A Variational Theory for Soft Shells

TL;DR

This paper introduces a hyperelastic theory for soft shells that separates deformation modes into stretching, drilling, and bending, emphasizing the unique role of drilling in soft matter shells and modeling soft elasticity with a multi-well energy functional.

Contribution

It proposes a novel variational framework that independently models the three deformation modes, including a quartic bending energy based on polar rotation, for soft shell materials.

Findings

The theory captures soft elasticity with a continuum of ground states.

The energy functional has a multi-well character, enabling modeling of complex deformation behaviors.

The approach highlights the significance of drilling mode in soft shells.

Abstract

Three general modes are distinguished in the deformation of a thin shell; these are stretching, drilling, and bending. Of these, the drilling mode is the one more likely to emerge in a soft matter shell (as compared to a hard, structural one), as it is ignited by a swerve of material fibers about the local normal. We propose a hyperelastic theory for soft shells, based on a separation criterion that envisages the strain-energy density as the sum of three independent pure measures of stretching, drilling, and bending. Each individual measure is prescribed to vanish on all other companion modes. The result is a direct, second-grade theory featuring a bending energy quartic in an invariant strain descriptor that stems from the polar rotation hidden in the deformation gradient (although quadratic energies are also appropriate in special cases). The proposed energy functional has a multi-well character, which fosters cases of soft elasticity (with a continuum of ground states) related to minimal surfaces.