Variational modeling of paperboard delamination under bending

We develop and analyze a variational model for multi-ply (i.e., multi-layered) paperboard. The model consists of a number of elastic sheets of a given thickness, which – at the expense of an energy per unit area – may delaminate. By providing an explicit construction for possible admissible deformations subject to boundary conditions that introduce a single bend, we discover a rich variety of energetic regimes. The regimes correspond to the experimentally observed: initial purely elastic response for small bending angle and the formation of a localized inelastic, delaminated hinge once the angle reaches a critical value. Our scaling upper bound then suggests the occurrence of several additional regimes as the angle increases. The upper bounds for the energy are partially matched by scaling lower bounds.