On Exponents of Thickness in Geometry Rigidity Inequality for Shells
Liang-Biao Chen, Peng-Fei Yao
TL;DR
The paper investigates thickness exponents in the geometry rigidity inequalities for shells, establishing $\mu(S)\leq 15/8$ for parabolic middle surfaces and $\mu(S)\leq 11/6$ for minimal surfaces with negative curvature or for negatively curved ruled surfaces. It shows that when $\mu(S)<2$, any $W^{2,2}$ isometry of the middle surface is rigid, yielding strong geometric rigidity beyond plates where $\mu(S)=2$. The approach combines $h$-uniform truncation lemmas, pointwise rigidity estimates for harmonic displacements, and Korn-type arguments to translate near-rigid deformations into global rigidity and to deduce the claimed exponents. These results sharpen the thickness–rigidity relation for shells and inform Gamma-convergence derivations of shell theories.
Abstract
We study exponents of thickness in Frieseck-James-Müller's inequalities for shells. We derive the following results: (a) the exponent of thickness $μ(S)\leq15/8$ if the middle surface $S$ is parabolic; (b) the exponent of thickness $μ(S)\leq11/6$ if the middle surface $S$ is a minimal surface with negative curvature; (c) the exponent of thickness $μ(S)\leq11/6$ if the middle surface $S$ is a ruled surface with negative curvature. The exponents of thickness in Frieseck-James-Müller's inequalities for thin shells represent the relationship between rigidity and thickness $h$ of a shell when the large deformations take place, i. e., the rigidity of the shell related to the thickness $h$ is $$Ch^{μ(S)}.$$ Thus the above results of $μ(S)<2$ show that those shells are strictly more rigid than plates since $μ(S)=2$ for plates. Moreover, we present another result which shows that when $μ(S)<2,$ any $W^{2,2}$ isometry of the middle surface is rigid.