As far as I know, elastic distortion (or elastic deformation or temporary distortion) is the case when an object is deformed by virtue of a cause and after the cause is removed, it regains its original shape in a finite amount of time. If it fails to attain its original shape in finite amount of time or takes infinite time it becomes plastic or permanent distortion.
Inelastic materials, simply put, are non elastic materials. They do not show a fixed trend of deformation vs applied force; in fact, they might not deform at all (rigid materials) or the deformation observed is not completely recoverable; on removal of the applied force, the material doesn't return to its original shape, but to a permanent deformed shape. Such materials are called Plastic materials.
A typical material like steel shows all these forms under different conditions of loading (applied force). For extremely low magnitudes of forces, it is practically rigid. Increasing magnitudes of force show a linear elastic response, while further increase show a non-linear, plastic response, till rupture occurs when the material breaks.
