Elastic-hardening

The initial slope is defined by the elastic modulus, $E$, after yield the hardening modulus $E_h$ governs as stress rises from ($\varepsilon_y, f_{yd}$) to ($\varepsilon_u, f_u$). For EN 1992 the hardening modulus is defined in terms of a hardening coefficient $k$ and the final point is ($\varepsilon_{uk} , kf_{yd}$) where the failure strain is reduced to $\varepsilon_{ud}$ (typically $0.9 \varepsilon_{uk}$).

The relationship between hardening modulus and hardening coefficient is:

$$E_h=\frac{(k-1)f_y} {\varepsilon_{uk}-f_y/E} \\$$

$$k=\frac{E_h(\varepsilon_{uk}-f_y)/E)}{f_y}{+1} \\$$

The material fails at $\varepsilon_{ud}$ where $\varepsilon_{ud}<\varepsilon_{uk}$. This is defined in Eurocode and related codes.