Metric Implications in the Kinematics of Surfaces

TL;DR

This paper explores how metric restrictions influence the energy modes in thin shells, providing insights into deformation constraints without relying on coordinate-based methods.

Contribution

It introduces a coordinate-free kinematic analysis of thin shells, linking metric restrictions to deformation energy contents and their physical implications.

Findings

Identifies three independent deformation energy modes: stretching, drilling, bending.

Analyzes the impact of metric restrictions on energy contents in shells.

Highlights potential hindrance of elastic response due to physical constraints.

Abstract

In the direct approach to continua in reduced space dimensions, a thin shell is described as a mathematical surface in three-dimensional space. An exploratory kinematic study of such surfaces could be very valuable, especially if conducted with no use of coordinates. Three energy contents have been identified in a thin shell, which refer to three independent deformation modes: stretching, drilling, and bending. We analyze the consequences for the three energy contents produced by metric restrictions imposed on the admissible deformations. Would the latter stem from physical constraints, the elastic response of a shell could be hindered in ways that might not be readily expected.