Variational modeling of paperboard delamination under bending

Sergio Conti; Patrick Dondl; Julia Orlik; Sergio Conti; Patrick Dondl; Julia Orlik

doi:10.3934/mine.2023039

Mathematics in Engineering

2023, Volume 5, Issue 2: 1-28. doi: 10.3934/mine.2023039

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Research article

Variational modeling of paperboard delamination under bending

1.
Institut für Angewandte Mathematik, Universität Bonn, 53115 Bonn, Germany
2.
Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, 79104 Freiburg i. Br., Germany
3.
Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM, 67663 Kaiserslautern, Germany

Received: 11 February 2022 Revised: 15 May 2022 Accepted: 03 June 2022 Published: 15 June 2022

We develop and analyze a variational model for laminated paperboard. The model consists of a number of elastic sheets of a given thickness, which – at the expense of an energy per unit area – may delaminate. By providing an explicit construction for possible admissible deformations subject to boundary conditions that introduce a single bend, we discover a rich variety of energetic regimes. The regimes correspond to the experimentally observed: initial purely elastic response for small bending angle and the formation of a localized inelastic, delaminated hinge once the angle reaches a critical value. Our scaling upper bound then suggests the occurrence of several additional regimes as the angle increases. The upper bounds for the energy are partially matched by scaling lower bounds.
- calculus of variations,
- energy scaling,
- nonlinear elasticity,
- delamination,
- paperboard
Citation: Sergio Conti, Patrick Dondl, Julia Orlik. Variational modeling of paperboard delamination under bending[J]. Mathematics in Engineering, 2023, 5(2): 1-28. doi: 10.3934/mine.2023039

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Abstract

We develop and analyze a variational model for laminated paperboard. The model consists of a number of elastic sheets of a given thickness, which – at the expense of an energy per unit area – may delaminate. By providing an explicit construction for possible admissible deformations subject to boundary conditions that introduce a single bend, we discover a rich variety of energetic regimes. The regimes correspond to the experimentally observed: initial purely elastic response for small bending angle and the formation of a localized inelastic, delaminated hinge once the angle reaches a critical value. Our scaling upper bound then suggests the occurrence of several additional regimes as the angle increases. The upper bounds for the energy are partially matched by scaling lower bounds.

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