# How to find ultimate stress given true stress - true strain relation?

I am given the following problem:

Consider a brass whose true (plastic) stress-true (plastic) strain is given by: $$\sigma_{T,p} = 530 \ \rm{MP}\times(\epsilon_{T,p})^{0.36}$$ To simplify the problem, assume that you can ignore both the elastic stress and strain. What is the ultimate tensile stress?

I can't seem to find a direct way to calculate the ultimate tensile stress (engineering stress at the onset of necking). The only relation that seems useful in this case is $$\epsilon_T = \ln(1+\epsilon_E)$$, which I used to calculate the engineering strain at the onset of necking. However, I'm not given a useful relation between engineering stress and strain.

• Look at the equation for the true plastic stress as a plastic strain that you were given. It shows that the true plastic stress increases monotonically with increasing plastic strain, which might seem to imply that the material has an infinite ultimate tensile stress. But, no, as the material is plastically strained, it necks so that the cross sectional area is decreasing at the same time that the true plastic stress is increasing. So there will be a maximum stress that the rod can support. You need a relationship between the plastic strain and that cross sectional area. – Samuel Weir Oct 8 '18 at 18:54

Seems to me that you are just missing this expression for true stress: $$σ' = σ(1+ε)$$
Which you can use in conjunction with: $$ε' = ln(1+ε)$$, to substitute in your strain-stress relation.