Strain tensor selection and the elastic theory of incompatible thin sheets

TL;DR

This paper proposes an alternative strain tensor formulation for incompatible elastic sheets, leading to simpler equations and new insights, especially for planar and uniaxial deformations, differing from the traditional metric deviation approach.

Contribution

It introduces a distance-based strain tensor formulation that simplifies equilibrium equations and aligns with classical elastica in uniaxial cases, offering new criteria for isometric immersions.

Findings

Linear, solvable equilibrium equations for flat incompatible sheets.

Uniaxial deformation results match classical elastica theory.

Spherical caps generally satisfy the equilibrium criterion except near boundaries.

Abstract

The existing theory of incompatible elastic sheets uses the deviation of the surface metric from a reference metric to define the strain tensor [Efrati et al., J. Mech. Phys. Solids {\bf 57}, 762 (2009)]. For a class of simple axisymmetric problems we examine an alternative formulation, defining the strain based on deviations of distances (rather than distances squared) from their rest values. While the two formulations converge in the limit of small slopes and in the limit of an incompressible sheet, for other cases they are found not to be equivalent. The alternative formulation offers several features which are absent in the existing theory. (a) In the case of planar deformations of flat incompatible sheets, it yields linear, exactly solvable, equations of equilibrium. (b) When reduced to uniaxial (one-dimensional) deformations, it coincides with the theory of extensible elastica; in particular, for a uniaxially bent sheet it yields an unstrained cylindrical configuration. (c) It gives a simple criterion determining whether an isometric immersion of an incompatible sheet is at mechanical equilibrium with respect to normal forces. For a reference metric of constant positive Gaussian curvature, a spherical cap is found to satisfy this criterion except in an arbitrarily narrow boundary layer.