Closed-form solution for large-deformation contact of soft protrusions

Large-deformation contact of soft hyperelastic protrusions arises in many mechanical systems, and accurate prediction of the contact response is essential for understanding their mechanical behavior. However, analytical descriptions remain limited beyond classical small-strain theories. Closed-form solutions are derived for the contact between soft hemispherical, semi-cylindrical, conical, and pyramidal protrusions and a rigid plane, yielding closed-form relations for contact load, contact area, and contact pressure distribution under finite deformation. An appropriate hyperelastic constitutive description is identified through combined experimental characterization and finite-element simulations, leading to the selection of a one-term Ogden model (N = 1), and the Ogden model parameters were determined as effective fitting parameters for the investigated compression/contact conditions. The proposed formulations are validated within the ranges of material properties, geometric parameters, and displacements considered in this study, and remain accurate well beyond the applicability limits of Hertzian and Sneddon-Love solutions. The resulting contact relations clarify the roles of geometry and material nonlinearity in load-area evolution and pressure localization. An application to microstructured flexible piezoresistive pressure sensors is presented to illustrate the effects of material properties and microstructural geometry on sensitivity, linearity, and measurement range. The proposed closed-form solutions extend existing contact theories to finite deformations of soft hyperelastic protrusions and provide an efficient theoretical framework for contact analysis and the optimized design of flexible pressure sensors.