abaqus经典声学分析例题a4-visco-harmonic-theory

1、Harmonic Viscoelasticity TheoryAppendix 4Copyright 2006 ABAQUS, Inc.A4.2OverviewClassical Linear ViscoelasticityHarmonic ExcitationModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Classical Linear ViscoelasticityCopyright 2006 ABAQUS, Inc.A4.4Classical Linear Viscoelasticity

2、Recall the stress relaxation equation:E (t -t ) de (t ) dt .t( )ot=dt - This equation is inadequate for Fourier transform methods since (for solids) E(t) 0 as t .E (t ) EIntroduce e (t ) =-1(dimensionless relaxation function). Here E is the long-term modulus. We see that e(t) 0 as t .Modeling Rubber

3、 and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A4.5Classical Linear Viscoelasticity Substituting for e(t) in the stress relaxation equation yieldse(t -t ) de (t ) dt to (t ) = E e (t ) + Edt -e(t -t ) ds (t ) dt .t= s(t ) +dt -Lettingt = t -t yields()dst -t0o (t ) = s(t ) -()e tt .ddtMod

4、eling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Harmonic ExcitationCopyright 2006 ABAQUS, Inc.A4.7Harmonic ExcitationConsider the application of a sinusoidal strain,e (t ) = e exp(iwt ).where w is the angular frequency.Substituting e (t) in the long-term stress relaxation equa

5、tion yields d dt(Ee exp(iw (t -t )dto (t ) = Eet-( )()e t0e (t )exp(-iwt )dt e (t ).w= E1+ i 0Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A4.8Harmonic ExcitationIn other words, s (t ) = E* (w )e (t )where E* (w ) = E (1+ iwe* (w )is the complex moduluse* (w )is the Four

6、ier transform of e(t ).andSincee* (w ) = Re(e* )+ i Im(e* ):E* (w ) = E (1- w Im(e* ) Es (w )storage modulusEw Re(e* ) El (w )loss modulus+ iModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A4.9Harmonic ExcitationPrevious formulas relating storage and loss moduli to the Four

7、ier transform of the dimensionless relaxation function define frequency domain viscoelasticity data:()Ew()sIm we= 1-*EE (w )data required by ABAQUSRe(we* ) =lEModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A4.10Harmonic ExcitationThe storage and loss moduli can be measured

8、 with excitation tests in the frequency domain. The data produced can be used not only in frequency domain analyses but also to derive short-term time domain data. If the time periods of interest are of the same order as the response time of the equipment used to measure relaxation or creep data, th

9、is may be the only way to obtain such data.Consider the complex modulus E* (w )an intermediate function:known (measured). CalculateE* (w )e(w ) =-1.EModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A4.11Harmonic ExcitationThe Fourier transform of the relaxation function is thene* (w ) = e(w ) ,iwfrom which we can do an inverse Fourier transform to obta