1、 Copyright 2006 ABAQUS, Inc. Time Domain Viscoelasticity Lecture 8 Modeling Rubber and Viscoelasticity with ABAQUS L8.2 Copyright 2006 ABAQUS, Inc. Overview Classical Linear Viscoelasticity Prony Series Representation Finite-Strain Viscoelasticity Relaxation and Creep Test Data Prony Series Data Aut
2、omatic Material Evaluation Usage Hints Copyright 2006 ABAQUS, Inc. Classical Linear Viscoelasticity Modeling Rubber and Viscoelasticity with ABAQUS L8.4 Copyright 2006 ABAQUS, Inc. Classical Linear Viscoelasticity Recall that classical means small-strain theory. In ABAQUS this means: relaxation/cree
3、p behavior defined by *VISCOELASTIC elasticity is defined by *ELASTIC Isotropic linear viscoelasticity is implemented in ABAQUS Modeling Rubber and Viscoelasticity with ABAQUS L8.5 Copyright 2006 ABAQUS, Inc. Classical Linear Viscoelasticity To generalize the viscoelastic equations to multiaxial str
4、ess states it is best to work with shear (deviatoric) and volumetric (dilatational) behavior: where S is the deviatoric stress tensor and p is the hydrostatic pressure. ( )() 00 0 0 1( ) ( ) dG tttd Gd t tt t =+-SSS ( )() 00 0 0 1( ) ( ) dK p tp tptd Kd t tt t =+- Modeling Rubber and Viscoelasticity
5、 with ABAQUS L8.6 Copyright 2006 ABAQUS, Inc. Classical Linear Viscoelasticity Definitions We decompose the total stress into shear and volumetric parts by: We decompose the total strain into shear and volumetric parts by: The elastic stress-strain relations decompose into: where are the deviatoric
6、and pressure stress states that would exist for the current strain state if the material were behaving purely elastically. 112233 11 ( )() 33 pptrsss=-= -= -+SIwhere.ssss 112233 1 ( )() 3 volvol treeeee=+=+eIwhere.eeee 0000 ( )( )( ) vol tGtp tKe=eand,S S 00 ( )( )tp tSand Copyright 2006 ABAQUS, Inc
7、. Prony Series Representation Modeling Rubber and Viscoelasticity with ABAQUS L8.8 Copyright 2006 ABAQUS, Inc. Prony Series Representation In ABAQUS the time-dependent behavior are represented in terms of a Prony series: material coefficients are material coefficients are G0 and K0 are determined fr
8、om 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 to specify An exception is void filled elastomers (elastic foams) in wh
9、ich there is generally significant volumetric relaxation. () / 0 1 ( )11 G i N p i i GGge t t t - = 骣 =-琪 琪 桫 () / 0 1 ( )11 K i N p i i KKke t t t - = 骣 =-琪 琪 桫 ( ) and( )GKtt ( ).Kt pG ii up to N pairs of gandt pK ii up to N pairs of kandt Modeling Rubber and Viscoelasticity with ABAQUS L8.9 Copyr
10、ight 2006 ABAQUS, Inc. Prony Series Representation In ABAQUS are specified in one of four different ways: Prony series curve fit from experimental stress relaxation test data: *VISCOLEASTIC, TIME=RELAXATION TEST DATA Prony series curve fit from experimental creep test data: *VISCOELASTIC, TIME=CREEP
11、 TEST DATA Prony series coefficients specified directly by the user: *VISCOELASTIC, TIME=PRONY Defined from frequency-dependent cyclic test data: *VISCOLEASTIC, TIME=FREQUENCY DATA ( )( )GKttand Modeling Rubber and Viscoelasticity with ABAQUS L8. 10 Copyright 2006 ABAQUS, Inc. The rate-independent e
12、lastic moduli are specified using *ELASTIC *ELASTIC, MODULI=INSTANTANEOUS E0, 0 or *ELASTIC, MODULI=LONG TERM E1, 1 ABAQUS will determine the appropriate G and K values from the user specified E and values. Prony Series Representation Modeling Rubber and Viscoelasticity with ABAQUS L8. 11 Copyright 2006 ABAQUS, Inc. Prony Series Representation If the user specifies long-term elastic moduli, ABAQUS will compute the instantaneous elastic moduli using the relaxation information where the are the Prony series coefficients. The choice of defining the elasticity in terms of instantaneous or lo
