Are there solid materials that begin as isotropic materials (i.e. isotropic elastic parameters) in their natural (i.e. unstressed) state, but that become anisotropic under sufficient stress/strain? What would it take to modify an existing material with isotropic elastic properties to have anisotropic elastic properties?
Deformation induced anisotropy is ubiquitous in materials, whether they are metals, polymers, foams, soils, rocks etc. A typical example is any rolled metal product.
If you wish to deal only with purely elastic deformations, you have to define what elastic properties mean and how you will measure these properties.
Suppose that you are interested only in small-strain elasticity in a single crystal, perfectly annealed and free of defects. If you deform the crystal, do the elastic properties of the crystal change depending on the direction of deformation? Keep in mind that no solid material is truly isotropic except for some glasses.
Assume that the speed of propagation of a wave in a particular direction of the crystal is a proxy for the elastic properties of the crystal in that direction. When you compress the crystal, the atomic spacing decreases and the wave takes a shorter time to travel, i.e., the elastic modulus increases in that direction and the crystal anisotropy increases. However, the induced anisotropy is small and not permanent. See, for example, https://www.sciencedirect.com/science/article/pii/0041624X63900039 and the Acousto-elastic effect.
Any permanent anisotropy requires an increase in entropy via plastic dissipation, microcrack growth etc.
For induced anisotropy in bulk metallic glasses, see https://www.sciencedirect.com/science/article/pii/S1359646211001072