Quadratic-stretch elasticity

TL;DR

This paper develops a nonlinear small-strain elastic theory based on quadratic Biot strains, enabling a clear separation of stretching and bending energies in shells, with simplified isotropic invariants and potential extensions to anisotropic materials.

Contribution

It introduces a quadratic-order expansion in Biot strains for nonlinear elasticity, simplifying calculations and clarifying energy separation in shell theories.

Findings

Provides a systematic quadratic expansion in Biot strains

Enables explicit expressions for stretches, rotations, and stresses

Discusses extensions to anisotropic elastic theories

Abstract

A nonlinear small-strain elastic theory is constructed from a systematic expansion in Biot strains, truncated at quadratic order. The primary motivation is the desire for a clean separation between stretching and bending energies for shells, which appears to arise only from reduction of a bulk energy of this type. An approximation of isotropic invariants, bypassing the solution of a quartic equation or computation of tensor square roots, allows stretches, rotations, stresses, and balance laws to be written in terms of derivatives of position. Two-field formulations are also presented. Extensions to anisotropic theories are briefly discussed.