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

Linear isotropic hardening

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

NB: the plastic modulus is defined as $h = \frac{E E_T}{E - E_T}$, where $E$ is the Young modulus and $E_T$ the tangent modulus.

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

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