7. Auxetic Solids in Polar and Spherical Coordinates

This chapter considers auxetic solids in such a form that is best analyzed using polar or spherical coordinate system. Specifically, this chapter considers the effect of auxeticity on stresses in rotating disks—both thin and thick—as well as the stresses in thick-walled cylinders and thick-walled spheres arising from internal and external pressure. Plotted results suggest that auxetic materials are advantageous over conventional ones for use as internally pressurized thick-walled cylinders, but inferior in comparison to conventional ones for application as thick-walled cylinders under external pressure. In the case of rotating thin disks with Poisson’s ratio of −1/3, the circumferential stress is independent of the radial distance, i.e. uniform throughout the entire disk, but not so for rotating disks with central hole. The maximum stress in a thin solid disk, as well as both the maximum radial and circumferential stresses in a thin disk with a central hole, decreases linearly as the Poisson’s ratio becomes more negative. Similar to thin rotating disks, the circumferential stress in thick rotating disks is independent from the radial distance when the Poisson’s ratio is −1/3; unlike thin disks, this circumferential stress is not uniform throughout the entire disk, as it varies along the disk thickness. In addition, the radial and circumferential stresses in a thick rotating disk is independent from the through thickness direction when the Poisson’s ratio is either 0 or −1. In the case of thick-walled spheres, the radial displacement is inversely proportional to the square of the radial distance if the Poisson’s ratio is 0.5, but varies linearly to the radial distance if the Poisson’s ratio is −1.