Nonlinear contact mechanics of soft elastic spheres under extreme compression

The contact of soft elastic spheres on substrates is a fundamental problem with significant relevance to fields such as bioengineering, robotics, micro-assembly, and wearables. Accurate analytical solutions for contact behaviors under extreme compression, particularly at compression ratios (compression displacement normalized by sphere radius) exceeding 10 %, are still lacking. This study investigates the contact mechanics of an elastic sphere against a rigid substrate (i.e., the flattening problem) under large deformations, integrating theoretical analysis, finite element analysis (FEA) simulations, and experiments. A finite-deformation theory framework for the flattening problem is proposed, accounting for finite-thickness and radial expansion effects. This framework facilitates analytical solutions for contact force, contact radius, and contact pressure. Systematic analysis of the three key sources of nonlinearity—geometry, material, and contact properties—reveals that geometric nonlinearity is the primary factor causing deviations in contact forces from the Hertzian theory. Based on these insights, explicit solutions for contact force, contact radius, and contact pressure are obtained using simple linear correction functions, achieving excellent agreement with FEA results. Experimental validation with Ecoflex samples demonstrates the high accuracy of these solutions at compression ratios up to 80 %. Additionally, their applicability to cellular mechanics is validated through precise predictions of contact forces reported in the literature for various cell types at compression ratios up to 75 %. This work provides an effective approach to addressing nonlinearities in the flattening problem, enabling accurate predictions of contact behavior under extreme compression. Our findings offer valuable guidelines for contact analysis and structural design involving soft elastomers.