A Geometric Mapping from Rectilinear Material Orthotropy to Isotropy: Insights to Plates and Shells

TL;DR

This paper introduces a geometric mapping that transforms orthotropic shells into isotropic ones, enabling easier analysis of their buckling and indentation responses, and clarifies the roles of material and geometric anisotropy.

Contribution

It presents a rescaling transformation for orthotropic shells, deriving exact formulas for buckling and indentation, and clarifies the influence of material and geometric factors on shell rigidity.

Findings

The mapping is valid when material orthotropy aligns with principal curvature directions.

Derived exact expressions for buckling pressure and indentation response.

Identified the geometric mean of elastic constants as a key stiffness measure.

Abstract

Orthotropic shell structures are ubiquitous in biology and engineering, from bacterial cell walls to reinforced domes. We present a rescaling transformation that maps an orthotropic shallow shell to an isotropic one with a different local geometry. The mapping is applicable to any shell section for which the material orthotropy directions match the principal curvature directions, assuming a commonly used form for the orthotropic shear modulus. Using the rescaling transformation, we derive exact expressions for the buckling pressure as well as the linear indentation response of orthotropic cylinders and general ellipsoids of revolution, which we verify against numerical simulations. Our analysis disentangles the separate contributions of geometric and material anisotropy to shell rigidity. In particular, we identify the geometric mean of orthotropic elastic constants as the key quantifier of material stiffness, playing a role akin to the Gaussian curvature which captures the geometric stiffness contribution. Besides providing insights into the mechanical response of orthotropic shells, our work rigorously establishes the validity of isotropic approximations to orthotropic shells and also identifies situations in which these approximations might fail.