It has been realised that the analytical model of 3-D chip curling plays a central role in the chip formation that would predict the chip forms/shapes efficiently for the aim of chip control. The purpose of the two-part paper is to investigate and model the 3-D chip curling processes in theory. In part I, an analytical modelling procedure is described in detail and a general mathematical model for predicting chip forms /shapes in 3-D cutting processes with complex chip breaker tools is presented on the basis of three basic chip curling patterns, i.e. up-curl, side-curl and twisting. In part II, the theoretical analysis and discussion have been done systematically based on the model presented. Therefore a group of criterions for the effects of three chip angular velocity components ωx, ωy, ωz, and chip flow angle η0 on the chip forms/shapes is derived. The varying tendency of the chip radius and chip pitch with varying ωx, ωy, ωz, and η0 is discussed based on the group of criterions. For better understanding, a group of 3-D diagrams is given to visualise the varying tendency of chip geometry. The 3-D analytical model and the group of criterions presented in part I and part II of the paper reveal new insights in to the nature of chip curling in 3-D machining in theory and are significant for the analytical prediction of the chip curling and the chip breaking, as well as the geometric design of the chip breaker/groove.

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