Indentation of geometrically exact beams

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We investigate arbitrarily large indentations of beams in two dimensions by a frictionless and rigid cylindrical punch whose axis is perpendicular to the plane of the beam. The beam is modeled as a geometrically exact beam that admits large displacements and rotations. Absence of both friction and inter-penetration between the contacting surfaces allows the normal contact pressure and the separation between the surfaces to be related by the Karush–Kuhn–Tucker (KKT) conditions. This is then implemented in a nonlinear finite element framework through both penalty and augmented lagrangian methods. We solve for the contact domain and the forces and moments acting within it, and relate them to the total applied force and the extent of indentation. We report results for several punch radii and beam thicknesses and consider both pinned and clamped beams. In this, we discuss the importance of nonlinear effects such as wrapping of the beam around the punch and the stiffening of the beam due to axial stretch. We demonstrate the importance of utilizing a nonlinear beam formulation by comparing our results with the predictions of theories predicated on small deformations. For example, while the latter claim loss of contact with the beam in the vicinity of the punch’s apex as the indentation proceeds, we find that the punch and the beam remain in contact.