The effects of pressure loads in the dimension reduction of elasticity models
Martin Kružík, Filippo Riva
TL;DR
This work derives 2D elasticity theories from 3D nonlinear elasticity under a constant-pressure live load using Γ-convergence, revealing that pressure couples nontrivially only in high-scaling von Kármán-type limits. For bending regimes the minimizers are unchanged by pressure, while the membrane regime suggests a π-independent limit up to a constant shift. The results are complemented by explicit calculations in isotropic cases and by a formal example supporting a conjectured π-independence of the membrane limit. The analysis systematically characterizes how live boundary loads influence dimension-reduced models across all scaling regimes, including precise corrections to energy functionals and recovery sequences. These findings have implications for modeling thin structures in fluid environments and for understanding flutter-type phenomena in dimension-reduced elasticity.
Abstract
We study the dimensional reduction from three to two dimensions in hyperelastic materials subject to a live load, modeled as a constant pressure force. Our results demonstrate that this loading has a significant impact in higher-order scaling regimes, namely those associated with von Kármán-type theories, where a nontrivial interplay arises between the elastic energy and the pressure term. In contrast, we rigorously show that in lower-order bending regimes, as described by Kirchhoff-type theories, the pressure load does not influence the minimizers. Finally, after identifying the corresponding $Γ$-limit, we conjecture that a similar independence from the pressure term persists in the most flexible membrane regimes.