MP/Source/HUANG_Const_Model.f90

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SUBROUTINE huang_const_model(dfgrad,matrix,nmatrix, &
     particle,nparticle,nparticletype,interfac,ninterfac,stress, strain, ilarge, ismall)


!-----  this program is for the constitutive modeling of solid propellants
!     accounting for the effect of particle/matrix interface debonding.  the
!     program can handle one or two types of particles that have the same
!     elastic moduli but different radii.
        
!-----  input: all variables above except the last few, i.e., 
!              dfgrad,matrix,nmatrix,particle,nparticle,
!              interfac,ninterfac
!-----  output: the last few variables, stress, strain, ilarge, ismall

!-----  arrays for input/output
  REAL*8 dfgrad(3,3),matrix(nmatrix), &
       particle(nparticle,nparticletype),interfac(ninterfac), &
       stress(3,3)

  REAL*8 :: ilarge, ismall

!       dfgrad: deformation gradient
!       nmatrix: number of matrix properties
!     matrix: properties of the matrix
!          matrix(1): young's modulus of the matrix
!          matrix(2): poisson's ratio of the matrix
!          matrix(3): volume fraction of the matrix
!     nparticletype: number of the particle types
!                    nparticletype is no more than 2 in this program!
!       nparticle: number particle properties for each type 
!       particle: properties of particles
!          particle(1,i): young's modulus of type-i particles
!          particle(2,i): poisson's ratio of type-i particles
!          particle(3,i): volume fraction of type-i particles
!          particle(4,i): radius of type-i particles
!       ninterfac: number of interface properties
!       interfac: properties of the interface
!          interfac(1): strength of the interface
!          interfac(2): linear modulus of the interface
!          interfac(3): softening modulus of the interface
!       stress: stress tensor
!   strain: strain tensor
!       ilarge, ismall: damage information for large and small particles
!       ilarge=1: large particles not failed
!       ilarge=2: large particles is failing
!       ilarge=3: large particles completely failed
!       ismall=1: small particles not failed
!       ismall=2: small particles is failing
!       ismall=3: small particles completely failed



!-----  arrays for the internal use in the subroutine
  REAL*8 alpha(nparticletype),alphaprime(nparticletype), &
       bulkp(nparticletype),shearp(nparticletype), &
       strain(3,3),stressprime(3,3),strainprime(3,3), &
       mstrainpt(5),mstresspt(5)

!-----  variables for internal use in the subroutine
  REAL*8 em,posm,bulkm,shearm,bulkcomposite,shearcomposite,strength, &
       kinthard,kintsoft,stressmean,strainmean

!-----  the green strain 
  DO i=1,3
     DO  j=1,3
        strain(i,j)= dfgrad(1,i)*dfgrad(1,j) &
             +dfgrad(2,i)*dfgrad(2,j) &
             +dfgrad(3,i)*dfgrad(3,j) 
        IF (i.EQ.j) THEN
           strain(i,j)=(strain(i,j)-1)*0.5d0
        ELSE
           strain(i,j)=strain(i,j)*0.5d0
        END IF
     ENDDO
  ENDDO

!-----  the mean and deviatoric strains
  strainmean=(strain(1,1)+strain(2,2)+strain(3,3))/3.d0
  DO  i=1,3
     DO  j=1,3
        IF (i.EQ.j) THEN 
           strainprime(i,j)=strain(i,j)-strainmean
        ELSE
           strainprime(i,j)=strain(i,j)
        END IF
     ENDDO
  ENDDO
                                
!-----  parameters for particles: alpha and alphaprime
  em   = matrix(1)
  posm = matrix(2)
  bulkm= em/(3.d0*(1.d0-2.d0*posm))
  shearm=em/(2.d0*(1.d0+posm))
  strength=interfac(1)
  kinthard=interfac(2)
  kintsoft=interfac(3)
  
  DO i=1,nparticletype
     bulkp(i)=particle(1,i)/(3*(1-2*particle(2,i)))
     shearp(i)=particle(1,i)/(2*(1+particle(2,i)))
     alpha(i)=3.d0*(1-posm)/(2*em) &
          /( 1.d0/(kinthard*particle(4,i)) &
          +1.d0/(3.d0*bulkp(i)) &
          +1.d0/(4.d0*shearm))
     alphaprime(i)=-3.d0*(1.d0-posm)/(2.d0*em) &
          /(-1.d0/(kintsoft*particle(4,i)) &
          +1.d0/(3.d0*bulkp(i)) &
          +1.d0/(4.d0*shearm))
     
  ENDDO


  c1=particle(3,1)
  c2=particle(3,2)
  a1=particle(4,1)
  a2=particle(4,2)
  c=c1+c2

  alpha1=alpha(1)
  alpha2=alpha(2)
  alphaprime1=alphaprime(1)
  alphaprime2=alphaprime(2)

!-----  critical particle radius
  aprime=1.d0/(1.d0/(4.d0*shearm)+1.d0/(3.d0*bulkp(1)))/kintsoft        

!-----  the mean stress and strain of the composite at the transition point 
!     between different stages.
!     path2:(i,i)->(ii,i)->(iii,i)->(iii,ii)->(iii,iii)
  mstrainpt(1)=0.d0
  mstresspt(1)=0.d0
!     (i,i)->(ii,i)
  mstrainpt(2)=strength/(2.d0*em*alpha1) &
       *(2.d0*(1.d0-2.d0*posm)-(1.d0+posm)*(-c+c2*alpha2+c1*alpha1))
  mstresspt(2)=strength*(1.d0-c+c2*alpha2+c1*alpha1)/alpha1
!     (ii,i)->(iii,i)
  mstrainpt(3)=strength*(1.d0/alpha1+1.d0/alphaprime1)  &
       *(2.d0*(1.d0-2.d0*posm)+(1.d0+posm)*(c-c2*alpha2))/(2.d0*em)      
  mstresspt(3)=strength*(1.d0/alpha1+1.d0/alphaprime1)*(1.d0-c+c2*alpha2)
!     (iii,i)->(iii,ii)
  mstrainpt(4)=strength*(2.d0*(1.d0-2.d0*posm)+c*(1.d0+posm)-(1.d0+posm)*c2*alpha2)  &
       /(2.d0*em*alpha2)
  mstresspt(4)=strength*(1.d0-c+c2*alpha2)/alpha2
!     (iii,ii)->(iii,iii)
  mstrainpt(5)=strength/(2.d0*em) &
       *(1/alpha2+1/alphaprime2)*(2*(1-2*posm)+c*(1+posm))
  mstresspt(5)=(1.d0-c)*strength*(1.d0/alpha2+1.d0/alphaprime2)

!     path1:(i,i)->(ii,i)->(ii,ii)->(iii,ii)->(iii,iii)
  IF (a1.LT.aprime) THEN
!     (ii,i)->(ii,ii) 
     mstrainpt(3)=strength/(2.d0*em)  &
          *(2.d0*(1.d0-2.d0*posm)/alpha2  &
          -(1.d0+posm)*(c+c1*alphaprime1*(1.d0/alpha1-1.d0/alpha2)  &
          -c/alpha2))        
     mstresspt(3)=strength  &
          *((1.d0-c)/alpha2+c+c1*alphaprime1 &
          *(1.d0/alpha1-1.d0/alpha2))     
!     (ii,ii)->(iii,ii)
     mstrainpt(4)=strength/(2.d0*em) &
          *((2.d0*(1.d0-2.d0*posm)+c*(1.d0+posm)) &
          *(1.d0/alpha1+1.d0/alphaprime1) &
          -(1.d0+posm)*c2*(1.d0-alphaprime2/alpha1 &
          -alphaprime2/alphaprime1 &
          +alphaprime2/alpha2))
     mstresspt(4)=strength*((1.d0-c)*(1.d0/alpha1+1.d0/alphaprime1) &
          +c2*(1.d0-alphaprime2/alpha1 &
          -alphaprime2/alphaprime1 &
          +alphaprime2/alpha2))
  END IF

!----  mean stress in the composite
  DO i=1,4
     IF (strainmean.GE.mstrainpt(i).AND.  &
          strainmean.LT.mstrainpt(i+1)) THEN
        stressmean=mstresspt(i)+(strainmean-mstrainpt(i)) &
             *(mstresspt(i+1)-mstresspt(i)) &
             /(mstrainpt(i+1)-mstrainpt(i))

                IF(i.EQ.1) THEN
                        ilarge=1
                        ismall=1
                ELSE IF(i.EQ.2) THEN
                        ilarge=2
                        ismall=1
                ELSE IF(i.EQ.3) THEN
                        IF(a1.LT.aprime) THEN
                                ilarge=2
                                ismall=2
                        ELSE
                                ilarge=3
                                ismall=1
                        ENDIF
                ELSE IF(i.EQ.4) THEN
                        ilarge=3
                        ismall=2
                ENDIF
     END IF
  ENDDO
  IF (strainmean.GE.mstrainpt(5)) THEN
     stressmean=mstresspt(5)*strainmean/mstrainpt(5)
                ilarge=3
                ismall=3
  END IF
  IF (strainmean.LT.0) THEN
     bulkcomposite=bulkm+c*(bulkp(1)-bulkm)*3.d0*(1-posm) &
          /(3.d0*(1.d0-posm)+(1.d0+posm) &
          *(bulkp(1)/bulkm-1.d0)*(1.d0-c))
     stressmean=3.d0*bulkcomposite*strainmean
                ilarge=1
                ismall=1
  END IF

!-----  deviatoric stress
  shearcomposite=shearm+c*(shearp(1)-shearm)*15.d0*(1.d0-posm) &
       /(15.d0*(1.d0-posm)+2.d0*(4.d0-5.d0*posm) &
       *(shearp(1)/shearm-1.d0)*(1.d0-c))
  DO i=1,3
     DO j=1,3
        stressprime(i,j)=2.d0*shearcomposite*strainprime(i,j)
     ENDDO
  ENDDO

!-----  update the stress
  DO i=1,3
     DO j=1,3
        IF (i.EQ.j) THEN 
           stress(i,j)=stressprime(i,j)+stressmean
        ELSE
           stress(i,j)=stressprime(i,j)
        END IF
     ENDDO
  ENDDO

END SUBROUTINE huang_const_model