Pre-buckling stress distribution and buckling load prediction in kirigami sheets: An analytical approach

Kirigami, the ancient art of creating intricate patterns by cutting and folding a thin sheet, offers exciting possibilities for engineers. For example, a kirigami sheet with a linear matrix of cuts perpendicular to an applied tensile load produces intricate out-of-plane buckling patterns. Depending on the ratios of cut length to longitudinal cut spacing (parallel to applied load) and transverse cut spacing (perpendicular to applied load), both symmetric and anti-symmetric buckling patterns can form. In addition, some sheets show a planar combination of, or multi-stability between, these two states. Prior research has described the phase diagram of these behaviours using 1D analytical methods, finite element (FE) simulations, and experiments. While FE modelling is effective for predicting buckling modes and loads, it provides less physical insight and is computationally more intensive than analytical methods. To fill this gap, we present a 2D analytical model based on patch loading theory for the pre-buckling stress distribution, the critical buckling force and the corresponding buckling modes of a kirigami sheet with a parallel matrix of cuts. FE models in the commercial FE package ABAQUS and an in-house FE implementation are used as a benchmark, demonstrating good correlation in the critical buckling forces at low computational cost. In addition, the mode transition behaviour between symmetric and anti-symmetric buckling modes is also explained by the model. Besides providing physical insight, our method enables rapid design and parameter sensitivity studies of this particular kirigami sheet.