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# Isotropic hardening

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

## LinearIsotropicHardening

#### Description

Linear isotropic hardening

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

#### Parameters

Name Metafor Code Dependency
Initial yield stress $\sigma^{el}$ IH_SIGEL TM/TO
Plastic Modulus $h$ IH_H TM/TO

## SaturatedIsotropicHardening

#### Description

Saturated isotropic hardening

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

#### Parameters

Name Metafor Code Dependency
Initial yield stress $\sigma^{el}$ IH_SIGEL TM/TO
$Q$ IH_Q TM/TO
$\xi$ IH_KSI TM/TO

## DoubleSaturatedIsotropicHardening

#### Description

Double saturated isotropic hardening

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

#### Parameters

Name Metafor Code Dependency
Initial yield stress $\sigma^{el}$ IH_SIGEL TM/TO
$Q_1$ IH_Q1 TM/TO
$\xi_1$ IH_KS1 TM/TO
$Q_2$ IH_Q2 TM/TO
$\xi_2$ IH_KS2 TM/TO

## RambergOsgoodIsotropicHardening

#### Description

Ramberg-Osgood isotropic hardening

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

#### Parameters

Name Metafor Code Dependency
Initial yield stress $\sigma^{el}$ IH_SIGEL TM/TO
$A$ IH_A TM/TO
$n$ IH_N TM/TO

## SwiftIsotropicHardening

#### Description

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

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

#### Parameters

Name Metafor Code Dependency
Initial yield stress $\sigma^{el}$ IH_SIGEL TM/TO
$B$ IH_B TM/TO
$n$ IH_N TM/TO

## KrupkowskyIsotropicHardening

#### Description

Krupkowsky isotropic hardening

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

#### Parameters

Name Metafor Code Dependency
Initial equivalent plastic strain IH_EVPL0 TM/TO
$K$ IH_K TM/TO
$n$ IH_N TM/TO

## Nl8pIsotropicHardening

#### Description

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

$$\begin {eqnarray*} \sigma_{vm} &=& \left(P_2-P_1\right)\, \left(1-\exp\left(-P_3\,\bar{\varepsilon}^{vp}\right)\right)\, + \, P_4\left(\bar{\varepsilon}^{vp}\right)^{P_5} \, \\ & & + \, P_1\left(1+P_6\,\bar{\varepsilon}^{vp}\right)^{P_7} \, + \, P_8\,\bar{\varepsilon}^{vp} \end{eqnarray*}$$ === Parameters === ^ Name ^ Metafor Code ^ Dependency ^ |$P_1$ | IH_P1 | TM/TO |

 $P_2$ IH_P2 TM/TO $P_3$ IH_P3 TM/TO $P_4$ IH_P4 TM/TO $P_5$ IH_P5 TM/TO $P_6$ IH_P6 TM/TO $P_7$ IH_P7 TM/TO $P_8$ IH_P8 TM/TO

## FunctIsotropicHardening

#### Description

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

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

#### Parameters

Name Metafor Code Dependency
Initial yield stress $\sigma^{el}$ IH_SIGEL GD

A Functions y=f(t) must be associated to IH_SIGEL (depending on generalized displacements GD).

## PowerIsotropicHardening

#### Description

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

This law is integrated with an iterative method.

#### Parameters

Name Metafor Code Dependency
$P_1$ IH_P1 TM/TO
$P_2$ IH_P2 TM/TO
$P_3$ IH_P3 TM/TO
$P_4$ IH_P4 TM/TO

## AutesserreIsotropicHardening

#### Description

“Smatch” isotropic hardening.

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

#### Parameters

Name Metafor Code Dependency
$P_1$ IH_P1 TM/TO
$P_2$ IH_P2 TM/TO
$P_3$ IH_P3 TM/TO
$P_4$ IH_P4 TM/TO
$P_5$ IH_P5 TM/TO

## GoijaertsIsotropicHardening

#### Description

“Goijaerts” isotropic hardening

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

#### Parameters

Name Metafor Code Dependency
$M_1$ IH_M1 TM/TO
$M_2$ IH_M2 TM/TO
$M_3$ IH_M3 TM/TO
$M_4$ IH_M4 TM/TO

## KocksMeckingIsotropicHardening

#### Description

“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

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

#### Parameters

Name Metafor Code Dependency
$\sigma_0$ IH_SIGEL TM/TO
$\beta$ KM_BETA TM/TO
$\Theta_{0}$ KM_THETA0 TM/TO
$\Theta_{IV}$ KM_THETA4 TM/TO