1、Time Domain ViscoelasticityLecture 8Copyright 2006 ABAQUS, Inc.L8.2OverviewClassical Linear Viscoelasticity Prony Series Representation Finite-Strain Viscoelasticity Relaxation and Creep Test Data Prony Series DataAutomatic Material EvaluationUsage HintsModeling Rubber and Viscoelasticity with ABAQU

2、SCopyright 2006 ABAQUS, Inc.Classical Linear ViscoelasticityCopyright 2006 ABAQUS, Inc.L8.4Classical Linear ViscoelasticityRecall that classical means small-strain theory.Isotropic linear viscoelasticity is implemented in ABAQUSModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, In

3、c.In ABAQUS this means: relaxation/creep behavior defined by *VISCOELASTIC elasticity is defined by *ELASTICL8.5Classical Linear ViscoelasticityTo generalize the viscoelastic equations to multiaxial stress states it is best to work with shear (deviatoric) and volumetric (dilatational) behavior: dG(t

4、 ) SS (t ) = S (t) + 1 (t -t ) dt00dtG00 dK (t ) pp (t ) = p (t) + 1 (t -t ) dt00dtK00where S is the deviatoric stress tensor and p is the hydrostatic pressure.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.6Classical Linear ViscoelasticityDefinitionsWe decompose the to

5、tal stress into shear and volumetric parts by:wherep = - 1 tr(s ) = - 1 (so = S - pI+ s+ s).11223333We decompose the total strain into shear and volumetric parts by:e = e + 1 e vol Ie vol= tr(e ) = (e+ e+ e).where1122333The elastic stress-strain relations decompose into:e vol ,S 0 (t) = G0 e(t)andp0

6、 (t) = K0where S0 (t) and p0 (t)are the deviatoric and pressure stress states thatwould exist for the current strain state if the material were behavingpurely elastically.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Prony Series RepresentationCopyright 2006 ABAQUS, Inc.L

7、8.8Prony Series RepresentationIn ABAQUS the time-dependent behavior G(t ) and K(t ) are represented in terms of a Prony series:G(t ) = G0 1- gi(1- e)material coefficients areN-t /t Gpig pand t Gup to N pairs ofiii=1K (t ) = K0 1- ki(1- ematerial coefficients areN-t /t Kpiup to N pairs of kand t Kpii

8、i=1G0 and K0 are determined from the elasticity definition. These are simply a sum of a series of exponential decays.For many materials, including solid elastomers, the relaxation behavior is dominated by shear relaxation. In these cases it is not necessary toK (t ).specify An exception is void fill

9、ed elastomers (elastic foams) in which there is generally significant volumetric relaxation.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.9Prony Series RepresentationIn ABAQUS G(t ) and K(t )of four different ways:are specified in oneProny series curve fit from experim

10、ental stress relaxation test data:*VISCOLEASTIC, TIME=RELAXATION TEST DATAProny series curve fit from experimentalcreep test data:*VISCOELASTIC, TIME=CREEP TEST DATAProny series coefficients specified directly by the user:*VISCOELASTIC, TIME=PRONYDefined from frequency-dependent cyclic test data:*VI

11、SCOLEASTIC, TIME=FREQUENCY DATAModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.10Prony Series RepresentationThe rate-independent elastic moduli are specified using *ELASTIC*ELASTIC,E0, n0or*ELASTIC,E1, n1MODULI=INSTANTANEOUSMODULI=LONGTERMABAQUS will determine the approp

12、riate G and K values from the user specified E and n values.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.11Prony Series RepresentationIf the user specifies long-term elastic moduli, ABAQUS will compute the instantaneous elastic moduli using the relaxation informationN

13、 pN k =1k =1G = G1-p K = K0 1-gk0kkg pp kand kwhere theare the Prony series coefficients.kThe choice of defining the elasticity in terms of instantaneous or long- term is a matter of convenience only. However, the MODULI parameter defaults to LONG TERM. Thus, if you enter instantaneous data but omit

14、 the INSTANTANEOUS parameter, it will adversely affect your solution.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Finite-Strain ViscoelasticityCopyright 2006 ABAQUS, Inc.L8.13Finite-Strain ViscoelasticityPolynomial Strain Energy FunctionEnergy function has the form:N N1i

15、+ j=1i=1() (- 3)+ij( Jel-1).2iU =CI- 3Iij12DiThe Prony series acts as a non-dimensionalized multiplier:()N Cij (t ) = C ij 1- gi1- e-t /t G0pii=1( 1 1N 1- ki1- e-t /t Kp=iDi (t )0Dii=11C0andwheredefine the instantaneous shear and volumeijresponse.D0iModeling Rubber and Viscoelasticity with ABAQUSCop

16、yright 2006 ABAQUS, Inc.L8.14Finite-Strain ViscoelasticityOgdens Strain Energy FunctionEnergy Function has the form:NN2m()1( Jel-1),aaa2il+ l2+ l3- 3+U = iiii1a 2Dii=1ii=1The Prony series acts as a non-dimensionalized multiplier:()N -t /t Gmi (t ) =m 0pi 1-gi1- eii=1() 1 1N 1- ki1- e-t /t Kp=iDi (t

17、)0Dii=11m 0wheredefine the instantaneous shear and volumeandiresponse.D0iModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.15Finite-Strain ViscoelasticityMarlow Strain Energy FunctionThe Prony series acts as a non-dimensionalized multiplier:()N -t /t GRdev0devpt ) = U1-g1-

18、 eU(iii=1i()N -t /t KRt ) = U0volp1-k1- eU(ivoli=1where U 0and U 0define the instantaneous shear and volume straindevenergy functions.volModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.16Finite-Strain ViscoelasticityArruda-Boyce and Van der Waals Strain Energy FunctionTh

19、e strain energy functions are shown in Appendix 2.The Prony series acts as a non-dimensionalized multiplier:()m 0N -t /t Gm (t ) =p1-gi1- eii=1() 1 1 N =0 1- ki1- e-t /t KpiD(t )Di=11m 0andwheredefine the instantaneous shear and volumeD0response.Modeling Rubber and Viscoelasticity with ABAQUSCopyrig

20、ht 2006 ABAQUS, Inc.L8.17Finite-Strain ViscoelasticityHyperfoam Energy FunctionThe energy function is:N2m()1aaa-a b+ l?- 3 + bJ垐l+ lU = i-1,iiii i123ela 2ii=1iNote the deviatoric and volumetric behavior is fully coupled.*VISCOELASTIC should obey g pp ; that is, the shear and volume= kkrelaxation rat

21、es should be equal.kThe relaxation behavior is governed by Prony series:)(N -t /timi (t ) = m i 1-0pgi1- ei=1Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.18Finite-Strain ViscoelasticityHyperfoam Energy Function (contd)When using viscoelasticity in conjunction with the

22、 hyperfoam material model:You may use *SHEAR TEST DATA to specify the viscoelastic behavior. ABAQUS will set the volume behavior equal to the shear.You may use *VOLUMETRIC TEST DATA to specify the viscoelastic behavior. ABAQUS will set the shear behavior equal to the volumetric.You may use *COMBINED

23、 TEST DATA to specify the viscoelastic behavior. You should make the shear and volume data the same.You may use TIME=PRONY and specify the Prony coefficients directly. If you give just theterms, then the terms will be set to thesame value (and vicgep-versa).p kkkModeling Rubber and Viscoelasticity w

24、ith ABAQUSCopyright 2006 ABAQUS, Inc.Relaxation and Creep Test DataCopyright 2006 ABAQUS, Inc.L8.20Relaxation and Creep Test DataRelaxation Test DataConsider a simple shear relaxation testHere g0 is the instantaneous (short-time) applied shear strain, andt (t) is the measured shear stress response.N

25、ote: Be careful that the short-time duration of the prescribed strain is consistent with the time scale of your linear elastic or hyperelastic material definition.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.21Relaxation and Creep Test DataRelaxation Test Data (contd)

26、The measured shear stress response is data pairs of (shear stress, time)t0 , time0t1 , time1t2 , time2t3 , time3Because this is linear viscoelasticity only one curve may be used. If your material is not exactly linear in its viscoelastic response, then test at an applied strain that is close to your

27、 component analysis strain level of interest.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.22Relaxation and Creep Test DataRelaxation Test Data (contd)Test data processing for use in ABAQUS:Simply normalize all the measured stress values by t0:t0 / t0 , time0t1 / t0 ,

28、time1t2 / t0 , time2t3 / t0 , time3:and input this data using the *SHEAR*VISCOELASTIC, TIME=RELAXATIONTEST DATA suboption ofTEST DATA.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.23Relaxation and Creep Test DataExample 1: Relaxation Test Data Usage*MATERIAL,NAME=*ELAS

29、TIC, MODULI=. (data lines)or *HYPERLEASTIC, MODULI=*VISCOELASTIC, TIME=RELAXATION TESTERRTOL = 0.01, NMAX = 13*SHEAR TEST DATA, SHRINF = 0.5DATA,1.0000,0.9695,0.9417,.0.00010.0010.002*VOLUMETRIC TESTDATA,VOLINF = 0.51.0000,0.9695,0.9417,.0.00010.0010.002nonlinearleastsquares fitspairs of g p andt G

30、iipairs of kandt KpiiModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.24Relaxation and Creep Test DataExample 1: Relaxation Test Data Usage (contd)Volumetric relaxation test data is optional; many solid materials exhibit insignificant volumetric relaxation behavior.Separa

31、te fits are performed on the shear and volumetric parts and combined into one set of Prony series parameters.SHRINF is the normalized shear stress (modulus) as time1; if a value for this parameter is specified it will act as a further constraint to enforce1- gi= gR ().pi=1VOLINF is the normalized pr

32、essure (modulus) as time1; if a value for this parameter is specified it will act as a further constraint to enforceNN1- ki p = kR ().i=1Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.25Relaxation and Creep Test DataExample 2: Relaxation Test Data Usage (combined test d

33、ata)If both the shear and volumetric relaxation tests are performed and the same time intervals are used in each, then the normalized experimentaldata can be specified using a single keyword: *COMBINEDTESTDATAModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.26Relaxation a

34、nd Creep Test DataExample 2: Relaxation Test Data Usage (combined test data, contd)*MATERIAL,NAME=*ELASTIC, MODULI=or*HYPERLEASTIC, MODULI= or*HYPERFOAM. (data lines)*VISCOELASTIC, TIME=RELAXATION TESTDATA,ERRTOL = 0.01, NMAX =*COMBINED TEST DATA,SHRINF = 0.5, VOLINF =130.50.99256,0.98525,0.97805,.0

35、.99256,0.98525,0.97805,single curve fitgroups ofg p, k p, and tiiiModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.27Relaxation and Creep Test DataCreep Test DataConsider a simple shear creep testHere t0 is the instantaneous (short-time) applied shear stress, and

36、g (t) is the measured shear strain response.Note: Be careful that the short-time duration of the prescribed stress is consistent with the time scale of your linear elastic or hyperelastic material definition.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.28Relaxation an

37、d Creep Test DataCreep Test Data (contd)The measured shear strain response is data pairs of (shear strain, time)g0 , time0g1 , time1g2 , time2g3 , time3Because this is linear viscoelasticity only one curve may be used. If your material is not exactly linear in its viscoelastic response then test at

38、an applied stress that is close to your component analysis stress level of interest.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.29Relaxation and Creep Test DataCreep Test Data (contd)Test data processing for use in ABAQUS:Simply normalize all the measured strain valu

39、es by g0 :g0 / g0 , time0g1 / g0 , time1g2 / g0 , time2g3 / g0 , time3and input this data using the *SHEAR*VISCOELASTIC, TIME=CREEP TESTTEST DATA suboption ofDATA.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.30Relaxation and Creep Test DataExample 3: Creep Test Data U

40、sage*MATERIAL, NAME=*ELASTIC, MODULI=OR*HYPERLEASTIC, MODULI=. (data lines)*VISCOELASTIC, TIME=CREEP TEST DATA,ERRTOL=0.01, NMAX=13*SHEAR TEST DATA, SHRINF=21.00747,1.01487,. 1.99619,0.10.2100.0*VOLUMETRIC TESTDATA, VOLINF=21.00747,1.01487,. 1.99619,0.10.2100.0nonlinearleastsquares fitspairs of g p

41、andt Giipairs of kandt KpiiModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.31Relaxation and Creep Test DataExample 3: Creep Test Data Usage (contd)Volumetric creep test data is optional; many solid materials exhibit insignificant volumetric creep/relaxation behavior.Sepa

42、rate fits are performed on the shear and volumetric parts and combined into one set of Prony series parameters.SHRINF is the normalized shear strain (compliance) as time1; if a value for this parameter is specified it will act as a further constraint toenforceN1- gi= gR ().pi=1VOLINF is the normaliz

43、ed volume strain (compliance) as time1; if a value for this parameter is specified it will act as a further constraint toenforceN1- ki p = kR ().i=1Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.32Relaxation and Creep Test DataExample 4: Creep Test Data Usage (combined

44、test data)If both the shear and volumetric creep tests are performed and the same time intervals are used in each, then the normalized experimental datacan be specified using a single keyword: *COMBINEDTESTDATAModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.33Relaxation

45、and Creep Test DataExample 4: Creep Test Data Usage (combined test data, contd)*MATERIAL, NAME=*ELASTIC,MODULI=or*HYPERLEASTIC,MODULI=*HYPERFOAMor. (data lines)*VISCOELASTIC, TIME = CREEP TESTDATA,ERRTOL = 0.01, NMAX =*COMBINED TEST DATA, SHRINF = 2, VOLINF = 2131.00747,1.01487,. 1.99619,1.00747,1.0

46、1487,9619,100.0single curve fitgroups ofg p, k p, and tiiiModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.34Relaxation and Creep Test Data Prony Series Curve Fit NotesIt is important that the proper number of terms in the Prony series be used Too few terms will

47、yield a poor fit.Too many terms may cause ill-conditioningTypically you need about the same number of Prony terms as you have decades of time data.The ERRTOL parameter on the *VISCOELASTIC option controls the accuracyof the fit, and thus the number of terms generated.It is the allowable average RMS

48、error in the least squares fit. The default value is 0.01.The NMAX parameter on the *VISCOELASTIC option controls specifies the maximum number of terms in the Prony series.Fit is performed from N=1 to N=NMAX until convergence is achieved for the lowest N with respect to ERRTOL .Visually check your f

49、it using unit-cube type analyses.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Prony Series DataCopyright 2006 ABAQUS, Inc.L8.36Prony Series DataProny Series Data An alternative to specifying test data is to enter the Prony coefficients directly.*MATERIAL, NAME=.*ELASTIC,

50、MODULI=.Or*HYPERLEASTIC,MODULI=. Or*HYPERFOAM. (data lines)*VISCOELASTIC,TIME=PRONY,g p, k1p , t11g p, k p , t 2t t t22123g p, k p , t333Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.37Prony Series DataProny Series DataRule of thumb is one-two logarithmic decades of ti

51、me per ti .For example, suppose we want to model the relaxation over the time spanfrom 0.1 seconds to 7200 seconds.0.1 1.0 seconds is one decade of time1 10 is 2 decades10 100 is 3 decades100 1000 is 4 decades1000 10,000 is 5 decadesWe can estimate that it will take 3-5 Prony series terms for a good fit to 5 decades of relaxation.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L8.38Prony Series DataChanging number of Prony Terms: N= 2, N= 3, N= 5N=3BetterN=2 Very PoorN=5 BestModeling R