DAMASK_EICMD/code/constitutive_j2.f90

! Copyright 2011-13 Max-Planck-Institut für Eisenforschung GmbH
!
! This file is part of DAMASK,
! the Düsseldorf Advanced MAterial Simulation Kit.
!
! DAMASK is free software: you can redistribute it and/or modify
! it under the terms of the GNU General Public License as published by
! the Free Software Foundation, either version 3 of the License, or
! (at your option) any later version.
!
! DAMASK is distributed in the hope that it will be useful,
! but WITHOUT ANY WARRANTY; without even the implied warranty of
! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
! GNU General Public License for more details.
!
! You should have received a copy of the GNU General Public License
! along with DAMASK. If not, see <http://www.gnu.org/licenses/>.
!
!--------------------------------------------------------------------------------------------------
! $Id$
!--------------------------------------------------------------------------------------------------
!> @author Franz Roters, Max-Planck-Institut für Eisenforschung GmbH
!> @author Philip Eisenlohr, Max-Planck-Institut für Eisenforschung GmbH
!> @brief material subroutine for isotropic (J2) plasticity
!> @details Isotropic (J2) Plasticity which resembles the phenopowerlaw plasticity without
!! resolving the stress on the slip systems. Will give the response of phenopowerlaw for an
!! untextured polycrystal
!--------------------------------------------------------------------------------------------------
module constitutive_j2
 use prec, only: &
   pReal,&
   pInt
 
 implicit none
 private
 integer(pInt),                       dimension(:),     allocatable,         public, protected :: &
   constitutive_j2_sizeDotState, &                                                                  !< number of dotStates
   constitutive_j2_sizeState, &                                                                     !< total number of microstructural variables
   constitutive_j2_sizePostResults                                                                  !< cumulative size of post results
   
 integer(pInt),                       dimension(:,:),   allocatable, target, public :: &
   constitutive_j2_sizePostResult                                                                   !< size of each post result output
   
 character(len=64),                   dimension(:,:),   allocatable, target, public :: &
   constitutive_j2_output                                                                           !< name of each post result output
 
 integer(pInt),                       dimension(:),     allocatable,         private :: &
   constitutive_j2_Noutput                                                                          !< number of outputs per instance
   
 real(pReal),                         dimension(:),     allocatable,         private :: &
   constitutive_j2_fTaylor, &                                                                       !< Taylor factor
   constitutive_j2_tau0, &                                                                          !< initial plastic stress
   constitutive_j2_gdot0, &                                                                         !< reference velocity
   constitutive_j2_n, &                                                                             !< Visco-plastic parameter
!--------------------------------------------------------------------------------------------------
! h0 as function of h0 = A + B log (gammadot) 
   constitutive_j2_h0, &
   constitutive_j2_h0_slopeLnRate, &
   constitutive_j2_tausat, &                                                                        !< final plastic stress
   constitutive_j2_a, &
   constitutive_j2_aTolResistance, &
!--------------------------------------------------------------------------------------------------
! tausat += (asinh((gammadot / SinhFitA)**(1 / SinhFitD)))**(1 / SinhFitC) / (SinhFitB * (gammadot / gammadot0)**(1/n))
   constitutive_j2_tausat_SinhFitA, &                                                               !< fitting parameter for normalized strain rate vs. stress function
   constitutive_j2_tausat_SinhFitB, &                                                               !< fitting parameter for normalized strain rate vs. stress function
   constitutive_j2_tausat_SinhFitC, &                                                               !< fitting parameter for normalized strain rate vs. stress function
   constitutive_j2_tausat_SinhFitD                                                                  !< fitting parameter for normalized strain rate vs. stress function

 enum, bind(c) 
   enumerator :: undefined_ID, &
                 flowstress_ID, &
                 strainrate_ID
 end enum
 integer(kind(undefined_ID)),         dimension(:,:),   allocatable,          private :: & 
   constitutive_j2_outputID                                                                         !< ID of each post result output

 public  :: &
   constitutive_j2_init, &
   constitutive_j2_stateInit, &
   constitutive_j2_aTolState, &
   constitutive_j2_LpAndItsTangent, &
   constitutive_j2_dotState, &
   constitutive_j2_postResults

contains


!--------------------------------------------------------------------------------------------------
!> @brief module initialization
!> @details reads in material parameters, allocates arrays, and does sanity checks
!--------------------------------------------------------------------------------------------------
subroutine constitutive_j2_init(fileUnit)
 use, intrinsic :: iso_fortran_env                                                                  ! to get compiler_version and compiler_options (at least for gfortran 4.6 at the moment)
 use debug, only: &
   debug_level, &
   debug_constitutive, &
   debug_levelBasic
 use math, only: &
   math_Mandel3333to66, &
   math_Voigt66to3333
 use IO, only: &
   IO_read, &
   IO_lc, &
   IO_getTag, &
   IO_isBlank, &
   IO_stringPos, &
   IO_stringValue, &
   IO_floatValue, &
   IO_error, &
   IO_timeStamp, &
   IO_EOF
 use material, only: &
   homogenization_maxNgrains, &
   phase_plasticity, &
   phase_plasticityInstance, &
   phase_Noutput, &
   PLASTICITY_J2_label, &
   PLASTICITY_J2_ID, &
   MATERIAL_partPhase
 use lattice  

 implicit none
 integer(pInt), intent(in) :: fileUnit
 
 integer(pInt), parameter :: MAXNCHUNKS = 7_pInt
 
 integer(pInt), dimension(1_pInt+2_pInt*MAXNCHUNKS) :: positions
 integer(pInt) :: phase, maxNinstance, instance,o, mySize
 character(len=65536) :: &
   tag  = '', &
   line = ''

 write(6,'(/,a)')   ' <<<+-  constitutive_'//PLASTICITY_J2_label//' init  -+>>>'
 write(6,'(a)')     ' $Id$'
 write(6,'(a15,a)') ' Current time: ',IO_timeStamp()
#include "compilation_info.f90"
 
 maxNinstance = int(count(phase_plasticity == PLASTICITY_J2_ID),pInt)
 if (maxNinstance == 0_pInt) return

 if (iand(debug_level(debug_constitutive),debug_levelBasic) /= 0_pInt) &
   write(6,'(a16,1x,i5,/)') '# instances:',maxNinstance
 
 allocate(constitutive_j2_sizeDotState(maxNinstance),                         source=1_pInt)
 allocate(constitutive_j2_sizeState(maxNinstance),                            source=1_pInt)
 allocate(constitutive_j2_sizePostResults(maxNinstance),                      source=0_pInt)
 allocate(constitutive_j2_sizePostResult(maxval(phase_Noutput), maxNinstance),source=0_pInt)
 allocate(constitutive_j2_output(maxval(phase_Noutput), maxNinstance))
          constitutive_j2_output = ''
 allocate(constitutive_j2_outputID(maxval(phase_Noutput),maxNinstance),       source=undefined_ID)
 allocate(constitutive_j2_Noutput(maxNinstance),                              source=0_pInt)
 allocate(constitutive_j2_fTaylor(maxNinstance),                              source=0.0_pReal)
 allocate(constitutive_j2_tau0(maxNinstance),                                 source=0.0_pReal)
 allocate(constitutive_j2_gdot0(maxNinstance),                                source=0.0_pReal)
 allocate(constitutive_j2_n(maxNinstance),                                    source=0.0_pReal)
 allocate(constitutive_j2_h0(maxNinstance),                                   source=0.0_pReal)
 allocate(constitutive_j2_h0_slopeLnRate(maxNinstance),                       source=0.0_pReal)
 allocate(constitutive_j2_tausat(maxNinstance),                               source=0.0_pReal)
 allocate(constitutive_j2_a(maxNinstance),                                    source=0.0_pReal)
 allocate(constitutive_j2_aTolResistance(maxNinstance),                       source=0.0_pReal)
 allocate(constitutive_j2_tausat_SinhFitA(maxNinstance),                      source=0.0_pReal)
 allocate(constitutive_j2_tausat_SinhFitB(maxNinstance),                      source=0.0_pReal)
 allocate(constitutive_j2_tausat_SinhFitC(maxNinstance),                      source=0.0_pReal)
 allocate(constitutive_j2_tausat_SinhFitD(maxNinstance),                      source=0.0_pReal)
 
 rewind(fileUnit)
 phase = 0_pInt
 do while (trim(line) /= IO_EOF .and. IO_lc(IO_getTag(line,'<','>')) /= material_partPhase)         ! wind forward to <phase>
   line = IO_read(fileUnit)
 enddo
 
 parsingFile: do while (trim(line) /= IO_EOF)                                                        ! read through sections of phase part
   line = IO_read(fileUnit)
   if (IO_isBlank(line)) cycle                                                                      ! skip empty lines
   if (IO_getTag(line,'<','>') /= '') then                                                          ! stop at next part
     line = IO_read(fileUnit, .true.)                                                               ! reset IO_read
     exit                                                                                           
   endif
   if (IO_getTag(line,'[',']') /= '') then                                                          ! next section
     phase = phase + 1_pInt                                                                         ! advance section counter
     if (phase_plasticity(phase) == PLASTICITY_J2_ID) then
       instance = phase_plasticityInstance(phase)
     endif
     cycle                                                                                          ! skip to next line
   endif
   if (phase > 0_pInt ) then; if (phase_plasticity(phase) == PLASTICITY_J2_ID) then                 ! one of my sections. Do not short-circuit here (.and. between if-statements), it's not safe in Fortran
     instance = phase_plasticityInstance(phase)                                                     ! which instance of my plasticity is present phase
     positions = IO_stringPos(line,MAXNCHUNKS) 
     tag = IO_lc(IO_stringValue(line,positions,1_pInt))                                             ! extract key
     select case(tag)
       case ('plasticity','elasticity','lattice_structure', &
             'covera_ratio','c/a_ratio','c/a', &
             'c11','c12','c13','c22','c23','c33','c44','c55','c66')
       case ('(output)')
         constitutive_j2_Noutput(instance) = constitutive_j2_Noutput(instance) + 1_pInt
         constitutive_j2_output(constitutive_j2_Noutput(instance),instance) = &
                                            IO_lc(IO_stringValue(line,positions,2_pInt))
         select case(IO_lc(IO_stringValue(line,positions,2_pInt)))
           case ('flowstress')
             constitutive_j2_outputID(constitutive_j2_Noutput(instance),instance) = flowstress_ID
           case ('strainrate')
             constitutive_j2_outputID(constitutive_j2_Noutput(instance),instance) = strainrate_ID
           case default
             call IO_error(105_pInt,ext_msg=IO_stringValue(line,positions,2_pInt)//' ('//PLASTICITY_J2_label//')')
         end select
       case ('tau0')
         constitutive_j2_tau0(instance)         = IO_floatValue(line,positions,2_pInt)
         if (constitutive_j2_tau0(instance) < 0.0_pReal) &
           call IO_error(211_pInt,ext_msg=trim(tag)//' ('//PLASTICITY_J2_label//')')
       case ('gdot0')
         constitutive_j2_gdot0(instance)        = IO_floatValue(line,positions,2_pInt)
         if (constitutive_j2_gdot0(instance) <= 0.0_pReal) &
           call IO_error(211_pInt,ext_msg=trim(tag)//' ('//PLASTICITY_J2_label//')')
       case ('n')
         constitutive_j2_n(instance)            = IO_floatValue(line,positions,2_pInt)
         if (constitutive_j2_n(instance) <= 0.0_pReal) &
           call IO_error(211_pInt,ext_msg=trim(tag)//' ('//PLASTICITY_J2_label//')')
       case ('h0')
         constitutive_j2_h0(instance)           = IO_floatValue(line,positions,2_pInt)
       case ('h0_slope','slopelnrate')
         constitutive_j2_h0_slopeLnRate(instance)  = IO_floatValue(line,positions,2_pInt)
       case ('tausat')
         constitutive_j2_tausat(instance)          = IO_floatValue(line,positions,2_pInt)
         if (constitutive_j2_tausat(instance) <= 0.0_pReal) &
           call IO_error(211_pInt,ext_msg=trim(tag)//' ('//PLASTICITY_J2_label//')')
       case ('tausat_sinhfita')
         constitutive_j2_tausat_SinhFitA(instance) = IO_floatValue(line,positions,2_pInt)
       case ('tausat_sinhfitb')
         constitutive_j2_tausat_SinhFitB(instance) = IO_floatValue(line,positions,2_pInt)
       case ('tausat_sinhfitc')
         constitutive_j2_tausat_SinhFitC(instance) = IO_floatValue(line,positions,2_pInt)
       case ('tausat_sinhfitd')
         constitutive_j2_tausat_SinhFitD(instance) = IO_floatValue(line,positions,2_pInt)
       case ('a', 'w0')
         constitutive_j2_a(instance)               = IO_floatValue(line,positions,2_pInt)
         if (constitutive_j2_a(instance) <= 0.0_pReal) &
           call IO_error(211_pInt,ext_msg=trim(tag)//' ('//PLASTICITY_J2_label//')')
       case ('taylorfactor')
         constitutive_j2_fTaylor(instance)         = IO_floatValue(line,positions,2_pInt)
         if (constitutive_j2_fTaylor(instance) <= 0.0_pReal) &
           call IO_error(211_pInt,ext_msg=trim(tag)//' ('//PLASTICITY_J2_label//')')
       case ('atol_resistance')
         constitutive_j2_aTolResistance(instance)  = IO_floatValue(line,positions,2_pInt)
         if (constitutive_j2_aTolResistance(instance) <= 0.0_pReal) &
           call IO_error(211_pInt,ext_msg=trim(tag)//' ('//PLASTICITY_J2_label//')')
       case default
         call IO_error(210_pInt,ext_msg=trim(tag)//' ('//PLASTICITY_J2_label//')')
     end select
   endif; endif
 enddo parsingFile

 instancesLoop: do instance = 1_pInt,maxNinstance
   outputsLoop: do o = 1_pInt,constitutive_j2_Noutput(instance)
     select case(constitutive_j2_outputID(o,instance))
       case(flowstress_ID,strainrate_ID)
         mySize = 1_pInt
       case default
     end select
  
     if (mySize > 0_pInt) then                                                                      ! any meaningful output found
       constitutive_j2_sizePostResult(o,instance) = mySize
       constitutive_j2_sizePostResults(instance) = &
       constitutive_j2_sizePostResults(instance) + mySize
     endif
   enddo outputsLoop
 enddo instancesLoop

end subroutine constitutive_j2_init


!--------------------------------------------------------------------------------------------------
!> @brief sets the initial microstructural state for a given instance of this plasticity
!> @details initial microstructural state is set to the value specified by tau0
!--------------------------------------------------------------------------------------------------
pure function constitutive_j2_stateInit(instance)
  
 implicit none
 real(pReal),   dimension(1)            :: constitutive_j2_stateInit
 integer(pInt),              intent(in) :: instance                                                 !< number specifying the instance of the plasticity
 
 constitutive_j2_stateInit = constitutive_j2_tau0(instance)

end function constitutive_j2_stateInit


!--------------------------------------------------------------------------------------------------
!> @brief sets the relevant state values for a given instance of this plasticity
!--------------------------------------------------------------------------------------------------
pure function constitutive_j2_aTolState(instance)

 implicit none
 real(pReal),   dimension(1)            :: constitutive_j2_aTolState
 integer(pInt),              intent(in) :: instance                                                 !< number specifying the instance of the plasticity


 constitutive_j2_aTolState = constitutive_j2_aTolResistance(instance)

end function constitutive_j2_aTolState


!--------------------------------------------------------------------------------------------------
!> @brief calculates plastic velocity gradient and its tangent
!--------------------------------------------------------------------------------------------------
pure subroutine constitutive_j2_LpAndItsTangent(Lp,dLp_dTstar99,Tstar_v,state,ipc,ip,el)
 use prec, only: &
   p_vec
 use math, only: &
   math_mul6x6, &
   math_Mandel6to33, &
   math_Plain3333to99, &
   math_deviatoric33, &
   math_mul33xx33
 use mesh, only: &
   mesh_NcpElems, &
   mesh_maxNips
 use material, only: &
   homogenization_maxNgrains, &
   material_phase, &
   phase_plasticityInstance

 implicit none
 real(pReal), dimension(3,3), intent(out) :: &
   Lp                                                                                               !< plastic velocity gradient
 real(pReal), dimension(9,9), intent(out) :: &
   dLp_dTstar99                                                                                     !< derivative of Lp with respect to 2nd Piola Kirchhoff stress

 real(pReal), dimension(6),   intent(in) :: &
   Tstar_v                                                                                          !< 2nd Piola Kirchhoff stress tensor in Mandel notation
 integer(pInt),               intent(in) :: &
   ipc, &                                                                                           !< component-ID of integration point
   ip, &                                                                                            !< integration point
   el                                                                                               !< element
 type(p_vec),                 intent(in) :: &
   state                                                                                            !< microstructure state

 real(pReal), dimension(3,3) :: &
   Tstar_dev_33                                                                                     !< deviatoric part of the 2nd Piola Kirchhoff stress tensor as 2nd order tensor
 real(pReal), dimension(3,3,3,3) :: &
   dLp_dTstar_3333                                                                                  !< derivative of Lp with respect to Tstar as 4th order tensor
 real(pReal) :: &
   gamma_dot, &                                                                                     !< strainrate
   norm_Tstar_dev, &                                                                                !< euclidean norm of Tstar_dev
   squarenorm_Tstar_dev                                                                             !< square of the euclidean norm of Tstar_dev
 integer(pInt) :: &
   instance, &
   k, l, m, n

 instance = phase_plasticityInstance(material_phase(ipc,ip,el))
 Tstar_dev_33 = math_deviatoric33(math_Mandel6to33(Tstar_v))                                        ! deviatoric part of 2nd Piola-Kirchhoff stress
 squarenorm_Tstar_dev = math_mul33xx33(Tstar_dev_33,Tstar_dev_33)
 norm_Tstar_dev = sqrt(squarenorm_Tstar_dev) 

 if (norm_Tstar_dev <= 0.0_pReal) then                                                              ! Tstar == 0 --> both Lp and dLp_dTstar are zero
   Lp = 0.0_pReal
   dLp_dTstar99 = 0.0_pReal
 else
   gamma_dot = constitutive_j2_gdot0(instance) &
             * (sqrt(1.5_pReal) * norm_Tstar_dev / constitutive_j2_fTaylor(instance) / state%p(1)) &
                                                  **constitutive_j2_n(instance)

   Lp = Tstar_dev_33/norm_Tstar_dev * gamma_dot/constitutive_j2_fTaylor(instance)

!--------------------------------------------------------------------------------------------------
! Calculation of the tangent of Lp
   forall (k=1_pInt:3_pInt,l=1_pInt:3_pInt,m=1_pInt:3_pInt,n=1_pInt:3_pInt) &
     dLp_dTstar_3333(k,l,m,n) = (constitutive_j2_n(instance)-1.0_pReal) * &
                                      Tstar_dev_33(k,l)*Tstar_dev_33(m,n) / squarenorm_Tstar_dev
   forall (k=1_pInt:3_pInt,l=1_pInt:3_pInt) &
     dLp_dTstar_3333(k,l,k,l) = dLp_dTstar_3333(k,l,k,l) + 1.0_pReal
   forall (k=1_pInt:3_pInt,m=1_pInt:3_pInt) &
     dLp_dTstar_3333(k,k,m,m) = dLp_dTstar_3333(k,k,m,m) - 1.0_pReal/3.0_pReal
   dLp_dTstar99 = math_Plain3333to99(gamma_dot / constitutive_j2_fTaylor(instance) * &
                                      dLp_dTstar_3333 / norm_Tstar_dev)
 end if

end subroutine constitutive_j2_LpAndItsTangent


!--------------------------------------------------------------------------------------------------
!> @brief calculates the rate of change of microstructure
!--------------------------------------------------------------------------------------------------
pure function constitutive_j2_dotState(Tstar_v,state,ipc,ip,el)
 use prec, only: &
   p_vec
 use math, only: &
   math_mul6x6
 use mesh, only: &
   mesh_NcpElems, &
   mesh_maxNips
 use material, only: &
   homogenization_maxNgrains, &
   material_phase, &
   phase_plasticityInstance
 
 implicit none
 real(pReal), dimension(1) :: &
   constitutive_j2_dotState
 real(pReal), dimension(6), intent(in):: &
   Tstar_v                                                                                          !< 2nd Piola Kirchhoff stress tensor in Mandel notation
 integer(pInt),             intent(in) :: &
   ipc, &                                                                                           !< component-ID of integration point
   ip, &                                                                                            !< integration point
   el                                                                                               !< element
 type(p_vec),               intent(in) :: &
   state                                                                                            !< microstructure state

 real(pReal), dimension(6) :: &
   Tstar_dev_v                                                                                      !< deviatoric part of the 2nd Piola Kirchhoff stress tensor in Mandel notation
 real(pReal) :: &
   gamma_dot, &                                                                                     !< strainrate
   hardening, &                                                                                     !< hardening coefficient
   saturation, &                                                                                    !< saturation resistance
   norm_Tstar_dev                                                                                   !< euclidean norm of Tstar_dev
 integer(pInt) :: &
   instance

 instance = phase_plasticityInstance(material_phase(ipc,ip,el))
!--------------------------------------------------------------------------------------------------
! norm of deviatoric part of 2nd Piola-Kirchhoff stress
 Tstar_dev_v(1:3) = Tstar_v(1:3) - sum(Tstar_v(1:3))/3.0_pReal
 Tstar_dev_v(4:6) = Tstar_v(4:6)
 norm_Tstar_dev = sqrt(math_mul6x6(Tstar_dev_v,Tstar_dev_v))

!--------------------------------------------------------------------------------------------------
! strain rate 
 gamma_dot = constitutive_j2_gdot0(instance) * (            sqrt(1.5_pReal) * norm_Tstar_dev & 
            / &!-----------------------------------------------------------------------------------
              (constitutive_j2_fTaylor(instance) * state%p(1)) ) ** constitutive_j2_n(instance)
 
!--------------------------------------------------------------------------------------------------
! hardening coefficient
 if (abs(gamma_dot) > 1e-12_pReal) then
   if (constitutive_j2_tausat_SinhFitA(instance) == 0.0_pReal) then
     saturation = constitutive_j2_tausat(instance)
   else
     saturation = (  constitutive_j2_tausat(instance) &
                   + ( log(  ( gamma_dot / constitutive_j2_tausat_SinhFitA(instance)&
                               )**(1.0_pReal / constitutive_j2_tausat_SinhFitD(instance))&
                            + sqrt(  ( gamma_dot / constitutive_j2_tausat_SinhFitA(instance) &
                                      )**(2.0_pReal / constitutive_j2_tausat_SinhFitD(instance)) &
                                   + 1.0_pReal ) &
                            ) & ! asinh(K) = ln(K + sqrt(K^2 +1))
                       )**(1.0_pReal / constitutive_j2_tausat_SinhFitC(instance)) &
                   / (  constitutive_j2_tausat_SinhFitB(instance) &
                      * (gamma_dot / constitutive_j2_gdot0(instance))**(1.0_pReal / constitutive_j2_n(instance)) &
                      ) &
                   )
   endif
   hardening = ( constitutive_j2_h0(instance) + constitutive_j2_h0_slopeLnRate(instance) * log(gamma_dot) ) &
               * abs( 1.0_pReal - state%p(1)/saturation )**constitutive_j2_a(instance) &
               * sign(1.0_pReal, 1.0_pReal - state%p(1)/saturation)
 else
   hardening = 0.0_pReal
 endif
 
 constitutive_j2_dotState = hardening * gamma_dot

end function constitutive_j2_dotState


!--------------------------------------------------------------------------------------------------
!> @brief return array of constitutive results
!--------------------------------------------------------------------------------------------------
pure function constitutive_j2_postResults(Tstar_v,state,ipc,ip,el)
 use prec, only: &
   p_vec
 use math, only: &
   math_mul6x6
 use mesh, only: &
   mesh_NcpElems, &
   mesh_maxNips
 use material, only: &
   homogenization_maxNgrains, &
   material_phase, &
   phase_plasticityInstance, &
   phase_Noutput

 implicit none
 real(pReal), dimension(6),  intent(in) :: &
   Tstar_v                                                                                          !< 2nd Piola Kirchhoff stress tensor in Mandel notation
 integer(pInt),              intent(in) :: &
   ipc, &                                                                                           !< component-ID of integration point
   ip, &                                                                                            !< integration point
   el                                                                                               !< element
 type(p_vec),                intent(in) :: &
   state                                                                                            !< microstructure state

 real(pReal), dimension(constitutive_j2_sizePostResults(phase_plasticityInstance(material_phase(ipc,ip,el)))) :: &
                                           constitutive_j2_postResults
 
 real(pReal), dimension(6) :: &
   Tstar_dev_v                                                                                      ! deviatoric part of the 2nd Piola Kirchhoff stress tensor in Mandel notation
 real(pReal) :: &
   norm_Tstar_dev                                                                                   ! euclidean norm of Tstar_dev
 integer(pInt) :: &
   instance, &
   o, &
   c
 
 instance = phase_plasticityInstance(material_phase(ipc,ip,el))
 
!--------------------------------------------------------------------------------------------------
! calculate deviatoric part of 2nd Piola-Kirchhoff stress and its norm
 Tstar_dev_v(1:3) = Tstar_v(1:3) - sum(Tstar_v(1:3))/3.0_pReal
 Tstar_dev_v(4:6) = Tstar_v(4:6)
 norm_Tstar_dev = sqrt(math_mul6x6(Tstar_dev_v,Tstar_dev_v))
 
 c = 0_pInt
 constitutive_j2_postResults = 0.0_pReal

 outputsLoop: do o = 1_pInt,phase_Noutput(material_phase(ipc,ip,el))
   select case(constitutive_j2_outputID(o,instance))
     case (flowstress_ID)
       constitutive_j2_postResults(c+1_pInt) = state%p(1)
       c = c + 1_pInt
     case (strainrate_ID)
       constitutive_j2_postResults(c+1_pInt) = &
                constitutive_j2_gdot0(instance) * (            sqrt(1.5_pReal) * norm_Tstar_dev & 
             / &!----------------------------------------------------------------------------------
              (constitutive_j2_fTaylor(instance) * state%p(1)) ) ** constitutive_j2_n(instance)
       c = c + 1_pInt
   end select
 enddo outputsLoop

end function constitutive_j2_postResults

end module constitutive_j2