The mathematical description of large bending deformations of thin elastic rods and plates leads to fourth order problems with nonlinear pointwise constraints that give rise to various free boundary phenomena. The free boundary may separate regions of trivial and large deformations, describe the contact zone in the presence of an obstacle, or be related to the occurrence of self-contact. We devise and analyse numerical methods that are capable of reliably capturing these effects under minimal regularity assumptions and which allow us to experimentally study topological transitions.
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