These problems are to provide self paced examples to develop skills in performing elastomer material curve fitting and finite element analysis using MSC.Marc and MSC.Marc Mentat.
Workshop data files are in the product directory ..mentat2008r1/examples/training/mar103 and usually coppied to your working directory eea/
wkshops_A/ or eea/wkshops_B/. Subfolders are:
uniaxial biaxial planar
Chapter 6: Workshop Problems Some MSC.Marc Mentat Hints and Shortcuts
Some MSC.Marc Mentat Hints and Shortcuts
1. Enter MSC.Marc Mentat to begin, Quit to stop
2. Mouse in Graphics: Left to pick, Right to accept pick 3. Mouse in Menu: Left to pick another menu or function,
Middle for help, Right to return to previous menu.
<cr> means keyboard return.
4. Save your work frequently. Go to FILES and select SAVE AS and specify a file name. Use SAVE from then on.
This will save the current MENTAT database to disk.
5. Dialog region at the lower left of screen displays current activity and prompts for input. Check this region
frequently to see if input is required.
6. Dynamic Viewing can be used to position the model in the graphics area. When activated, the mouse buttons:
Left – translates the model Right – zooms in/out
Middle – rotates in 3D
Use RESET VIEW and FILL to return to original view.
Be sure to turn off DYNAMIC VIEW before picking in the graphics area.
Model 1: Uniaxial Stress Specimen Chapter 6: Workshop Problems
Model 1: Uniaxial Stress Specimen
Objective: To model an elastomeric material under a uniaxial stress deformation mode.
To focus on curve fitting elastomeric test data, a fully runnable procedure file is provided that will build and (and run) an initial model. However, the model contains only a trivial neo-Hookean material model with C10 = 0.5. It will be your job to modify the model by reading in the test data and curve fitting it using various material models.
In a terminal window, use the cd command to move to the wkshops_A/uniaxial or the wkshops_B/uniaxial directory.
Type “mentat” to start the MSC.Marc Mentat program, then starting from the main menu proceed as follows:
UTILS
PROCEDURES EXECUTE
pick the file named uni_neo05.proc OK
OK
This will produce and run a uniaxial stress model. Please familiarize
yourself with this model. Look at the BC’s, the material specification, the contact bodies and contact table, and the loadcase.
Chapter 6: Workshop Problems Model 1: Uniaxial Stress Specimen
After the procedure file is finished the final picture on your screen will look like this.
Here is a brief summary of the uniaxial model we have created:
• A single brick element, full integration, Herrmann.
• Boundary conditions on
x=0 & y=0 faces to prevent free translation in space.
• Material model is neo-Hookean with C10 = 0.5
• Rigid contact surfaces are used to impose deformation.
lower rigid body, cbody2, is stationary.
upper rigid body, cbody3, is moved so as to first push, then pull, the brick element.
• Loading is performed in 40 equal time increments. Increment 10 is full compression of 50%, increment 30 is full extension of 200%, increment 40 returns the brick to it’s original configuration.
Now let’s look at the results of this analysis before curve fitting our uniaxial test data.
Model 1: Uniaxial Stress Specimen Chapter 6: Workshop Problems
All of the postprocessing functions are accessed from RESULTS, which is located on the topmost MAIN menu. We are especially interested in
deformed shape plots and XY plots of stress vs. strain.
MAIN
Chapter 6: Workshop Problems Model 1: Uniaxial Stress Specimen
Now let’s generate the stress-strain plot that the MSC.Marc analysis has calculated. When we curve fit the actual test data, this analysis stress-strain curve should match the curve fit response exactly.
HISTORY PLOT
Since the original area is one, and since the original length in the
z-direction is one, the above plot is the engineering stress versus the engineering strain for a uniaxial stress specimen with neo-Hookean behavior. We use the Body 2 force just to get the sign correct.
Another way of getting engineering stress-strain output is to use the user subroutine PRINCA.F. This is a plotv routine that calculates principal values of engineering stress & strain as well as principal stretch ratio. If available try re-running this analysis with the princa.f routine.
Q: Why is it ok to use a one element model for this problem?
A: ____________________________________________________
Model 1: Uniaxial Curve Fit Chapter 6: Workshop Problems
Model 1: Uniaxial Curve Fit
Using this model file, go to the material definition stage and redefine the material by reading the uniaxial data, filename st_18.data, and proceed to re-run the problem using neo-Hookean, Mooney 2-term, Mooney 3-term, and Ogden 2-term fits.
MATERIAL PROPERTIES
EXPERIMENTAL DATA FITTING TABLES
READ RAW
FILTER: type *.data pick file st_18.data, OK
Chapter 6: Workshop Problems Model 1: Uniaxial Curve Fit
Make the table type experimental_data, and associate this data with the uniaxial button. Your screen should look similar to the one below, and we are ready to start curve fitting the data.
TABLE TYPE
experimental_data, OK, RETURN UNIAXIAL
table2
Model 1: Uniaxial Curve Fit Chapter 6: Workshop Problems
Choose the neo-Hookean curve fitting routine and base the curve fit on just uniaxial data. The compute button will compute the model
coefficients. By default, responses for many modes are plotted. The single neo-Hookean coefficient, C10, is 0.265.
ELASTOMERS
NEO-HOOKEAN UNIAXIAL
COMPUTE, OK SCALE AXES
PLOT OPTIONS
SIMPLE SHEAR, RETURN (this turns off simple shear)
Chapter 6: Workshop Problems Model 1: Uniaxial Curve Fit
Comments:
We have just fit a neo-Hookean model using only uniaxial data.
MSC.Marc Mentat by default shows the model’s response in all major modes of deformation. This is very important. You should always know your model’s response to each mode of deformation.
Look again at the previous stress-strain plot. Notice the relative magnitude of the responses. Uniaxial is the lowest magnitude, the planar shear is higher, and the biaxial response is the highest. This is typical of most elastomers. See, for example, the stress-strain plot on the front cover of these notes.
Always start fitting with simple models first. If a simple model captures the curvature of the test data, use it! Proceed to higher order and more complex models only as needed.
Go back and use the EXTRAPOLATION feature and replot the
neo-Hookean results from -0.5 to 2.0 strain. It is very important to look at the model’s response over a wide range of strain, including both tension and compression. We are looking for stability limits (maxima in the stress-strain curve). Mooney form models with all positive coefficients
guarantee stability in all modes, for all strain. The simpler the material model, the higher probability it will be stable over a wider strain range.
Later, after curve fitting several choices of models and selecting the best one, we will re-run our simple analysis.
Model 1: Uniaxial Curve Fit Chapter 6: Workshop Problems
Here’s how to use the extrapolation feature to extend the strain range over which we plot the model’s response. We see that our neo-Hookean model is stable for all deformation modes.
NEO-HOOKEAN
EXTRAPOLATION EXTRAPOLATE
LEFT BOUND, enter -0.5, <cr>
RIGHT BOUND, enter 2.0, <cr>, OK COMPUTE, OK
SCALE AXES
Chapter 6: Workshop Problems Model 1: Uniaxial Curve Fit
Now fit a Mooney 2-term material model. Turn the extrapolation feature off for now. The Mooney coefficients are C10 = 0.074 and C01 = 0.280.
Positive coefficients guarantee stability. Notice the relative magnitudes now – the biaxial stiffness is about 4 times the earlier material model. Of course, the fit to the uniaxial data is better, with more terms this model can capture a higher curvature in the stress-strain data.
MOONEY(2)
EXTRAPOLATION
EXTRAPOLATE, OK (we want to turn it off) COMPUTE, OK
SCALE AXES
Model 1: Uniaxial Curve Fit Chapter 6: Workshop Problems
Now fit a Mooney 3-term material model. The Mooney coefficients are C10 = -0.735, C01 = 1.21, and C11 = 0.194. The uniaxial response is fantastic! The presence of a negative coefficient means that the material model might be unstable. We need to visually determine the stability range of the model. Note that the peak stress for the biaxial response has gone from 1.0 (neo-Hookean), to 4.5 (Mooney 2-term), to 36 (Mooney 3-term). Which one is correct?
MOONEY(3)
COMPUTE, OK SCALE AXES
Chapter 6: Workshop Problems Model 1: Uniaxial Curve Fit
Comments:
Which biaxial fit is correct? Well, we don’t know because we haven’t (yet) performed a biaxial test. This is the great difficulty with the Mooney form and Ogden form material models – they are just curve fits. There is no “rubber physics” embedded in these equations. As we see here, a curve fit to uniaxial data will have a good response for that mode of
deformation. But the responses for the other modes of deformation are all over the map. A rule of thumb based on observations of natural rubber and some other elastomers is that the tensile equi-biaxial response should be about 1.5 to 2.5 times the uniaxial tension response. We have seen many instances of higher order Mooney and Ogden models (using only uniaxial data) returning biaxial responses that are far too high. These are clearly bad material models.
Try playing with the POSITIVE COEFFICIENTS option to see how much the responses change.
For the curve fitting examples, you may need to toggle certain things on & off to better view and understand the computed fit. Keep these features in mind throughout all of these exercises:
• EXTRAPOLATION on/off
• PLOT OPTIONS, PREDICTED MODES
(select subsets of UNIAXIAL, BIAXIAL, PLANAR SHEAR)
• PLOT OPTIONS, LIMITS, YMAX, etc.
Model 1: Uniaxial Curve Fit Chapter 6: Workshop Problems
Now fit an Ogden 2-term material model. The uniaxial response is very good, but the biaxial response is now even higher than the Mooney 3-term. Ogden coefficients come in pairs, the moduli are and the exponents are . If each and have the same sign then stability is guaranteed. If a is positive and its corresponding is negative (or vice versa) then the material model might be unstable. Thus we may need to visually determine the stability range of the model.
OGDEN
COMPUTE, OK
μi
αi μi αi
μi αi
Chapter 6: Workshop Problems Model 1: Uniaxial Curve Fit
Comments:
We are now finished with the curve fitting portion of this uniaxial exercise. We see that the Mooney 3-term and Ogden 2-term fit the
uniaxial test data very well. However, we are concerned (or should be!) that the equi-biaxial response for some models (M 3-term, O 2-term) are too high and could make the material model overly stiff if that mode of deformation exists in our analysis. We need equi-biaxial test data to get a better fit to that mode.
Let’s run this uniaxial analysis with the Ogden 3-term model.
We select the curve fit model by pressing the APPLY button. Now go back and view the material model. Submit the analysis, then we will
post-process and show the analysis calculated stress-strain curve.
OGDEN
# OF TERMS = 3, OK COMPUTE, APPLY, OK
PLOT OPTIONS (turn off all – leave uniaxial only) COPY TO GEN. XY PLOTTER
RETURN (thrice)
MECH. MATERIALS TYPE, MORE
OGDEN (look at the material properties) OK
FILES
SAVE AS ogden3, OK MAIN
Model 1: Uniaxial Curve Fit Chapter 6: Workshop Problems
Now go to postprocessing and generate the engineering stress-strain curve (we did this earlier with the original model). We will also save the analysis generated stress-strain curve to an external file for comparison to the test data.
MAIN
COPY TO GEN. XY PLOTTER SAVE type ogden3.tab
This last command saves the table to an external file named ogden3.tab (.tab is just to remind us that it is table data).
Chapter 6: Workshop Problems Model 1: Uniaxial Curve Fit
To compare the two stress-strain curves, we will use MSC.Marc Mentat’s generalized plotter feature.
UTILS
GENERALIZED XY PLOT FIT
SHOW IDS = 0
Model 1: Uniaxial Curve Fit Chapter 6: Workshop Problems
Zoom in and tilt the plot and you will notice three curves:
the data, the fit, and the response of our model.
Note that the model must follow the hyperelastic material model (Ogden(3)) exactly.
Dat a Ogden(
3) fit Respon
se
sestr SgnrieeingnE
Engineering Strain
Chapter 6: Workshop Problems Model 1: Uniaxial Curve Fit
One may also use xmgr to read the file ogden3.tab that was generated in MSC.Marc Mentat. From a terminal window type:
xmgr st_18.data ogden3.tab
A graphics screen will appear in which the experimental data is shown in black and the analysis generated stress-strain curve is shown in red. Of course, the test data only extends to about 100% strain whereas we performed our analysis out to 200% strain.
Model 1: Uniaxial Curve Fit Chapter 6: Workshop Problems
Stress and Strain Measures
In order to plot the engineering stress and strain measures in this example, we plotted “Pos Z cbody3” versus “Force Z cbody2” and because the original length and cross-sectional area are unity, “Pos Z cbody3” versus
“Force Z cbody2” is the engineering strain versus the engineering stress.
Since a total Lagrangian formulation is being used, the stress and strain measures (or Lagrangian measures) on the post file are Cauchy stress and Green-Lagrange strain which are different than the engineering measures.
In this section, we shall convert the Lagrangian measures to engineering measures using the copy to clipboard feature available on the PC version of Mentat.
COLLECT GLOBAL DATA (this collects all the data) NODE/VARIABLES
ADD GLOBAL VAR.
Pos Z cbody3 Force Z cbody2 FIT, RETURN
Chapter 6: Workshop Problems Model 1: Uniaxial Curve Fit
Stress and Strain Measures
Now we shall repeat the above for the stress and strain values on the post file (Lagrangian measures).
NODE/VARIABLES CLEAR CURVES ADD VARIABLE
Comp 33 of Total Strain Comp 33 of Cauchy Stress FIT, RETURN
COPY TO CLIPBOARD
With the plotted values stored in the clipboard, paste the clipboard into the worksheet starting in column, the top of the worksheet should look like:
Save this Excel file as neohookean05_job1.t16.xls.
Pos Z cbody3 Force Z cbody2 Comp 33 of Total Strain Node 8 Comp 33 of Cauchy Stress Node 8
0 0 0 0
-0.05 -0.158024 -0.04875 -0.150124
-0.1 -0.334548 -0.095 -0.3011
-0.15 -0.534053 -0.13875 -0.453959
-0.2 -0.762458 -0.18 -0.609991
-0.25 -1.02772 -0.21875 -0.770832
-0.3 -1.34074 -0.255 -0.938581
-0.35 -1.71677 -0.28875 -1.11599
-0.4 -2.17766 -0.32 -1.30671
-0.45 -2.75563 -0.34875 -1.51575
-0.5 -3.4998 -0.375 -1.75011
-0.375 -1.9349 -0.304688 -1.20941
-0.25 -1.02772 -0.21875 -0.770831
Model 1: Uniaxial Curve Fit Chapter 6: Workshop Problems
Stress and Strain Measures
Now we can plot the two different strain and stress measures in Excel as:
This plot allows us to clearly see the difference between the two measures and notice that for small values of strain, the difference becomes very small.
Strain and Stress Measures
-6
Engineering Strain Versus Engineering Stress [MPa]
Green Lagrange Strain Versus Cauchy Stress [MPa]
Chapter 6: Workshop Problems Model 1: Uniaxial Curve Fit
Stress and Strain Measures
All of these measures are related, and we now will show how to convert from the Green-Lagrange strain and Cauchy stress to engineering values for this particular problem. The uniaxial direction in the model is in the
“z” or direction so we will use the 33 component of stress and strain.
Letting and be the 33 component of Green-Lagrange strain and Cauchy stress and be the engineering measures, respectively, we have for this deformation mode the following relations:
and
The above formulas come from the definition of Green-Lagrange strain, (see Appendix A on page 250) where is the
deformation gradient that is determined from the stretch ratios (see
“Summary of All Modes” on page 89)
From incompressibility we have and then
E33 t33
Model 1: Uniaxial Curve Fit Chapter 6: Workshop Problems
Stress and Strain Measures
The Excel file can be used to verify the conversion as:
Where columns E and F show the formulas to convert from the total Lagrangian to engineering measures of stress and strain, and columns E and F are identical to columns A and B, respectively. This file,
neohookean05_job1.t16.xls, is also available in the uniaxial directory.
A B C D =SQRT(2*C3+1) - 1 =D3/(1+E3)
Pos Z cbody3 Force Z cbody2 Comp 33 of Total Strain NComp 33 of Cauchy Stres Convert E33 to ε33 Convert t33 to σ33
0.00 0.00 0.00 0.00 0.00 0.00
-0.05 -0.16 -0.05 -0.15 -0.05 -0.16
-0.10 -0.33 -0.10 -0.30 -0.10 -0.33
-0.15 -0.53 -0.14 -0.45 -0.15 -0.53
-0.20 -0.76 -0.18 -0.61 -0.20 -0.76
-0.25 -1.03 -0.22 -0.77 -0.25 -1.03
-0.30 -1.34 -0.26 -0.94 -0.30 -1.34
-0.35 -1.72 -0.29 -1.12 -0.35 -1.72
-0.40 -2.18 -0.32 -1.31 -0.40 -2.18
-0.45 -2.76 -0.35 -1.52 -0.45 -2.76
-0.50 -3.50 -0.38 -1.75 -0.50 -3.50
-0.38 -1.93 -0.30 -1.21 -0.38 -1.94
-0.25 -1.03 -0.22 -0.77 -0.25 -1.03
-0.13 -0.43 -0.12 -0.38 -0.13 -0.43
0.00 0.00 0.00 0.00 0.00 0.00
0.13 0.33 0.13 0.38 0.13 0.33
0.25 0.61 0.28 0.76 0.25 0.61
0.38 0.85 0.45 1.16 0.38 0.85
0.50 1.06 0.63 1.58 0.50 1.06
0.63 1.25 0.82 2.02 0.63 1.25
0.75 1.42 1.03 2.49 0.75 1.42
0.88 1.59 1.26 2.98 0.87 1.59
1.00 1.75 1.50 3.50 1.00 1.75
1.13 1.90 1.76 4.04 1.12 1.90
1.25 2.05 2.03 4.62 1.25 2.05
1.38 2.20 2.32 5.22 1.37 2.20
1.50 2.34 2.63 5.85 1.50 2.34
1.63 2.48 2.95 6.50 1.62 2.48
1.75 2.62 3.28 7.19 1.75 2.62
1.88 2.75 3.63 7.91 1.87 2.75
2.00 2.89 4.00 8.66 2.00 2.89
1.80 2.67 3.42 7.48 1.80 2.67
1.60 2.45 2.88 6.37 1.60 2.45
1.40 2.23 2.38 5.34 1.40 2.22
1.20 1.99 1.92 4.38 1.20 1.99
1.00 1.75 1.50 3.50 1.00 1.75
0.80 1.49 1.12 2.68 0.80 1.49
0.60 1.21 0.78 1.93 0.60 1.21
0.40 0.89 0.48 1.25 0.40 0.89
0.20 0.51 0.22 0.61 0.20 0.51
0.00 0.00 0.00 0.00 0.00 0.00
Chapter 6: Workshop Problems Model 1: Uniaxial Curve Fit
Stress and Strain Measures
Summarizing the various stress and strain measures used we have:
In our uniaxial example, these measures are related as:
Stress and Strain Measures Stress Measure Strain Measure Curve Fitting Engineering Engineering Analysis
Total Lagrange Cauchy Green-Lagrange
Updated Lagrange Cauchy Logarithmic
Stress and Strain Measures Stress Measure Strain Measure Curve Fitting
Analysis
Total Lagrange Updated Lagrange
σ33 ε33
t33 = (1 +ε33)σ33 E33 = (1 +ε33)2– 1
t33 = (1 +ε33)σ33 E33 = ln(1+ε33)
Model 1C: Tensile Specimen with Continuous Damage Chapter 6: Workshop Problems
Model 1C: Tensile Specimen with Continuous Damage
Objective: To model an elastomeric material under a cyclical uniaxial deformation mode subjected to damage accumulated from continuously varying strain cycles. For instance, looking at the test data below, we notice that upon repeated cycling the peak stress decays.
0.0 0.2 0.4 0.6 0.8 1.0
Engineering Strain [1]
0.0 0.2 0.4 0.6 0.8 1.0
Engineering Stress [Mpa]
Tensile Data
Continuous Damage
Chapter 6: Workshop Problems Model 1C: Tensile Specimen with Continuous Damage
In this workshop problem, we will simulate this behavior using the continuous damage model discussed in Appendix B. To clarify the behavior let’s plot the peak stress versus the cycle number as shown below.
If our application experiences, this kind of behavior then we may wish to simulate this continuous damage. We would start by doing any normal hyperelastic curve fit. However, we would use the 1st cycle of the stress
0.0 2.0 4.0 6.0 8.0 10.0
Cycle Number
0.90 0.95 1.00 1.05 1.10
Engineering Stress [Mpa]
Tensile Data
Continuous Damage for Engineering Strain = 1.00
Model 1C: Tensile Specimen with Continuous Damage Chapter 6: Workshop Problems
From NT (Windows 2000) just click on the uni_neo05.proc file or from unix Type “mentat” to start the MSC.Marc Mentat program, then starting from the main menu proceed as follows:
UTILS
PROCEDURES EXECUTE
pick the file named uni_neo05.proc OK
MAIN
This will produce and run a uniaxial stress model. Using this model file, we will go to the material definition stage and redefine the material by reading the uniaxial data, filename st_1st.tab, damage data, st_cont.tab, loading data st_load.tab and proceed to re-run the problem using an Ogden 1-term fit with continuous damage.
MATERIAL PROPERTIES
EXPERIMENTAL DATA FITTING TABLES
READ
NORMAL
FILTER: type st*
pick file st_1st.tab, OK (different data from st_18.data)
Chapter 6: Workshop Problems Model 1C: Tensile Specimen with Continuous Damage
Your screen should look similar to the one below
While we are here let’s read some more tables.
READ
NORMAL
FILTER: type st*
pick file st_cont.tab pick file st_load.tab RETURN
Now we are ready to start curve fitting the data.
Model 1C: Tensile Specimen with Continuous Damage Chapter 6: Workshop Problems
ELASTOMERS MORE
CONTINUOUS DAMAGE CONSTANT
NUMBER OF TERMS = 2
FREE ENERGY = 1.07 (this is just the 1st peak stress) COMPUTE
APPLY, OK, RETURN
OGDEN
UNIAXIAL
Chapter 6: Workshop Problems Model 1C: Tensile Specimen with Continuous Damage
RETURN (twice)
Let’s review the material properties to check that the curve fit has been properly applied to the selected material.
MAIN
MATERIAL PROPERTIES MORE
OGDEN, DAMAGE EFFECTS - RUBBER, OK OK
Model 1C: Tensile Specimen with Continuous Damage Chapter 6: Workshop Problems
Now we can complete the model and run the analysis. The remaining item to finish is to attach a table to the contact body to cycle the loading several times from a strain of 0 to a strain of 1.
MAIN CONTACT
CONTACT BODIES EDIT (pick cbody3)
RIGID POSITION
(Z) TABLE (pick table st_load)
Chapter 6: Workshop Problems Model 1C: Tensile Specimen with Continuous Damage
STATIC
TOTAL LOADCASE TIME = 940
# STEPS = 20 OK
MAIN FILES
SAVE AS ogden_damage OK MAIN
JOBS
RUN, SUBMIT1, MONITOR, OK MAIN
COPY TO GEN. XY PLOTTER
COPY TO GEN. XY PLOTTER