Calculate the Engineering Ultimate Tensile Strength and the maximum load in tension testing of an annealed copper specimen with an original diameter of 5 mm. Assume that the strain hardening coefficient for annealed copper is 0.50 and that the true stress at necking is 290 MPa.

Explanation:

Original diameter = 5mm

d= 0.005m. r=0.0025m

True stress (σ) = 290MPa

Coefficient of annealing copper (n)=0.5

True stress is given as force/area

σ=F/A

Ao=πr^2

Ao=1.963×10^-5m2

F=σ ×A

F=290×10^6 ×π×(0.0025)^2

F= 5694.13N

Given that

σ=Kε^n

290=Kε^0.5

n=ε=0.5 at necking

Therefore

290=K×0.5^0.5

K=410.12MPa

ε=In(Ao/A)

0.5=In(π×0.0025^2/A)

exp(0.5)=1.96×10^-5/A

A=1.96×10^-5/exp(0.5)

A= 1.2×10^-5m2

This is the new area

Then the Ultimate Tensile Stress

σ=F/A

σ=5694.13/1.2×10^-5

σ=4.75×10^8Pa

σ=475MPa

Therefore,

The ultimate tensile stress is 475MPa.

The maximum tension is

Fo/Ao=F/A

F=(Fo/Ao) ×A

F=(5694.13/1963×10^-5)×1.2×10^-5

F= 3479.99

F=3480N.

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