<commit_before>﻿Strain Transfer Across Grain Boundaries
=======================================

.. include:: includes.rst

The strain transfer across grain boundaries can be defined by the four following mechanisms (see Figure 1) [#LimRaj_1985_1]_, [#SuttonBalluffi_1995]_, [#Zaefferer_2003]_ and [#Priester_2013]_ :

 1) direct transmission with slip systems having the same Burgers vector, and the grain boundary is transparent to dislocations (no strengthening effect) (Figure 1-a);

 2) direct transmission, but slip systems have different Burgers vector (leaving a residual boundary dislocations) (Figure 1-b);

 3) indirect transmission, and slip systems have different Burgers vector (leaving a residual boundary dislocations) (Figure 1-c);

 4) no transmission and the grain boundary acts as an impenetrable boundary, which implies stress accumulations, localized rotations, pile-up of dislocations... (Figure 1-d).

.. figure:: ./_pictures/Schemes_SlipTransmission/slip_transfer.png
   :scale: 40 %
   :align: center
   
   *Figure 1 : Possible strain transfer across grain boundaries (GB) from Sutton and Balluffi.*
   
Several authors proposed slip transfer parameters from modellings or experiments for the last 60 years.
A non-exhaustive list of those criteria is given in the next part of this work, including geometrical parameter, stress and energetic functions, and recent combinations of the previous parameters.

Geometrical Criteria
--------------------
Based on numerous investigations of dislocation-grain boundary interactions, quantitative geometrical expressions describing the slip transmission mechanisms have been developed. 
A non-exhaustive list of geometrical criteria is detailed subsequently.
The geometry of the slip transfer event is most of the time described by the scheme given Figure 2.
:math:`\kappa` is the angle between slip directions, :math:`\theta` is the angle between the two slip plane intersections with the grain boundary,
:math:`\psi` is the angle between slip plane normal directions, :math:`\gamma` is the angle between the direction of incoming slip and the plane normal of outgoing slip,
and :math:`\delta` is between the direction of outgoing slip and the plane normal of incoming slip.
:math:`n`, :math:`d` and :math:`l` are respectively the slip plane normals, slip directions and the lines of intersection of the slip plane and the grain boundary.
:math:`\vec b` is the Burgers vector of the slip plane and :math:`\vec b_r` is the residual Burgers vector of the residual dislocation at the grain boundary.
The subscripts :math:`in` and :math:`out` refer to the incoming and outgoing slip systems, respectively.
  
.. figure:: ./_pictures/Schemes_SlipTransmission/slip_transmission_scheme2.png
   :scale: 45 %
   :align: center
   
   *Figure 2 : Geometrical description of the slip transfer.*
   
* :math:`N` **factor from Livingston and Chalmers in 1957** [#LivingstonChalmers_1957]_ 

    .. math:: N = (n_{in} \cdot n_{out})*(d_{in} \cdot d_{out}) + (n_{in} \cdot d_{out})*(n_{out} \cdot d_{in})
        :label: n_factor
        
    .. math:: N = \cos(\psi)\cdot\cos(\kappa) + \cos(\gamma)\cdot\cos(\delta)
        :label: n_factor_angle
        
  Many authors referred to this criterion to analyze slip transmission [#HauserChamlers_1961]_, [#Davis_1966]_, [#HookHirth_1967_1]_, [#HookHirth_1967_2]_, [#Shen_1986]_, [#Shen_1988]_, [#Lee_1990_1]_, [#Lee_1990_2]_, [#Clark_1992]_ and [#Ueda_2002]_.
  Pond et al. proposed to compute this geometric criteria for hexagonal metals using Frank's method [#Pond_1986]_.      

  The |matlab| function used to calculate the N factor is: `N_factor.m <https://github.com/stabix/stabix/tree/master/slip_transfer/N_factor.m>`_
  
* :math:`LRB` **factor from Shen et al. in 1986** [#Shen_1986]_ and [#Shen_1988]_

    .. math:: LRB = (l_{in} \cdot l_{out})*(d_{in} \cdot d_{out})
        :label: LRB_factor
        
    .. math:: LRB = \cos(\theta)\cdot\cos(\kappa)
        :label: LRB_factor_angle
        
  The original notation of this :math:`LRB` factor is :math:`M`, but unfortunately this notation is often used for Taylor factor [#Bieler_2014]_ .
  Pond et al. proposed to compute this geometric criteria for hexagonal metals using Frank's method [#Pond_1986]_. 
  Recently, Spearot and Sangid have plotted this parameter in function of the misorientation of the bicrystal using atomistic simulations [#SpearotSangid_2014]_.
  
  [#Lee_1989]_, [#Lee_1990_1]_, [#Lee_1990_2]_, [#Clark_1992]_, [#Kehagias_1995]_, [#Kehagias_1996]_, [#Ashmawi_2001]_, [#Gemperle_2004]_, [#Gemperlova_2004]_ and [#Shi_2011]_ mentioned in their respective studies this geometrical parameter as a condition for slip transmission.
  
  The inclination of the grain boundary (:math:`\beta`) is required to evaluate this factor and the :math:`LRB` or :math:`M` factor should be maximized.
  
  The |matlab| function used to calculate the LRB factor is: `LRB_parameter.m <https://github.com/stabix/stabix/tree/master/slip_transfer/LRB_parameter.m>`_

* :math:`m'` **parameter from Luster and Morris in 1995** [#LusterMorris_1995]_
        
    .. math:: m' = (n_{in} \cdot n_{out})*(d_{in} \cdot d_{out}) 
        :label: m_prime
        
    .. math:: m' = \cos(\psi)\cdot\cos(\kappa)
        :label: m_prime_angle

  Many authors found that this :math:`m'` parameter which takes into account the degree of coplanarity of slip systems, is promising to predict slip transmission [#WangNgan_2004]_, [#WoNgan_2004]_, [#Britton_2009]_, [#Bieler_2009]_, [#Bieler_2014]_ and [#Guo_2014]_.
  Both :math:`m'` and :math:`LRB` can be easily assessed in computational experiments [#Bieler_2014]_ .
  This :math:`m'` factor should be maximized (1 means grain boundary is transparent and 0 means grain boundary is an impenetrable boundary).
  
  .. figure:: ./_pictures/Schemes_SlipTransmission/mprime_values.png
   :scale: 30 %
   :align: center
   
   *Figure 3 : Distribution of m' parameter in function of angles values.*
   
  .. figure:: ./_pictures/Schemes_SlipTransmission/mprime_values_vs_misorientation.png
   :scale: 65 %
   :align: center
   
   *Figure 4 : Distribution of m' parameter calculated for prismatic <a> slip systems in function of  misorientation angle.*
   
   A resistance factor of the grain boundary can be described by the following equation : 
   
    .. math:: GB_resfac = 1 - m'
        :label: resistance_factor
        
  This factor is equal to 0 (grain boundary transparent if the :math:`m'` parameter is equal to 1 (when grains perfectly aligned).
   
  The |matlab| function used to calculate the m' parameter is: `mprime.m <https://github.com/stabix/stabix/tree/master/slip_transfer/mprime.m>`_
     
* :math:`\vec b_r` **the residual Burgers vector** [#Marcinkowski_1970]_, [#Bollmann_1970]_, [#LimRaj_1985_1]_, [#LimRaj_1985_2]_, [#Clark_1989]_, [#Lee_1990_2]_ and [#Clark_1992]_.
    .. math:: \vec b_r = g_{in}\cdot\vec b_{in} - g_{out}\cdot\vec b_{out}
        :label: residual_burgers_vector
        
  The magnitude of this residual Burgers vector should be minimized.
  
  Shirokoff et al., Kehagias et al. and Kacher et al. used the residual Burgers vector as a criterion to analyse slip transmission in cp-Ti (HCP) [#Shirokoff_1993]_, [#Kehagias_1995]_, [#Kehagias_1996]_ and [#KacherRobertson_2014]_,
  Lagow et al. in Mo (BCC) [#Lagow_2001]_, Gemperle et al. and Gemperlova et al. in FeSi (BCC) [#Gemperle_2004]_ and [#Gemperlova_2004]_, Kacher et al. in 304 stainless steel (FCC) [#KacherRobertson_2012]_,
  and Jacques et al. for semiconductors [#Jacques_1990]_.
  
  Patriarca et al. demonstrated for BCC material the role of the residual Burgers vector in predicting slip transmission, by analysing strain field across GBs determined by digital image correlation [#Patriarca_2013]_. 
  
  Misra and Gibala used the residual Burgers vector to analyze slip across a fcc/bcc interphase boundary [#MisraGibala_1999]_.
  
  The |matlab| function used to calculate the residual Burgers vector is: `residual_Burgers_vector.m <https://github.com/stabix/stabix/tree/master/slip_transfer/residual_Burgers_vector.m>`_
 
* **The misorientation or disorientation** (:math:`\Delta g` or :math:`\Delta g_d`)  [#AustChen_1954]_, [#ClarkChalmers_1954]_ and [#WoNgan_2004]_

  It has been observed during first experiments of bicrystals deformation in 1954, that the yield stress and the rate of work hardening increased with the orientation difference between the crystals [#AustChen_1954]_ and [#ClarkChalmers_1954]_.
  
  Some authors demonstrated a strong correlation between misorientation between grains in a bicrystal and the grain boundary energy through crystal plasticity finite elements modelling and
  molecular dynamics simulations [#SuttonBalluffi_1995]_, [#Ma_2006]_, [#Li_2009]_, [#Bachurin_2010]_, [#Sangid_2011]_ and [#Sangid_2012]_. Some authors studied the stability of grain boundaries by the calculations
  of energy difference vs. misorientation angle through the hexagonal c-axis/a-axis [#Faraoun_2006]_.
  
  The misorientation and disorientation equations are given in the `crystallographic properties of a bicrystal. <http://stabix.readthedocs.org/en/latest/bicrystal_definition.html>`_
 
  The |matlab| function used to calculate the misorientation angle is: `misorientation.m <https://github.com/stabix/stabix/tree/master/crystallo/misorientation.m>`_

Stress Criteria
---------------
* **Schmid Factor** (:math:`m`) [#Reid_1973]_, [#Seal_2012]_ and [#Abuzaid_2012]_

  The Schmid's law can be expressed by the following equation:
  
    .. math:: \tau^{i} = \sigma : {S_0}^{i}
        :label: schmid_factor
        
    .. math:: {S_0}^{i} = d \otimes n
        :label: schmid_matrix
  
  :math:`\sigma` is an arbitrary stress state and :math:`\tau^{i}` the resolved shear stress.
  :math:`{S_0}^{i}` is the Schmid matrix defined by the dyadic product of the slip plane normals :math:`n` and the slip directions :math:`d` of the slip system :math:`i`.
  The Schmid factor, :math:`m`, is defined as the ratio of the resolved shear stress :math:`\tau^{i}` to a given uniaxial stress.
  
  Knowing the value of the highest Schmid factor of a given slip system for both grains in a bicrystal, Abuzaid et al. [#Abuzaid_2012]_ proposed the following criterion :

    .. math:: m_{GB} = m_{in} + m_{out}
        :label: schmid_factor_gb
        
  The subscripts :math:`GB`, :math:`in` and :math:`out` refer to the grain boundary, and the incoming and outgoing slip systems, respectively.
  This GB Schmid factor (:math:`m_{GB}`) factor should be maximized.
  
  The |matlab| function used to calculate the Schmid factor is: `resolved_shear_stress.m <https://github.com/stabix/stabix/tree/master/crystal_plasticity/resolved_shear_stress.m>`_
  
* **Generalized Schmid Factor** (:math:`GSF`) [#Reid_1973]_ and [#Bieler_2014]_

  The generalized Schmid factor, which describes the shear stress on a given slip system, can be
  computed from any stress tensor :math:`\sigma` based on the Frobenius norm of the tensor.

    .. math:: GSF = d \cdot g \sigma g^T \cdot n^T
        :label: generalized_schmid_factor
    
  :math:`n` and :math:`d` are respectively the slip plane normals and the slip directions of the slip system. The superscript :math:`T` is the transpose notation and :math:`g` is the orientation matrix for a given crystal.
        
  The |matlab| function used to calculate the generalized Schmid factor is: `generalized_schmid_factor.m <https://github.com/stabix/stabix/tree/master/crystal_plasticity/generalized_schmid_factor.m>`_
        
* **Resolved Shear Stress** (:math:`\tau`) [#Lee_1989]_, [#Lee_1990_1]_, [#Lee_1990_2]_, [#Clark_1992]_, [#Lagow_2001]_, [#Bieler_2009]_,  [#Dewald1_2007]_, [#Dewald2_2007]_ and [#Dewald3_2011]_
  
  The resolved shear stress :math:`\tau` acting on the outgoing slip system from the piled-up dislocations should be maximized. This criterion considers the local stress state.
  
  The resolved shear stress on the grain boundary should be minimized.
  
  For Shi and Zikry, the ratio of the resolved shear stress to the reference shear stress of the outgoing slip system (stress ratio) should be greater than a critical value (which is approximately 1) [#Shi_2011]_.
  
  For Li et al. and Gao et al. the resolved shear stress acting on the incoming dislocation on the slip plane must be larger than the critical penetration stress.
  From the energy point of view, only when the work by the external force on the incoming dislocation is greater than the summation of the GB energy
  and strain energy of GB dislocation debris, it is possible that the incoming dislocation can penetrate through the GB [#Li_2009]_ and [#Gao_2011]_.
  
  It is possible to assess the shear stress from the geometrical factor :math:`N` (Livingston and Chamlers) :
    <commit_msg>Update slip_transmission_toolbox.rst<commit_after>﻿Strain Transfer Across Grain Boundaries
=======================================

.. include:: includes.rst

The strain transfer across grain boundaries can be defined by the four following mechanisms (see Figure 1) [#LimRaj_1985_1]_, [#SuttonBalluffi_1995]_, [#Zaefferer_2003]_ and [#Priester_2013]_ :

 1) direct transmission with slip systems having the same Burgers vector, and the grain boundary is transparent to dislocations (no strengthening effect) (Figure 1-a);

 2) direct transmission, but slip systems have different Burgers vector (leaving a residual boundary dislocations) (Figure 1-b);

 3) indirect transmission, and slip systems have different Burgers vector (leaving a residual boundary dislocations) (Figure 1-c);

 4) no transmission and the grain boundary acts as an impenetrable boundary, which implies stress accumulations, localized rotations, pile-up of dislocations... (Figure 1-d).

.. figure:: ./_pictures/Schemes_SlipTransmission/slip_transfer.png
   :scale: 40 %
   :align: center
   
   *Figure 1 : Possible strain transfer across grain boundaries (GB) from Sutton and Balluffi.*
   
Several authors proposed slip transfer parameters from modellings or experiments for the last 60 years.
A non-exhaustive list of those criteria is given in the next part of this work, including geometrical parameter, stress and energetic functions, and recent combinations of the previous parameters.

Geometrical Criteria
--------------------
Based on numerous investigations of dislocation-grain boundary interactions, quantitative geometrical expressions describing the slip transmission mechanisms have been developed. 
A non-exhaustive list of geometrical criteria is detailed subsequently.
The geometry of the slip transfer event is most of the time described by the scheme given Figure 2.
:math:`\kappa` is the angle between slip directions, :math:`\theta` is the angle between the two slip plane intersections with the grain boundary,
:math:`\psi` is the angle between slip plane normal directions, :math:`\gamma` is the angle between the direction of incoming slip and the plane normal of outgoing slip,
and :math:`\delta` is between the direction of outgoing slip and the plane normal of incoming slip.
:math:`n`, :math:`d` and :math:`l` are respectively the slip plane normals, slip directions and the lines of intersection of the slip plane and the grain boundary.
:math:`\vec b` is the Burgers vector of the slip plane and :math:`\vec b_\text r` is the residual Burgers vector of the residual dislocation at the grain boundary.
The subscripts 'in' and 'out' refer to the incoming and outgoing slip systems, respectively.
  
.. figure:: ./_pictures/Schemes_SlipTransmission/slip_transmission_scheme2.png
   :scale: 45 %
   :align: center
   
   *Figure 2 : Geometrical description of the slip transfer.*
   
* :math:`N` **factor from Livingston and Chalmers in 1957** [#LivingstonChalmers_1957]_ 

    .. math:: N = (\vec n_\text{in} \cdot \vec n_\text{out})(\vec d_\text{in} \cdot \vec d_\text{out}) + (\vec n_\text{in} \cdot \vec d_\text{out})(\vec n_\text{out} \cdot \vec d_\text{in})
        :label: n_factor
        
    .. math:: N = \cos\psi\cos\kappa + \cos\gamma\cos\delta
        :label: n_factor_angle
        
  Many authors referred to this criterion to analyze slip transmission [#HauserChamlers_1961]_, [#Davis_1966]_, [#HookHirth_1967_1]_, [#HookHirth_1967_2]_, [#Shen_1986]_, [#Shen_1988]_, [#Lee_1990_1]_, [#Lee_1990_2]_, [#Clark_1992]_ and [#Ueda_2002]_.
  Pond et al. proposed to compute this geometric criteria for hexagonal metals using Frank's method [#Pond_1986]_.      

  The |matlab| function used to calculate the N factor is: `N_factor.m <https://github.com/stabix/stabix/tree/master/slip_transfer/N_factor.m>`_
  
* :math:`LRB` **factor from Shen et al. in 1986** [#Shen_1986]_ and [#Shen_1988]_

    .. math:: LRB = (\vec l_\text{in} \cdot \vec l_\text{out})(\vec d_\text{in} \cdot \vec d_\text{out})
        :label: LRB_factor
        
    .. math:: LRB = \cos\theta\cos\kappa
        :label: LRB_factor_angle
        
  The original notation of this :math:`LRB` factor is :math:`M`, but unfortunately this notation is often used for the Taylor factor [#Bieler_2014]_.
  Pond et al. proposed to compute this geometric criteria for hexagonal metals using Frank's method [#Pond_1986]_. 
  Recently, Spearot and Sangid have plotted this parameter as a function of the misorientation of the bicrystal using atomistic simulations [#SpearotSangid_2014]_.
  
  [#Lee_1989]_, [#Lee_1990_1]_, [#Lee_1990_2]_, [#Clark_1992]_, [#Kehagias_1995]_, [#Kehagias_1996]_, [#Ashmawi_2001]_, [#Gemperle_2004]_, [#Gemperlova_2004]_ and [#Shi_2011]_ mentioned in their respective studies this geometrical parameter as a condition for slip transmission.
  
  The inclination of the grain boundary (:math:`\beta`) is required to evaluate this factor and the :math:`LRB` or :math:`M` factor should be maximized.
  
  The |matlab| function used to calculate the LRB factor is: `LRB_parameter.m <https://github.com/stabix/stabix/tree/master/slip_transfer/LRB_parameter.m>`_

* :math:`m'` **parameter from Luster and Morris in 1995** [#LusterMorris_1995]_
        
    .. math:: m' = (\vec n_\text{in} \cdot \vec n_\text{out})(\vec d_\text{in} \cdot \vec d_\text{out}) 
        :label: m_prime
        
    .. math:: m' = \cos\psi\cos\kappa
        :label: m_prime_angle

  Many authors found that this :math:`m'` parameter, which takes into account the degree of coplanarity of slip systems, is promising to predict slip transmission [#WangNgan_2004]_, [#WoNgan_2004]_, [#Britton_2009]_, [#Bieler_2009]_, [#Bieler_2014]_ and [#Guo_2014]_.
  Both :math:`m'` and :math:`LRB` can be easily assessed in computational experiments [#Bieler_2014]_.
  This :math:`m'` factor should be maximized (1 means grain boundary is transparent and 0 means grain boundary is an impenetrable boundary).
  
  .. figure:: ./_pictures/Schemes_SlipTransmission/mprime_values.png
   :scale: 30 %
   :align: center
   
   *Figure 3 : Distribution of m' parameter in function of angles values.*
   
  .. figure:: ./_pictures/Schemes_SlipTransmission/mprime_values_vs_misorientation.png
   :scale: 65 %
   :align: center
   
   *Figure 4 : Distribution of m' parameter calculated for prismatic <a> slip systems in function of  misorientation angle.*
   
   A resistance factor of the grain boundary can be described by the following equation : 
   
    .. math:: GB_\text{resfac} = 1 - m'
        :label: resistance_factor
        
  This factor is equal to 0 (grain boundary transparent if the :math:`m'` parameter is equal to 1 (when slip perfectly aligns).
   
  The |matlab| function used to calculate the m' parameter is: `mprime.m <https://github.com/stabix/stabix/tree/master/slip_transfer/mprime.m>`_
     
* :math:`\vec b_\text r` **the residual Burgers vector** [#Marcinkowski_1970]_, [#Bollmann_1970]_, [#LimRaj_1985_1]_, [#LimRaj_1985_2]_, [#Clark_1989]_, [#Lee_1990_2]_ and [#Clark_1992]_.
    .. math:: \vec b_r = \vec g_\text{in}\cdot\vec b_\text{in} - \vec g_\text{out}\cdot\vec b_\text{out}
        :label: residual_burgers_vector
        
  The magnitude of this residual Burgers vector should be minimized.
  
  Shirokoff et al., Kehagias et al., and Kacher et al. used the residual Burgers vector as a criterion to analyse slip transmission in cp-Ti (hex) [#Shirokoff_1993]_, [#Kehagias_1995]_, [#Kehagias_1996]_ and [#KacherRobertson_2014]_,
  Lagow et al. in Mo (bcc) [#Lagow_2001]_, Gemperle et al. and Gemperlova et al. in FeSi (bcc) [#Gemperle_2004]_ and [#Gemperlova_2004]_, Kacher et al. in 304 stainless steel (fcc) [#KacherRobertson_2012]_,
  and Jacques et al. for semiconductors [#Jacques_1990]_.
  
  Patriarca et al. demonstrated for BCC material the role of the residual Burgers vector in predicting slip transmission, by analysing strain field across GBs determined by digital image correlation [#Patriarca_2013]_. 
  
  Misra and Gibala used the residual Burgers vector to analyze slip across a fcc/bcc interphase boundary [#MisraGibala_1999]_.
  
  The |matlab| function used to calculate the residual Burgers vector is: `residual_Burgers_vector.m <https://github.com/stabix/stabix/tree/master/slip_transfer/residual_Burgers_vector.m>`_
 
* **The misorientation or disorientation** (:math:`\Delta g` or :math:`\Delta g_\text d`)  [#AustChen_1954]_, [#ClarkChalmers_1954]_ and [#WoNgan_2004]_

  It has been observed during first experiments of bicrystals deformation in 1954, that the yield stress and the rate of work hardening increased with the orientation difference between the crystals [#AustChen_1954]_ and [#ClarkChalmers_1954]_.
  
  Some authors demonstrated a strong correlation between misorientation between grains in a bicrystal and the grain boundary energy through crystal plasticity finite elements modelling and
  molecular dynamics simulations [#SuttonBalluffi_1995]_, [#Ma_2006]_, [#Li_2009]_, [#Bachurin_2010]_, [#Sangid_2011]_ and [#Sangid_2012]_. Some authors studied the stability of grain boundaries by the calculations
  of energy difference vs. misorientation angle through the hexagonal c-axis/a-axis [#Faraoun_2006]_.
  
  The misorientation and disorientation equations are given in the `crystallographic properties of a bicrystal. <http://stabix.readthedocs.org/en/latest/bicrystal_definition.html>`_
 
  The |matlab| function used to calculate the misorientation angle is: `misorientation.m <https://github.com/stabix/stabix/tree/master/crystallo/misorientation.m>`_

Stress Criteria
---------------
* **Schmid Factor** (:math:`m`) [#Reid_1973]_, [#Seal_2012]_ and [#Abuzaid_2012]_

  The Schmid's law can be expressed by the following equation:
  
    .. math:: \tau^i = \sigma : {S_0}^i
        :label: schmid_factor
        
    .. math:: {S_0}^i = \vec d^i \otimes \vec n^i
        :label: schmid_matrix
  
  :math:`\sigma` is an arbitrary stress state and :math:`\tau^i` the resolved shear stress on slip system :math:`i`.
  :math:`{S_0}^i` is the Schmid matrix defined by the dyadic product of the slip plane normals :math:`\vec n` and the slip directions :math:`\vec d` of the slip system :math:`i`.
  The Schmid factor, :math:`m`, is defined as the ratio of the resolved shear stress :math:`\tau^{i}` to a given uniaxial stress.
  
  Knowing the value of the highest Schmid factor of a given slip system for both grains in a bicrystal, Abuzaid et al. [#Abuzaid_2012]_ proposed the following criterion :

    .. math:: m_\text{GB} = m_\text{in} + m_\text{out}
        :label: schmid_factor_gb
        
  The subscripts 'GB', 'in', and 'out' refer to the grain boundary, and the incoming and outgoing slip systems, respectively.
  This GB Schmid factor (:math:`m_\text{GB}`) factor should be maximized.
  
  The |matlab| function used to calculate the Schmid factor is: `resolved_shear_stress.m <https://github.com/stabix/stabix/tree/master/crystal_plasticity/resolved_shear_stress.m>`_
  
* **Generalized Schmid Factor** (:math:`GSF`) [#Reid_1973]_ and [#Bieler_2014]_

  The generalized Schmid factor, which describes the shear stress on a given slip system, can be
  computed from any stress tensor :math:`\sigma` based on the Frobenius norm of the tensor.

    .. math:: GSF = \vec d \cdot g \sigma g^T \cdot \vec n^T
        :label: generalized_schmid_factor
    
  :math:`\vec n` and :math:`\vec d` are respectively the slip plane normals and the slip directions of the slip system. The superscript :math:`T` is the transpose notation and :math:`g` is the orientation matrix for a given crystal.
        
  The |matlab| function used to calculate the generalized Schmid factor is: `generalized_schmid_factor.m <https://github.com/stabix/stabix/tree/master/crystal_plasticity/generalized_schmid_factor.m>`_
        
* **Resolved Shear Stress** (:math:`\tau`) [#Lee_1989]_, [#Lee_1990_1]_, [#Lee_1990_2]_, [#Clark_1992]_, [#Lagow_2001]_, [#Bieler_2009]_,  [#Dewald1_2007]_, [#Dewald2_2007]_ and [#Dewald3_2011]_
  
  The resolved shear stress :math:`\tau` acting on the outgoing slip system from the piled-up dislocations should be maximized. This criterion considers the local stress state.
  
  The resolved shear stress on the grain boundary should be minimized.
  
  For Shi and Zikry, the ratio of the resolved shear stress to the reference shear stress of the outgoing slip system (stress ratio) should be greater than a critical value (which is approximately 1) [#Shi_2011]_.
  
  For Li et al. and Gao et al. the resolved shear stress acting on the incoming dislocation on the slip plane must be larger than the critical penetration stress.
  From the energy point of view, only when the work by the external force on the incoming dislocation is greater than the summation of the GB energy
  and strain energy of GB dislocation debris, it is possible that the incoming dislocation can penetrate through the GB [#Li_2009]_ and [#Gao_2011]_.
  
  It is possible to assess the shear stress from the geometrical factor :math:`N` (Livingston and Chamlers) :
    
{'fraction_non_alphanumeric': 0.10043550314101823, 'fraction_numerical': 0.03206536401125371, 'mean_word_length': 4.284725050916497, 'pattern_counts': {'":': 0, '<': 21, '<?xml version=': 0, '>': 21, 'https://': 14, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'dir': 'git-commits-cleaned', 'id': '5303902', 'n_tokens_mistral': 8773, 'n_tokens_neox': 7496, 'n_words': 3012}