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====== Yield Stress ======

The class ''YieldStress'' manages the yield stress used in the plastic criterion, whether plastic (isotropic hardening), visco-plastic (Perzyna as additive, Cowper-Symonds as multiplicative or ZerilliArmstrong, JohnsonCook, ... as flow stress models)


$$ 
\sigma_{yield}  = \sigma_{yield} (\bar{\varepsilon}^{vp}, \dot{\bar{\varepsilon}}^{vp}, grainSize, ...)
$$

The laws implemented in Metafor are described below

===== IsotropicHardening =====

In order to simplify most of the test cases, the ''IsotropicHardening'' classes ([[isohard]]) derive from the ''YieldStress'' class.

$$ 
\sigma_{yield}  = \sigma_{isoHard}( \bar{\varepsilon}^{vp})
$$
Consequently, isotropic hardening is directly referred in the material as a ''YieldStress''.


===== GsIsoHYieldStress =====

Yield Stress defined by an isotropic hardening:

$$ 
\sigma_{yield}  = \sigma_{isoHard}( \bar{\varepsilon}^{vp}) + \sigma_{grainSize}( \bar{\varepsilon}^{vp}, \dot{\bar{\varepsilon}}^{vp}, grainSize )
$$

=== Parameters === 

^          Name                                        ^  Metafor Code  ^ Dependency         ^
|Number of the isotropic hardening law                 |  ''IH_NUM''    |                    |
|Number of the grain size evolution law                |  ''GS_NUM''    |                    |

===== PerzynaYieldStress =====

Yield Stress defined by an isotropic hardening and a visco-plastic Perzyna yield stress:

$$
\sigma_{yield}  = \sigma_{isoHard}( \bar{\varepsilon}^{vp}) + K (\dot{\bar{\varepsilon}}^{vp})^M \left(\bar{\varepsilon}^{vp}\right)^N
$$

=== Parameters === 

^          Name                           ^  Metafor Code  ^ Dependency         ^
|Number of the isotropic hardening law    |  ''IH_NUM''    |                    |
|$K $                                     |  ''PERZYNA_K'' |  ''TM/TO''         | 
|$M $                                     |  ''PERZYNA_M'' |  ''TM/TO''         |
|$N $                                     |  ''PERZYNA_N'' |  ''TM/TO''         |



===== GsPerzynaYieldStress =====

Yield Stress defined by an isotropic hardening, a visco-plastic Perzyna yield stress and a "grain size" stress:

$$ 
\sigma_{yield}  = \sigma_{isoHard}( \bar{\varepsilon}^{vp}) + K (\dot{\bar{\varepsilon}}^{vp})^M \left(\bar{\varepsilon}^{vp}\right)^N + \sigma_{grainSize}( \bar{\varepsilon}^{vp}, \dot{\bar{\varepsilon}}^{vp}, grainSize )
$$

=== Parameters === 

^          Name                           ^  Metafor Code   ^ Dependency         ^
|Number of the isotropic hardening law    |  ''IH_NUM''     |                    |
|Number of the grain size evolution law   |  ''GS_NUM''     |                    |
|$K $                                     |  ''PERZYNA_K''  |  ''TM/TO''         | 
|$M $                                     |  ''PERZYNA_M''  |  ''TM/TO''         |
|$N $                                     |  ''PERZYNA_N''  |  ''TM/TO''         |



===== JohnsonCookYieldStress =====

Johnson-Cook visco-plastic law:

\begin{multline}
\sigma_{yield}= \left( A+B \left( \bar{\varepsilon}^{vp} \right)^n \right) 
\left(1+C\ln\left(\dfrac{\dot{\bar{\varepsilon}}^{vp}}{\dot{\varepsilon}_0}\right)+C_2\left(\ln\left(\dfrac{\dot{\bar{\varepsilon}}^{vp}}{\dot{\varepsilon}_0}\right)\right)^2\right)  \\
\left( 1- \left( \dfrac{T-T_{room}}{T_{melt}-T_{room}} \right)^m \right) 
\end{multline}


=== Parameters ===

^          Name                                        ^  Metafor Code  ^ Dependency         ^
|$A $                                 |    ''JC_A''     |      ''TM/TO''       |
|$B $                                 |    ''JC_B''     |      ''TM/TO''       | 
|$n $                                 |    ''JC_N''     |      ''TM/TO''       |
|$m $                                 |    ''JC_M''     |      ''TM/TO''       | 
|Room temperature $T_{room}$          |  ''JC_TROOM''   |      ''TM/TO''       |
|Melting temperature   $T_{melt}$     |  ''JC_TMELT''   |      ''TM/TO''       | 
|$C $                                 |    ''JC_C''     |      ''TM/TO''       |
|$C_2 $                               |   ''JC_C2''     |      ''TM/TO''       |
|$\dot{\varepsilon}_0$                |  ''JC_EPSP0''   |      ''TM/TO''       |
 
===== JohnsonCookMecYieldStress =====

Isotherm version of the Johnson-Cook visco-plastic law :

$$
\sigma_{yield}= \left( A+B \left( \bar{\varepsilon}^{vp} \right)^n \right) 
\left(1+C\ln\left(\dfrac{\dot{\bar{\varepsilon}}^{vp}}{\dot{\varepsilon}_0}\right)+C_2\left(\ln\left(\dfrac{\dot{\bar{\varepsilon}}^{vp}}{\dot{\varepsilon}_0}\right)\right)^2\right) 
$$


=== Parameters ===

^          Name                       ^  Metafor Code  ^ Dependency         ^
|$A $                                 |    ''JC_A''     |      ''TM/TO''       |
|$B $                                 |    ''JC_B''     |      ''TM/TO''       | 
|$n $                                 |    ''JC_N''     |      ''TM/TO''       |
|$C $                                 |    ''JC_C''     |      ''TM/TO''       |
|$C_2 $                               |   ''JC_C2''     |      ''TM/TO''       |
|${\dot\varepsilon}_{0}$              |  ''JC_EPSP0''   |      ''TM/TO''       |

===== PowJohnsonCookYieldStress =====

Power version of the Johnson-Cook visco-plastic law (PP implementation):

$$
\sigma_{yield}= \left( A+B \left( \bar{\varepsilon}^{vp} \right)^n \right) 
\left(1+\dfrac{\dot{\bar{\varepsilon}}^{vp}}{\dot{\varepsilon}_0}\right)^C \left( 1- \left( \dfrac{T-T_{room}}{T_{melt}-T_{room}} \right)^m \right)  
$$

=== Parameters ===

^          Name                       ^  Metafor Code  ^ Dependency         ^
|$A $                                 |    ''JC_A''     |      ''TM/TO''       |
|$B $                                 |    ''JC_B''     |      ''TM/TO''       | 
|$n $                                 |    ''JC_N''     |      ''TM/TO''       |
|$m $                                 |    ''JC_M''     |      ''TM/TO''       | 
|Room temperature $T_{room}$          |  ''JC_TROOM''   |      ''TM/TO''       |
|Melting temperature $T_{melt}$       |  ''JC_TMELT''   |      ''TM/TO''       | 
|$C $                                 |    ''JC_C''     |      ''TM/TO''       |
|$\dot{\varepsilon}_0$                |  ''JC_EPSP0''   |      ''TM/TO''       |
 
===== ZerilliArmstrongYieldStress =====

Zerilli-Armstong visco-plastic law:

$$
\sigma_{yield} = \sigma_0 + C_5 \left( \bar{\varepsilon}^{vp} \right) ^{n_1} + C_2 \left( \bar{\varepsilon}^{vp} \right)^{n_2} \exp \left( -C_3 T + C_4 T \ln \dot{\bar{\varepsilon}}^{vp}  \right) 
$$

For FCC metals, $C_5=0$.
For BCC metals, $n_2=0$.

:!: Careful: this law is thermomechanical, and the temperature is the absolute one. To have proper results, finite elements must be thermomechanical, and so must be the integration scheme.

=== Parameters ===

^  Name      ^  Metafor Code   ^ Dependency           ^
|$\sigma_0$  |  ''ZA_SIGMA0''  |  ''TM/TO''           |
|$C_5$       |   ''ZA_C5''     |  ''TM/TO''           | 
|$n_1$       |   ''ZA_N1''     |  ''TM/TO''           | 
|$C_2$       |   ''ZA_C2''     |  ''TM/TO''           |
|$n_2$       |   ''ZA_N2''     |  ''TM/TO''           |
|$C_3$       |   ''ZA_C3''     |  ''TM/TO''           |
|$C_4$       |   ''ZA_C4''     |  ''TM/TO''           |


===== CowperSymondsYieldStress =====

Cowper-Symonds visco-plastic law.

$$
\sigma_{yield}= \sigma_0 \left( 1 + \left(\dfrac{\dot{\overline{\varepsilon}}^{vp}}{D}\right)^{\frac{1}{p}} \right)
$$

where $\sigma_0$ is the static yield stress, whose evolution is often given by a piecewise linear function.


=== Parameters ===

^          Name         ^  Metafor Code  ^ Dependency         ^
|Number of the hardening law    |    ''IH_NUM''   |                      |
|$D $                  |    ''CS_D''     |   ''TM/TO''          |
|$p $                  |    ''CS_P''     |   ''TM/TO''          |


===== ViscoKocksMeckingYieldStress =====

=== Description ===

Visco-plastic extension of the "Kocks-Mecking" isotropic hardening law.

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

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

where $\sigma_{y}^{tr} $ is a transition yield stress, computed to guarantee continuity between both phases:

$$
\sigma_{y}^{tr} = \sigma_{y}^{0} + \sigma_{v}\frac{(\Theta_{0}-\Theta_{IV})}{\Theta_{0}} 
$$

and where $\bar{\varepsilon}^{vp}_{tr}$ is the corresponding plastic strain:

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

The viscous component of the yield stress is hidden in the calculation of the saturation stress:

$$
\sigma_{v} = \sigma_{v0} \left ( \frac{\dot{\bar{\varepsilon}}^{vp}}{\dot{\bar{\varepsilon}}^{vp}_{0}} \right ) ^{\left (\frac{kT}{A} \right )}
$$

$k$ : Boltzman's constant = $1.381e^{-23} \mbox{J}/\mbox{K}$

$T$ : Temperature K (CAREFUL in Kelvin : requires the definition of the temperature in the material)

$A$ : Activation energy (material constant)

${\dot{\bar{\varepsilon}}^{vp}_{0}}$ : reference plastic strain rate (= $1.e7$)

=== Parameters ===

^         Name ^  Metafor Code       ^ Dependency         ^
|$\sigma_0$    |   ''IH_SIGMA0''     |  ''TM/TO''           |
|$\Theta_{0}$  |   ''KM_THETA0''     |  ''TM/TO''           | 
|$\Theta_{IV}$ |   ''KM_THETA4''     |  ''TM/TO''           |
|$\sigma_{v0}$ |   ''KM_SIGV0''      |  ''-''           |
|${\dot{\bar{\varepsilon}}^{vp}_{0}}$ |   ''KM_DEVPL0''     |  ''-''           |
|$k$ : Boltzman's constant |   ''KM_BOLTZMANN''   |  ''-''          |
|$A$                         |   ''KM_A''           |  ''-''          |


===== ViscoKocksMecking2YieldStress =====

=== Description ===

Second visco-plastic extension of the "Kocks-Mecking" isotropic hardening law, to match the Ta6V better.

No fourth step

The basic equation is still:
$$
\sigma_{y} = \sigma_{y}^{0} + \sigma_{v} \left [1-exp \left (-\frac{\Theta_{0}}{\sigma_{v}} \bar{\varepsilon}^{vp}\right ) \right ]
$$

Both plastic strain rate and temperature dependency are hidden in the calculation of the Voce stress $\sigma_{v}$. If this Voce stress is plotted as a function of the $g$ parameter, defined as:

$$ g = \frac{kT}{\mu b^3} \ln \left ( \frac{\dot{\bar{\varepsilon}}^{vp}_{0}}{\dot{\bar{\varepsilon}}^{vp}} \right ) $$  

where

$b$ : is the norm of the Burgers vector \\
$k$ : is the Boltzmann's constant  \\
(for Ta6V, $ \frac{k}{b^3} ~ 1.135 $, in "$\mbox{Mpa}$")\\
$T$ : Temperature K (CAREFUL in Kelvin : requires the definition of the temperature in the material) \\
$\mu$ : Elastic shear modulus of the material (requires a new definition of Young's modulus and Poisson ratio in the KM2 law!!!) \\
$\dot{\bar{\varepsilon}}^{vp}_{0}$: reference plastic strain rate (= $1.e7$) \\

the following figure is obtained: 

{{ :doc:user:kocksmecking2.png?direct&200 |}}

KocksMecking usually predicts a linear evolution (KM data). However, the Ta6V trial data are the colored dots. The evolution is more in broken lines, defined as follow: 

$G12$ and $G23$ are defined as the transition values between the three areas: 

$$ 
\frac{\sigma_{v}}{\mu} = A1 \; g  + B1 \;\;\;si\;\;\;       g < G12 \\
\frac{\sigma_{v}}{\mu} = A2 \; g  + B2 \;\;\;si\;\;\; G12 < g < G23 \\
\frac{\sigma_{v}}{\mu} = A3 \; g  + B3 \;\;\;si\;\;\; G23 < g
$$

Finally, the apparition of plasticity was observed to depend on strain rate and temperature (here, the temperature dependency is explicitly defined, but it could be done implicitly):

$$\sigma_0 = A + B T + (C + D T) \ln \left (\dot{\bar{\varepsilon}}^{vp}\right ) $$ 


=== Parameters ===

^          Name    ^  Metafor Code  ^ Dependency         ^
|$ A $         |   ''IH_SIGEL_A''      |      ''-''          |
|$ B $         |   ''IH_SIGEL_B''      |      ''-''          |
|$ C $         |   ''IH_SIGEL_C''      |      ''-''          |
|$ D $         |   ''IH_SIGEL_D''      |      ''-''          |
|$ A1 $        |   ''KM2_SIGVOCE_A1''  |      ''-''          |
|$ B1 $        |   ''KM2_SIGVOCE_B1''  |      ''-''          |
|$ G12 $       |   ''KM2_SIGVOCE_G12''  |      ''-''          |
|$ A2 $        |   ''KM2_SIGVOCE_A2''  |      ''-''          |
|$ B2 $        |   ''KM2_SIGVOCE_B2''  |      ''-''          |
|$ G23 $       |   ''KM2_SIGVOCE_G23''  |      ''-''          |
|$ A3 $        |   ''KM2_SIGVOCE_A3''  |      ''-''          |
|$ B3 $        |   ''KM2_SIGVOCE_B3''  |      ''-''          |
|$\frac{k}{b^3}$ | ''KM2_BOLTZMANN_BURGER3''  |  ''-''          |
|Young's Modulus         |   ''KM2_ELASTIC_MODULUS''      |  ''-''          |
|Poisson ratio   |   ''KM2_POISSON_RATIO''        |  ''-''          |
|$\Theta_{0}$  |   ''KM_THETA0''       |    ''TM/TO''        | 
|${\dot{\bar{\varepsilon}}^{vp}_{0}}$ |   ''KM_DEVPL0''    |  ''-''           |


===== SellarsTegartYieldStress =====

Yield Stress used for hot rolling defined by  :
  * a viscous power term : 
$$
S0 = \sqrt{3}  \; KK (\sqrt{3}  \;\dot{\bar{\varepsilon}}^{vp})^{M0}
$$
  * a viscous asinh Term : 
$$
Ss = As  \; asinh((\frac{\dot{\bar{\varepsilon}}^{vp}}{Zs})^{Ms})
$$
  * for the Yield Stress computed by (using a Voce hardening term): 
$$
\sigma_{yield}  = S0 + (Ss - S0) (1-e^{-C \; \bar{\varepsilon}^{vp}})^{R}
$$

=== Parameters === 

^          Name                           ^  Metafor Code  ^ Dependency         ^
|$KK $                                     |  ''SELLARSTEGART_KK'' |  ''TM/TO''         | 
|$M0 $                                     |  ''SELLARSTEGART_M0'' |  ''TM/TO''         |
|$As $                                     |  ''SELLARSTEGART_AS'' |  ''TM/TO''         |
|$Zs $                                     |  ''SELLARSTEGART_MS'' |  ''TM/TO''         | 
|$Ms $                                     |  ''SELLARSTEGART_ZS'' |  ''TM/TO''         |
|$C $                                      |  ''SELLARSTEGART_R'' |  ''TM/TO''         |
|$R $                                      |  ''SELLARSTEGART_C'' |  ''TM/TO''         |


===== PythonYieldStress =====

User defined Yield Stress by a pythonDirector :
Python Director allows user to define their own Yield Stress law. Four functions has to be defined in the Python Class : a constructor (__init__), a destructor (__del__) that must never be called, and the computation functions : getYieldStress (returning YieldStress) and getYieldHardening (returning h).
See the example below of a Perzyna law : 
<code>

class MyYieldStress(PythonYieldStress):
    def __init__(self, _no, _svm0, _h, _K, _m):
        print("MyYieldStress: __init__")
        PythonYieldStress.__init__(self,_no)
        self.svm0 = _svm0
        self.h    = _h
        self.K    = _K
        self.m    = _m        
    def __del__(self):
        print("MyIsoH : __del__")
        print("callToDestructor of MyYieldStress not allowed. Add MyIsoH.__disown__()")
        input('')
        exit(1)
    def getYieldStress(self, evpl, deltaEvpl, dTime, yieldGPState, pLaw):
        #print "MyYieldStress compute Yield Stress"
        if dTime > 0.0:
            dEvpl = deltaEvpl/dTime
        else:
            dEvpl = 0.0
        sigH = self.svm0+evpl*self.h
        sigV = self.K*pow(dEvpl, self.m)
        #yieldGPState.setVpStress(sigV)
        return sigH+sigV
    def getYieldHardening(self, epl, deltaEvpl, dTime, yieldGPState, pLaw) :
        if dTime > 0.0:
            dEvpl = deltaEvpl/dTime
            h = self.h + (self.K*pow(dEvpl, self.m-1))/dTime
        else:
            dEvpl = 0.0
            h = self.h
        return h
</code>