Singular Points and Singular Curves in von K\'arm\'an Elastic Surfaces

TL;DR

This paper extends the classical von Kármán equations to model elastic surfaces with singularities like points and curves, capturing phenomena such as crumpling, folding, and defects in thin elastic materials.

Contribution

It introduces a generalized framework for von Kármán equations that accommodates piecewise smooth fields with singularities, enabling analysis of complex deformation patterns.

Findings

Modeling of singular deformation and stress in elastic surfaces.

Application to conical deformations, folds, and defect-induced incompatibilities.

Framework captures singular behaviors at points and curves.

Abstract

Mechanical fields over thin elastic surfaces can develop singularities at isolated points and curves in response to constrained deformations (e.g., crumpling and folding of paper), singular body forces and couples, distributions of isolated defects (e.g., dislocations and disclinations), and singular metric anomaly fields (e.g., growth and thermal strains). With such concerns as our motivation, we model thin elastic surfaces as von K{\'a}rm{\'a}n plates and generalize the classical von K{\'a}rm{\'a}n equations, which are restricted to smooth fields, to fields which are piecewise smooth, and can possibly concentrate at singular curves, in addition to being singular at isolated points. The inhomogeneous sources to the von K{\'a}rm{\'a}n equations, given in terms of plastic strains, defect induced incompatibility, and body forces, are likewise allowed to be singular at isolated points and curves in the domain. The generalized framework is used to discuss the singular nature of deformation and stress arising due to conical deformations, folds, and folds terminating at a singular point.