abaqus经典声学分析例题l7-visco

1、Viscoelastic Material BehaviorLecture 7Copyright 2006 ABAQUS, Inc.L7.2OverviewTime Domain Response Linear Viscoelasticity Temperature Dependence Frequency Domain ResponseHysteresis and DampingModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Time Domain ResponseCopyright 2006

2、 ABAQUS, Inc.L7.4Time Domain ResponseDefinitionCertain materials are rate-dependent and behave elastically. When unloaded, they return to their undeformed state.These materials are called viscoelastic.ExamplesPolymers such as plasticsGlass Rubber FoamsSolid rocket propellantsModeling Rubber and Visc

3、oelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L7.5Time Domain ResponseFor prescribed stress (force), these materials creepCreep test measures strain (displacement) response as function of time while stress (force) is held constant on the specimen.prescribed stressModeling Rubber and Viscoelastic

4、ity with ABAQUSCopyright 2006 ABAQUS, Inc.L7.6Time Domain ResponseCreepAlso occurs in metals,Typically not recoverable (inelastic)Creep material model is viscoplastic, not viscoelasticSignificant at high temperature (with respect to the melting point)Creep of polymers is significant starting at low

5、temperatures ( -200 oC)For viscoelastic materials full elastic recovery occurs upon unloadingModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L7.7Time Domain ResponseFor prescribed strains these materials exhibit stress relaxationStress relaxation test measures the stress (f

6、orce) response as function of time while strain (displacement) is held constant on the specimenprescribed strainModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L7.8Time Domain ResponseStress Relaxation and RecoveryViscous fluids, such as glass, polymers at high temperature

7、and unvulcanized elastomers will relax to zero stress and will not recover when the applied strain is released.Viscoelastic solids, such as polymers at lower temperatures, and vulcanized elastomers will relax asymptotically to a nonzero stress level. Upon release of the applied strain, they will par

8、tially recover elastically (immediately) and fully recover viscously over time.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Linear ViscoelasticityCopyright 2006 ABAQUS, Inc.L7.10Linear ViscoelasticityOne-Dimensional IdealizationLinear and finite-strain viscoelasticity ar

9、e idealized as series pairs of springs and dashpots in parallel with a springGeneralized Maxwell ModelThe number of dashpots is equal to the number of terms in the Prony series representing the stress response (the number of terms needed to fit the test data for the time domain of interest).Every “n

10、etwork” (spring-dashpot pair) experiences the same total strain.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L7.11Linear ViscoelasticityLinear Viscoelasticity in ABAQUSThe dashpots strain rate is proportional to stressThe spring response may be linear or nonlinear:For “c

11、lassical” linear viscoelasticity the springs are linear. This implies a linear elastic material model in ABAQUS For finite-strain viscoelasticity the springs are nonlinear. This implies a hyperelastic or hyperfoam material model in ABAQUSModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006

12、ABAQUS, Inc.e cr = A s ,where A =1ViscosityL7.12Linear ViscoelasticityHow do I know if my material exhibits “linear” viscoelasticity?From a practical perspective, one tests the validity of “linear” viscoelasticity by testing at multiple load levels and comparing (overlaying) the normalized response

13、plots. Data for a silicone rubber:Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L7.13Linear Viscoelasticity Response of not-as-linear viscoelastic elastomerThe material shown below is tested over a bit larger range of strain and the viscoelastic response of the material i

14、s less linear as indicated by the variations in the normalized stress relaxation curves. One must make a judgment call as to which relaxation curve to use.Stress RelaxationStress Relaxation20% Strain40% Strain10.760% Strain80% Strain100% Strain0.60.90.50.80.40.30.70.20.60.100.505001000Time (secs)150

15、0200005001000Time (secs)15002000Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Stress (MPa)Stress Normalized 20% Strain40% Strain60% Strain80% Strain100% StrainL7.14Linear ViscoelasticityCreep response for linear viscoelasticityHere is the creep response for a perfectly li

16、near viscoelastic material loaded to 1, 2, and 4 MPa.If these curves were normalized by the instantaneous strain they would perfectly overlay one another.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L7.15Linear ViscoelasticityCreep response for nonlinear viscoelasticityT

17、he dashed lines depict the creep response for a material that does not obey “linear” viscoelasticity.This kind of general nonlinear viscoelastic cannot be modeled in ABAQUS with the *VISCOELASTIC material option.Your material may behave nearly linear over a more narrow range of loading.Modeling Rubb

18、er and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L7.16Linear Viscoelasticity“Classical” linear viscoelasticity:Small-strain theory with linear elastic response.Implies use of a linear elastic material model in ABAQUS.Experiments demonstrate that this model is accurate for many materials

19、at small strains (say 0.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L7.22Frequency Domain ResponseComplex ModulusIt is convenient to separate the viscoelastic response into “in-phase” and “out-of-phase” components.= g0 sinwt= s0 sin(wt+d)= s0 (sinwt cosd + coswt sind)=

20、s0 (sinwt cos d + sin (wt+90) sind)strainstressout-of-phase stressin-phase stressModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L7.23Frequency Domain ResponseComplex Modulus (contd) The complex shear modulus is denoted G*or G *(w).shear stressG*(w) = Complex Shear Modulus

21、(w ) =shear strains 0ei(wt+d )seG (w ) =*=eiwtg0= s 0cosd + i s 0sindg 0g 0= Gs + i GlModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L7.24Frequency Domain ResponseComplex Modulus (contd) Storage Modulus, Gs := s 0cosdGsg0 Characterizes the in-phase shear modulus Loss Modul

22、us, Gl := s 0sin dGlg0 Characterizes the out-of-phase shear modulusModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L7.25Frequency Domain ResponseComplex Modulus (contd)For a harmonic loading of elastomers the storage and loss moduli typically look something like this:Modeli

23、ng Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L7.26Frequency Domain ResponseComplex Modulus (contd)For unfilled rubbers the storage and loss moduli are dependent on frequency only.Gl= tandThe ratio:is commonly referred to as “tan delta”GsFor unfilled rubbers this ratio is often

24、 nearly a constant (over some frequency range of interest).Typical value for natural rubber is 0.2.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L7.27Frequency Domain ResponseComplex Modulus (contd)For filled rubbers the storage and loss moduli are usually dependent on th

25、e strain amplitudeThe X-axis in these figures is the shear amplitude.Storage modulusLoss modulusModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Hysteresis and DampingCopyright 2006 ABAQUS, Inc.L7.29Hysteresis and DampingViscoelastic materials dissipate energy. In the case o

26、f cyclic loadings, this is termed hysteresis; it arises from the frictional sliding of the long molecules across one another.In other cases we refer to the energy dissipation characteristic as damping.Energy lost due to viscoelastic behavior is output in ABAQUS using:CENER:dissipation energy; elemen

27、t integration pointvariabledissipation energy; whole element variable dissipation energy per unit volume; whole element variabledissipation energy; whole model variableELCD:ECDDEN:ALLCD:Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L7.30Hysteresis and DampingEnergy dissipation through hysteresis is represented by the area between the loading and unloading curves in a load-deformation cycle, and occurs with all rubbers.The complementary property is resilience, which is a measure of the energy returned. Fillers in the rubber will increase hysteres