E&P: Intro Formulae

Question 1

Show that true strain and engineering strain are related by:

𝜀𝑡 = ln(𝜀 + 1)

Answer 1

Engineering strain
𝜀 =𝑑𝐿/𝐿𝑜
=
(𝐿 − 𝐿𝑜)/𝐿𝑜
=
(𝐿/𝐿𝑜)− 1
𝜀 + 1 =(𝐿/𝐿𝑜)

True strain
𝜀𝑡 = 𝑙𝑛(𝐿/𝐿𝑜)

Therefore
𝜀𝑡 = 𝑙𝑛(𝜀 + 1)

Question 2

Assuming plastic deformation is a constant-volume process, show that true stress and
engineering stress are approximately related by:
𝜎𝑡 = 𝜎(𝜀 + 1)

Answer 2

Engineering Stress
𝜎 =𝐹/𝐴𝑜

True stress
𝜎𝑡 =𝐹/𝐴

Constant volume assumption:
𝐴𝑜𝐿𝑜 = 𝐴𝐿

𝐴𝑜/𝐴 = 𝐿/𝐿𝑜 = 𝜀 + 1

Since

𝜀 =𝑑𝐿/𝐿𝑜
=
(𝐿 − 𝐿𝑜)/𝐿𝑜
=
(𝐿/𝐿𝑜)− 1
𝜀 + 1 =(𝐿/𝐿𝑜)

𝜎𝑡 =(𝐹/𝐴𝑜)*𝐴𝑜/𝐴 = 𝜎(𝐿/𝐿𝑜)

= 𝜎(𝜀 + 1)

Question 3

A two bar structure has the following dimensions:

L1 = 300mm, A1 = 200mm^2
L2 = 400mm, A2 = 400mm^2

Which of the bars fail first?

Answer 3

Bar one because there is a larger strain, therefore a greater stress.

Question 4

What does 0.2% Proof Stress mean

Answer 4

0.2% plastic strain has occured

Question 5

What is the formula for the limit load?

Answer 5

F = yield stress * (A1+A2)