A mechanical model for thin sheet straight cutting in the presence of an elastic support

Abstract

We study the mechanics of sheet straight cutting in terms of a linear elastic fracture mechanics (LEFM) problem for a infinite thin elastic Kirchhoff plate partly supported by a Winkler foundation. The plate features a semi-infinite crack that is located at the edge of the supported zone and that is subjected to shear and bending loads, representing the action of the cutting tool (e.g. scissors blades). The fact that the plate is only partly supported by the foundation significantly complicates the analysis for it creates a non-symmetric framework, both locally and globally. Yet, a semi-analytical solution is obtained through casting the matrix Wiener–Hopf problem in terms of a pair of convolution integral equations defined on a semi-infinite domain. Stress intensity factors (SIFs) are obtained which converge to the known limits for a symmetric and skew-symmetric free plate. This analysis reveals the fundamental role played by the support in affecting the SIFs in an opposing manner, by enhancing/decreasing the symmetric/skew-symmetric components. Consequently, changing the support stiffness is capable of shifting the failure mechanism, from bending to shear. This observation may be taken advantage of when cutting materials which are more sensitive to either of these failure mechanisms. Also, it proves that the role of the support cannot be neglected when developing mechanical models of any cutting process.