Plastic strain ($e=e^{\mathrm{e}}+e^{\mathrm{p}}$)

In small-strain elastoplasticity, total infinitesimal strain tensor splits into an elastic part and a generally irreversible plastic part. The elastic constitutive equation determines stress from $e^{\mathrm{e}}$, while a yield condition and a plastic flow law determine the evolution of $e^{\mathrm{p}}$. Removing external load need not remove plastic strain, so using Hooke's law on total strain after yielding is insufficient.

A plastic flow rule prescribes the direction and rate of plastic straining once a yield condition is active. In a plastic-potential model its direction is the stress gradient of $g$, multiplied by a nonnegative plastic multiplier. An associated flow rule takes $g$ equal to the yield function. The stress-strain split, loading conditions and hardening evolution are needed along with this equation to close an elastoplastic model.