushing force. The maximum crushing force depends on the strength, shape and size of the rock, the amount and velocity of the imposed displacement of the working surfaces of the crusher on the rock and the conditions of the contact surfaces. From experience it is known that the strength of the material and, therefore, the crushing forces depend on the size of the specimen. This dependence is the so called size effect. The determination is based on compression tests. The force development is recorded during the compression of the specimen and for every individual test the maximum force is determined. The occurring maximum crushing forces of several rocks with the same diameter comminuted under the same conditions can be assessed by a mean value and a variance. The relationship between the mean values of the cross section related maximum crushing forces FAmax and the corresponding sizes d of the specimens can then be described by mathematical functions. They are derived from different approaches (phenomenological, statistical and statistical-physical methods). The phenomenological methods are subjective and yield functions depending on the assumptions of a person, whereas the statistical and statistical-physical methods are based on an evaluated function and the optimization of the parameters.