黏弹性

ABAQUSAnalysisUr'sManual17.7.1Timedomainviscoelasticity

Products:ABAQUS/StandardABAQUS/ExplicitABAQUS/CAE

References

















“Materiallibrary:overview,”Section16.1.1

“Elasticbehavior:overview,”Section17.1.1

“Frequencydomainviscoelasticity,”Section17.7.2

*VISCOELASTIC

*SHEARTESTDATA

*VOLUMETRICTESTDATA

*COMBINEDTESTDATA

*TRS

Overview

Thetimedomainviscoelasticmaterialmodel:

describesisotropicrate-dependentmaterialbehaviorformaterials

inwhichdissipativelossprimarilycaudby“viscous”

(internaldamping)effectsmustbemodeledinthetimedomain;

assumesthattheshear(deviatoric)andvolumetricbehaviorsare

independentinmultiaxialstressstates(exceptwhenudforanelastomericfoam);



canbeudonlyinconjunctionwith“Linearelasticbehavior,”

Section17.2.1;“Hyperelasticbehaviorofrubberlikematerials,”

Section17.5.1;or“Hyperelasticbehaviorinelastomericfoams,”

Section17.5.2,todefinetheelasticmaterialproperties;



isactiveonlyduringatransientstaticanalysis(“Quasi-static

analysis,”Section6.2.5),atransientimplicitdynamicanalysis

(“Implicitdynamicanalysisusingdirectintegration,”Section

6.3.2),anexplicitdynamicanalysis(“Explicitdynamicanalysis,”

Section6.3.3),asteady-statetransportanalysis(“Steady-state

transportanalysis,”Section6.4.1),afullycoupled

temperature-displacementanalysis(“Fullycoupledthermal-stress

analysis,”Section6.5.4),oratransient(consolidation)coupled

porefluiddiffusionandstressanalysis(“Coupledporefluid

diffusionandstressanalysis,”Section6.7.1);



canbeudinlarge-strainproblems;and

canbecalibratedusingtime-dependentcreeptestdata,

time-dependentrelaxationtestdata,orfrequency-dependentcyclictestdata.



Definingtheshearbehavior

TimedomainviscoelasticityisavailableinABAQUSforsmall-strain

applicationswheretherate-independentelasticresponcanbedefined

withalinearelasticmaterialmodelandforlarge-strainapplications

wheretherate-independentelasticresponmustbedefinedwitha

hyperelasticorhyperfoammaterialmodel.

注释：小应变简化为弹性，大应变超弹性或超弹泡沫

Smallstrain（小应变）

Considerasheartestatsmallstrain,inwhichatimevaryingshearstrain,

,ponistheshearstressTheviscoelasticmaterialmodeldefines

as

.

where事迹材料 isthetime-dependent“shearrelaxationmodulus”（剪切松弛模量）

thatcharacterizesthematerial'nstitutivebehaviorcanbeillustrated

byconsideringarelaxationtestinwhichastrainissuddenlyappliedtoaspecimen

inningoftheexperiment,whenthe

strainissuddenlyapplied,istakenaszerotime,sothat（t=0,初始应变突然施加）

coelasticmaterialmodelis“long-termelastic”

（长期弹性）inthenthat,afterhavingbeensubjectedtoaconstantstrainfora

verylongtime,,

.

as

Theshearrelaxationmoduluscanbewrittenindimensionlessform:

where

istheinstantaneousshearmodulus（瞬时剪切模量）,sothat

theexpressionforthestresstakestheform

Thedimensionlessrelaxationfunctionhasthelimitingvalues

.

Largestrain（大应变）

Theequationforthestresscanbetransformedbyusingintegrationby

parts:

and

Itisconvenienttowritethiquationintheform

where

rmallowsa

straightforwardgeneralization（简单概括）tononlinearelasticdeformationsbyreplacingthelinearelasticrelation

withthenonlinearelasticityrelation

.Thisgeneralizationyieldsalinearviscoelasticitymodel,inthenthat

thedimensionlessstressrelaxationfunctionisindependentofthemagnitudeofthe

deformation.

Intheaboveequationtheinstantaneousstress,,appliedattime

influencesthestress,,ore,tocreateaproper

finite-strainformulation,itisnecessarytomapthestressthatexisted

ABAQUSthisisdonebymeansofamixed“push-forward”transformationwiththerelativedeformationgradient

:

Toensurethatthestressremainssymmetric,ABAQUSustheintegralform:

wmac强制退出 hereisthedeviatoricpartoftheKirchhoffst贪婪拼音 ress.基尔莫夫应力偏量

Thefinite-straintheoryisdescribedinmoredetailin“Finite-strain

viscoelasticity,”Section4.8.2oftheABAQUSTheoryManual.

Definingthevolumetricbehavior定义体

Thevolumetricbehaviorcanbewritteninaformthatissimilartotheshearbehavior:

wherepisthehydrostaticpressure,（静水压力）

bulkmodulus,

volumestrain.

Theaboveexpansionappliestosmallaswellasfinitestrainsincethe

volumestrainsaregenerallysmallandthereisnoneedtomapthepressure

fromtimetotimet.

istheinstantaneoulastic

isthe

isthedimensionlessbulkrelaxationmodulus,and

Temperatureeffects温度效应

Theeffectoftemperature,,onth演讲自我介绍 ematerialbehaviorisintroduced

throughthedependenceoftheinstantaneousstress,,ontemperature

ressionforthelinear-elasticshearstressisrewrittenas

wheretheinstantaneousshearmodulus

,and

istemperaturedependent,

isthereducedtime,definedby

ducedtimeconceptfor

temperaturedependenceisusuallyreferredtoasthermo-rheologicallysimple(TRS)

heshiftfunctionisapproximatedbythe

Williams-Landell-Ferry(WLF)form:

where,

isthereferencetemperatureatwhichtherelaxationdataaregivenand

,

deformationchangeswillbeelastic,badontheinstantaneousmoduli.

Thereducedtimeconceptisalsoudforthevolumetricbehaviorandthe

itionalinformationontheWLFequation,

e“Viscoelasticity,”Section4.8.1oftheABAQUSTheoryManual.

Numericalimplementation数值执行结果

ABAQUSassumesthattheviscoelasticmaterialisdefinedbyaPronyriexpansionofthedimensionlessrelaxationmodulus:

whereN,

,and

,

,tutionin

thesmall-strainexpressionfortheshearstressyields

where

The

areinterpretedasstatevariablesthatcontrolthestressrelaxation,and

isthe“creep”strain:thedifferencebetweenthetotalmechanicalstrainandthe

instantaneoulasticstrain(thestressdividedbytheinstantaneoulasticmodulus).InABAQUS/Standard

isavailableasthecreepstrainoutputvariableCE

(“ABAQUS/Standardoutputvariableidentifiers,”Section4.2.1).

AsimilarPronyriexpansionisudforthevolumetricrespon,whichisvalidforbothsmall-andfinite-strainapplications:

where

ABAQUSassumesthat.

ThePronyriexpansion,incombinationwiththefinite-strain

expressionfortheshearstress,producesthefollowinglarge-strainshearmodel:

where

The

areinterpretedasstatevariablesthatcontrolthestressrelaxation.

Iftheinstantaneousmaterialbehaviorisdefinedbylinearelasticityorhyperelasticity,thebulkandshearbehaviorcanbedefined

r,iftheinstantaneousbehaviorisdefinedbythe

hyperfoammodel,thedeviatoricandvolumetricconstitutivebehaviorare

coupledanditisnecessarytouthesamerelaxationdataforbothbehaviors.

Inalloftheaboveexpressionstemperaturedependenceisreadilyintroducedbyreplacing

by

and

by

.

Determinationofviscoelasticmaterialparameters定义粘弹性材料参数

Theaboveequationsareudtomodelthetime-dependentshearand

axationparameters

canbedefinedinoneoffourways:directspecificationoftheProny

riesparameters,inclusionofcreeptestdata,inclusionofrelaxation

testdata,orinclusionoffrequency-dependentdataobtainedfrom

atureeffectsareincludedin

thesamemannerregardlessofthemethodudtodefinetheviscoelasticmaterial.

ABAQUS/CAEallowsyoutoevaluatethebehaviorofviscoelasticmaterials

byautomaticallycreatingresponcurvesbadonexperimentaltestdata

elasticmaterialcanbeevaluatedonlyifitis

definedinthetimedomainandincludeshyperelasticand/orelastic

“Evaluatinghyperelasticandviscoelasticmaterial

behavior,”Section12.4.6oftheABAQUS/CAEUr'sManual.

Directspecification（直接参数）ThePronyriesparameters

,

,andcanbedefineddirectlyfor

snorestrictiononthenumberof

termsthatcanbeud.（应用数目没有限制）Ifarelaxationtimeis

associatedwithonlyoneofthetwomoduli,leavetheotheroneblankor

enterazero.（如果松弛时间仅于两个模量的一个有关联，则另外一个设为1

或0）Thedatashouldbegiveninascendingorder（升序）oftherelaxation

beroflinesofdatagivendefinesthenumberofterms甲醛中毒表现 in

thePronyries,modelisudinconjunctionwiththe

hyperfoam（超弹泡沫材料模型）materialmodel,thetwomodulusratioshavetobethesame(

).

InputFileUsage:

*VISCOELASTIC,TIME=PRONY

Thedatalineisrepeatedasoftenasneededtodefinethe

first,cond,third,nthePronyries.

ABAQUS/CAEUs

Propertymodule:materialeditor:Mechanical

ElasticityViscoelastic:Domain:TimeandTime:

age:

Prony

Enterasmanyrowsofdatainthetableasneededtodefine

thefirst,cond,third,nthePronyries.

Creeptestdata

Ifcreeptestdataarespecified,ABAQUSwillcalculatethetermsinthe

malizedshearandbulkcompliancesaredefinedas

whereand

istheshearcompliance,

isthetotalshearstrain,

isthe

istheconstant

.

istheconstantshearstressinashearcreeptest;

isthetotalvolumetricstrain,and

,

volumetriccompliance,

Thecreepdataareconvertedtorelaxationdatathroughtheconvolutionintegrals

ABAQUSthenusthenormalizedshearmodulus

modulus

parameters.

Usingtheshearandvolumetrictestdataconcutively

andnormalizedbulk

inanonlinearleast-squaresfittodeterminethePronyries

Thesheartestdataandvolumetrictestdatacanbeudconcutively

todefinethenormalizedshearandbulkcompliancesasfunctionsoftime.

Aparateleast-squaresfitisperformedoneachdatat;andthetwo

derivedtsofPronyriesparam上海十大古镇排名 eters,intoonetofparameters,

InputFileUsage:

.

and

,aremerged

stoptionis

eofthecondandthirdoptionsis

required.

*VISCOELASTIC,TIME=CREEPTESTDATA

*SHEARTESTDATA

*VOLUMETRICTESTDATA

ABAQUS/CAEUs

Propertymodule:materialeditor:Mechanical

ElasticityViscoelastic:Domain:TimeandTime:

age:

Creeptestdata

Inaddition,lectoneorbothofthefollowing:TestData

TestData

Usingthecombinedtestdata

ShearTestData

VolumetricTestData

Alternatively,thecombinedtestdatacanbeudtospecifythe

normalizedshearandbulkcompliancessimultaneouslyasfunctionsoftime.

Asingleleast-squaresfitisperformedonthecombinedtoftestdatatodetermineonetofPronyriesparameters,

InputFileUsage:

ABAQUS/CAEUs

Propertymodule:materialeditor:Mechanical

ElasticityViscoelastic:Domain:Time,Time:

age:

Creeptestdata,andTestDataCombinedTestData

Relaxationtestdata

Aswithcreeptestdata,ABAQUSwillcalculatethePronyriesparametersautomaticallyfromrelaxationtestdata.

Usingtheshearandvolumetrictestdataconcutively

.

Ubothofthefollowingoptions:

*VISCOELASTIC,TIME=CREEPTESTDATA

*COMBINEDTESTDATA

Again,thesheartestdataandvolumetrictestdatacanbeud

concutivelytodefinetherelaxationmoduliasfunctionsoftime.A

paratenonlinearleast-squaresfitisperformedoneachdatat;and

thetwoderivedtsofPronyriesparameters,aremergedintoonetofparameters,

InputFileUsage:

.

and

,

stoptionis

eofthecondandthirdoptionsis

required.

*VISCOELASTIC,TIME=RELAXATIONTESTDATA

*SHEARTESTDATA

*VOLUMETRICTESTDATA

ABAQUS/CAEUs

Propertymodule:materialeditor:Mechanical

ElasticityViscoelastic:Domain:TimeandTime:

age:

Relaxationtestdata

Inaddition,lectoneorbothofthefollowing:TestData

TestData

Usingthecombinedtestdata

ShearTestData

VolumetricTestData

Alternatively,thecombinedtestdatacanbeudtospecifythe

e

least-squaresfitisperformedonthecombinedtoftestdatatodetermineonetofPronyriesparameters,

InputFileUsage:

*VISCOELASTIC,TIME=RELAXATIONTESTDATA

*COMBINEDTESTDATA

Ubothofthefollowingoptions:

.

ABAQUS/CAEUs

Propertymodule:materialeditor:Mechanical

ElasticityViscoelastic:Domain:Time,Time:

age:

Relaxationtestdata,andTestDataCombinedTest

Data

Frequency-dependenttestdata

ThePronyriestermscanalsobecalibratedusingfrequency-dependent

caABAQUSusanalyticalexpressionsthatrelate

expressionsfortheshearmoduli,obtainedbyconvertingthePronyries

termsfromthetimedomaintothefrequencydomainbymakinguofFouriertransforms,canbewrittenasfollows:

where

isthestoragemodulus,

isthelossmodulus,

isthe

angularfrequency,

expressionsareudinanonlinearleast-squaresfittodeterminethePronyries

parametersfromthestorageandlossmodulicyclictestdataobtainedatMfrequenciesbyminimizingtheerrorfunction

:

where

and

arethetestdataand

and

,respectively,arethe

ressionsforthebulkmoduli,

and

,arewrittenanalogously.

Thefrequencydomaindataaredefinedintabularformbygivingthereal

andimaginarypartsofand—whereisthecircularfrequency—as

functionsoffrequencyincyclespertime.

istheFouriertransform

ofthenondimensionalshearrelaxationfunction

thefrequency-dependentstorageandlossmoduliand

,therealandimaginarypartsof

and

,

.Given,

,

arethengivenas

whereandarethelong-termshearandbulkmodulideterminedfrom

theelasticorhyperelasticproperties.

InputFileUsage:

*VISCOELASTIC,TIME=FREQUENCYDATA

ABAQUS/CAEUs

Propertymodule:materialeditor:Mechanical

ElasticityViscoelastic:Domain:TimeandTime:

age:

Frequencydata

CalibratingthePronyriesparameters

Youcanspecifytwooptionalparametersrelatedtothecalibrationof

Pronyriesparametersforviscoelasticmaterials:theerrortolerance

ortoleranceistheallowableaverageroot-mean-square

errorofdatapointsintheleast-squaresfit,anditsdefaultvalueis

aximumnumberoftermsNinthePronyries,andits

default(andmaximum)willperformtheleast-squares

fitfromtountilconvergenceisachievedforthelowest

Nwithrespecttotheerrortolerance.

Thefollowingaresomeguidelinesfordeterminingthenumberoftermsin

ntheguidelines,youcanchoo

.

Thereshouldbeenoughdatapairsfordeterminingalltheparameters

,assumingthatNisthenumberof

Pronyriesterms,thereshouldbeatotalofatleastdata

pointsinsheartestdata,datapointsinvolumetrictestdata,

datapointsincombinedtestdata,如何添加脚注 anddatapointsinthe

frequencydomain.



ThenumberoftermsinthePronyriesshouldbetypicallynotmore

thanthenumberoflogarithmic“decades”spannedbythetestdata.



Thenumberoflogarithmic“decades”isdefinedas

,where

and

arethemaximumandminimum

timeinthetestdata,respectively.

InputFileUsage:

*VISCOELASTIC,ERRTOL=

error_tolerance

,NMAX=

ABAQUS/CAEUs

Propertymodule:materialeditor:Mechanical

ElasticityViscoelastic:Domain:Time;Time:

age:

Creeptestdata,Relaxationtestdata,orFrequency

data;MaximumnumberoftermsinthePronyries:

;andAllowableaverageroot-mean-square

error:

error_tolerance

Temperatureeffects

Regardlessofthemethodudtodefinetheviscoelasticbehavior,

thermo-rheologicallysimpletemperatureeffectscanbeincludedby

ft

functioncanbedefinedbytheWilliams-Landell-Ferry(WLF)approximation.

InputFileUsage:

*TRS,DEFINITION=WLF

InABAQUS/Explicituthefollowingoption:

*TRS

ABAQUS/CAEUs

Propertymodule:materialeditor:Mechanical

ElasticityViscoelastic:Domain:Time,Time:any

age:

method

,andSuboptionsTrs:Shiftfunction:WLF

UrsubroutinespecificationinABAQUS/Standard

InABAQUS/Standarduthefollowingoption:

InABAQUS/Standardtheshiftfunctioncanalternativelybespecifiedin

ursubroutineUTRS.

InputFileUsage:

*TRS,DEFINITION=USER

ABAQUS/CAEUs

Propertymodule:materialeditor:Mechanical

ElasticityViscoelastic:Domain:Time,Time:any

age:

method

,andSuboptionsTrs:Shiftfunction:Ur

subroutineUTRS

Definingtherate-independentpartofthematerialrespon

Inallcalasticmodulimustbespecifiedtodefinethe

-strainlinear

elasticbehaviorisdefinedbyanelasticmaterialmodel(“Linearelastic

behavior,”Section17.2.1),andlarge-deformationbehaviorisdefinedby

ahyperelastic(“Hyperelasticbehaviorofrubberlikematerials,”Section

17.5.1)orhyperfoam(“Hyperelasticbehaviorinelastomericfoams,”

Section17.5.2)e-independentelasticityforany

ofthemodelscanbedefinedintermsofeitherinstantaneoulastic

iceofdefiningtheelasticity

intermsofinstantaneousorlong-termmoduliisamatterofconvenienceonly;itdoesnothaveaneffectonthesolution.

Theeffectiverelaxationmoduliareobtainedbymultiplyingthe

instantaneoulasticmoduliwiththedimensionlessrelaxationfunctions

elasticmaterials

Forlinearelasticmaterialbehavior

and

whereandaretheinstantaneousshearandbulkmodulideterminedfrom

and

:

thevaluesoftheur-definedinstantaneoulasticmoduli

and

.

Iflong-termelasticmoduliaredefined,theinstantaneousmoduliaredeterminedfrom

Hyperelasticmaterials

Forhyperelasticmaterialbehaviortherelaxationcoefficientsare

appliedeithertotheconstantsthatdefinetheenergyfunctionor

polynomialfunctionandits

particularcas(reducedpolynomial,Mooney-Rivlin,neo-Hookean,andYeoh)

fortheOgdenfunction

fortheArruda-BoyceandVanderWaalsfunctions

andfortheMarlowfunction

Forthecoefficientsgoverningthecompressiblebehaviorofthepolynomial

modelsandtheOgdenmodel

fortheArruda-BoyceandVanderWaalsfunctions

and国土面积排名前十的国家 fortheMarlowfunction

Iflong-termelasticmoduliaredefined,theinstantaneousmoduliaredeterminedfrom

whiletheinstantaneousbulkcompliancemoduliareobtainedfrom

fortheMarlowfunctionswehave

Elastomericfoams

Forelastomericfoammaterialbehaviortheinstantaneousshearandbulk

relaxationcoefficientsareassumedtobeequalandareappliedtothe

materialconstantsintheenergyfunction:

Ifonlytheshearrelaxationcoefficientsarespecified,thebulk

relaxationcoefshearandbulkrelaxation

coefficientsarespecifiedandtheyareunequal,ABAQUSissuesanerrormessage.

Iflong-termelasticmoduliaredefined,theinstantaneousmoduliaredeterminedfrom

Materialresponindifferentanalysisprocedures

Thetime-domainviscoelasticmaterialmodelisactiveduringthefollowingprocedures:













transientstaticanalysis(“Quasi-staticanalysis,”Section

6.2.5),

transientimplicitdynamicanalysis(“Implicitdynamicanalysis

usingdirectintegration,”Section6.3.2),

explicitdynamicanalysis(“Explicitdynamicanalysis,”Section

6.3.3),

steady-statetransportanalysis(“Steady-statetransport

analysis,”Section6.4.1),

fullycoupledtemperature-displacementanalysis(“Fullycoupled

thermal-stressanalysis,”Sect学科英文 ion6.5.4),and

transient(consolidation)coupledporefluiddiffusionandstress

analysis(“Coupledporefluiddiffusionandstressanalysis,”

Section6.7.1).

Viscoelasticmaterialresponisalwaysignoredinastaticanalysis.

Itcanalsobeignoredinacoupledtemperature-displacementanalysisor

inasoilsconsolidationanalysisbyspecifyingthatnocreepor

viscoelasticresponisoccurringduringthestepevenifcreepor

viscoelasticmaterialpropertiesaredefined(e“Fullycoupled

thermal-stressanalysis,”Section6.5.4,or“Coupledporefluiddiffusion

andstressanalysis,”Section6.7.1).Inthecasitisassumedthat

theloadingisappliedinstantaneously,sothattheresultingrespon

correspondstoanelasticsolutionbadoninstantaneoulasticmoduli.

ABAQUS/Standardalsoprovidestheoptiontoobtainthefullyrelaxed

long-termelasticsolutiondirectlyinastaticorsteady-statetransport

g-term

cousdampingstress(theinternal

stressassociatedwitheachoftheProny-riesterms)areincread

graduallyfromtheirvaluesatthebeginningofthesteptotheirlong-termvaluesattheendofthestepifthelong-termvalueisspecified.

Materialoptions

Theviscoelasticmaterialmodelmustbecombinedwithanelasticmaterial

edwiththeisotropiclinearelasticitymodel(“Linear

elasticbehavior,”Section17.2.1)todefineclassical,linear麻杏石甘散 ,small-strain,viscoelasticbehaviororwiththehyperelastic

(“Hyperelasticbehaviorofrubberlikematerials,”Section17.5.1)or

hyperfoam(“Hyperelasticbehaviorinelastomericfoams,”Section17.5.2)

modelstodefinelarge-deformation,nonlinear,viscoelasticbehavior.

Theelasticpropertiesdefinedforthemodelscanbetemperaturedependent.

Viscoelasticitycannotbecombinedwithanyoftheplasticitymodelsor

“Combiningmaterial

behaviors,”Section16.1.3,formoredetails.

Elements

Thetimedomainviscoelasticmaterialmodelcanbeudwithany

stress/displacementorcoupledtemperature-displacementelementinABAQUS.

Output

InadditiontothestandardoutputidentifiersavailableinABAQUS

(“ABAQUS/Standardoutputvariableidentifiers,”Section4.2.1,and

“ABAQUS/Explicitoutputvariableidentifiers,”Section4.2.2),the

followingvariableshavespecialmeaninginABAQUS/Standardifviscoelasticityisdefined:

EE

CE

Thestrainmeasuresareudtoapproximatethestrainenergy,SENER,andthe

viscousdissipation,pproximationsmayleadtounderestimationof

thestrainenergyandoverestimationoftheviscousdissipationsincetheeffectsof

nbeparticularly

noticeableinthecaofnon-monotonicloading.

Forthecaoflarge-strainviscoelasticity,ABAQUS/Explicitdoesnot

performthecomputationoftheviscousdissipationforperformance

d,thecontributionofviscousdissipationisincludedin

thestrainenergyoutput,SENER;uently,

specialcaremustbeexercidwheninterpretingstrainenergyresults

oflarge-strainviscoelasticmaterialsinABAQUS/Explicitsincetheyincludeviscousdissipationeffects.

Elasticstraincorrespondingtothestressstateattime

tandtheinstantaneoulasticmaterialproperties.

Equivalentcreepstraindefinedasthedifferencebetweenthetotalstrainandtheelasticstrain.