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SUBROUTINES.FOR
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C
C The following Fortran 77 code is developed by Seyed Shayan Sajjadinia and is shared under the MIT License.
C
C If this research data is useful for your work, kindly please consider citing the relavant paper [https://doi.org/10.1177/0954411919854011].
C
C Please read the paper for more information about the theory and implementation details.
C
C
SUBROUTINE SDVINI(STATEV,COORDS,NSTATV,NCRDS,NOEL,NPT,
1 LAYER,KSPT)
C
INCLUDE 'ABA_PARAM.INC'
C
DIMENSION STATEV(NSTATV),COORDS(NCRDS)
DOUBLE PRECISION DEPTH,RPHI,G(10),RTHETA
C
PARAMETER (ZERO=0.D0,ONE=1.D0,TWO=2.D0,TEN=10.D0,FOUR=4.D0,
1 CONS1=5.235987755983D0,PI=3.14159265359D0,CONS2=2.61799387799D0)
C
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
C THIS SUBROUTINE CODE IS GENERATED TO CONTROL THE NON-HOMOGENEOUS AND VARIABLE MATERIAL PARAMETERS VIA STATEV
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
C
C STATEV(1) CONTROLES THE TYPE OF MATERIAL MODELED:
C STATEV(1)=1 ==>> TOTALLY HEALTHY CARTILAGE # relavant publication for more details: [http://dx.doi.org/10.1177/0954411919854011]
C STATEV(1)=2 ==>> DEGRADED CARTILAGE WITHOUT FIBRILAR GEOMETRILAL ABNORMALITIES # relavant publication for more details: [http://dx.doi.org/10.1177/0954411919854011]
C STATEV(1)=3 ==>> DEGRADED CARTILAGE WITH FIBRILARATION IN DEEP ZONE # relavant publication for more details: [http://dx.doi.org/10.13140/RG.2.2.32634.44488/1]
C STATEV(1)=4 ==>> DEGRADED CARTILAGE WITH LOW FIBRILAR ROTATION # relavant publication for more details: [http://dx.doi.org/10.24200/SCI.2020.51785.2362]
C STATEV(1)=5 ==>> DEGRADED CARTILAGE WITH HIGH FIBRILAR ROTATION # relavant publication for more details: [http://dx.doi.org/10.24200/SCI.2020.51785.2362]
C
STATEV(1)=1
DEPTH=-10D0 ! DEPTH OF THE CARTILAGE
DEPTH=DBLE(COORDS(2)/DEPTH) ! NOMALIZATION OF DEPTH
C
C VARIBLE PARAMETER DUE TO ANISOTROPIC AND NON-HOMOGENEOUS NATURE OF THE SOLID MATRIX:
C
STATEV(2)=DBLE(1.4*(DEPTH**TWO)-1.1*DEPTH+0.59) ! DEPTH-DEPENDENT FIBER CONSTANT
IF (STATEV(1).EQ.4) THEN
CALL RANDOM-NUMBER(RPHI)
RPHI=RPHI*PI/TWO
ELSE
IF (DEPTH.GT.0.3) THEN
STATEV(4)=ONE
STATEV(5)=ZERO
ELSEIF (DEPTH.LE.0.3) THEN
RPHI=DBLE(CONS1*DEPTH)
STATEV(4)=DBLE(SIN(RPHI))
IF (STATEV(1).EQ.5) THEN
RTHETA=DBLE((PI/FOUR)-(CONS2*DEPTH))
ELSEIF (STATEV(1).EQ.6) THEN
RTHETA=DBLE((PI)-(FOUR*CONS2*DEPTH))
ENDIF
STATEV(5)=DBLE(COS(RPHI))
STATEV(6)=ZERO
IF (STATEV(1).EQ.(5.OR.6)) THEN
STATEV(5)=DBLE(SIN(RTHETA)*STATEV(5))
STATEV(6)=DBLE(COS(RTHETA)*STATEV(5))
ENDIF
ENDIF
IF (STATEV(1).EQ.1) THEN ! DEPTH-DEPENDENT SOLID MATERIAL CONSTANT
STATEV(3)=DBLE(0.1+0.2*DEPTH)
ELSE
STATEV(3)=DBLE(0.05+0.2*DEPTH)
ENDIF
ENDIF
G(1)=0.005D0
G(2)=0.01D0
G(3)=0.025D0
G(4)=0.035D0
G(5)=0.042D0
G(6)=0.048D0
G(7)=0.053D0
G(8)=0.058D0
G(9)=0.06D0
G(10)=0.06D0
STATEV(7)=INT(DEPTH*9)+1
STATEV(8)=G(STATEV(10)) ! GAG DEPTH-DEPENDENT MATERIAL CONSTANT
STATEV(9)=DEPTH
C THESE INITIALIZED STATE VARIBLES ARE ALSO USED FOR VERIFICATIONS
DO i = 10, NSTATV
STATEV(i)=ZERO
ENDDO
C
C TAFFETANI NEO-HOOKEAN MODEL (FOR VALIDATON)
C G(1)=0.005D0
C G(2)=0.008D0
C G(3)=0.015D0
C G(4)=0.05D0
C G(5)=0.08D0
C G(6)=0.1D0
C G(7)=0.3D0
C G(8)=0.65D0
C G(9)=0.8D0
C G(10)=0.8D0
C STATEV(9)=G(INT(DEPTH*9)+1)
C
RETURN
END
C
C
C
C
SUBROUTINE FLOW(H,SINK,U,KSTEP,KINC,TIME,NOEL,NPT,COORDS,
1 JLTYP,SNAME)
C
INCLUDE 'ABA_PARAM.INC'
DIMENSION TIME(2), COORDS(3)
CHARACTER*80 SNAME
H=1
SINK=0
IF ((COORDS(1).LE.1.25).AND.(COORDS(1).GE.-1.25)) THEN
H=0
ENDIF
RETURN
END
C
C
C
C
C
SUBROUTINE UMAT(STRESS,STATEV,DDSDDE,SSE,SPD,SCD,
1 RPL,DDSDDT,DRPLDE,DRPLDT,
2 STRAN,DSTRAN,TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED,CMNAME,
3 NDI,NSHR,NTENS,NSTATV,PROPS,NPROPS,COORDS,DROT,PNEWDT,
4 CELENT,DFGRD0,DFGRD1,NOEL,NPT,LAYER,KSPT,JSTEP,KINC)
C
INCLUDE 'ABA_PARAM.INC'
C
CHARACTER*80 CMNAME
DIMENSION STRESS(NTENS),STATEV(NSTATV),
1 DDSDDE(NTENS,NTENS),DDSDDT(NTENS),DRPLDE(NTENS),
2 STRAN(NTENS),DSTRAN(NTENS),TIME(2),PREDEF(1),DPRED(1),
3 PROPS(NPROPS),COORDS(3),DROT(3,3),DFGRD0(3,3),DFGRD1(3,3),
4 JSTEP(4)
C
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
C LOCAL PARAMETERS AND VARIABLES
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
C
DOUBLE PRECISION TRANF(3,3),DET,IDENT(3,3),INDEX(2,6),ALPHA1,
1 ALPHA2,IGAGD,NVEC0(27),RH,W1,W2,W3,W4,W5,W6,STRG(NTENS,NTENS),
2 EPS,FV1(3),HH,NEWV1(3),NSTR(NTENS),STR,LANDA,DELTAV(NTENS),
3 BVEC(NTENS),C,NS0,E1MP,E2MP,K1MP,CSTR,DFGRD(3,3),EP,STRS(NTENS),
4 STATE(NSTATV),GAG,VV(NTENS)
INTEGER i, j, k, l,r, m, n, K1, K2, K3, K4, K5, K6,KKK,DDS,FF
PARAMETER (ZERO=0.D0,ONE=1.D0,TWO=2.D0,THREE=3.D0,FOUR=4.D0,
1 TT=7.D0,SIX=6.D0,HALF=5.D-1,FFD=0.57735026919D0,
2 R2P2=0.7071067811865D0,R3P3=0.5773502691896D0,TEN=10D0,
3 CONST=0.4342944819D0)
C
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
C INITIALIZATION
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
C
DDS=1 ! THIS VARIABLE CONTROLS THE DDSDDE FORMULATION.
ALPHA1=STATEV(8) ! DEPTH-DEPENDENT GAG CONSTANT
ALPHA2=3.22D0 ! GAG CONSTANT
C=3.009D0 ! FIBRILLAR RELATIVE DENSITY CONSTANT
RH=STATEV(2) ! TOTAL FIBRILLAR DENSITY
NS0=STATEV(3) ! DEPTH-DEPENDENT SOLID VOLUME FRACTION CONSTANT
DO i = 1, 3
DELTAV(i)=ONE ! KRONKER-DELTA IN VOIGT-NOTATION
DO j = 1, 3
IDENT(j,i) = ZERO ! KRONKER-DELTA 2ND ORDER TENSOR
TRANF(j,i) = ZERO ! TRANSOPSE OF DEFORMATION GRADIENT TENSOR (DFGRD1) IN THE END OF THE INCREMNT
ENDDO
IDENT(i,i) = ONE
ENDDO
DO i=4,NTENS ! NTENS CONTROLS THE DIMENTIONALITY OF THE CODE
DELTAV(i)=ZERO
ENDDO
DO i = 1,NTENS
STRESS(i)=ZERO ! CAUCHY STRESS TENSOR THAT SHOULD BE UPDATED
STRS(i)=ZERO ! FIBRILLAR STRESS TENSOR
DO j = 1,NTENS
DDSDDE(j,i)=ZERO ! JACOBIAN MATRIX TENSOR THAT SHOULD BE UPDATED.
ENDDO
ENDDO
INDEX(1,1)=1 ! INDEX ARRAY ARE USED TO CIRCOMVENT ASSIGNING EQUAL COMPONENTS DUE TO SYMMYTRY OF HIYER ORDER TENSORS
INDEX(2,1)=1
INDEX(1,2)=2
INDEX(2,2)=2
INDEX(1,3)=3
INDEX(2,3)=3
INDEX(1,4)=1
INDEX(2,4)=2
INDEX(1,5)=1
INDEX(2,5)=3
INDEX(1,6)=2
INDEX(2,6)=3
C
NVEC0(1)=STATEV(5) ! NVEC ARRAY ARE THE DEFINED INITAIAL UNIT VECTOR DIRECTIONS ALONG FIBRILS.
NVEC0(2)=STATEV(4)
NVEC0(3)=STATEV(6)
C
NVEC0(4)=-STATEV(5)
NVEC0(5)=STATEV(4)
NVEC0(6)=-STATEV(6)
C
NVEC0(7)=ONE
NVEC0(8)=ZERO
NVEC0(9)=ZERO
C
NVEC0(10)=ZERO
NVEC0(11)=ONE
NVEC0(12)=ZERO
C
NVEC0(13)=ZERO
NVEC0(14)=ZERO
NVEC0(15)=ONE
C
NVEC0(16)=FFD ! FFD SIGNIFIES 45 DEGREE
NVEC0(17)=FFD
NVEC0(18)=FFD
C
NVEC0(19)=-FFD
NVEC0(20)=FFD
NVEC0(21)=FFD
C
NVEC0(22)=FFD
NVEC0(23)=-FFD
NVEC0(24)=FFD
C
NVEC0(25)=FFD
NVEC0(26)=FFD
NVEC0(27)=-FFD
C
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
C DDSDDE DERIVATION VIA PERTURBATION METHOD (INITIALIZATION)
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
C
IF (DDS.EQ.1) THEN
KKK=1
EP=10D-8
DO i = 1,3
DO j = 1,3
DFGRD(j,i)=DFGRD1(j,i)
ENDDO
ENDDO
DO K6=1,NSTATV
STATE(K6)=STATEV(K6)
ENDDO
i=1
j=1
80 CONTINUE
DO K1=1,3
DO K2=1,3
DFGRD1(K2,K1)=DFGRD(K2,K1)+(IDENT(K2,i)*DFGRD(j,K1)+IDENT(K2,j)
1 *DFGRD(i,K1))*EP/TWO
ENDDO
ENDDO
90 CONTINUE
DO K6=1,NTENS
STRESS(K6)=ZERO
ENDDO
DO K6=1,NSTATV
STATEV(K6)=STATE(K6)
ENDDO
ENDIF
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
C STRESS CALCULATIONS
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
CALL TRANSPOSE(DFGRD1,TRANF)
CALL VMATMUL(DFGRD1,TRANF,NTENS,BVEC) ! BVEC IS THE LEFT CAUCHY-GREEN OR FINGER DEFORMATION TENSOR
CALL DETERMINANT(DFGRD1,DET) ! DET IS THE DETERMINANT OF DEFORMATION GRADIENT IN THE END OF THE INCREMENT.
C
HH=NS0*(RH*C)/(TWO*C+TT) ! CNTRIBUTION OF OTHER CONSTANTS ON FIBRILAR STRESS
E1MP=4.63D0 ! LINEAR MATERIAL CONSTANT OF FIBRILLAR PART
E2MP=3670D0 ! NONLINEAR MATERIAL CONSTANT OF FIBRILLAR PART
IF (STATEV(1).NE.1) THEN
E2MP=3670/FOUR ! FOR OA
ENDIF
E1MP=E1MP*HH
E2MP=E2MP*HH
DO i=0,8
IF (i.EQ.2) THEN ! SECONDARY FIBRIL HAVE LOWER DENSITY BY C CONSTANT.
E1MP=E1MP/C
E2MP=E2MP/C
ENDIF
DO m = 1,3
FV1(m)=ZERO ! FV VETOR IS THE INNER PRODUCT OF DFGRD1 AND NVEC0
DO n = 1,3
FV1(m)=DFGRD1(m,n)*NVEC0(3*i+n)+FV1(m)
ENDDO
ENDDO
LANDA=SQRT(FV1(1)**TWO+FV1(2)**TWO+FV1(3)**TWO) ! LANDA IS THE ELONGATION
EPS=LOG(LANDA) ! EPS IS THE FIBRIL LOGARITMIC STRIN
IF (EPS.GT.ZERO) THEN
DO n=1,3
NEWV1(n)=FV1(n)/LANDA ! NEWV1 IS THE CURRENT FIBRIL DIRECTION
ENDDO
STR=(E1MP+E2MP*EPS)*EPS*LANDA/DET ! STR IS THE LOCAL FIBRIL STRESS
DO K6=1,NTENS
K3=INDEX(1,K6)
K4=INDEX(2,K6)
VV(K6)=NEWV1(K3)*NEWV1(K4) ! VV IS THE DYADIC PRODUCT OF CURRENT DIRECTION VECTORS THAT IS THE STRUCTRAL VECTOR
STRS(K6)=STR*VV(K6) ! STRS IS THE GLOBAL FIBRIL STRESS
STRESS(K6)=STRESS(K6)+STRS(K6)
ENDDO
IF (DDS.NE.1) THEN ! EXACT DDSDDE IMPLEMETATION FOR FIBRILLAR PART
W1=(LANDA/DET)*E2MP*EPS
W3=(ONE/EPS)-ONE
DO K6=1,NTENS
DO K5=1,NTENS
DDSDDE(K5,K6)=(W1*VV(K5)+W3*STRS(K5))*VV(K6)+DDSDDE(K5,K6)
ENDDO
ENDDO
ENDIF
ENDIF
ENDDO
STATEV(10)=STRESS(2) ! S22 STRESS OF FIBRILLAR PART
C
GM=0.723D0 ! GM IS THE NEO-HOOKEAN CONSTANT FOR PG CONRIBUTION
GM=GM*NS0*(ONE-RH) ! CONTRIBUTION OF OTHER CONSTANTS
W5=GM/DET
W6=((LOG(DET)/SIX)*(((THREE*NS0/(DET-NS0))
1 *((DET*LOG(DET)/(DET-NS0))-TWO))-FOUR)+(DET**(TWO/THREE)))*W5
DO K6=1,NTENS
NSTR(K6)=-DELTAV(K6)*W6+BVEC(K6)*W5
C NSTR(K6)=STATEV(9)*(BVEC(K6)-DELTAV(K6))/DET ! TEFFANI NEO-HOOKEAN MODEL (FOR VALIDATION)
ENDDO
STATEV(11)=NSTR(2) ! S22 STRESS OF NON-FIBRILLAR PART
GAG=ALPHA1*(DET**(-ALPHA2))
STATEV(12)=GAG ! S22 STRESS OF GAG PART
DO K6=1,3
STRESS(K6)=NSTR(K6)-GAG+STRESS(K6)
ENDDO
DO K6=4,NTENS
STRESS(K6)=NSTR(K6)+STRESS(K6)
ENDDO
C
IF (DDS.NE.1) THEN ! OTHER PARTS OF EXACT DDSDDE
W3=(GM/TWO)*((FOUR*(DET**(TWO/THREE))-FOUR+(THREE*NS0/(DET-NS0))*
1 (((DET*LOG(DET))/(DET-NS0))-TWO))/(THREE*DET)+((LOG(DET)-ONE)*
2 NS0*LOG(DET))/((DET-NS0)**TWO))
CALL TENF(STRESS,NTENS,STRG) ! STRG IS THE TENSOR FORM OF THE STRESS VECTOR
W2=GAG*(ALPHA2-ONE)
DO K6=1,NTENS
K3 = INDEX(1,K6)
K4 = INDEX(2,K6)
DO K5=1,NTENS
K1 = INDEX(1,K5)
K2 = INDEX(2,K5)
DDSDDE(K5,K6)=HALF*(IDENT(K4,K1)*STRG(K3,K2)+IDENT(K3,K2)*
1 STRG(K1,K4)+IDENT(K3,K1)*STRG(K4,K2)+IDENT(K4,K2)*STRG(K1,K3))
2 +W6*(IDENT(K1,K3)*IDENT(K2,K4)+IDENT(K1,K4)*IDENT(K2,K3))
3 -W3*IDENT(K1,K2)*IDENT(K3,K4)+IDENT(K1,K2)*IDENT(K3,K4)*W2
4 +(IDENT(K1,K4)*IDENT(K2,K3)+IDENT(K1,K3)*IDENT(K2,K4))*GAG
5 +DDSDDE(K5,K6)
ENDDO
ENDDO
ENDIF
C
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
C DDSDDE DERIVATION VIA PERTURBATION METHOD (THE SECOND PART)
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
C
IF (DDS.EQ.1) THEN
IF (KKK.LT.NTENS) THEN
DO K5=1,NTENS
DDSDDE(K5,KKK)=STRESS(K5)*DET
ENDDO
KKK=KKK+1
i=INDEX(1,KKK)
j=INDEX(2,KKK)
GO TO 80
ENDIF
IF (KKK.EQ.NTENS) THEN
DO K5=1,NTENS
DDSDDE(K5,KKK)=STRESS(K5)*DET
ENDDO
DO K1=1,3
DO K2=1,3
DFGRD1(K2,K1)=DFGRD(K2,K1)
ENDDO
ENDDO
KKK=KKK+1
GO TO 90
ENDIF
W2=ONE/EP
W1=W2/DET
DO K6=1,NTENS
DO K5=1,NTENS
DDSDDE(K5,K6)=W1*DDSDDE(K5,K6)-W2*STRESS(K5)
ENDDO
ENDDO
ENDIF
C
RETURN
END
C
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
C THE FOLLOWING SUBROUTINES ARE INTERNAL TO THE UMAT SUBROUTINE
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
C
C TRANSPOSE(A)
C
SUBROUTINE TRANSPOSE(A,AT)
INTEGER i , j
DOUBLE PRECISION A(3,3), AT(3,3)
Do i = 1 , 3
DO j = 1 , 3
AT(j,i) = A(i,j)
ENDDO
ENDDO
RETURN
END
C
C DETERMINANT(A)
C
SUBROUTINE DETERMINANT(A,DET)
DOUBLE PRECISION A(3,3), DET
DET=A(1,1)*A(2,2)*A(3,3)-A(1,1)*A(2,3)*A(3,2)
1 -A(2,1)*A(1,2)*A(3,3)+A(2,1)*A(1,3)*A(3,2)
2 +A(3,1)*A(1,2)*A(2,3)-A(3,1)*A(1,3)*A(2,2)
RETURN
END
C
C INNER PRODUCT OF TWO MATRICES IN VOIGT-NOTATION
C
SUBROUTINE VMATMUL(A,B,N,C)
DOUBLE PRECISION A(3,3), B(3,3), C(N)
INTEGER i , j
C(1)=A(1,1)*B(1,1)+A(1,2)*B(2,1)+A(1,3)*B(3,1)
C(2)=A(2,1)*B(1,2)+A(2,2)*B(2,2)+A(2,3)*B(3,2)
C(3)=A(3,1)*B(1,3)+A(3,2)*B(2,3)+A(3,3)*B(3,3)
C(4)=A(1,1)*B(1,2)+A(1,2)*B(2,2)+A(1,3)*B(3,2)
IF (N.EQ.6) THEN
C(5)=A(1,1)*B(1,3)+A(1,2)*B(2,3)+A(1,3)*B(3,3)
C(6)=A(2,1)*B(1,3)+A(2,2)*B(2,3)+A(2,3)*B(3,3)
ENDIF
RETURN
END
C
C MATRIX FORM OF A VECTOR
C
SUBROUTINE TENF(V,N,M)
DOUBLE PRECISION V(N),M(3,3)
M(1,1)=V(1)
M(2,1)=V(4)
IF (N.EQ.6) THEN
M(3,1)=V(5)
ELSE
M(3,1)=0
ENDIF
M(1,2)=V(4)
M(2,2)=V(2)
IF (N.EQ.6) THEN
M(3,2)=V(6)
M(1,3)=V(5)
M(2,3)=V(6)
ELSE
M(3,2)=0
M(1,3)=0
M(2,3)=0
ENDIF
M(3,3)=V(3)
RETURN
END
C
END