A new and basic analytical model of three-dimensional cutting is proposed by assuming multiple thin shear planes with either the maximum shear stress or minimum energy principle. The three-dimensional cutting process with an arbitrarily shaped cutting edge in a flat rake face is formulated with simple vector equations in order to understand and quickly simulate the process. The cutting edge and workpiece profile are discretized and expressed by their position vectors. Two equations among three unknown vectors, which show the directions of shear, chip flow, and resultant cutting force, are derived from the geometric relations of velocities and forces. The last vector equation required to solve the three unknown vectors is obtained by applying either the maximum shear stress or minimum energy principle. It is confirmed that the directions and the cutting forces simulated by solving the proposed vector equations agree with experimentally measured data. Furthermore, the developed model is applied to consider the three-dimensional cutting mechanics, i.e., how the chip is formed in the three-dimensional cutting with compressive stress acting between the discrete chips, as an example.

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