abaqus经典声学分析例题l11-hysteresis

1、Hysteresis in ElastomersLecture 11Copyright 2006 ABAQUS, Inc.L11.2OverviewSummary of Viscoelasticity Hysteresis in Elastomers ABAQUS UsageExampleModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Summary of ViscoelasticityCopyright 2006 ABAQUS, Inc.L11.4Summary of Viscoelastic

2、ity“Classical” linear viscoelasticity: small-strain theory in which the instantaneous stress is proportional to the strain. Experiments demonstrate that this model is accurate for many materials at small strains ( 0.01).“Finite-strain” linear viscoelasticity: hyperelastic or hyperfoam theory in whic

3、h the relaxation rate is proportional to the stress. Simplest model for viscoelasticity at large strains. Small amount of experimental data required to calibrate model.For many materials the relaxation rate is proportional to the stress and the viscoelastic models are appropriate; however, there are

4、 many materials that do not exhibit this proportional behavior.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L11.5Summary of ViscoelasticityIn filled and some unfilled rubbers the creep or relaxation rate is notproportional to the stress.Typically, creep and stress relaxa

5、tion are more pronounced at higher stress levels.In addition, at higher stress levels creep and stress relaxation occur faster initially and reach a plateau more slowly than with viscoelasticity.This leads to hysteresis-type behavior in cyclic loading, where the amount of hysteresis increases with l

6、oading amplitude but is relatively independent of the cycling frequency.This kind of general nonlinear, finite-strain, time-dependent behavior is what the hysteresis model attempts to capture.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Hysteresis in ElastomersCopyright

7、2006 ABAQUS, Inc.L11.7Hysteresis in ElastomersMany rubbers are known to be rate-dependent and to exhibit hysteresis upon cyclic loading.The figure at right shows a typical hysteresis response (uniaxial compression at constant strain rate) for a filled rubber subjected to different final strains (fro

8、m Bergstrom and Boyce1).1. Bergstrom, J.S., and M.C. Boyce, “Constitutive Modeling of the Large Strain Time- Dependent Behavior of Elastomers,” Journal of the Mechanics and Physics of Solids, vol. 46, pp. 931-954, 1998.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L11.8Hy

9、steresis in ElastomersThe data show: Repeatability of the results No permanent set after one completed load cycleThe figure at right shows a typical strain rate dependence during uniaxial compression to a fixed strain level.2.00strain rates: -0.001/s, -0.01/s,-0.05/s, -0.2/s1.60increasing strain rat

10、e1.200.80Chloroprene rubber (15 pph)0.400.00.00.200.400.600.80True strain (compressive)Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.True stress (-MPa)L11.9Hysteresis in ElastomersBergstrom and Boyce developed a large strain (400% is not uncommon), time-dependent constitu

11、tive model for elastomeric materials.They observed the following in experiments with carbon-black-filled Chloroprene rubber subjected to different time-dependent strain histories:Both filled and unfilled elastomers show significant amounts of hysteresis during cyclic loading.The amount of carbon bla

12、ck particles does not strongly influence the normalized amount of hysteresis.Both filled and unfilled elastomers are strain-rate dependent, and the rate dependence is higher during loading than unloading.At fixed strain the stress approaches the same equilibrium level with relaxation time whether lo

13、ading or unloading.They then derived a phenomenological constitutive model, which is implemented in ABAQUS (*HYSTERESIS).Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L11.10Hysteresis in ElastomersComponents in the modelnetwork BElastic and creep strains are large and of

14、comparable magnitude.Creep response only for shear distortional behavior; the volumetric response is purely elastic.Nonlinear dependence on strain work AThe hysteresis model decomposes the mechanical behavior into two parts: an equilibrium or purely elastic response (network A) and a time-de

15、pendent deviation from equilibrium (network B).The figure shows a one-dimensional idealization.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L11.11Hysteresis in ElastomersCreep test: The following plot shows the normalized strain versus time for four different stresses us

16、ing the hysteresis model.The strain for each case is normalized with respect to the instantaneous strain.2.00nominal stress - 4 nominal stress - 3 nominal stress - 2 nominal stress - 11.501.000.50The material reaches a strain plateau much more slowly than with viscoelasticity.0.00.00.501.001.50Time2

17、.002.503.00Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Normalized strainL11.12Hysteresis in ElastomersStress relaxation test: The following plot shows the normalized stress versus time for four different strains using the hysteresis model.The stress for each case is nor

18、malized with respect to the instantaneous stress.1.00nominal strain - 4 nominal strain - 3 nominal strain - 2 nominal strain - 10.800.600.400.20The stress reaches a plateau much more slowly than with viscoelasticity.0.00.00.501.001.50Time2.002.503.00Modeling Rubber and Viscoelasticity with ABAQUSCop

19、yright 2006 ABAQUS, Inc.Normalized stressL11.13Hysteresis in Elastomers The *HYPERELASTIC option defines the response of network A; the spring response is nonlinear. The *HYSTERESIS option defines the response of network B; the effective creep strain rate in network B is given by the expression= A(l

20、cr -1)C s m .e cr The positive exponent m, generally greater than 1, characterizes the (scalar) effective stress dependence of the effective creep strain rate. The exponent C, restricted to the interval -1, 0, characterizes the creep strain dependence (through the creep stretch lcr) on the creep str

21、ain rate. The nonnegative constant A maintains dimensional consistency in the equation.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L11.14Hysteresis in ElastomersIn addition to these material constants the hysteresis model is characterized by a stress scaling factor, S,

22、that defines the ratio of the stress carried by network B to the stress carried by network A under instantaneous loading; i.e., identical elastic stretching in both networks.Typical values of the constants above (Bergstrom and Boyce, 1998):5(sec)-1 (MPa)-m , m = 4, C = -1.0.S = 1.6, A =(3 )mUsage: t

23、he above four values in the given order are entered on the data line for the *HYSTERESIS option.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L11.15Hysteresis in ElastomersRestrictionsHysteresis is active in the following procedures only:*STATIC*VISCO*DYNAMICThe model req

24、uires the *HYPERELASTIC option to define the elasticbehavior.Hysteresis can be used only with elements that permit hyperelastic materials; thus, is can be used only in large-strain problems.Hybrid elements can be used only when the accompanying hyperelasticity definition is incompressible.Modeling R

25、ubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L11.16Hysteresis in ElastomersRestrictions (contd)The hysteresis material properties cannot be temperature dependent; however, the elastic material properties can be temperature dependent.The model does not model “Mullins effect” or the

26、 softening of an elastomer when it is first subjected to loading.Before material properties are measured, the rubber should be stretched repeatedly to operating strain levels.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.ABAQUS UsageCopyright 2006 ABAQUS, Inc.L11.18ABAQUS

27、 UsageThe elasticity of the model is defined by using the *HYPERELASTIC option. The MODULI parameter may be set to either LONG TERM (to define the long-term behavior of the material; default setting) or INSTANTANEOUS (to define the instantaneous behavior).The stress scaling factor and the creep para

28、meters for network B areinput directly on the data line of the *HYSTERESIS option.Both the *HYPERELASTIC option and *HYSTERESIS option must be used together in the material definition.The hysteresis material model creates unsymmetric stiffness matrices, so ABAQUS/Standard uses unsymmetric matrix sto

29、rage and solution by default.Typical values of the material parameters are given in the ABAQUS/Standard Users Manual.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.L11.19ABAQUS UsageDefining hysteresis in ABAQUS/CAEModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.ExampleCopyright 2006 ABAQUS, Inc.L11.21ExampleThis example is taken from the ABAQUS Verification Manual. The material being modeled is Chloroprene rubber (15 pph carbon black filler).Material modelThe rubber is modeled with the Arruda-Boyce hyperelasticity m