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Based on theory of ultrasonic nondestructive testing on surface fatigue damage of metal components, the wave law of ultrasonic nonlinearity caused by fatigue is studied. When there are lattice defects in metal material, second-order nonlinear coefficient β changes during ultrasonic propagation. According to the point, the system of nonlinear ultrasonic testing is build. The change trends of harmonic amplitudes and ultrasonic coefficients are measured during fatigue bending testing of materials such as 45 steel, 2024 aluminum alloy and 304 stainless steel. The results shows: in elastic phase, the ratios of harmonic and fundamental waves monotonically increase with fatigue life, and in plastic phase, deformations appear and micro-cracks expand into macro-cracks in materials, the ratios firstly decrease and then increase with fatigue life. However the quadratic sums of nonlinear coefficient are approximately linear with the fatigue life. Therefore, when the relationship between the quadratic sums and fatigue life is known, it can be used to characterize fatigue state of metal materials.

The search of sphericity evaluation is a time-consuming work. The minimum circumscribed sphere (MCS) is suitable for the sphere with the maximum material condition. An algorithm of sphericity evaluation based on the MCS is introduced. The MCS of a measured data point set is determined by a small number of critical data points according to geometric criteria. The vertices of the convex hull are the candidates of these critical data points. Two theorems are developed to solve the sphericity evaluation problems. The validated results show that the proposed strategy offers an effective way to identify the critical data points at the early stage of computation and gives an efficient approach to solve the sphericity problems.

Among the four methods (minimum zone sphere, minimum circumscribed sphere, maximum inscribed sphere, and least square sphere), only the minimum zone sphere complies with ANSI and ISO standards and has the minimum sphericity error value. Evaluation of sphericity error is formulated as a non-differentiable unconstrained optimization problem and hard to handle. The minimum circumscribed sphere and the maximum inscribed sphere are all easily solved by iterative comparisons, so the relationship between the minimum zone sphere and the minimum circumscribed sphere, the maximum inscribed sphere is proposed to solve efficiently the minimum zone problem. The relationship is implemented and validated with the data available in the literature.

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