The experimental determination of elastomeric material constants depends greatly on the deformation state, specimen geometry, and what is measured.
Chapter 5: Material Test Data Fitting Major Modes of Deformation
Major Modes of Deformation Uniaxial Tension
Biaxial Tension
(equivalent strain as uniaxial compression)1
2
3 λ1 = λ2 = λ λ2 = λ3 = 1 ⁄ λ2
λ1 = λ2 = λ λ3 = 1 ⁄ λ2
Major Modes of Deformation Chapter 5: Material Test Data Fitting
Major Modes of Deformation (cont.) Planar Tension, Planar Shear, Pure Shear
Simple Shear
λ1 = λ λ2 = 1 λ3 = 1 ⁄ λ
1
2
3
Chapter 5: Material Test Data Fitting Major Modes of Deformation
Major Modes of Deformation (cont.)
Volumetric (aka Hydrostatic, Bulk Compression)
F F
Confined Hydrostatic
Compression
Compression
Confined Compression Test (UniVolumetric) Chapter 5: Material Test Data Fitting
Confined Compression Test (UniVolumetric)
Strain State:
Stress State:
For this deformation state we have ,
and the uniaxial strain is equal to the volumetric strain or
. The bulk modulus becomes
MSC.Marc Mentat uses the pressure, , versus a “uniaxial equivalent” of the volumetric strain namely,
, to determine the bulk
F L, λ1 = 1 λ2 = 1 λ3 = L L⁄ 0
σ1 = σ2 = σ3 = – F A⁄ o = p
λ1λ2λ3 = V V⁄ 0 = L L⁄ 0
0.000 0.010 0.020 0.030 0.040
Equivalent Uniaxial Strain [1]
0.0
For Mentat Curve Fitting
1
Chapter 5: Material Test Data Fitting Hydrostatic Compression Test
Hydrostatic Compression Test
Strain State:
Stress State:
For this strain state we have
and since
the uniaxial strain becomes one third the volumetric strain or .
The bulk modulus becomes
Again MSC.Marc Mentat uses the pressure, , versus a “uniaxial
F L,
Summary of All Modes Chapter 5: Material Test Data Fitting
Summary of All Modes
Mode:
Chapter 5: Material Test Data Fitting General Guidelines
General Guidelines
Its just curve fitting!
No Polymer physics as basis Don’t use too high order fit
Remember polynomial fit lessons (high school?)
Number of Data Points
Don’t use too many Regularize if needed Add/Subtract points if needed
Weighting of Points
Range and Scope of Data
Check fit outside range of data
Check fit in other modes of deformation – scope
Mooney, Ogden Limitations Chapter 5: Material Test Data Fitting
Mooney, Ogden Limitations
Phenomenological models – not material “law”
These models are mathematical forms, nothing more
Summary of phenomenological models given by Yeoh (1995)
“Rivlin and Saunders (1951) have pointed out that the agreement between experimental tensile data and the Mooney-Rivlin
equation is somewhat fortuitous. The Mooney-Rivlin model obtained by fitting tensile data is quite inadequate in other modes of deformation, especially compression.”
Using only uniaxial tension data is dangerous!
Mooney model in MSC.Marc allows no compressibility
Ogden model does allow compressibility
Chapter 5: Material Test Data Fitting Visual Checks
Visual Checks
Extrapolations can be dangerous
Always visually check your model’s predicted response
Check it outside of the data’s range (see below) Check it outside the test’s scope
Predicted Response
DATA
Real Material Predicted
Response
Real Material
σ
dσ dε 0• >
dσ dε 0• <
ε
Material Stability Chapter 5: Material Test Data Fitting
Material Stability
Unstable material model -> numerical difficulties in FEA
Druckers stability postulate, Graphically:
Remember effects of Newton-Raphson and strain range
dσ dε• > 0
σ
ε
dσ11 •dε11 > 0 dσ11• dε11 < 0
Chapter 5: Material Test Data Fitting Future Trends
Future Trends
Statistical Mechanics Models
Based on single-chain polymer chain physics
Build up to network level using non-gaussian statistics
8 Chain model by Arruda-Boyce (1993)
2 parameter model, can be expressed in terms of I1
Paper: “A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials”, J. Mech. Phys.
Solids, V41 N2, pp 389-412.
Also similar is the Gent model (1996)
Paper: “A new Constitutive Relation for Rubber”, Rubber Chem. and Technology, v. 69, pp 59-61.
Claim: alleviates need to gather test data from
multiple modes
Adjusting Raw Data Chapter 5: Material Test Data Fitting
Adjusting Raw Data
The stress strain response of the three modes of deformation are shown below as taken from the laboratory equipment. In its raw form
it is not ready to be fit to a hyperelastic material model. It needs to
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.5 1.0 1.5 2.0
Equal Biaxial
EngineeringStress[Mpa]
The Raw Data (4 points/sec)
Engineering Strain [1]
Pure Shear Tension
Chapter 5: Material Test Data Fitting Adjusting Raw Data
Adjusting Raw Data (cont.)
The raw data is adjusted as shown below by taking the 18th upload cycle.
In doing this Mullins effect is ignored. This 18th upload cycle
then needs to be shifted such that the curve passes through the origin.
Remember hyperelastic models must be elastic and have their stress
0.0 0.2 0.4 0.6 0.8 1.0
Adjusting The Raw Data
Shift to the Origin
Equal Biaxial Shifted
Adjusting Raw Data Chapter 5: Material Test Data Fitting
Adjusting Raw Data (cont.)
There is nothing special about taking the upload cycle, for instance the curve fitting may be done on the download path or both upload and download paths as shown below. The intended application can help you
decide upon the most appropriate way to adjust the data prior to fitting the hyperelastic material models.
0 1
0 1
uniaxial/experiment uniaxial/neo_hookean
1 1
0
0 Engineering Strain [1]
Engineering Stress [Mpa]
Fit to upload
& download Fit to upload
Chapter 5: Material Test Data Fitting Consider All Modes of Deformation
Consider All Modes of Deformation
The plot below illustrates the danger in curve fitting only the tensile data, namely the other modes may become too stiff. This is why MSC.Marc Mentat always draws the other modes even when no experimental data is present.
Below, a 3-term Ogden provides a great fit to the tensile data, but spoils the other modes. This can be avoided by looking for a balance between the various deformation modes.
The Three Basic Strain States Chapter 5: Material Test Data Fitting
The Three Basic Strain States
After shifting each mode to pass through the origin, the final curves are shown below. Very many elastomeric materials have this basic shape of the three modes, with uniaxial, shear, and biaxial having
increasing stress for the same strain, respectively. Knowledge of this and the actual shape above where say at a strain of 80%, the ratio of equal biaxial to uniaxial stress is about 2 (i.e., 1.3/0.75 = 1.73) will become very
0.0 0.2 0.4 0.6 0.8 1.0
The Three Basic Strain States
General Elastomer Trends
Equal Biaxial Pure Shear Tension
Chapter 5: Material Test Data Fitting Curve Fitting with MSC.Marc Mentat
Curve Fitting with MSC.Marc Mentat
Objective: Fit experimental data of Mooney or Ogden materials with MSC.Marc Mentat. Begin at the main menu.
MATERIAL PROPERTIES
Curve Fitting with MSC.Marc Mentat Chapter 5: Material Test Data Fitting
Curve Fitting with MSC.Marc Mentat (cont)
The resulting display of the material model is similar to this.
The numerical coefficients for the model are shown in the pop-up menu. Use the APPLY button to copy these coefficients to your material model.
Notice that the uniaxial, biaxial, planar shear and simple shear modes are shown, where the uniaxial mode matches the
material input. To turn some modes off, or make other display modifications go to PLOT OPTIONS.
PLOT OPTIONS
SIMPLE SHEAR (this toggles it off) PLANAR SHEAR (this toggles it off) RETURN
SCALE AXES
Chapter 5: Material Test Data Fitting Curve Fitting with MSC.Marc Mentat
Curve Fitting with MSC.Marc Mentat (cont)
Objective: Fit experimental data of Viscoelastic materials with MSC.Marc Mentat. Begin at the main menu.
MATERIAL PROPERTIES
Curve Fitting with MSC.Marc Mentat Chapter 5: Material Test Data Fitting
Curve Fitting with MSC.Marc Mentat (cont) Mooney-Rivlin fitting is linear, uses least squares fitting
The least squares error is given by either:
The and are relative or absolute respectively is the total number of data points
is the calculated stress
is the measured engineering stress
Relative error is the default
errorR 1 σicalc
errorR errorA Ndata
σicalc σimeasured
Chapter 5: Material Test Data Fitting Curve Fitting with MSC.Marc Mentat
Curve Fitting with MSC.Marc Mentat (cont)
Ogden fitting is nonlinear, uses downhill-simplex method
Downhill-simplex method is an iterative method
Uses a number of start points Continues until:
is set using CONVERGENCE TOLERANCE
can be set with the ERROR LIMIT button
abs error( max – errormin) abs error( max +errormin)
--- tol ---2
<
tol
errormin
Curve Fitting with MSC.Marc Mentat Chapter 5: Material Test Data Fitting
Curve Fitting with MSC.Marc Mentat (cont) Viscoelastic fitting is linear, uses least
squares fitting
A Prony series (exponential decay) is fit to the test data
The least squares error is given by:
For a good fit, the number of Prony series terms
should equal the number of time decades in the
test data
Chapter 5: Material Test Data Fitting Curve Fitting with MSC.Marc Mentat