Extension, torsion and expansion of an incompressible, hemitropic Cosserat circular cylinder

Abstract

Constitutive equations for the stress and couple stress on an incompressible, hemitropic, constrained Cosserat material are derived, and the theory is applied to study the problem of finite extension, torsion and expansion of a circular cylinder. As in the theory of isotropic simple elastic materials, it is found that the deformation is controllable by application of only a normal force and a tosional moment at the cylinder ends. It is shown that in general the well known universal relation between the torsional stiffness and the axial force for incompressible, isotropic simple materials in the limit as the twist goes to zero does not exist for incompressible, hemitropic Cosserat materials. However, for a special and unusual class of hemitropic materials, the same universal formula is found to hold for a certain reduced torsional stiffness. The main problem is solved completely for incompressible, hemitropic, linearly elastic, Cosserat materials; and certain additional special features of the Kelvin-Poynting. type, which here appear to the first order in the amount of twist of the cylinder, are derived and discussed in relation to experimentally observed composite material behavior.

Elastodynamic Transformation Cloaking for Non-Centrosymmetric Gradient Solids

“…Noncentrosymmetric solids can be modeled in the setting of generalized continuum mechanics and have been studied by many researchers [Cheverton and Beatty, 1981, Lakes and Benedict, 1982, Lakes, 2001, Sharma, 2004, Liu et al, 2012, Ieşan and Quintanilla, 2016, Böhmer et al, 2020. Papanicolopulos [2011] studied chirality in 3D isotropic gradient elasticity under the assumption of small strains.…”

Chirality in the plane

“…This type of chirality was related to the gradient of rotation, which led to the existence of torsion. Based on the assertion that hyperelastic Cosserat materials are hemitropic (SO(3)-right-invariant) if and only if the strain energy is hemitropic, a set of hemitropic strain invariants was given in [9]. Many attempts were made to understand the mechanism behind the loss of chirality and in constructing a generic two-dimensional chiral configuration without referring to higher dimensions.…”

Geometrically nonlinear Cosserat elasticity in the plane: applications to chirality

“…It turns out that these three terms identically vanish when the first Cosserat planar problem is considered 2 . For instance, based on representation theorems [41], a total of 20 invariants were discussed, five of which are chiral according to our formulation in Appendix A. However, a direct calculation verifies that all chiral terms vanish when the deformation gradient is confined to the plane, see Eq.…”