The IsotropicHardening class manages all isotropic hardening laws in Metafor, which are described below.

Linear isotropic hardening

$$ \sigma_{vm} = \sigma^{el} + h\, \bar{\varepsilon}^{vp} $$

Saturated isotropic hardening

$$ \sigma_{vm} = \sigma^{el} + Q\left(1-\exp\left(-\xi \bar{\varepsilon}^{vp}\right)\right) $$

Double saturated isotropic hardening

$$ \sigma_{vm} = \sigma^{el} + Q_1\left(1-\exp\left(-\xi_1 \bar{\varepsilon}^{vp}\right)\right) + Q_2\left(1-\exp\left(-\xi_2 \bar{\varepsilon}^{vp}\right)\right) $$

Ramberg-Osgood isotropic hardening

$$ \sigma_{vm} = \sigma^{el} \left(1+A\, \bar{\varepsilon}^{vp}\right)^{\frac{1}{n}} $$

Swift isotropic hardening (a more common formulation of Ramberg - Osgood)

$$ \sigma_{vm} = \sigma^{el} +B \left(\bar{\varepsilon}^{vp}\right)^{n} $$

Krupkowski isotropic hardening

$$ \sigma_{vm} = K \left(\bar{\varepsilon}^{vp}_ {0} + \bar{\varepsilon}^{vp}\right)^{n} $$

Nonlinear isotropic hardening with 8 parameters. First one implemented, can be used to do almost everything.

$$ \sigma_{vm} = \left(P_2-P_1\right) \left(1-\exp\left(-P_3\bar{\varepsilon}^{vp}\right)\right) + P_4\left(\bar{\varepsilon}^{vp}\right)^{P_5} + $$ $$ + P_1\left(1+P_6\bar{\varepsilon}^{vp}\right)^{P_7} + P_8\bar{\varepsilon}^{vp} $$

Piecewise linear isotropic hardening. A function is associated to the yield stress.

$$ \sigma_{vm} = \sigma^{el} \, * \, f\left(\bar{\varepsilon}^{vp}\right) $$

An Functions y=f(t) must be associated to IH_SIGEL (depending on Field(IF_EPL)).

$$ \sigma_{vm}= P_1 \left[ P_2 \sigma_{vm} + P_3 \overline{\varepsilon}^{vp} \right] ^{P_4} $$

This law is integrated with an iterative method.

“Smatch” isotropic hardening.

$$ \sigma_{vm}= \left( P_1 + P_2 \overline{\varepsilon}^{vp} \right) \left( 1 - P_3 \exp \left( -P_4 \overline{\varepsilon}^{vp} \right) \right) + P_5 $$

“Goijaerts” isotropic hardening

$$ \sigma_{vm}= \sigma_{el} + M_1 \left( 1-\exp(-\frac{\overline{\varepsilon}^{vp}}{M_2})\right) + M_3 \sqrt{\overline{\varepsilon}^{vp}} + M_4 \overline{\varepsilon}^{vp} $$

“Kocks-Mecking” isotropic hardening

$$ \sigma_{y} = \sigma_{y}^{0} + \frac{\Theta_{0}}{\beta} [ 1-exp(-\beta \bar{\varepsilon}^{vp}) ] \;\;\; si \;\;\; \bar{\varepsilon}^{vp} < \bar{\varepsilon}^{vp}_{tr} $$

$$ \sigma_{y} = \sigma_{y}^{tr} + \Theta_{IV} \left( \bar{\varepsilon}^{vp} - \bar{\varepsilon}^{vp}_{tr}\right) \;\;\; si \;\;\; \bar{\varepsilon}^{vp} >\bar{\varepsilon}^{vp}_{tr} $$

where the transition yield stress between stages 3 and 4 is defined as

$$ \sigma_{y}^{tr} = \sigma_{y}^{0} + \frac{\Theta_{0}-\Theta_{IV}}{\beta} $$

and the corresponding yield strain as

$$ \bar{\varepsilon}^{vp}_{tr} = \frac{1}{\beta} \ln \left(\frac{\Theta_{0}}{\Theta_{IV}}\right) $$