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Singular Points and Singular Curves in von Kármán Elastic Surfaces

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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 Kármán plates and generalize the classical von Kármán 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 Kármán 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.

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Acknowledgements

AG acknowledges the financial support from SERB (DST) Grant No. CRG/2018/002873 titled “Micromechanics of Defects in Thin Elastic Structures”.

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  1. Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, 208016, India

    Animesh Pandey & Anurag Gupta

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  1. Animesh Pandey
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  2. Anurag Gupta
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Contributions

A.P. contributed towards formulating the solution methodology in addition to preparing proofs and detailed calculations. A.P. wrote the main manuscript text. A.G. designed the problem and planned the manuscript. All authors reviewed the manuscript.

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Correspondence to Anurag Gupta.

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This article was originally accepted for publication in the special issue ‘Soft Matter Elasticity’. It was prematurely published in Journal of Elasticity, volume 150, issue 2, August 2022, pp. 367–399 (https://doi.org/10.1007/s10659-022-09918-z), due to a mistake at the publisher’s side. It has been republished in volume 153, issues 4-5, July 2023, dedicated to the special issue on Soft Matter Elasticity (https://doi.org/10.1007/s10659-022-09969-2).

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Pandey, A., Gupta, A. Singular Points and Singular Curves in von Kármán Elastic Surfaces. J Elast 153, 681–713 (2023). https://doi.org/10.1007/s10659-022-09969-2

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