Writing a VUMAT. Appendix 3. Overview

Transcription

1 ABAQUS/Expliit: Advaned Topis Appendix 3 Writing a VUMAT ABAQUS/Expliit: Advaned Topis A3.2 Overview Introdution Motivation Steps Required in Writing a VUMAT Example: VUMAT for Kinemati Hardening Plastiity

2 ABAQUS/Expliit: Advaned Topis Introdution ABAQUS/Expliit: Advaned Topis A3.4 Introdution ABAQUS/Expliit has an interfae that allows you to implement general onstitutive equations. In ABAQUS/Expliit the user-defined material model is implemented in user subroutine VUMAT. Use VUMAT when none of the existing material models inluded in the ABAQUS/Expliit material library aurately represents the behavior of the material to be modeled.

3 ABAQUS/Expliit: Advaned Topis A3.5 Introdution These interfaes make it possible to define any (proprietary) onstitutive model of arbitrary omplexity. User-defined material models an be used with any ABAQUS/Expliit strutural element type. Multiple user materials an be implemented in a single VUMAT routine and an be used together. In this leture the implementation of material models in VUMAT will be disussed and illustrated with an example. ABAQUS/Expliit: Advaned Topis Motivation

4 ABAQUS/Expliit: Advaned Topis A3.7 Motivation Proper testing of advaned onstitutive models to simulate experimental results often requires omplex finite element models. Complex material modeling Speial analysis problems our if the onstitutive model simulates material instabilities and loalization phenomena. The material model developer should be onerned only with the development of the material model and not with the development and maintenane of the FE software. Developments unrelated to material modeling Porting problems with new systems Long-term program maintenane of user-developed ode ABAQUS/Expliit: Advaned Topis Steps Required in Writing a VUMAT

5 ABAQUS/Expliit: Advaned Topis A3.9 Steps Required in Writing a VUMAT Proper definition of the onstitutive equation, whih requires one of the following: Expliit definition of stress (Cauhy stress for large-strain appliations) Definition of the stress rate only (in orotational framework) Furthermore, it is likely to require: Definition of dependene on time, temperature, or field variables Definition of internal state variables, either expliitly or in rate form ABAQUS/Expliit: Advaned Topis A3.10 Steps Required in Writing a VUMAT Transformation of the onstitutive rate equation into an inremental equation using a suitable integration proedure: Forward Euler (expliit integration) Bakward Euler (impliit integration) Midpoint method

6 ABAQUS/Expliit: Advaned Topis A3.11 Steps Required in Writing a VUMAT This is the hard part! Forward Euler (expliit) integration methods are simple but have a stability limit, ε < ε stab, where ε stab is usually less than the elasti strain magnitude. For expliit integration the time inrement must be ontrolled. For impliit or midpoint integration, the algorithm is more ompliated and often requires loal iteration. However, there is usually no stability limit. An inremental expression for the internal state variables must also be obtained. ABAQUS/Expliit: Advaned Topis A3.12 Steps Required in Writing a VUMAT Coding the VUMAT. Follow FORTRAN 77 or C onventions. Make sure that the ode an be vetorized. Make sure that all variables are defined and initialized properly. Use ABAQUS utility routines as required. Assign enough storage spae for state variables with the DEPVAR option.

7 ABAQUS/Expliit: Advaned Topis A3.13 Steps Required in Writing a VUMAT Verifying the VUMAT with a small (one element) input file. 1 2 Run tests with all displaements presribed to verify the integration algorithm for stresses and state variables. Suggested tests inlude: Uniaxial Uniaxial in oblique diretion Uniaxial with finite rotation Finite shear Compare test results with analytial solutions or standard ABAQUS material models, if possible. If the above verifiation is suessful, apply to more ompliated problems. ABAQUS/Expliit: Advaned Topis

8 ABAQUS/Expliit: Advaned Topis A3.15 The following ats as the interfae to a VUMAT in whih kinemati hardening plastiity is defined. *MATERIAL, NAME=KINPLAS *USER MATERIAL, CONSTANTS=4 30.E6, 0.3, 30.E3, 40.E3 *DEPVAR 5 *INITIAL CONDITIONS, TYPE=SOLUTION Data line to speify initial solutiondependent variables ABAQUS/Expliit: Advaned Topis A3.16 The user subroutine, whih must be kept in a separate file, is invoked with the ABAQUS exeution proedure, as follows: abaqus job=... user=... The user subroutine must be invoked in a restarted analysis beause user subroutines are not saved on the restart file.

9 ABAQUS/Expliit: Advaned Topis A3.17 Additional notes: Solution-dependent state variables an be output with identifiers SDV1, SDV2, et. Contour, path, and X Y plots of SDVs an be plotted in ABAQUS/Viewer. Inlude only a single VUMAT subroutine in the analysis. If more than one material must be defined, test on the material name in the VUMAT routine and branh. ABAQUS/Expliit: Advaned Topis A3.18 The VUMAT subroutine header is shown below: subroutine vumat( Read only (unmodifiable) variables- 1 nblok, ndir, nshr, nstatev, nfieldv, nprops, lanneal, 2 steptime, totaltime, dt, mname, oordmp, harlength, 3 props, density, strainin, relspinin, 4 tempold, strethold, defgradold, fieldold, 5 stressold, stateold, enerinternold, enerinelasold, 6 tempnew, strethnew, defgradnew, fieldnew, write only (modifiable) variables - 7 stressnew, statenew, enerinternnew, enerinelasnew) inlude 'vaba_param.in'

10 ABAQUS/Expliit: Advaned Topis A3.19 dimension props(nprops), density(nblok), oordmp(nblok), 1 harlength(nblok), strainin(nblok, ndir+nshr), 2 relspinin(nblok, nshr), tempold(nblok), 3 strethold(nblok, ndir+nshr),defgradold(nblok,ndir+nshr+nshr), 4 fieldold(nblok, nfieldv), stressold(nblok, ndir+nshr), 5 stateold(nblok, nstatev), enerinternold(nblok), 6 enerinelasold(nblok), tempnew(nblok), 7 strethnew(nblok, ndir+nshr),defgradnew(nblok,ndir+nshr+nshr), 8 fieldnew(nblok, nfieldv), stressnew(nblok,ndir+nshr), 9 statenew(nblok, nstatev), enerinternnew(nblok), 1 enerinelasnew(nblok) harater*80 mname ABAQUS/Expliit: Advaned Topis A3.20 VUMAT variables The following quantities are available in VUMAT, but they annot be redefined: Stress, streth, and SDVs at the start of the inrement Relative rotation vetor and deformation gradient at the start and end of an inrement and strain inrement Total and inremental values of time, temperature, and user-defined field variables at the start and end of an inrement Material onstants, density, material point position, and a harateristi element length Internal and dissipated energies at the beginning of the inrement Number of material points to be proessed in a all to the routine (NBLOCK) A flag indiating whether the routine is being alled during an annealing proess

11 ABAQUS/Expliit: Advaned Topis A3.21 The following quantities must be defined: Stress and SDVs at the end of an inrement The following variables may be defined: Internal and dissipated energies at the end of the inrement Many of these variables are equivalent or similar to those in UMAT. Complete desriptions of all parameters are provided in the VUMAT setion in Chapter 25 of the ABAQUS Analysis User s Manual. ABAQUS/Expliit: Advaned Topis A3.22 The header is usually followed by dimensioning of loal arrays. It is good pratie to define onstants via parameters and to inlude omments. parameter( zero = 0.d0, one = 1.d0, two = 2.d0, three = 3.d0, 1 third = one/three, half = 0.5d0, two_thirds = two/three, 2 three_halfs = 1.5d0) The parameter assignments yield aurate floating point onstant definitions on any platform.

12 ABAQUS/Expliit: Advaned Topis A3.23 VUMAT onventions Stresses and strains are stored as vetors. For plane stress elements: σ 11, σ 22, σ 12. For plane strain and axisymmetri elements: σ 11, σ 22, σ 33, σ 12. For three-dimensional elements: σ 11, σ 22, σ 33, σ 12,, σ 23, σ 31. For three-dimensional elements, this storage sheme is different than that for ABAQUS/Standard. The shear strain is stored as tensor shear strains, 1 ε12 = γ12. 2 ABAQUS/Expliit: Advaned Topis A3.24 The deformation gradient is stored similar to the way in whih symmetri tensors are stored. For plane stress elements: F 11, F 22, F 12, F 21. For plane strain and axisymmetri elements: F 11, F 22, F 33, F 12, F 21. For three-dimensional elements: F 11, F 22, F 33, F 12, F 23, F 31, F 21, F 32, F 13.

13 ABAQUS/Expliit: Advaned Topis A3.25 VUMAT formulation aspets Vetorized interfae In VUMAT the data are passed in and out in large bloks (dimension nblok). nblok typially is equal to 64 or 128. Eah entry in an array of length nblok orresponds to a single material point. All material points in the same blok have the same material name and belong to the same element type. This struture allows vetorization of the routine. A vetorized VUMAT should make sure that all operations are done in vetor mode with nblok the vetor length. In vetorized ode branhing inside loops should be avoided. Element type based branhing should be outside the nblok loop. ABAQUS/Expliit: Advaned Topis A3.26 Corotational formulation The onstitutive equation is formulated in a orotational framework, based on the Green-Naghdi rate. The inremental rotation is obtained from the total rotation T F = R U with the expression Ω = R R. The strain inrement is obtained with Hughes-Winget. Other measures an be obtained from the deformation gradient. The relative spin inrement ω Ω is also provided. The quantity ω orresponds to the Jaumann rate. In ABAQUS it is used in ertain instanes: e.g., solid elements using the built-in linear elasti and plasti material models.

14 ABAQUS/Expliit: Advaned Topis A3.27 The user must define the Cauhy stress: this stress reappears during the next inrement as the old stress. There is no need to rotate tensor state variables. A rotation is needed, however, if a rate other than the Green- Naghdi rate is desired. For example, to use the Jaumann rate, evaluate the expression defined by Hughes and Winget for the rotation inrement using the relative spin inrement: R = I ( ω Ω ) + ( ) 2 I ω Ω 2 Then, rotate all old tensor quantities before performing onstitutive updates. For example, for the stress tensor: stressold stressnew t t = t T σ + R σ R + σ ABAQUS/Expliit: Advaned Topis A3.28 VUMATs and hyperelastiity Hyperelasti onstitutive equations relate the Cauhy stress σ to the deformation gradient F through the left Cauhy-Green deformation tensor B. Using F for hyperelasti onstitutive models in a VUMAT presents some diffiulties, however, beause ABAQUS/Expliit uses a orotational system whih automatially aounts for rigid body rotations. The deformation gradient that is passed into the VUMAT is referred to a fixed basis assoiated with the original onfiguration. It also inorporates the rotations reall the deformation gradient an be written as F = RU, where R is the rotation tensor and U is the streth tensor.

15 ABAQUS/Expliit: Advaned Topis A3.29 Thus, to avoid inluding the effets of the rotations twie, hyperelasti onstitutive models implemented in a VUMAT should be formulated in terms of the streth tensor U. This allows you to obtain the orotational Cauhy stress diretly. For example, for neo-hookean hyperelastiity: 2 C 1 2 σ = 10 B tr( B) I + ( 1) J 3 D J I B = B J 23,. Substituting F = RU into the above expressions yields: 2 C tr( ) ( 1) J 3 D J T σ = R U U I + I R, where U = U J. 1 1 The orotational stress is the quantity ontained within the urly brakets: orot σ = C10 U tr( U ) I + ( J 1) I. J 3 D 1 ABAQUS/Expliit: Advaned Topis Example: VUMAT for Kinemati Hardening Plastiity

16 ABAQUS/Expliit: Advaned Topis A3.31 Example: VUMAT for Kinemati Hardening Plastiity Governing equations Elastiity: el el ij ij kk ij σ = λδ ε + 2µε, or in a Jaumann (orotational) rate form: J el el ij ij& kk s & ij & σ = λδ ε + µε. The Jaumann rate equation is integrated in a orotational framework: J el el ij ij kk ij σ = λδ ε + 2µ ε. ABAQUS/Expliit: Advaned Topis A3.32 Example: VUMAT for Kinemati Hardening Plastiity Plastiity: Yield funtion: 3 2 ( S )( S ) α α σ = 0. ij ij ij ij y Equivalent plasti strain rate: & pl 2 pl pl ε = & εij & εij. 3 Plasti flow law: pl 3 pl & εij = ( Sij αij ) & ε σ y. 2 Prager-Ziegler (linear) kinemati hardening: 2 pl & αij = h& εij. 3

17 ABAQUS/Expliit: Advaned Topis A3.33 Example: VUMAT for Kinemati Hardening Plastiity Integration proedure We first alulate the equivalent stress based on purely elasti behavior (elasti preditor), 0 ( )( ) pr 3 pr o pr pr o σ = Sij αij Sij α ij, Sij = Sij + 2µ eij. 2 Plasti flow ours if the elasti preditor is larger than the yield stress. The bakward Euler method is used to integrate the equations, ( S ) 3 ε pl pr o pl pr ij = ij αij ε σ. 2 After some manipulation we obtain a losed form expression for the equivalent plasti strain inrement, pl pr ( y ) ( h 3 ) ε = σ σ + µ. ABAQUS/Expliit: Advaned Topis A3.34 Example: VUMAT for Kinemati Hardening Plastiity This leads to the following update equations for the shift tensor, the stress, and the plasti strain: pl pl 3 pl αij = ηijh ε, εij = ηij ε, 2 o 1 pr pr o pr σ ij = αij + αij + ηijσ y + δijσ kk, ηij = ( Sij αij ) σ. 3 The integration proedure for kinemati hardening is desribed in the ABAQUS Analysis User s Manual. The appropriate oding is shown on the following pages.

18 ABAQUS/Expliit: Advaned Topis A3.35 Example: VUMAT for Kinemati Hardening Plastiity Coding for kinemati hardening plastiity VUMAT J2 Mises plastiity with kinemati hardening for plane strain ase. The state variables are stored as: state(*, 1) = bak stress omponent 11 state(*, 2) = bak stress omponent 22 state(*, 3) = bak stress omponent 33 state(*, 4) = bak stress omponent 12 state(*, 5) = equivalent plasti strain e = props(1) xnu = props(2) yield = props(3) hard = props(4) elasti onstants twomu = e / ( one + xnu ) thremu = three_halfs * twomu sixmu = three * twomu alamda = twomu * ( e - twomu ) / ( sixmu - two * e ) term = one / ( twomu * ( one + hard/thremu ) ) on1 = sqrt( two_thirds ) ABAQUS/Expliit: Advaned Topis A3.36 Example: VUMAT for Kinemati Hardening Plastiity If steptime equals to zero, assume the material pure elasti and use initial elasti modulus if( steptime.eq. zero ) then do i = 1, nblok C C Trial Stress trae = strainin (i, 1) + strainin (i, 2) + strainin (i, 3) stressnew(i, 1)=stressOld(i, 1) + alamda*trae 1 + twomu*strainin(i,1) stressnew(i, 2)=stressOld(i, 2) + alamda*trae 1 + twomu*strainin(i, 2) stressnew(i, 3)=stressOld(i, 3) + alamda*trae 1 + twomu*strainin(i,3) stressnew(i, 4)=stressOld(i, 4) 1 + twomu*strainin(i, 4) end do else

19 ABAQUS/Expliit: Advaned Topis A3.37 Example: VUMAT for Kinemati Hardening Plastiity C C Plastiity alulations in blok form C do i = 1, nblok C Elasti preditor stress trae = strainin(i, 1) + strainin(i, 2) + strainin(i, 3) sig1= stressold(i, 1) + alamda*trae + twomu*strainin(i, 1) sig2= stressold(i, 2) + alamda*trae + twomu*strainin(i, 2) sig3= stressold(i, 3) + alamda*trae + twomu*strainin(i, 3) sig4= stressold(i, 4) + twomu*strainin(i, 4) C Elasti preditor stress measured from the bak stress s1 = sig1 - stateold(i, 1) s2 = sig2 - stateold(i, 2) s3 = sig3 - stateold(i, 3) s4 = sig4 - stateold(i, 4) C Deviatori part of preditor stress measured from the bak stress smean = third * ( s1 + s2 + s3 ) ds1 = s1 - smean ds2 = s2 - smean ds3 = s3 - smean C Magnitude of the deviatori preditor stress differene dsmag = sqrt( ds1**2 + ds2**2 + ds3**2 + two*s4**2 ) ABAQUS/Expliit: Advaned Topis A3.38 Example: VUMAT for Kinemati Hardening Plastiity Chek for yield by determining the fator for plastiity, zero for elasti, one for yield radius = on1 * yield fayld = zero if( dsmag - radius.ge. zero ) fayld = one Add a protetive addition fator to prevent a divide by zero when DSMAG is zero. If DSMAG is zero, we will not have exeeded the yield stress and FACYLD will be zero. dsmag = dsmag + ( one - fayld ) Calulated inrement in gamma ( this expliitly inludes the time step) diff = dsmag - radius dgamma = fayld * term * diff

20 ABAQUS/Expliit: Advaned Topis A3.39 Example: VUMAT for Kinemati Hardening Plastiity Update equivalent plasti strain deqps = on1 * dgamma statenew(i, 5) = stateold(i, 5) + deqps Divide DGAMMA by DSMAG so that the deviatori stresses are expliitly onverted to tensors of unit magnitude in the following alulations dgamma = dgamma / dsmag Update bak stress fator = hard * dgamma * two_thirds statenew(i, 1) = stateold(i, 1) + fator * ds1 statenew(i, 2) = stateold(i, 2) + fator * ds2 statenew(i, 3) = stateold(i, 3) + fator * ds3 statenew(i, 4) = stateold(i, 4) + fator * s4 ABAQUS/Expliit: Advaned Topis A3.40 Example: VUMAT for Kinemati Hardening Plastiity Update stress fator = twomu * dgamma stressnew(i, 1) = sig1 - fator * ds1 stressnew(i, 2) = sig2 - fator * ds2 stressnew(i, 3) = sig3 - fator * ds3 stressnew(i, 4) = sig4 - fator * s4 Update the speifi internal energy - stresspower = half * ( 1 ( stressold(i, 1)+stressNew(i, 1) )*strainin(i, 1) 2 + ( stressold(i, 2)+stressNew(i, 2) )*strainin(i, 2) 3 + ( stressold(i, 3)+stressNew(i, 3) )*strainin(i, 3) 4 + two*( stressold(i, 4)+stressNew(i, 4) )*strainin(i, 4) ) enerinternnew(i) = enerinternold(i) 1 + stresspower/density(i)

21 ABAQUS/Expliit: Advaned Topis A3.41 Example: VUMAT for Kinemati Hardening Plastiity Update the dissipated inelasti speifi energy - smean = third* (stressnew(i, 1)+stressNew(i, 2) 1 + stressnew(i, 3)) equivstress = sqrt( three_halfs 1 * ( (stressnew(i, 1)-smean)**2 2 + (stressnew(i, 2)-smean)**2 3 + (stressnew(i, 3)-smean)**2 4 + two * stressnew(i, 4)**2 ) ) plastiworkin = equivstress * deqps enerinelasnew(i) = enerinelasold(i) 1 + plastiworkin / density(i) end do end if return end ABAQUS/Expliit: Advaned Topis A3.42 Example: VUMAT for Kinemati Hardening Plastiity Remarks In the datahek phase, VUMAT is alled with a set of fititious strains and a TOTALTIME and STEPTIME both equal to 0.0. A hek is done on the user s onstitutive relation, and an initial stable time inrement is determined based on alulated equivalent initial material properties. You should ensure that elasti properties are used in this all to VUMAT; otherwise, too large an initial time inrement may be used, leading to instability. A warning message is printed to the status (.sta) file, informing the user that this hek is being performed.

22 ABAQUS/Expliit: Advaned Topis A3.43 Example: VUMAT for Kinemati Hardening Plastiity Speial oding tehniques are used to obtain vetorized oding. All small loops inside the material routine are unrolled. The same ode is exeuted regardless of whether the behavior is purely elasti or elasti plasti. Speial are must be taken to avoid divides by zero. No external subroutines are alled inside the loop. The use of loal salar variables inside the loop is allowed. The ompiler will automatially expand these loal salar variables to loal vetors. Iterations should be avoided. If iterations annot be avoided, use a fixed number of iterations and do not test on onvergene.