Evolution of pinch-and-swell structures in a power-law layer

Analytical and finite element (FEM) solutions for the necking of a single power-law layer up to large finite amplitude are obtained. Continuous necking of the layer produces pinch-and-swell structures. The layer is either a free plate or embedded in a homogeneous medium. An analytical solution for finite amplitude necking based on the assumption that plane sections remain plane (PSRP) agrees well with the FEM result for a layer power-law stress exponent n ≤ 5 and for a ratio of layer to medium effective viscosities m ≥ 100. FEM simulations for embedded layers verify that PSRP for m ≥ 20. The presented numerical experiments generate localized necking and pinch-and-swell structures similar to natural ones for n ≥ 5 and m > 20. Additional weakening mechanisms, such as strain softening, although likely to be operative in nature, are not required to generate natural pinch-and-swell structures. FEM experiments with random perturbation of the layer interfaces show that even with strong necking instability the layer is thinned at the swell as well as at the necks, affecting strain estimation from pinch-and-swell geometry.