Hyperelastic Membranes with Implicitly Defined, Continuously Embedded Fibers

2026-08-06

AI Summary

This study addresses the challenging problem of mechanically modeling fibers embedded within isotropic hyperelastic curved membranes by proposing an efficient computational framework grounded in first principles of continuum mechanics. The approach implicitly represents fiber geometry via a zero-level-set formulation and couples fibers with the membrane matrix within a geometrically nonlinear setting using coordinate-free tangential differential operators. This method achieves, for the first time, a continuous implicit embedding of fibers by synergistically integrating surface finite elements with immersed-domain concepts, yielding a novel hybrid numerical scheme. Numerical experiments demonstrate high-order convergence under smooth fields, significantly enhancing both accuracy and computational efficiency, and enabling high-fidelity simulations of complex fiber-reinforced membrane structures.

Abstract

A novel mechanical model and corresponding finite element method for anisotropic, hyperelastic, curved membranes are proposed. Hyperelastic fibers are embedded into the otherwise isotropic membrane, being relevant, for example, in reduced models for biological tissues and textiles. The geometrically nonlinear mechanics is formulated based on first principles of continuum mechanics (finite strain theory). The employed differential operators are formulated in a coordinate-free manner, through a framework known as tangential differential calculus. This enables a (semi-)implicit description of the fiber geometry through the intersection of level sets of some scalar function with the explicitly defined membrane surface. The mechanical model of the implicit fibers is then coupled to the mechanics of the membrane. For the numerical analysis, finite elements are applied such that the resulting scheme is a hybrid between classical Surface FEM and fictitious domain methods. For smooth physical fields, higher-order convergence rates are obtained and confirm the success of the numerical method.

Problem

• hyperelastic membranes
• embedded fibers
• anisotropic materials
• implicit geometry
• finite element method

Innovation

• hyperelastic membranes
• implicitly defined fibers
• tangential differential calculus
• level set embedding
• hybrid finite element method