The engineering stress-strain response of a hot-rolled structural steel is idealized by two straight segments: a linear elastic rise from the origin to yield at MPa and strain, followed by a straight line to fracture at MPa and strain. The modulus of toughness is most nearly:
- (A)
- (B)
- (C)
- (D)
Show worked solution
Answer: (C)
Nearly all of the area under the curve lies in the plastic trapezoid, so the modulus of toughness is about .
Modulus of toughness is the total energy per unit volume absorbed to fracture, which is the area under the engineering stress-strain curve; the idealized curve splits into an elastic triangle and a plastic trapezoid.
Yield strain is small, so the elastic triangle contributes a negligible share of the total.
Stress rises linearly from yield to fracture, so the plastic region is a trapezoid whose height is the mean of the two stress levels.
FE Reference Handbook — Materials Science and Engineering: Engineering Stress-Strain Curve, Modulus of Toughness
Why the other choices appear
- (A)Counts only the elastic triangle, , which is the modulus of resilience rather than the toughness.
- (B)Treats the plastic region as a triangle of height equal to the ultimate strength, .
- (D)Takes the area as a full rectangle of ultimate strength times fracture strain, .