A Strain Compatibility Criterion for Three-Dimensional Chip Curling

The final geometry of a three-dimensional chip is commonly described by a combination of geometric parameters, including curl radius, pitch, and axis orientation. However, parameter combinations that are mathematically admissible do not necessarily correspond to material deformations that are kinematically realizable. For regular helical chips formed under stable contact conditions between the chip-breaker and the chip, this study proposes a path–geometry strain compatibility criterion for the chip outer surface. A path-based surface strain model is first developed based on the steady-state sweeping process of the tool–chip separation line and the trajectories of material points along the bottom edge of a local cross-section. A finite-deformation mapping from a straight-strip reference configuration to the final helical configuration is then established in the local cross-sectional frame and the Frenet–Serret frame, yielding a geometry-based surface strain model. The two descriptions employ a common strain measure along the same material direction, and a strain compatibility index is defined from their difference. Experiments were conducted on FV520B precipitation-hardened martensitic stainless steel under three heat-treatment conditions using 12 chip-breaker geometries. High-speed imaging and micro-computed tomography (Micro-CT) were combined to characterize the chip outflow process and the three-dimensional curl geometry. The results show that, larger compatibility residuals were associated with the combination of stronger constraint imposed by the chip-breaker surface and greater resistance to plastic deformation. The proposed index is insensitive to the absolute geometric scale and primarily constrains combinations of angular parameters and the pitch-to-radius ratio. The proposed criterion can serve as a physically based screening condition for candidate three-dimensional curl geometries and can be coupled with existing geometric, mechanical, and chip-breaking models. Its applicability is limited to stable curling processes.