Surface-Polyconvex Models for Soft Elastic Solids

TL;DR

The paper introduces surface-polyconvex constitutive models for soft solids with surface energy, establishing a mathematically well-posed variational framework that guarantees the existence of minimizers when coupled with polyconvex bulk energies. By deriving a hierarchy of surface models—fluid-like, isotropic, and anisotropic—and implementing them in a finite element setting, the authors validate the approach against benchmark problems such as liquid bridges and Rayleigh–Plateau-type instabilities, highlighting deformation-dependent surface effects absent in traditional surface-tension models. The results reveal that surface polyconvexity yields positive semi-definite surface stresses and precludes stress-free reference configurations, while enabling deformation-sensitive responses and rich instability behavior. The work provides a rigorous, computationally tractable path for modeling surface elasticity in soft materials, with potential applications in soft robotics, bioengineering, and materials design, and suggests future directions toward coupled bulk–surface interactions and curvature-dependent surface energies.

Abstract

Soft solids with surface energy exhibit complex mechanical behavior, necessitating advanced constitutive models to capture the interplay between bulk and surface mechanics. This interplay has profound implications for material design and emerging technologies. In this work, we set up variational models for bulk-surface elasticity and explore a novel class of surface-polyconvex constitutive models that account for surface energy while ensuring the existence of minimizers. These models are implemented within a finite element framework and validated through benchmark problems and applications, including, e.g., the liquid bridge problem and the Rayleigh-Plateau instability, for which the surface energy plays the dominant role. The results demonstrate the ability of surface-polyconvex models to accurately capture surface-driven phenomena, establishing them as a powerful tool for advancing the mechanics of soft materials in both engineering and biological applications.

Figures (13)

• Figure 1: Scheme of cylinder.
• Figure 2: Deformed shapes of liquid bridge for the cases $\tilde{\Psi}_s = \gamma\hat{J}$ and $\tilde{\Psi}_s = \alpha||\hat{\bm{F}}||$ and no bulk material.
• Figure 3: Deformed shapes of liquid bridge for the cases $\tilde{\Psi}_s = \gamma\hat{J}$ and $\tilde{\Psi}_s = \alpha||\hat{\bm{F}}||$ and no bulk material obtained using 3D FE simulation.
• Figure 4: Illustration of the individual steps done to simulate the deformation of a prestretched cylinder.
• Figure 5: Deformed shapes of liquid bridge with prestretch $\lambda_{1,0}=1.5$ evaluated using FE analysis for the cases $\tilde{\Psi}_s = \gamma\hat{J}$ and $\tilde{\Psi}_s = \alpha||\hat{\bm{F}}||$ and no bulk material compared to the analytically evaluated deformed shapes without prestretch.

Theorems & Definitions (1)

• Theorem 3.1