Question

Consider a rod of diameter 10 mm, which follows a true stress-strain relationship given as follows: 0₁ = 50+307, where o, -true stress and s- true strain A tensile force of 100 N is applied at the ends such that the rod stretches from 10 mm to 12 mm, where it fails. Determine the difference between the true ultimate strength (o,), and engineering ultimate strength (o,)..​

Answer 1

The difference between the true ultimate strength (σu) and engineering ultimate strength (σe) is 355.73 MPa.

Let's determine the difference between the true ultimate strength (σu) and engineering ultimate strength (σe) given the true stress-strain relationship σ₁ = 50+307 and a tensile force of 100 N applied at the ends that stretches the rod from 10 mm to 12 mm, where it fails.

Steps to solve:

1. Define the variables:

σ₁: True stress

ε: True strain

σu: True ultimate strength

σe: Engineering ultimate strength

F: Tensile force

A₀: Initial cross-sectional area

L₀: Initial length

Lf: Final length

2. Given information:

σ₁ = 50+307

F = 100 N

A₀ = π(10/2)² = 78.54 mm²

L₀ = 10 mm

Lf = 12 mm

3. Calculate true strain:

ε = ln(Lf/L₀) = ln(12/10) = 0.182

4. Calculate true ultimate stress:

σu = σ₁ at ε = 0.182

σu = 50+307 = 357 MPa

5. Calculate engineering ultimate stress:

σe = F/A₀

σe = 100 N / 78.54 mm² = 1.27 MPa

6. Calculate the difference:

Δσ = σu - σe

Δσ = 357 MPa - 1.27 MPa = 355.73 MPa .

Therefore , the difference between the true ultimate strength (σu) and engineering ultimate strength (σe) is 355.73 MPa.