Closed-form solutions for contact of hyperelastic hemisphere and forward design of flexible piezoresistive pressure sensors

Flexible piezoresistive pressure sensors (FPPS) are used in soft robotics and wearable systems, where sensitivity, pressure range, and linearity must be properly tuned. These requirements are often addressed through multilayer microstructured architectures, in which resistance varies with the effective contact area generated under compression. Most designs, however, remain empirical because analytical models for the contact response of hyperelastic micro-protrusions are rather limited. This study develops a forward design framework for multilayer flexible piezoresistive pressure sensors based on closed-form solutions for the compression of hyperelastic hemispherical microstructures. Uniaxial and hemispherical compression tests on polydimethylsiloxane (PDMS) are used to identify a suitable hyperelastic constitutive model. Closed-form solutions are then derived for the contact between a hyperelastic hemisphere and a rigid flat indenter. The applicability of these solutions to the contact of hemispheres with different hyperelastic materials and geometries is examined through experiments and finite element simulations. The resulting solutions for the contact load and contact area are then used to design a four-layer sensor architecture that satisfies prescribed sensor targets. By relating pressure-dependent contact-area evolution to resistance change, the framework provides a systematic basis for balancing sensor performance over the prescribed pressure range. The approach is validated using carbon-black/polydimethylsiloxane composite microstructures and resistance-pressure measurements under compression. Sensors designed for low, medium, and high pressure ranges confirm the predictive capability of the method and quantify the attainable sensitivity within the considered design space. For a sensor operating up to 10 kPa with an initial shear modulus of 0.528 MPa, a sensitivity of 280 kPa⁻¹ is achieved.