True stress and true strain relations

True stress $\sigma_T$ is the load divided by the instantaneous cross-sectional area $A_i$; true strain $\varepsilon_T$ is the sum of incremental strains, each based on the instantaneous length.

$$\sigma_T = \frac{P}{A_i}, \qquad \varepsilon_T = \int_{L_0}^{L_i}\frac{dL}{L} = \ln\frac{L_i}{L_0}$$

Derivation

Engineering strain is $e = \dfrac{L_i - L_0}{L_0}$, so $\dfrac{L_i}{L_0} = 1 + e$. Therefore

$$\varepsilon_T = \ln(1 + e)$$

Plastic deformation conserves volume: $A_0 L_0 = A_i L_i$, so $A_i = \dfrac{A_0}{1+e}$. Then

$$\sigma_T = \frac{P}{A_i} = \frac{P}{A_0}(1+e) = \sigma\,(1 + e)$$

Validity

Both equations hold only up to the start of necking (uniform deformation). After necking the strain is no longer uniform, so volume constancy cannot be applied over the whole gauge length. Beyond necking, $\sigma_T$ is found from the measured neck area, and $\varepsilon_T = \ln(A_0/A_i)$.

Flow curve

In the plastic region $\sigma_T = K\varepsilon_T^{\,n}$, where $K$ is the strength coefficient and $n$ the strain-hardening exponent (0.1-0.5 for metals). Necking begins at $\varepsilon_T = n$.

Why true stress keeps rising

After the UTS the engineering stress falls, because the load falls while the original area $A_0$ is still used in the calculation. But the neck area shrinks even faster than the load, so the load per actual area ($\sigma_T$) keeps increasing, due to strain hardening, until fracture.