[Paper Review] Linear Strain Tensors and Optimal Exponential of thickness in Korn's Inequalities for Hyperbolic Shells

Peng‐Fei Yao|arXiv (Cornell University)|Jul 29, 2018

Composite Structure Analysis and Optimization19 references4 citations

TL;DR

This paper establishes the optimal thickness scaling of $ h^{4/3} $ in the first Korn inequality for hyperbolic shells without assuming a global principal coordinate system on the middle surface. By proving $ L^2 $ regularity for solutions to linear strain equations on non-characteristic regions and leveraging rigidity of the strain tensor, the authors derive the sharp Korn constant using a direct method and interpolation inequalities, generalizing prior results that required restrictive coordinate assumptions.

ABSTRACT

We perform a detailed analysis of the solvability of linear strain equations on hyperbolic surfaces to obtain $L^2$ regularity solutions. Then the rigidity results on the strain tensor of the middle surface are implied by the $L^2$ regularity for non-characteristic regions. Finally, we obtain the optimal constant in the first Korn inequality scales like $h^{4/3}$ for hyperbolic shells, generalizing the assumption that the middle surface of the shell is given by a single principal system in the literature.

Motivation & Objective

• To establish the optimal thickness scaling of $ h^{4/3} $ in the first Korn inequality for hyperbolic shells.
• To remove the restrictive assumption of a global single principal coordinate system on the middle surface, which was previously required in the literature.
• To prove $ L^2 $ regularity for solutions to linear strain equations on non-characteristic regions of hyperbolic surfaces.
• To derive rigidity results for the strain tensor on the middle surface, which are essential for bounding the Korn constant.
• To construct a direct solution method for the linear strain equation that avoids the complex computations used in prior works.

Proposed method

• Introduces a direct method to solve the linear strain equation $ Υ(y) = U $ on hyperbolic surfaces, avoiding reliance on principal coordinates.
• Establishes $ L^2 $ regularity of solutions to the linear strain equation on non-characteristic regions via energy estimates and PDE analysis.
• Applies a rigidity lemma for the strain tensor on the middle surface, showing that vanishing strain implies isometric displacement under suitable conditions.
• Uses interpolation inequalities to bridge $ L^2 $ and $ H^1 $ norms, enabling control of the displacement in terms of the strain.
• Employs a local principal coordinate system (existing by theorems in Riemannian geometry) as a foundation for constructing the Ansatz in the proof.
• Combines the rigidity result with the interpolation inequality and energy estimates to derive the sharp $ h^{4/3} $ scaling in Korn’s inequality.

Experimental results

Research questions

• RQ1 What is the optimal thickness scaling of the first Korn inequality for hyperbolic shells when the middle surface lacks a global principal coordinate system?
• RQ2 Can $ L^2 $ regularity for solutions to the linear strain equation be established on non-characteristic regions of a hyperbolic surface without assuming a single principal coordinate system?
• RQ3 How can the rigidity of the strain tensor on the middle surface be proven under weaker regularity assumptions to support Korn inequality estimates?
• RQ4 Can the $ h^{4/3} $ scaling in Korn’s inequality for hyperbolic shells be derived without the restrictive assumption of a single principal coordinate system?
• RQ5 What is the role of the Bochner technique and interpolation inequalities in obtaining sharp Korn constants for hyperbolic shells?

Key findings

• The optimal constant in the first Korn inequality for hyperbolic shells scales like $ h^{4/3} $, confirming the sharp scaling previously conjectured under restrictive assumptions.
• The $ L^2 $ regularity of solutions to the linear strain equation is established on non-characteristic regions of hyperbolic surfaces, enabling the derivation of rigidity results.
• The rigidity of the strain tensor on the middle surface is proven under $ L^2 $ regularity, showing that vanishing strain implies isometric displacement up to a rigid motion.
• The assumption of a global single principal coordinate system—previously required in the literature—is removed, generalizing the result to arbitrary $ C^3 $ hyperbolic surfaces.
• The proof constructs a valid Ansatz using local principal coordinates and combines it with interpolation and energy estimates to derive the sharp $ h^{4/3} $ scaling.
• The paper provides a counterexample showing that no single principal coordinate system exists globally on a specific hyperbolic surface, justifying the need to remove this assumption.