text: # Finite Strain Crystal Plasticity Slip Rate Residual
  
  !syntax description /Materials/FiniteStrainCPSlipRateRes
  
  ## Description
  
  !alert warning
  The `FiniteStrainCPSlipRateRes` model is maintained but not actively developed.
  Use of the [FiniteStrainUObasedCP](/FiniteStrainUObasedCP.md) crystal plasticity
  system is recommend instead.
  
  In `FiniteStrainCPSlipRateRes` the convergence of the crystal plasticity strain
  and stress response is determined with respect to the slip rate on each slip
  system of the crystal plasticity model. In contrast, in the material
  [FiniteStrainCrystalPLasticity](/FiniteStrainCrystalPlasticity.md) determines
  the crystal strain and stress response with response to the residual calculated
  by the second Piola-Kirchoff stress increment.
  
  ### Constitutive Model Definition
  
  Constitutive models are used to calculate the plastic slip rate.
  In this crystal plasticity material the slip rate is modeled as a power law:
  \begin{equation}
    \label{eqn:powerLawSlipRate}
    \dot{\gamma}^{\alpha} = \dot{\gamma}_o \left| \frac{\tau^{\alpha}}{g^{\alpha}} \right|^{1/m} sign \left( \tau^{\alpha} \right)
  \end{equation}
  where $\dot{\gamma}_o$ is a reference slip rate, $\tau^{\alpha}$ is the applied
  shear stress on each slip system $\alpha$, $g^{\alpha}$ is the slip system
  strength, or resistance to slip, and $m$ is the strain rate sensitivity
  exponent. The strength of each slip system is solved with an iterative process
  as a function of the slip increment
  \begin{equation}
    \label{eqn:slipStrengthEvolution}
    g^{\alpha} =  g_o + \Delta \gamma^{\alpha} q^{\alpha \beta} h _o \left| 1 - \frac{g^{\alpha}}{g_{sat}}  \right|^a sign \left( 1 - \frac{g^{\alpha}}{g_{sat}} \right)
  \end{equation}
  where $q^{\alpha \beta}$ is a hardening coefficient matrix that accounts for the
  different in self and latent hardening, [eqn:selfLatentQMatrix], $h_o$
  is an initial hardening term, $g_{sat}$ is a constant saturated hardening value,
  and $a$ is the hardening exponent [!citep](kalidindi1992). The self and latent
  hardening of the crystal is defined for an FCC system as
  \begin{equation}
    \label{eqn:selfLatentQMatrix}
    q^{\alpha \beta} = \begin{Bmatrix}
                         1.0 & q   & q   & q  \\
                         q   & 1.0 & q   & q  \\
                         q   & q   & 1.0 & q  \\
                         q   & q   & q   & 1.0
                       \end{Bmatrix}
  \end{equation}
  where $q$ is a constant value of latent hardening among non-coplanar slip
  systems. In [eqn:selfLatentQMatrix] the slip systems which share the
  same slip plane normal (e.g. $[\bar{1}11]$) are coplanar and grouped together
  with a latent hardening rate of unity [!citep](kalidindi1992). Each matrix entry in
  [eqn:selfLatentQMatrix] represents the interaction among two different
  coplanar slip system groups, that is a total of six slip systems
  [!citep](kalidindi1992).
  
  
  ## Units Assumed in the Crystal Plasticity Materials
  
  The simulation domain for crystal plasticity models is resolved on the order of
  individual crystal grains, and, as such, the mesh size is small. Although MOOSE
  itself is dimension agnostic, the crystal plasticity models are implemented in
  the +mm-MPa-s unit system+. This dimension system choice impacts the
  input files in the following manner:
  
  - Mesh dimensions should be constructed in mm
  - Elastic constant values (e.g. Young's modulus and shear modulus) are entered in MPa
  - Initial slip system strength values are entered in MPa
  - Simulation times are given in s
  - Strain rates and displacement loading rates are given in 1/s and mm/s, respectively
  
  In physically based models, which maybe based on this class, initial densities
  of crystal defects (e.g. dislocations, point defects) should be given in
  1/mm$^2$ or 1/mm$^3$
  
  !syntax parameters /Materials/FiniteStrainCPSlipRateRes
  
  !syntax inputs /Materials/FiniteStrainCPSlipRateRes
  
  !syntax children /Materials/FiniteStrainCPSlipRateRes
  
  !bibtex bibliography
  