Symmetry considerations for material surfaces

Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching

“…The response of the fabric, therefore, conforms to orthotropic symmetry relative to the fiber axes on the reference plane. This is an example of homogenous symmetry in the general theory of material symmetry for second-grade material surfaces [20] (see also in Ref. [21]).…”

“…Of central importance in this work are the geodesic curvatures η l and η m of the fibers, defined by [10] l α l ,α = η l p + κ l n and m α m ,α = η m q + κ m n, (20) with p = n × l and q = n × m.…”

Local Symmetry Group in the General Theory of Elastic Shells

“…where J (P) is given by (30). The assumption that the constitutive relation is insensitive to the change of κ into κ means that the strain energy densities W and W should coincide, that is…”

“…The group operation • used here is analogous to but not identical with the one introduced by Murdoch and Cohen [30] for the material surface with one deformable director. Our symmetry group also differs from the one of 3D micropolar elastic material introduced by Eringen and Kafadar [19], because we take into account that some arguments of W are pseudotensors and our group consists of more narrow subgroups as compared with the 3D case.…”

Coexistent Fluid-Phase Equilibria in Biomembranes with Bending Elasticity

“…Here, however, the analysis is complicated by the sensitivity to curvature conferred by the resistance of lipids to changes in their relative alignment. The issue of material symmetry for elastic surfaces with curvature energy received definitive treatment in Murdoch and Cohen's [9] extension of Noll's theory. This work was adapted in [6] to derive a general theory for fluid bilayers as a special case of that for elastic surfaces.…”