Questions: Toughness, Ductility, and Brittle Behavior

Question 1

Material A has a yield strength of 1,200 MPa and 2% elongation at fracture. Material B has a yield strength of 600 MPa and 25% elongation. Which material has higher toughness?

• A. Material A, because higher yield strength always means more energy absorbed before fracture
• B. Material B, because toughness is the area under the entire stress-strain curve, and B's large plastic deformation region stores far more energy despite lower strength
• C. They are equal in toughness because they trade strength for ductility in equivalent amounts
• D. Material A, because toughness equals yield strength divided by strain at fracture

Answer: (B)

Toughness is the area under the stress-strain curve — it requires both strength and ductility. A rough estimate for Material A: ~0.5 × 1,200 MPa × 0.02 ≈ 12 MJ/m³. For Material B: ~0.5 × 600 MPa × 0.25 ≈ 75 MJ/m³. Material B is roughly 6× tougher despite lower strength, because its large plastic deformation region contributes far more area. This directly illustrates that a strong-but-brittle material can have lower toughness than a weaker-but-ductile one.

Question 2

A glass window pane can withstand tensile stresses up to ~700 MPa under ideal conditions but fractures at ~0.1% strain. Structural steel yields at 400 MPa but sustains 20% elongation before fracture. Which material is tougher?

• A. Glass, because it has a higher ultimate tensile strength and can withstand greater stress
• B. Steel, because toughness depends on the area under the full stress-strain curve, and steel's massive plastic deformation region dominates even though its strength is lower
• C. They are equally tough because glass trades strain for stress in exact proportion to steel
• D. Glass, because elastic energy storage is more efficient than plastic deformation for energy absorption

Answer: (B)

Glass fractures before any plastic deformation, giving it a triangular, nearly elastic stress-strain curve with tiny area (~0.5 × 700 MPa × 0.001 ≈ 0.35 MJ/m³). Steel's large plastic deformation region (~0.5 × 400 MPa × 0.20 ≈ 40 MJ/m³) gives it roughly 100× more toughness. This is why glass shatters catastrophically under impact while steel bends — strength does not determine impact resistance; toughness does.

Question 3

A material that simultaneously achieves higher strength and higher ductility than another is definitively tougher, but increasing strength alone through cold working can actually reduce toughness.

• T. True
• F. False

Answer: True

When both strength and ductility increase, toughness (area under the curve) must increase. However, cold working raises yield strength and ultimate tensile strength by introducing dislocations that impede further deformation — but this same dislocation density reduces ductility, often dramatically. The gain in strength is outweighed by the loss in elongation, shrinking the area under the stress-strain curve and reducing toughness. Most strengthening mechanisms exhibit this strength-ductility tradeoff.

Question 4

Resilience and toughness both measure energy absorption capacity, so a highly resilient material is also necessarily highly tough.

• T. True
• F. False

Answer: False

Resilience and toughness measure fundamentally different things. Resilience is the area under the elastic portion of the stress-strain curve only — the energy stored and fully released upon unloading without permanent deformation. Toughness is the total area including the plastic region, representing energy absorbed up to fracture (which is dissipated, not returned). A hard spring steel can have very high resilience (large elastic range, high yield stress) but relatively low toughness (fractures with little plastic deformation). The two properties are optimized for different applications.

Question 5

Explain why hardened tool steel — which has higher yield strength than annealed (soft) steel — can have lower toughness. What happens to the stress-strain curve during hardening, and why does this reduce toughness?

Model answer: Hardening (by heat treatment, quenching, or work hardening) introduces microstructural barriers — martensite, dislocations, or precipitates — that raise the yield stress. However, these same features prevent dislocation movement, drastically reducing the material's ability to undergo plastic deformation. On the stress-strain curve, hardening raises the yield point and ultimate tensile strength but truncates the plastic region — the curve reaches fracture at much lower elongation. Toughness, being the area under the entire curve, decreases because the gain in height (strength) is more than offset by the loss in width (ductility). Hardened tool steel is brittle: it resists yielding well but cannot absorb much energy before catastrophic fracture.

This tradeoff is why structural applications requiring impact resistance (landing gear, pressure vessels, crash structures) use high-alloy steels or titanium alloys engineered to retain ductility at elevated strength — not maximally hardened steels. Materials selection is never a single-axis optimization.