The correct answer is Plasticity.

- Plasticity is the mechanical property that defines a material's capacity to undergo permanent deformation when subjected to stress.
- When a material is stressed beyond its elastic limit, it does not return to its original shape after the load is removed; instead, it deforms permanently.
- This characteristic is crucial in many engineering and medical applications where materials need to withstand forces without fracturing.

To clarify the other options:
- Ductility refers specifically to a material's ability to undergo significant plastic deformation under tensile stress, often associated with stretching into a wire.
- Elasticity is the property of a material to return to its original shape after the stress is removed, meaning the deformation is reversible.
- Hardness refers to a material's resistance to deformation, particularly permanent indentation or scratching.

Understanding plasticity is important in medical device design and surgical tools, where materials must often withstand forces that cause permanent shape changes without breaking.

Reference: Mechanics of Materials, Volume 1, Chapter 3, Page 45