L5.22
Modeling Rubber and Viscoelasticity with Abaqus
Abaqus Test Data Usage
• Typical Usage
*MATERIAL, NAME=VITON
*HYPERELASTIC, POLYNOMIAL, N=1, TEST DATA INPUT
*UNIAXIAL TEST DATA 0.00,0.00
0.03,0.02 data lines are
0.15,0.10 nominal stress, nominal strain 0.23,0.20
...
*PLANAR TEST DATA ...
*BIAXIAL TEST DATA ...
*VOLUMETRIC TEST DATA (to define optional compressibility)
...
*EXPANSION (to define optional CTE) ...
*VISCOELASTICITY (to define optional time-dependency)
...
Suboptions of
*HYPERELASTIC
L5.23
Modeling Rubber and Viscoelasticity with Abaqus
Abaqus Test Data Usage
• Volumetric
*VOLUMETRIC TEST DATA (to define optional compressibility) pressure_1, volume_ratio_1
pressure_2, volume_ratio_2
• Volumetric information should be specified for cases where the material does not have room to shear – that is, cases where the material is highly confined.
• For many elastomers a bulk modulus of approximately 2000MPa is reasonable.
• For highly confined applications, it is better to input a D1 of
0.001
than to leave it unspecified (and therefore incompressible)L5.24
Modeling Rubber and Viscoelasticity with Abaqus
Abaqus Test Data Usage
• Thermal Expansion
*EXPANSION
**(to define optional CTE) Alpha1, temp1
Alpha2, temp2 ...
• Defines the volumetric CTE
(coefficient of Thermal Expansion) for the material.
• Abaqus uses a total, or secant, measure from a reference temperature.
L5.25
Modeling Rubber and Viscoelasticity with Abaqus
Abaqus Test Data Usage
• Test data smoothing
• The test data input option provides a data-smoothing capability that is recommended
• Useful in situations where the test data do not vary smoothly
• Avoids potential
convergence problems during the analysis
• User can control the smoothing process
• This capability is particularly useful with the Marlow model when the data is scattered.
L5.26
Modeling Rubber and Viscoelasticity with Abaqus
Abaqus Test Data Usage
• Test data usage with the Marlow model
• For uniaxial, biaxial, and planar modes, either tension or compression data can be specified.
• Tension data determines the strain energy potential, which in turn determines the
compression behavior, and vice versa.
• When used with 1-D elements (beams, rebars, and trusses), data from both tension and compression tests can be specified together.
L5.27
Modeling Rubber and Viscoelasticity with Abaqus
Abaqus Test Data Usage
• The volumetric behavior for the Marlow model can be defined in one of the following ways:
• Volumetric test data
• Lateral test data in the uniaxial, biaxial, or planar mode
• These data options allow users to specify the lateral behavior along with the primary behavior. Lateral strains define the volumetric response
• Effective Poisson’s ratio
• Incompressibility is assumed if none of the above specified.
L5.28
Modeling Rubber and Viscoelasticity with Abaqus
Abaqus Test Data Usage
• In addition, for the volumetric mode, both hydrostatic tension and hydrostatic compression data can be specified.
• More commonly, only hydrostatic compression data are available.
Abaqus assumes that the hydrostatic pressure is an antisymmetric function of the nominal volumetric strain,
e
vol, aboute
vol= 0.Choosing a Strain Energy Function
L5.30
Modeling Rubber and Viscoelasticity with Abaqus
Choosing a Strain Energy Function
• The importance of using multiple types of test data when calibrating the models is discussed next.
• The recommended selection procedure is then summarized.
• In each case, the models are listed in order of preference
• The suggested approach considers physically motivated models first.
• Tips:
• Use simple models first.
• Keep the order, N, as low as possible to describe the data.
L5.31
Modeling Rubber and Viscoelasticity with Abaqus
Choosing a Strain Energy Function
• Importance of Multiple Types of Tests
• Generally, when data from multiple experimental tests are available, the Van der Waals and Ogden strain energy functions are more accurate in fitting the stress-strain curves.
• When limited amounts of test data exist for calibration, for instance, just uniaxial test data, the use of the Van der Waals, Ogden, full polynomial models can be quite dangerous.
• When using limited test data stay with the I1only models – Marlow, Arruda-Boyce, Van der Waals with
= 0, reduced polynomial (Neo-Hookean, Yeoh).L5.32
Modeling Rubber and Viscoelasticity with Abaqus
Choosing a Strain Energy Function
• Using only Uniaxial Test data
• The following group of slides show a comparison of the various strain energy functions when calibrated with only uniaxial test data. The other modes of test data (planar, equibiaxial) will be shown for reference.
• The test data were taken from Treloar (―Stress-strain data for vulcanized rubber under various types of deformations,‖ Trans. Faraday Society, 40, 1944) for uniaxial tension, biaxial tension, and planar tension.
• For each slide we show the fit to only uniaxial tension data. By doing so we can show that I1based models in general do ok, while I1and I2 models can give very poor results when fit to only uniaxial tension test data.
Abbreviations: ST = Simple Tension PT = Planar Tension EB = Equibiaxial Tension
L5.33
Modeling Rubber and Viscoelasticity with Abaqus
Choosing a Strain Energy Function
• Treloar test data
• Stress in MPa
• Focus on the relationship between 3 modes.
• This is a common semi-quantitative relationship
• PT slightly higher than ST
• EB 50% to 100% higher than ST
L5.34
Modeling Rubber and Viscoelasticity with Abaqus
Choosing a Strain Energy Function
• Marlow Model (I1based model)
• Uniaxial data represented exactly; other modes are represented reasonably well.
Curve Fits to only Uniaxial Test Data
L5.35
Modeling Rubber and Viscoelasticity with Abaqus
Choosing a Strain Energy Function
• Neo-Hookean and Yeoh Models (I1based models)
• Other modes are represented reasonably well.
Curve Fits to only Uniaxial Test Data
L5.36
Modeling Rubber and Viscoelasticity with Abaqus
Choosing a Strain Energy Function
• Arruda-Boyce Model (I1based model)
• Prediction similar to Neo-Hookean model
Curve Fits to only Uniaxial Test Data
A-B coefficients from paper Curve Fit in Abaqus
L5.37
Modeling Rubber and Viscoelasticity with Abaqus
Choosing a Strain Energy Function
• Full Polynomial and Ogden models (I1and I2based models)
• Other modes are extremely overly stiff (very poor with limited test data).
Curve Fits to only Uniaxial Test Data
L5.38
Modeling Rubber and Viscoelasticity with Abaqus
Choosing a Strain Energy Function
• Van der Waals models (
controls the I1and I2bases)• For limited test data the Van der Waals model changes dramatically.
Curve Fits to only Uniaxial Test Data
set by curve fitting
set to 0.0 (edit input file)
L5.39
Modeling Rubber and Viscoelasticity with Abaqus
• Using Full Datasets
• Typically the simplest models (I1based) will not improve very much as additional test data is used in the curve fitting process.
• The fewer the model parameters, the less likely the additional modes of deformation test data will improve the fit.
• The higher order (N) I1and I2based models will improve dramatically as additional test data is used for curve fitting.
• In the following slides we will repeat the fits shown earlier, but this time all 3 sets of Treloar data will be used as a basis for the curve fits.
Choosing a Strain Energy Function
L5.40
Modeling Rubber and Viscoelasticity with Abaqus
Choosing a Strain Energy Function
• Neo-Hookean and Yeoh Models (I1based models)
• Not much change over earlier limited data fit
Curve Fits using all Data Sets
L5.41
Modeling Rubber and Viscoelasticity with Abaqus
Choosing a Strain Energy Function
• Arruda-Boyce Model (I1based model)
Curve Fits using all Data Sets
A-B coefficients from paper Curve Fit in Abaqus
L5.42
Modeling Rubber and Viscoelasticity with Abaqus
Choosing a Strain Energy Function
• Full Polynomial and Ogden models (I1and I2based models)
• These fits improve dramatically over limited data case Curve Fits using all Data Sets
L5.43
Modeling Rubber and Viscoelasticity with Abaqus
• Van der Waals models (
controls I1and I2bases)• Using all data, Van der Waals gives good fit. No need to set
= 0. Curve Fits using all Data Setsset by curve fitting set to 0.0 (edit input file)
Choosing a Strain Energy Function
L5.44
Modeling Rubber and Viscoelasticity with Abaqus
Choosing a Strain Energy Function
• Summary: Selection procedure for strain energy functions
• Limited test data: small strain data
• Neo-Hookean model
• Limited test data: good detailed data for one kind of test (e.g., good uniaxial data)
• Marlow model
• Limited test data: initial modulus and stretch limit (and possibly a few extra data points)
• Arruda-Boyce
• Van der Waals with
= 0• Reduced polynomial (e.g., Yeoh) model
• Predicted behavior in other modes of straining will be plausible, but not necessarily accurate.
• Avoid using the Ogden and Full Polynomial models with limited test data.
L5.45
Modeling Rubber and Viscoelasticity with Abaqus
Choosing a Strain Energy Function
• Summary (cont'd)
• Full test suite of data (i.e., multi-axial data)
• Van der Waals model (
0)
• Ogden model
• The Full Polynomial model may be OK, but generally it doesn’t fit data as well as the Ogden model.
• It is better to use this model with data that have already been calibrated.
UHYPER
L5.47
Modeling Rubber and Viscoelasticity with Abaqus
UHYPER
• UHYPER syntax
• You may define your own elastomer behavior through the use of a user subroutine called UHYPER. You provide Fortran code to define the energy function, U, and first and second derivatives of Uwith respect to Ī1and Ī2.
• To invoke its use, the Abaqus input file looks like this:
*MATERIAL, NAME=...
*HYPERELASTIC, USER, TYPE=..., PROPERTIES=...
*EXPANSION (to define optional CTE) ...
*VISCOELASTICITY (to define optional time-dependency) ...
L5.48
Modeling Rubber and Viscoelasticity with Abaqus
UHYPER
• Defining UHYPER in Abaqus/CAE:
Mullins Effect
L5.50
Modeling Rubber and Viscoelasticity with Abaqus
Mullins Effect
• Stress softening in certain filled rubbers occurs due to damage associated with straining
• The results depicted in the figure show evidence of progressive damage (with cycles), with the response stabilizing after a few cycles
• The results also show evidence of permanent set and viscoelasticity
Courtesy: Axel Physical Testing Services
e
0 permanent setDamage: Unloading and further reloading follows different path characterized by stress softening
Progressive damage indicated by reduced stress with fixed strain loading cycles
Hysteresis: loading and unloading for a given cycle follow different paths—
energy is dissipated with each cycle Dashed line is the primary
curve (given by hyperelastic material model)
L5.51
Modeling Rubber and Viscoelasticity with Abaqus
Mullins Effect
• Idealized response—Abaqus model
• Does not model progressive damage during the first few cycles
• Does not take into account permanent set and
viscoelasticity
Energy dissipated once (damage);
no subsequent hysteresis or progressive damage
No permanent set or viscoelasticity
L5.52
Modeling Rubber and Viscoelasticity with Abaqus
Mullins Effect
• The material definition consists of two parts:
• Define the primary behavior using a hyperelastic material model.
• Test data, strain energy density function coefficients, or user subroutine UHYPERcan be used to define the primary behavior.
• Define the damage behavior using the *MULLINS EFFECT option.
L5.53
Modeling Rubber and Viscoelasticity with Abaqus
Mullins Effect
• The material parameters related to damage can be specified directly
• Alternatively, these parameters can be determined by Abaqus based on calibration of unloading-reloading test data
• Test data from one or more of the primary modes of deformation (uniaxial, biaxial, and planar) can be specified
• For a specific deformation mode, unloading-reloading test data from multiple maximum strain levels can be specified by repeated use of the appropriate test data option
• User subroutine UMULLINSis available in Abaqus/Standard
• This allows you to define the damage variable directly
• The Mullins effect model cannot be used with viscoelasticity or hysteresis.
L5.54
Modeling Rubber and Viscoelasticity with Abaqus
Mullins Effect
• Output variables:
• DMENER: Damage dissipation density at an integration point
• ELDMD: Damage dissipation in an element
• EDMDDEN: Damage dissipation per unit volume in an element
• ALLDMD: Total damage dissipation in the whole model (or over a user-specified element set)
L5.55
Modeling Rubber and Viscoelasticity with Abaqus
Mullins Effect
• Example: Calibration of test data
• Uniaxial test data to define the primary behavior
• Uniaxial unloading-reloading data from three different strain levels (stabilized cycles)
• The Abaqus model replaces stabilized cycle at each strain level with a single curve that represents both loading and unloading
L5.56
Modeling Rubber and Viscoelasticity with Abaqus
Mullins Effect
• Example: Load-deflection of a stationary solid rubber disk
• Rigid surface displaced up against fixed disk
• Unloaded
• Reloaded to deformation levels that are higher than the first loading
• Above deformation pattern constitutes two loading cycles
L5.57
Modeling Rubber and Viscoelasticity with Abaqus
Mullins Effect
Unload at constant damage
Unload/reload at constant damage
Dissipate more energy