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UHYPER_Shrimali_Lefevre_Lopez-Pamies.for
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!**********************************************************************
! Legal notice: UHYPER_Shrimali_Lefevre_Lopez-Pamies.for (Windows)
!
! Copyright (C) 2018 Bhavesh Shrimali ([email protected])
! Victor Lefèvre ([email protected])
! Oscar Lopez-Pamies ([email protected])
!
! This ABAQUS UHYPER subroutine implements the hyperelastic energy
! density derived in [1] for the macroscopic elastic response of
! non-Gaussian elastomers weakened by an isotropic and non-percolative
! distribution of equiaxed pores. This result is valid for any choice
! of I1-based incompressible energy density characterizing
! the non-Gaussian isotropic elastic response of the underlying
! elastomer. The present subroutine is implemented for the
! choice of strain energy density proposed in [2].
!
! This program 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.
!
! This program 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 this program. If not, see https://www.gnu.org/licenses/
!
!**********************************************************************
! Usage:
!
! The subroutine is to be used as an compressible USER hyperelastic
! model with 5 material properties, e.g.,
! *HYPERELASTIC, USER, TYPE=COMPRESSIBLE, PROPERTIES=5
! in the input (.inp) file.
!
! The 5 materials properties for the model to be provided as input to
! the subroutine via the PROPS array are listed in the table below:
! AMU1 = PROPS(1) ! PARAMETER #1 OF THE ELASTOMER
! ALPHA1 = PROPS(2) ! EXPONENT #1 OF THE ELASTOMER
! AMU2 = PROPS(3) ! PARAMETER #2 OF THE ELASTOMER
! ALPHA2 = PROPS(4) ! EXPONENT #2 OF THE ELASTOMER
! AF0 = PROPS(5) ! INITIAL POROSITY
!
! The two material parameters AMU1, AMU2 characterizing the elastic
! behavior of the underlying elastomer are non-negative real numbers
! (AMU1 >= 0, AMU2 >= 0) with strictly positive sum (AMU1 + AMU2 > 0).
! The two exponents ALPHA1, ALPHA2 are non-zero real numbers
! (ALPHA1 ≠ 0, ALPHA2 ≠ 0) leading to a strongly elliptic strain
! energy (see eq. (22) in [2]). This is left to the user to check.
!
! The initial porosity (AF0) must satisfy 0 <= AF0 <= 1.
!
! As expected from physical considerations, this macroscopic energy
! remains finite so long the determinant of the deformation
! gradient (AJ) satisfies the condition AJ - 1 + AF0 > 0. This
! inequality constraint is enforced through a MOREAU-YOSIDA
! regularization (see, e.g. [3]). The underlying weight (ANU) is set
! here by default to ANU = 1.0e15. The subroutine issues two kinds
! of messages regarding the constraint:
! -- a WARNING message when AJ - 1 + AF0 < 1e-9 which allows the job
! to carry on
! -- an ERROR message when AJ - 1 + AF0 < -0.01 and TERMINATES the job
! In both cases, please treat the results with caution and check that
! the current local porosity given by (AJ - 1 + AF0) / AJ
! remains positive; see below on how to request it.
!
! The porosity in the deformed configuration (AJ - 1 + AF0) / AJ
! is required to be output (to check the results for instance,
! see above) as a solution-dependent state variable (SDV), e.g.,
! using the following lines in the input (.inp) file:
! *DEPVAR
! 1
! 1, Porosity, Current local porosity
! The solution-dependent state variable may be initialized using the
! following lines in the input (.inp) file:
! *INITIAL CONDITIONS, TYPE=SOLUTION
! <some element set>, AF0
!
!**********************************************************************
! Additional information:
!
! This subroutine creates a solution-dependent state variable for
! for the current porosity (see above) but does not create predefined
! field variables.
!
! Examples can be found in the article posted in the SIMULIA Learning
! Community: https://r1132100503382-eu1-3dswym.3dexperience.3ds.com/
! #community:39/post:mRdxC3xkRzajJ6LVk0SgwA
!
! Please consult the ABAQUS Documentation for additional references
! regarding the use of compressible USER hyperelastic models with
! the UHYPER subroutine and the use of solution-dependent
! state variables.
!
! Due the near-incompressible nature of this model at low porosities,
! the use of hybrid elements is recommended.
!
!**********************************************************************
! References:
!
! [1] Shrimali, B., Lefèvre, V., Lopez-Pamies, O. 2019. A simple
! explicit homogenization solution for the macroscopic elastic
! response of isotropic porous elastomers. J. Mech. Phys. Solids
! 122, 364--380.
! [2] Lopez-Pamies, O., 2010. A new I1-based hyperelastic model for
! rubber elastic materials. C. R. Mec. 338, 3--11.
! [3] Parikh, N., Boyd, S., 2013. Proximal algorithms. Found. Trends
! Optim. 1, 123--231.
!
!**********************************************************************
!
SUBROUTINE UHYPER(BI1,BI2,AJ,U,UI1,UI2,UI3,TEMP,NOEL,
1 CMNAME,INCMPFLAG,NUMSTATEV,STATEV,NUMFIELDV,FIELDV,
2 FIELDVINC,NUMPROPS,PROPS)
!
INCLUDE 'ABA_PARAM.INC'
#INCLUDE <SMAASPUSERSUBROUTINES.HDR>
!
CHARACTER*80 CMNAME
DIMENSION U(2),UI1(3),UI2(6),UI3(6),STATEV(*),FIELDV(*),
1 FIELDVINC(*),PROPS(*)
!
! STDB_ABQERR AND GET_THREAD_ID INITIALIZATION
!
DIMENSION INTV(1),REALV(3)
CHARACTER*8 CHARV(1)
CHARACTER*100 STRING1, STRING2, STRING3
CHARACTER*300 STRING
!
DATA LWRITE /1/
!
INTEGER MYTHREADID
!
INTV(1)=0
REALV(1)=0.
REALV(2)=0.
REALV(3)=0.
CHARV(1)=''
!
MYTHREADID = GET_THREAD_ID()
!
! INPUT CHECKS
!
IF (MYTHREADID.EQ.0) THEN
IF (INCMPFLAG.EQ.1) THEN
STRING1='INCOMPRESSIBILITY FLAG IS 1. THE MODEL IS COMPRES'
STRING2='SIBLE. SET USER TYPE=COMPRESSIBLE.'
STRING = TRIM(STRING1) // TRIM(STRING2)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
ELSE IF (NUMSTATEV.NE.1) THEN
INTV(1)=NUMSTATEV
STRING1='RECEIVED REQUEST FOR %I SOLUTION-DEPENDENT STATE'
STRING2=' VARIABLES. THE SUBROUTINE CREATES 1 SOLUTION'
STRING3='-DEPENDENT STATE VARIABLE.'
STRING = TRIM(STRING1) // TRIM(STRING2) // TRIM(STRING3)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
ELSE IF (NUMFIELDV.NE.0) THEN
INTV(1)=NUMFIELDV
STRING1='RECEIVED REQUEST FOR %I PREDEFINED FIELD VARI'
STRING2='ABLES. THE SUBROUTINE DOES NOT CREATE PREDEFINED'
STRING3=' FIELD VARIABLES.'
STRING = TRIM(STRING1) // TRIM(STRING2) // TRIM(STRING3)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
ELSE IF (NUMPROPS.NE.5) THEN
INTV(1)=NUMPROPS
STRING1='RECEIVED %I MATERIAL PROPERTIES. THE SUBROUTINE'
STRING2=' REQUIRES 5 MATERIAL PROPERTIES.'
STRING = TRIM(STRING1) // TRIM(STRING2)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
END IF
END IF
!
! MATERIAL PARAMETERS
!
AMU1 = PROPS(1) ! PARAMETER #1 OF THE ELASTOMER
ALPHA1 = PROPS(2) ! I1 EXPONENT #1 OF THE ELASTOMER
AMU2 = PROPS(3) ! PARAMETER #2 OF THE ELASTOMER
ALPHA2 = PROPS(4) ! I1 EXPONENT #2 OF THE ELASTOMER
AF0 = PROPS(5) ! INITIAL POROSITY
!
! PARTIAL MATERIAL PARAMETERS CHECKS
!
IF (((AMU1.LT.0.).OR.(AMU2.LT.0.).OR.(AMU1+AMU2.LE.0.))
1 .AND.(MYTHREADID.EQ.0)) THEN
REALV(1)=AMU1
REALV(2)=AMU2
REALV(3)=AMU1+AMU2
STRING1='RECEIVED AMU1 = %R AND AMU2 = %R, AMU1 + AMU2 = %R.'
STRING2=' THE PARAMETERS AMU1 AND AMU2 MUST BE NON-NEGATIVE'
STRING3=' AND AMU1 + AMU2, MUST BE GREATER THAT ZERO.'
STRING = TRIM(STRING1) // TRIM(STRING2) // TRIM(STRING3)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
END IF
!
IF (((ALPHA1.EQ.0.).OR.(ALPHA2.EQ.0.)).AND.
1 (MYTHREADID.EQ.0)) THEN
REALV(1)=ALPHA1
REALV(2)=ALPHA2
STRING1='RECEIVED ALPHA1 = %R AND ALPHA2 = %R.'
STRING2=' THE EXPONENTS ALPHA1 AND ALPHA2 MUST BE NON-ZERO.'
STRING = TRIM(STRING1) // TRIM(STRING2)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
END IF
!
IF (AF0.LT.0.) THEN
IF (MYTHREADID.EQ.0) THEN
REALV(1)=AF0
STRING1='RECEIVED AF0 = %R. THE INITIAL POROSITY IS NEGATIVE.'
STRING = TRIM(STRING1)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
ELSE
CALL XIT
END IF
ELSE IF (AF0.GT.1.) THEN
IF (MYTHREADID.EQ.0) THEN
REALV(1)=AF0
STRING1='RECEIVED AF0 = %R. THE INITIAL POROSITY'
STRING2=' IS GREATER THAN 1.'
STRING = TRIM(STRING1) // TRIM(STRING2)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
ELSE
CALL XIT
END IF
END IF
!
! CONSTRAINT AJ + AF0 - 1 > 0 CHECK: WARNING MESSAGE
!
IF (AJ+AF0.LT.1.0 .AND. LWRITE.EQ.1 ) THEN
LWRITE = 0
INTV(1)=NOEL
STRING1='THE POROSITY IS NEGATIVE IN ELEMENT #%I.'
STRING2=' TREAT THE RESULTS WITH CAUTION.'
STRING = TRIM(STRING1) // TRIM(STRING2)
CALL STDB_ABQERR(-1,STRING,INTV,REALV,CHARV)
END IF
!
! CONSTRAINT AJ + AF0 - 1 > 0 CHECK: JOB TERMINATION
!
IF (AJ+AF0.LE.0.99) THEN
INTV(1)=NOEL
STRING1='THE POROSITY IS NEGATIVE IN ELEMENT #%I.'
STRING2=' JOB TERMINATED.'
STRING = TRIM(STRING1) // TRIM(STRING2)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
END IF
!
! RECURRING RATIOS AND FACTORS
!
AOT = 1./3.
ATT = 2./3.
AFT = 4./3.
AST = 7./3.
ATET = 10./3.
AFN = 4./9.
AOMAC = 1.-AF0
ACT = AJ-AOMAC ! AJ + AF0 - 1
!
! ENFORCE LOWER LIMIT TO ACT FOR THE EVALUATION OF ASI1
!
ACT_MIN = 1.0e-9
ACT0 = MAX(ACT, ACT_MIN)
DACT0DJ = 1.0
IF (ACT.LT.ACT_MIN) DACT0DJ = 0.0
!
! FIRST INVARIANT AI1 = F.F AND PARTIAL DERIVATIVES
!
AI1=BI1*AJ**ATT
DAI1DBI1=AJ**ATT
DAI1DAJ=ATT*BI1*AJ**(-AOT)
DAI1DBI1AJ=ATT*AJ**(-AOT)
DAI1DAJAJ=-AOT*ATT*BI1*AJ**(-AFT)
DAI1DBI1AJAJ=-AOT*ATT*AJ**(-AFT)
DAI1DAJAJAJ=AFT*AOT*ATT*BI1*AJ**(-AST)
!
! EQ (19) IN [1] AND PARTIAL DERIVATIVES
!
ASI1=3.*AOMAC*(AI1-3.)/(3.+2.*AF0) + (3.*(2.*AJ-1. -
1 AOMAC*(2.*AF0+3.*AJ**ATT)*AJ**AOT/(3.+2.*AF0) -
2 AF0**AOT*AJ**AOT*(2.*ACT0-AF0)/ACT0**AOT))/AJ**AOT
!
DASI1DAI1=3.*AOMAC/(3.+2.*AF0)
!
DASI1DAJ=AJ**(-AFT)+(2.*(3.+7.*AF0))/((3.+2.*AF0)*AJ**AOT) -
1 AF0**AOT*(4.*ACT0+AF0)*DACT0DJ/ACT0**AFT
!
DASI1DAJAJ=(2.*AF0**AOT*(AF0+ACT0)*DACT0DJ**2.0/ACT0**AST
1 -2./AJ**AST - (3.+7.*AF0)/((3.+2.*AF0)*AJ**AFT))*ATT
!
DASI1DAJAJAJ=(7./AJ**ATET + 2.*(3.+7.*AF0)/((3.+2.*AF0)*AJ**AST) -
1 AF0**AOT*(4.*ACT0+7.*AF0)*DACT0DJ**3.0/ACT0**ATET)*AFN
!
! ARGUMENT IN EQ (18) IN [1] AND PARTIAL DERIVATIVES
!
AII1=ABS(ASI1)/AOMAC+3.
!
DAII1DASI1=1./AOMAC
!
DAII1DBI1=DAII1DASI1*DASI1DAI1*DAI1DBI1
!
DAII1DAJ=DAII1DASI1*(DASI1DAI1*DAI1DAJ+DASI1DAJ)
!
DAII1DAJAJ=DAII1DASI1*(DASI1DAI1*DAI1DAJAJ+DASI1DAJAJ)
!
DAII1DBI1AJ=DAII1DASI1*DASI1DAI1*DAI1DBI1AJ
!
DAII1DBI1AJAJ=DAII1DASI1*DASI1DAI1*DAI1DBI1AJAJ
!
DAII1DAJAJAJ=DAII1DASI1*(DASI1DAI1*DAI1DAJAJAJ+DASI1DAJAJAJ)
!
! NON-GAUSSIAN HYPERELASTIC MODEL FOR THE ELASTOMER [2]
! AND DERIVATIVES
!
AP1 = AMU1*3.**(1.-ALPHA1)*0.5
AP2 = AMU2*3.**(1.-ALPHA2)*0.5
!
PSI = AP1/ALPHA1*(AII1**ALPHA1-3.**ALPHA1)+
1 AP2/ALPHA2*(AII1**ALPHA2-3.**ALPHA2)
!
DPSI = AP1*AII1**(ALPHA1-1.)+AP2*AII1**(ALPHA2-1.)
!
DDPSI = AP1*(ALPHA1-1.)*AII1**(ALPHA1-2.)+
1 AP2*(ALPHA2-1.)*AII1**(ALPHA2-2.)
!
DDDPSI = AP1*(ALPHA1-1.)*(ALPHA1-2.)*AII1**(ALPHA1-3.)+
1 AP2*(ALPHA2-1.)*(ALPHA2-2.)*AII1**(ALPHA2-3.)
!
! MOREAU-YOSIDA REGULARIZATION FOR THE CONSTRAINT ACT>0
!
ANU=1.0e15
!
AMYR=ANU/2*(ABS(ACT)-ACT)**2.
!
DAMYRDAJ=ANU*2*(ACT-ABS(ACT))
!
DAMYRDAJAJ = 0
IF (ACT.LE.0) DAMYRDAJAJ = ANU*4.
!
DAMYRDAJAJAJ=0.
!
! MACROSCOPIC STRAIN ENERGY DENSITY FUNCTION FOR
! POROUS ELASTOMER [1]
!
U(1) = AOMAC*PSI + AMYR
U(2) = 0.
!
! FIRST PARTIAL DERIVATIVES
!
UI1(1) = AOMAC*DPSI*DAII1DBI1
UI1(2) = 0.
UI1(3) = AOMAC*DPSI*DAII1DAJ + DAMYRDAJ
!
! SECOND PARTIAL DERIVATIVES
!
UI2(1) = AOMAC*DDPSI*DAII1DBI1**2.
UI2(2) = 0.
UI2(3) = AOMAC*DDPSI*DAII1DAJ**2.+
1 AOMAC*DPSI*DAII1DAJAJ + DAMYRDAJAJ
UI2(4) = 0.
UI2(5) = AOMAC*DDPSI*DAII1DBI1*DAII1DAJ+
1 AOMAC*DPSI*DAII1DBI1AJ
UI2(6) = 0.
!
! THIRD PARTIAL DERIVATIVES
!
UI3(1) = AOMAC*DDDPSI*DAII1DBI1**2.*DAII1DAJ+
1 2.*AOMAC*DDPSI*DAII1DBI1*DAII1DBI1AJ
UI3(2) = 0.
UI3(3) = 0.
UI3(4) = AOMAC*DDDPSI*DAII1DBI1*DAII1DAJ**2.+
1 2.*AOMAC*DDPSI*DAII1DBI1AJ*DAII1DAJ+
2 AOMAC*DDPSI*DAII1DBI1*DAII1DAJAJ+
3 AOMAC*DPSI*DAII1DBI1AJAJ
UI3(5) = 0.
UI3(6) = AOMAC*DDDPSI*DAII1DAJ**3.+
1 3.*AOMAC*DDPSI*DAII1DAJ*DAII1DAJAJ+
2 AOMAC*DPSI*DAII1DAJAJAJ + DAMYRDAJAJAJ
!
! SOLUTION-DEPENDENT STATE VARIABLE
! CURRENT POROSITY
!
STATEV(1)=ACT/AJ
!
RETURN
END
!
!**********************************************************************