Comparison of the Fold and Cusp Catastrophe Models for Tensile Cracking and Sliding Rockburst

The failure modes of rockburst in catastrophe theory play an essential role in both theoretical analysis and practical applications. The tensile cracking and sliding rockburst is studied by analyzing the stability of the simplified mechanical model based on the fold catastrophe model. Moreover, the theory of mechanical system stability, together with an engineering example, is introduced to verify the analysis accuracy. Additionally, the results of the fold catastrophe model are compared with that of the cusp catastrophe model, and the applicability of two catastrophe models is discussed. The results show that the analytical results of the fold catastrophe model are consistent with the solutions of the mechanical systems stability theory. Moreover, the critical loads calculated by two catastrophe models are both less than the sliding force, which conforms to the actual situations. Nevertheless, the critical loads calculated from the cusp catastrophe model are bigger than those obtained by the fold catastrophe model. In conclusion, a reasonable result of the critical load can be obtained by the fold catastrophe model rather than the cusp catastrophe model. Moreover, the fold catastrophe model has a much wider application. However, when the potential function of the system is a high-order function of the state variable, the fold catastrophe model can only be used to analyze local parts of the system, and using a more complex catastrophe model such as the cusp catastrophe model is recommended.

Diffraction Catastrophes

This chapter discusses the mathematics of diffraction catastrophes, focusing on the wavefront associated with a monochromatic wave with Q(x, y, f (x, y)) as a point on the wavefront surface and P (X, Y, Z) as a movable observation point. Optical caustics refer to surfaces (in space) and curves (in the plane) where light rays are focused. Caustics can be classified mathematically as the elementary catastrophes of singularity theory. In the plane, the classification gives two singularities: smooth caustic curves, which are “fold” catastrophes, and points where two fold caustics meet on opposite sides of a common tangent, which are “cusp” catastrophes. After providing an overview of the basic geometry of the fold and cusp catastrophes, the chapter considers the Fresnel integrals and the fold diffraction catastrophe. In particular, it examines the rainbow as a fold catastrophe, Airy integral in the complex plane, and the nature of the notation Ai(X).