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Indentation of an elastic arch on a frictional substrate: Pinning, unfolding and snapping

We investigate the morphology and mechanics of a naturally curved elastic arch loaded at its center and frictionally supported at both ends on a flat, rigid substrate. Through systematic numerical simulations, we classify the observed behaviors of the arch into three distinct types of configurations in terms of the arch geometry and the coefficient of static friction with the substrate. A linear theory is developed based on a planar elastica model combined with Amontons-Coulomb's frictional law, which quantitatively explains the numerically constructed phase diagram. The snapping transition of a loaded arch in a sufficiently large indentation regime, which involves a discontinuous force jump, is numerically observed. The proposed model problem allows a fully analytical investigation and demonstrates a rich variety of mechanical behaviors owing to the interplay between elasticity, geometry, and friction. This study provides a basis for understanding more common but complex systems, such as a cylindrical shell subjected to a concentrated load and simultaneously supported by frictional contact with surrounding objects.