Plastic, brittle, and rigid idealizations – compare plastic zones near cracks/notches Which statement about the size of the plastic zone (or plastic region) is correct for idealized material behaviors?

Introduction / Context:
In fracture and yielding, the extent of plasticity ahead of stress raisers controls ductility and energy absorption. Materials are often idealized as ductile, brittle, or rigid to discuss limiting behavior.

Given Data / Assumptions:

• Ductile = significant plastic deformation before fracture.
• Brittle = negligible plastic deformation before fracture.
• Rigid = no deformation until failure (idealization).

Concept / Approach:
Ductile materials undergo yielding in a finite region near stress concentrations, creating a relatively large plastic zone. Brittle materials exhibit little to no yielding, so the plastic zone is tiny or absent. A perfectly rigid material is an abstraction with zero strain up to failure, hence no plastic zone.

Step-by-Step Solution:
Map each idealization to expected plastic region size.Ductile → large plastic zone; brittle → negligible; rigid → none.Therefore all three individual statements are correct, making the combined option correct.

Verification / Alternative check:
Fracture mechanics indicates higher fracture toughness and larger plastic zones for ductile metals versus glass/ceramics (brittle).

Why Other Options Are Wrong:

• Any single choice omits portions of the correct set.
• “None” contradicts well-established definitions.

Common Pitfalls:
Assuming “no plastic zone” means “no yielding ever”; ductile/brittle are spectrums and depend on temperature and strain rate.

Final Answer:
All of the above