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Abstract
From the deformation of soft gels to the stability of lipid bilayers and emulsions, surface tension forces critically shape the mechanics of ultra-soft solids and liquid membranes. However, conventional mesh-based methods often break down under large mesh distortions. The material point method (MPM) handles large strains effectively, yet most MPM formulations approximate surfaces implicitly – typically using level-set methods – which introduces auxiliary equations to solve. In this work, we present the first physically consistent, sharp-interface formulation of surface tension in MPM that represents surface geometry explicitly. We extend the Convected Particle Domain Interpolation (CPDI) framework of MPM to define 2D and 3D Convected Surface Domain Interpolation (CSDI) shape functions. These zero-thickness surface elements, attached to the boundary of CPDI material points, allow direct incorporation of surface tension into the variational formulation. We further introduce the membrane point method, which uses CSDI to simulate zero-thickness liquid membranes with surface-only momentum balance. We present an open-source explicit surface-enriched MPM implementation that verifies this approach for fluids and hyperelastic bulks. Through several numerical examples—including minimal surfaces, pendant droplet, and soft body contact with surface tension—we demonstrate the accuracy and robustness of our method under large deformations. These results establish our framework as a foundational tool for elastocapillary and membrane modeling in soft matter mechanics.
