Derivation of a homogenized bendingtorsion theory for rods with microheterogeneous prestrain
Abstract
In this paper we investigate rods made of nonlinearly elastic, compositematerials that feature a microheterogeneous prestrain that oscillates (locally periodic) on a scale that is small compared to the length of the rod. As a main result we derive a homogenized bendingtorsion theory for rods as $\Gamma$limit from 3D nonlinear elasticity by simultaneous homogenization and dimension reduction under the assumption that the prestrain is of the order of the diameter of the rod. The limit model features a spontaneous curvaturetorsion tensor that captures the macroscopic effect of the microheterogeneous prestrain. We device a formula that allows to compute the spontaneous curvaturetorsion tensor by means of a weighted average of the given prestrain. The weight in the average depends on the geometry of the composite and invokes correctors that are defined with help of boundary value problems for the system of linear elasticity. The definition of the correctors depends on a relative scaling parameter $\gamma$, which monitors the ratio between the diameter of the rod and the period of the composite's microstructure. We observe an interesting sizeeffect: For the same prestrain a transition from flat minimizers to curved minimizers occurs by just changing the value of $\gamma$. Moreover, in the paper we analytically investigate the microstructureproperties relation in the case of isotropic, layered composites, and consider applications to nematic liquidcrystalelastomer rods and shape programming.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 arXiv:
 arXiv:1903.08290
 Bibcode:
 2019arXiv190308290B
 Keywords:

 Mathematics  Analysis of PDEs;
 74B20;
 74K10;
 35B27;
 74Q05
 EPrint:
 Minor changes in introduction and bibliographiy. Pictures added