Derivations of two one-dimensional models for transversely curved shallow shells: one leads to relaxation

2025/07/31 by Paroni Roberto, Roberto, Paroni, Picchi Scardaoni Marco +1
Engineering · #Dynamics and Control of Mechanical Systems #Structural Analysis and Optimization #Vibration and Dynamic Analysis

paper · pdf · doi:10.48550/arxiv.2507.23545

Abstract

We study the Γ-limit of sequences of variational problems for straight, transversely curved shallow shells, as the width of the planform ε goes to zero. The energy is of von Kármán type for shallow shells under suitable boundary conditions. What distinguishes the various regimes is the scaling of the stretching energy ∼ ε^2β, with β a positive number. We derive two one-dimensional models as β ranges in (0, 2]. Remarkably, boundary conditions are essential to get compactness. We show that for β∈ (0, 2) the Γ-limit leads to relaxation: the limit membrane energy vanishes on compression. For β=2 there is no relaxation, and the limit model is a nonlinear energy coupling four kinematical descriptors in a nontrivial way. As special cases of the latter limit model, a nonlinear Vlasov torsion theory and a nonlinear Euler-Bernoulli beam theory can be deduced.

Citations

• Rigorous analysis of shape transitions in frustrated elastic ribbons
• Numerical simulation of a fine-tunable Föppl-von Kármán model for foldable and bilayer plates
• Kinematically incompatible Föppl-von Kármán plates: analysis and numerics
• Geometric dependence of curvature-induced rigidity
• One-dimensional von Kármán models for elastic ribbons
• Thin-walled beams with a cross-section of arbitrary geometry: derivation of linear theories starting from 3D nonlinear elasticity

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