A geometric mapping from material orthotropy to isotropy: from square beef to rectangular tofu

Material orthotropy means that the two fundamental elastic constants, Young's modulus and Poisson's ratio, are different along orthogonal directions; two daily-life examples are beef with muscle fibers aligned in one direction (rectilinear orthotropy) and wood with annual rings (curvilinear orthotropy). We present here a rescaling transformation which can map a rectilinearly orthotropic material into an isotropic one with a different local geometry: metaphorically, a square beef → a rectangular tofu (isotropic). The newly found rescaling transformation justifies the fact that material orthotropy was generally ignored in studies of, for instance, biological membranes; also, it can be used to study mechanical properties of structures that are made of orthotropic materials. We demonstrate the use of the rescaling transformation in the context of an indentation problem for spherical shells and establish that in the absence of pressure, an orthotropic sphere's equator (beef-like) and two poles (wood-like) have the same indentation stiffness, just like an isotropic sphere. For long cylinders, the transformation can also be applied, but in a slightly different manner. We show that the indentation stiffness of an orthotropic cylinder depends on the degree of anisotropy in a fundamentally different way than the spherical case, because the cylinder can deform isometrically.