Modeling and simulation of thin sheet folding

Abstract

The article addresses the mathematical modeling of the folding of a thin elastic sheet along a prescribed curved arc. A rigorous model reduction from a general hyperelastic material description is carried out under appropriate scaling conditions on the energy and the geometric properties of the folding arc in dependence on the small sheet thickness. The resulting two-dimensional model is a piecewise nonlinear Kirchhoff plate bending model with a continuity condition at the folding arc. A discontinuous Galerkin method and an iterative scheme are devised for the accurate numerical approximation of large deformations.

Error Estimates For A Linear Folding Model

“…The interior penalty discontinuous Galerkin method turns out to be a practical candidate for this purpose as gradient jumps of the deformation may be neglected along the interface, thereby allowing for the simulation of foldable configurations. A corresponding large deformation model has recently been derived via dimension reduction by Bartels, Bonito and Hornung [6]. The authors adapt arguments from the seminal work of Friesecke, James and Müller [18] to account for the presence of a folding arc and follow ideas of Bartels [4] and Bonito, Nochetto and Ntogkas [9] for the numerical realization.…”

A Blake-Zisserman-Kirchhoff theory for plates with soft inclusions

“…Here the Lipschitz domain Ω ⊂ R 3 is the reference configuration of the body, y ∈ W 1,2 (Ω, R 3 ) (say) is a deformation mapping whose elastic energy is given in terms of a stored energy function curve [8] and the derivation of a general 'Griffith-Euler-Bernoulli theory' for thin brittle beams from a nonlinear 2d Griffith functional achieved in [58].…”

“…In a secondary step these are then further approximated with the help of a bulk relaxation argument. The validity of a minimal droplet condition is then examined for arbitrary norms with the help of local estimates for the volume of tubular neighborhoods of the full crack set J r ∪J ∇r ∪D, see (8), that amounts to require an outer Minkowski-content measurability condition (see comments and remarks after Theorem 3.3).…”