A cylindrical steel specimen with an original diameter of 12.8 mm is tensile tested tofracture and found to have an engineering fracture strength of 460 MPa. If its cross-sectional diameter at fracture is 10.7 mm, determine the true stress at fracture.

To determine the true stress at fracture, we can use the formula:

True Stress = Engineering Stress * (1 + Engineering Strain)

First, let's calculate the engineering stress:

Engineering Stress = Fracture Strength = 460 MPa

Next, we need to calculate the engineering strain. The engineering strain is given by the formula:

Engineering Strain = (Change in Diameter) / (Original Diameter)

Change in Diameter = Original Diameter - Diameter at Fracture = 12.8 mm - 10.7 mm = 2.1 mm

Engineering Strain = (2.1 mm) / (12.8 mm)

Now, we can calculate the true stress:

True Stress = Engineering Stress * (1 + Engineering Strain) = 460 MPa * (1 + (2.1 mm / 12.8 mm))

To convert the engineering strain from mm to a dimensionless value, we need to divide it by the original diameter:

True Stress = 460 MPa * (1 + (2.1 mm / 12.8 mm)) / (12.8 mm)

Simplifying the equation:

True Stress = 460 MPa * (1 + 0.164) / 12.8

Calculating the true stress:

True Stress = 460 MPa * 1.164 / 12.8 = 41.953125 MPa

Therefore, the true stress at fracture is approximately 41.95 MPa.