  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.
