Strain tensor selection and the elastic theory of incompatible thin sheets

Abstract

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 {\bf 57}, 762 (2009)]. 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.

24 pages

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Cited by in corpus (8)

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• Dilation-invariant bending of elastic plates, and broken symmetry in shells
• Stretch formulations and the Poynting effect in nonlinear elasticity
• Quadratic-stretch elasticity
• Some observations on variational elasticity and its application to plates and membranes
• Energies for elastic plates and shells from quadratic-stretch elasticity
• Shear stresses in fluid and solid membranes with bending elasticity
• Assorted remarks on bending measures and energies for plates and shells, and their invariance properties