Bauschinger effect

Remarks

Example: one-dimensional linear kinematic hardening has the yield condition $\abs{\sigma - \alpha} \le \sigma_y$ with back stress $\alpha = H \varepsilon^p$ for a constant $H > 0$. After loading to $\sigma_f = \sigma_y + \alpha$, reverse yield starts at $\sigma = \alpha - \sigma_y = \sigma_f - 2\sigma_y$. Its size $\abs{\sigma_f - 2\sigma_y}$ is less than $\sigma_f$ because $\sigma_f > \sigma_y$, and when $\sigma_f > 2\sigma_y$ reverse yielding even starts while the stress is still tensile. Non-example: linear isotropic hardening has the yield condition $\abs{\sigma} \le \sigma_y + H \varepsilon^p_{\mathrm{acc}}$, where $\varepsilon^p_{\mathrm{acc}}$ is the accumulated plastic strain. Its elastic range grows symmetrically, so reverse yield starts at $\sigma_r = \sigma_f$ and there is no Bauschinger effect. In several dimensions kinematic hardening uses a yield function $f(\sigma - \alpha) \le 0$ with a back stress tensor $\alpha$, so that the yield surface translates in stress space.