====== 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''         | 

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

===== 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_KSI1''   |  ''TM/TO''         | 
|$Q_2$                      |   ''IH_Q2''    |  ''TM/TO''         | 
|$\xi_2$                    |   ''IH_KSI2''   |  ''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 ===

Krupkowski isotropic hardening

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

=== Parameters ===

^          Name                                                   ^  Metafor Code  ^ Dependency         ^
| Initial equivalent plastic strain $\bar{\varepsilon}^{vp}_{0}$  |  ''IH_EVPL0''  |  ''TM/TO''         |
| strength coefficient  $K$                                       |    ''IH_K''    |  ''TM/TO''         | 
| strain hardening exponent $n$                                   |    ''IH_N''    |  ''TM/TO''         | 

===== Nl8pIsotropicHardening =====

=== Description ===

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


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

=== 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''  |  ''IF_EPL''         |

An [[doc:user:general:fonctions]] must be associated to ''IH_SIGEL'' (depending on ''Field(IF_EPL)'').


===== 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''           |

===== Python =====

User defined Isotropic Hardening by a pythonDirector :

Python Director allows user to define their own Isotropic Hardening law. Five functions has to be defined in the Python Class : 
  * a constructor (<code>__init__</code>),
  * a destructor (<code>__del__</code>) that must never be called, 
  * computeSvm  (epl, pLaw)
  * computeH (epl, pLaw)
  * computePotential (epl, pLaw) for hyperElastics models
See the example below of a Linear Isotropic Hardening : 
<code>

class MyIsoH(PythonIsotropicHardening):
    def __init__(self, _no, _svm0, _h):
        print("MyIsoH : __init__")
        PythonIsotropicHardening.__init__(self,_no)
        self.svm0 = _svm0
        self.h    = _h
        print("no = ", _no)
        print("self.svm0 = ", self.svm0)
        print("self.h = ", self.h)
        print("MyIsoH : __init__ finished")
        print("computeSvm(0.0) = " , self.computeSvm(0.0, None))
    def __del__(self):
        print("MyIsoH : __del__")
        print("callToDestructor of MyIsoH not allowed. Add MyIsoH.__disown__()")
        input('')
        exit(1)
    def computeSvm(self, epl, pLaw) :
        #print "MyIsoH compute SVM"
        return self.svm0+epl*self.h
    def computeH(self, epl, pLaw) :
        #print "MyIsoH compute H"
        return self.h
    def computePotential(self, epl, pLaw) :
        #print "MyIsoH compute Potential"
        return (self.svm0+self.h*epl*0.5)*epl
</code>