Quadratic-stretch elasticity

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.

Energies for elastic plates and shells from quadratic-stretch elasticity

“…This paper provides a detailed derivation of bending measures and energies for elastic plates and shells that were proposed in a companion paper [1] through physical arguments related to a sensible definition of "pure stretching" of a surface, and whose properties and advantages were discussed therein. The present results follow from dimensional reduction of a general isotropic bulk elastic energy quadratic in Biot strains [2][3][4]. The use of such strains as primitive quantities in elasticity emerged in the field of rod mechanics [5], and its advantages for small-strain elasticity theories in soft matter were recently discussed in great detail by Oshri and Diamant [6] and subsequently by Wood and Hanna [7], both in the context of thin bodies.…”

“…The Cauchy-Green deformation tensors and their associated (Green-Lagrange or Euler-Almansi) strains involving metric differences are already quadratic in stretch, meaning that an energy quadratic in these strains will be quartic in stretch. In order to construct our primitive bending energy, we will instead employ a strain measure linear in stretch, guided by previous works [4][5][6][7]. This measure is the Biot strain E B = − I, a referential tensor whose present counterpart, with the same eigenvalues, is the Bell strain − I.…”

“…Note that these three-dimensional quantities, which we will evaluate on the surface, should not be confused with the corresponding surface Biot and Bell strains U − A A and V − a a that will appear later in our reduced energy. Our use of Biot rather than Bell is an arbitrary choice that makes use of prior derivations [4], however, when we wish to use the curvature tensor of the deformed surface in our description of bending, we will find that the left surface stretch V appears naturally.…”

“…While formulations of three-dimensional elasticity in terms of stretches and rotations can be difficult or require approximations [4], certain simplifications are possible for two-dimensional bodies that allow us to express the energy purely in terms of the components of the present and referential metric and curvature tensors, which are easily computed from derivatives of position.…”

“…The basis of the present derivation is a bulk elasticity theory formed by systematic expansion in Biot strains or similar quantities linear in stretch [4], including low-order terms not present in neo-Hookean elasticity or theories constructed from metric differences (Green-Lagrange or Euler-Almansi strains). The use of stretch tensors requires consideration of a rotation tensor or tensor square root operations, and in three dimensions these introduce significant complications not present when using metric components.…”