2016/06/30 by Oz Oshri, Haim Diamant
Engineering · Mathematics · Physics and Astronomy · #Advanced Materials and Mechanics #Boundary (topology) #Classical mechanics #Compressibility #Curvature #Elasticity and Material Modeling #Gaussian #Gaussian curvature #Geometry #Infinitesimal strain theory #Limit (mathematics) #Mathematical analysis #Mathematics #Mean curvature #Mechanics #Physics #Structural Analysis and Optimization #Surface (topology) #Tensor (intrinsic definition) #cond-mat.soft
paper · pdf · doi:10.1103/physreve.95.053003
published as Phys. Rev. E 95, 053003 (2017) · 24 pages
arxiv created 2017/01/23 · openalex publication_date 2017/05/16 · arxiv updated 2017/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The existing theory of incompatible elastic sheets uses the deviation of the surface metric from a reference metric to define the strain tensor [Efrati et al., J. Mech. Phys. Solids 57, 762 (2009)JMPSA80022-509610.1016/j.jmps.2008.12.004]. For a class of simple axisymmetric problems we examine an alternative formulation, defining the strain based on deviations of distances (rather than distances squared) from their rest values. While the two formulations converge in the limit of small slopes and in the limit of an incompressible sheet, for other cases they are found not to be equivalent. The alternative formulation offers several features which are absent in the existing theory. (a) In the case of planar deformations of flat incompatible sheets, it yields linear, exactly solvable, equations of equilibrium. (b) When reduced to uniaxial (one-dimensional) deformations, it coincides with the theory of extensible elastica; in particular, for a uniaxially bent sheet it yields an unstrained cylindrical configuration. (c) It gives a simple criterion determining whether an isometric immersion of an incompatible sheet is at mechanical equilibrium with respect to normal forces. For a reference metric of constant positive Gaussian curvature, a spherical cap is found to satisfy this criterion except in an arbitrarily narrow boundary layer.