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UEL_Cohesive.for
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UEL_Cohesive.for
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C
C ABAQUS user subroutine for a 2D cohesive zone model, based on the paper by
C Y.F. Gao and A.F. Bower, "A simple technique for avoiding convergence
C problems in finite element simulations of crack nucleation and growth on
C cohesive interfaces," Modelling Simul. Mater. Sci. Eng. 12, 453-463, 2004.
C
C The Fortran code and the user's manual are last modified on December 9, 2004.
C
C Y.F. Gao & A.F. Bower, Divsion of Engineering, Brown University
C
C=========================== SUBROUTINE UEL ===================
C
SUBROUTINE UEL(RHS,STIF,SVARS,ENERGY,NDOFEL,NRHS,NSVARS,
& PROPS,NPROPS,COORDS,MCRD,NNODE,U,DU,V,A,JTYPE,TIME,DTIME,
& KSTEP,KINC,JELEM,PARAMS,NDLOAD,JDLTYP,ADLMAG,PREDEF,NPREDF,
& LFLAGS,MLVARX,DDLMAG,MDLOAD,PNEWDT,JPROPS,NJPROP,PERIOD)
C
INCLUDE 'ABA_PARAM.INC'
C IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DIMENSION RHS(MLVARX,*),STIF(NDOFEL,NDOFEL),PROPS(*)
DIMENSION SVARS(*),ENERGY(8),COORDS(MCRD,NNODE),U(NDOFEL)
DIMENSION DU(MLVARX,*),V(NDOFEL),A(NDOFEL),TIME(2),PARAMS(*)
DIMENSION JDLTYP(MDLOAD,*),ADLMAG(MDLOAD,*),DDLMAG(MDLOAD,*)
DIMENSION PREDEF(2,NPREDF,NNODE),LFLAGS(*),JPROPS(*)
C
C Only relevant variables are commented. Refer to ABAQUS manual for others.
C
C Variables with intent(out)
C RHS Residual Vector
C STIF Stiffness matrix
C
C Variables with intent(in)
C PROPS Element property array for the Xu-Needleman model
C PROPS(1) SIGMA_max
C PROPS(2) Delta_n
C PROPS(3) Delta_t
C PROPS(4) Q
C PROPS(5) R
C COORDS Nodal coordinate array: COORD(J,N) is jth coord of nth node
C U Total accumulated DOF array. Contains accumulated displacements,
C ordered as (u_i^1, u_i^2)
C DU Incremental displacements, ordered as
C DU(2*(N-1)+I,1) = u_i^n
C
C NNODE No. nodes on element.
C NDOFEL No. degrees of freedom for element
C NPROPS No. real valued element properties
C MLVARX Dimensioning variable.
C NRHS No. RHS vectors.
C MCRD Largest of max value of COORDINATES parameter or active DOF <3.
C
C NNODE=4 Four-point element (4,3 should collapse onto 1,2 respectively)
C
C 4----3
C | |
C 1----2
C
C NDOFEL=8 Two-dimensional, d.o.f=4*2
C NPROPS=5 Five parameters passed in
C MLVARX=1
C NRHS=1
C MCRD=2 (X,Y)
C
DIMENSION W(4),PQ(4)
DIMENSION F(10),DF(10)
DIMENSION GAP(2),TRACT(2),DTDG(2,2)
DIMENSION RNM(2),TANGENT(2),UREL(2)
DIMENSION VGAP(2),VREL(2)
NINTP=2
C
C The above arrays are used inside the subroutine
C NINTP No. integration points
C PQ(NINTP) Local coordinates of integration points
C W(NINTP) Weights of integration points
C F(NNODE) Shape function
C DF(NNODE) Derivative of shape function w.r.t local coordinate
C
DO I = 1,NDOFEL
RHS(I,1)=0.D0
DO J = 1,NDOFEL
STIF(J,I) = 0.D0
END DO
END DO
C
C Set up integration points and weights
CALL KINTPT(PQ,W,NNODE,NINTP)
C
DO I=1,NINTP
C
C Shape functions and derivatives
CALL KSHAPE(NNODE,PQ(I),F)
CALL DSHAPE(NNODE,PQ(I),DF)
C
C Compute Normal and tangent vectors to boundary at int pt.
C Plane of interface defined as average of upper and lower
C surfaces (hopefully coincident, but you never know...)
TANGENT(1)=.5D0*(COORDS(1,2)-COORDS(1,1)
& +COORDS(1,3)-COORDS(1,4))
TANGENT(2)=.5D0*(COORDS(2,2)-COORDS(2,1)
& +COORDS(2,3)-COORDS(2,4))
CALL KUNITV(TANGENT,DET)
DET=.5D0*DET
RNM(1)=-TANGENT(2)
RNM(2)=TANGENT(1)
C
C Relative displacement, in global coords
UREL(1) = 0.D0
UREL(2) = 0.D0
VREL(1)=0.D0
VREL(2)=0.D0
DO N = 1,NNODE/2
N2 = N+NNODE/2
DO J = 1,2
UREL(J) = UREL(J) + F(N2)*(U(2*(N2-1)+J))
& - F(N)*(U(2*(N-1)+J))
VREL(J) = VREL(J) + F(N2)*(V(2*(N2-1)+J))
& - F(N)*(V(2*(N-1)+J))
END DO
END DO
C
C GAP(1) is normal separation, GAP(2) is separation in tan dirn
GAP(1) = RNM(1)*UREL(1)+RNM(2)*UREL(2)
GAP(2) = TANGENT(1)*UREL(1)+TANGENT(2)*UREL(2)
VGAP(1)=RNM(1)*VREL(1)+RNM(2)*VREL(2)
VGAP(2)=TANGENT(1)*VREL(1)+TANGENT(2)*VREL(2)
C
CALL SEPLAW(PROPS,GAP,VGAP,TRACT,DTDG,DTIME)
C
SIG = 1.D0
DO N = 1,NNODE
IF (N.GT.NNODE/2) SIG = -1.D0
DO K = 1,2
RHS(2*(N-1)+K,1) = RHS(2*(N-1)+K,1)
& + SIG*F(N)*( TRACT(1)*RNM(K)
& + TRACT(2)*TANGENT(K))*W(I)*DET
END DO
END DO
SIGN = 1.D0
DO N = 1,NNODE
IF (N.GT.NNODE/2) SIGN = -1.D0
SIGM = 1.D0
DO M=1,NNODE
IF (M.GT.NNODE/2) SIGM = -1.D0
DO KN = 1,2
DO KM = 1,2
ICOL = 2*(M-1) + KM
IROW = 2*(N-1) + KN
C
STIF(ICOL,IROW) = STIF(ICOL,IROW)
& +( (DTDG(1,1)*RNM(KN)+DTDG(2,1)*TANGENT(KN))*RNM(KM)
& +(DTDG(1,2)*RNM(KN)+DTDG(2,2)*TANGENT(KN))*TANGENT(KM) )
& *SIGM*SIGN*F(M)*F(N)*W(I)*DET
END DO
END DO
C
END DO
END DO
C
END DO
C
RETURN
END
C
C===================== SUBROUTINE SEPLAW =====================
C
SUBROUTINE SEPLAW(PROPS,GAP,VGAP,TRACT,DTDGAP,DTIME)
C
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
C
DIMENSION PROPS(5)
DIMENSION GAP(2),VGAP(2),TRACT(2),DTDGAP(2,2)
C
C Subroutine to specify traction-separation law for
C a debonding interface.
C
C Currently coded for Xu-Needleman constitutive law, i.e.
C X.P. Xu and A. Needleman, J. Mech. Phys. Solids, 42, 1397-1434.
C
C PROPS(1) SIGMA_max
C PROPS(2) Delta_n
C PROPS(3) Delta_t
C PROPS(4) Q
C PROPS(5) R
C
C GAP(1) Normal separation
C GAP(2) Tangential separation
C TRACT(1) Normal traction
C TRACT(2) Tangential traction
C
C
C -- Work of separation
SEPWRK = DEXP(1.D0)*PROPS(1)*PROPS(2)
DN = PROPS(2)
DT = PROPS(3)
Q = PROPS(4)
R = PROPS(5)
C
C
C1 = ( 1.D0-DEXP(-GAP(2)*GAP(2)/(DT*DT)) )
C1 = C1 * (1.D0-Q)/(R-1.D0) * (R - GAP(1)/DN)
C2 = (GAP(1)/DN)*DEXP(-GAP(2)*GAP(2)/(DT*DT))
TRACT(1) = (SEPWRK/DN)*DEXP(-GAP(1)/DN)*(C2+C1)
C
C1 = Q + (R-Q)/(R-1.D0)*(GAP(1)/DN)
C1 = C1 * DEXP(-GAP(1)/DN) * DEXP(-GAP(2)*GAP(2)/(DT*DT))
C1 = C1 * 2.D0*(DN/DT)*(SEPWRK/DN)
TRACT(2) = C1*GAP(2)/DT
C
C1 = (1.D0-Q)/(R-1)*(1.D0-DEXP(-GAP(2)*GAP(2)/(DT*DT)))
C1 = C1 * (R+1.D0-GAP(1)/DN)
C1 = (1.D0-GAP(1)/DN)*DEXP(-GAP(2)*GAP(2)/(DT*DT)) - C1
DTDGAP(1,1) = (SEPWRK/(DN*DN))*DEXP(-GAP(1)/DN) * C1
C
C1 = Q + (GAP(1)/DN) * (R-Q)/(R-1.D0)
C1 = C1 * DEXP(-GAP(1)/DN)*DEXP(-GAP(2)*GAP(2)/(DT*DT))
C1 = 2.D0*(SEPWRK/(DT*DT)) * C1
DTDGAP(2,2) = C1*(1.D0-2.D0*(GAP(2)*GAP(2)/(DT*DT)))
C
C1 = -GAP(1)/DN + (1.D0-Q)/(R-1.D0)*(R-GAP(1)/DN)
C1 = C1 * DEXP(-GAP(1)/DN)*DEXP(-GAP(2)*GAP(2)/(DT*DT))
C1 = 2.D0*(SEPWRK/(DT*DN))*C1
DTDGAP(1,2) = (GAP(2)/DT)*C1
DTDGAP(2,1) = DTDGAP(1,2)
C
C ZETA is the fictitious viscosity used to regularize the instability
C problem. In our papere, \zeta_n is ZETA*PROPS(1) here. Simply
C setting ZETA=0.D0 (or delete the following three lines
C gives the standard Xu-Needleman model.
if (dtime>0.d0) then
ZETA=0.0001D0
TRACT(1)=TRACT(1)+ZETA*PROPS(1)*VGAP(1)/DN
DTDGAP(1,1)=DTDGAP(1,1)+ZETA*PROPS(1)/DN/DTIME
endif
C
RETURN
END
C
C
C========================== SUBROUTINE INTPTS =================================
C
SUBROUTINE KINTPT(PQ,W,KNODE,NINTP)
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
C
DIMENSION PQ(*),W(*)
C
C Subroutine to initialize integration points PQ and weights W
C NINTP = No. integration points (DEFAULT=2)
C KNODE = No. of nodes on element (DEFAULT=4)
C
C Integration points for 2D interface elements
C
C 4----3
C | |
C | |
C 1----2
c
IF (NINTP.EQ.2) THEN
CN=0.5773502691896260D0
PQ(1)=-CN
PQ(2)=CN
W(1)=1.D0
W(2)=1.D0
ENDIF
C
RETURN
END
C
C=========================== SUBROUTINE SHAPE ===============================
C
SUBROUTINE KSHAPE(KNODE,PQ,F)
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
C
DIMENSION F(*)
C
C
IF(KNODE.EQ.4) THEN
F(1)=.5D0*(1-PQ)
F(2)=.5D0*(1+PQ)
F(3)=F(2)
F(4)=F(1)
ENDIF
RETURN
END
C
C============================== SUBROUTINE DSHAPE ====================
C
SUBROUTINE DSHAPE(KNODE,PQ,DF)
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
C
DIMENSION DF(*)
C
C
IF(KNODE.EQ.4) THEN
DF(1)=-.5D0
DF(2)=.5D0
DF(3)=DF(2)
DF(4)=DF(1)
ENDIF
RETURN
END
C
C========================== SUBROUTINE KUNITV ===================
C
SUBROUTINE KUNITV(A,AMAG)
DOUBLE PRECISION A(2), AMAG
C
C Normalize vector A and return its magnitude as AMAG
C
AMAG = DSQRT(A(1)*A(1)+A(2)*A(2))
A(1) = A(1)/AMAG
A(2) = A(2)/AMAG
C
RETURN
END