Experimental research on strength distribution of brittle materials under quasi-static condition

In order to quantitatively analyze the relationship between the selection function and material sizes and properties in the mathematical model of crushing, the quasi-static uniaxial compression tests were carried out on typical brittle materials K9 glass and ceramic spheres of five sizes to determine the crushing force and breakage energy during crushing.Three common statistical models, i.e., Weibull model, Lognormal model and Logistic model, were selected to fit the particle strength (crushing force, crushing stress, breakage energy and breakage specific energy) respectively, and the appropriate statistical model was selected as the selection function of particle breakage.The quantitative relationship between particle strength and particle size and material properties under different definitions was studied.The experimental results show that under quasi-static loading, the strength of K9 glass and ceramic spheres showed obvious scatter, and statistical model was needed to describe the strength distribution of particles.By comparing Logistic model, Lognormal model and Weibull model, it was found that the fitting accuracy of the first two models was better than that of Weibull model, but considering the complexity of mathematical form and the physical meaning of model parameters, Logistic model was chosen to describe the strength distribution of particles.Dispersion degree of distribution D in Logistic model was related to material properties and weakly functional to particle size.Therefore, when Logistic model was used, D was be fixed and errors could be accepted.F₅₀ (the fracture probability is 50% corresponding to the crushing force) and E₅₀ (the fracture probability is 50% corresponding to the breakage energy) in the model were proportional to the particle size, while σ₅₀ (the fracture probability is 50% corresponding to the crushing stress) and E_m50 (the fracture probability is 50% corresponding to the breakage specific energy) decreased in power function with the increase of particle size.The relationship between breakage energy and crushing force was only related to material properties, not to particle size.It could be used to distinguish different materials.The relationship between crushing stress and breakage specific energy was only related to material properties, not to particle size, in the double logarithmic coordinate system, the linear slope was approximately 0.6.