EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
Determination Of An Elastic Potential For A Thin Rubber Sheet
EN 227 Final ProjectDonald Ward & Brian Burke
12/3/03
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
ØØOverview Overview
•Introduction•Testing Procedure•Experimental Setup
•Load application•Deformation characterization
•Material Response•Material symmetry•Incompressibility•Elastic Potential
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
ØØTesting Procedure Testing Procedure •Examine microstructure for clues on material response (e.g. preferential direction, fiber direction, porosity, etc.)
•Uniaxial Test
•Examine material response in several directions looking for Material Symmetry Group
•Biaxial Test
•Measure all three principle stretches
•Known tractions
•Compare results to known elastic potentials
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
ØØExperimental Setup Experimental Setup
•Self-centering•Pulleys help align the system and ensure that stresses do not contain any shear components•No support for moment loading•Clip and pin minimize slip
Biaxial System
2
1
Pulleys
Load lines
Clip
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
Computer Imaging of Deformation
• Take digital image of un-deformed configuration.
• Convert image into text file with an intensity value for each pixel.
• Search the values for high intensity sources (black grid lines)
• Store these pixels
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
95
145
195
245
295
345
70 90 110 130 150 170 190 210 230 25095
145
195
245
295
345
110 130 150 170 190 210
Center(high pixelcount)
5 0 100 150 200 250 300 350
50
100
150
200
250
•From the saved pixels search for those pixels that have a considerable number of neighbors in the 1 and 2 directions•The pixel positions are then averaged to determine the centersfor the corners •This process is done for the deformed and un-deformed images•The images are correlated through the center which contains many more intense pixels than the rest
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
•Micrographs of material show a randomdistribution of material implying isotropyin the plane•A second observation made is that the material appears somewhat porous this possible implies compressibility but other results show that the material is more
closely represented asincompressible
ØØMicrostructure Microstructure
100 mm
50 mm 20 mm
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
ØØMaterial Symmetry Group Material Symmetry Group
• Check angular dependence of load response
•Three samples cut from same sheet
•Response measured under uni-axial loading
•Arrows indicate direction of load
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
ØØMaterial Symmetry Group (cont.)Material Symmetry Group (cont.)
e1
e2 1 2 3
A
B
C
Data examined along horizontal and vertical segments
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
ØØMaterial Symmetry Group (cont.)Material Symmetry Group (cont.)
3 3.5 4 4.52
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8Sample-1;Uni-Axial Load:100g; y 2 vs. x 2
m m
mm
Line 1Line 2Line 3
•Good agreement in principle directions
•No off-diagnol components in deformation gradient
1.8 2 2.2 2.4 2.6 2.8 31.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Sample-1;Uni-Axial Load:100g; y1 vs. x
1
mm
mm
Line ALine BLine C
3 3.5 4 4.51.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Sample-1;Uni-Axial Load:100g; y1 vs. x
2
m m
mm
Line 1Line 2Line 3
1.8 2 2.2 2.4 2.6 2.8 32
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8Sample-1;Uni-Axial Load:100g; y2 vs. x 1
m m
mm
Line ALine BLine C
y1 vs. x1 y1 vs. x2
y2 vs. x1 y2 vs. x2
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
ØØMaterial Symmetry Group (cont.)Material Symmetry Group (cont.)
•Response along each line segment averaged for each sample
•The average response for each material direction plotted on same axis
•Slopes matched!
•Material seems isotropic in plane
3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 52
2.5
3
3.5
4
4.5
Average responses of all samples, Load:100; y2 vs. x
2
mm
mm
Sample 1Sample 2Sample 3
λ1
1
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
ØØBiaxial Experiment Biaxial Experiment –– Typical ResultsTypical Results
2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.22.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
4.2Sample-1;Bi-Axial Load:2k; y1 vs. x1
m m
mm
Line ALine BLine CAvg. Res.
3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.42.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
4.2Sample-1;Bi-Axial Load:2k; y1 vs. x2
m m
mm
Line 1Line 2Line 3
2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.24.8
5
5.2
5.4
5.6
5.8
6
6.2
6.4
6.6Sample-1;Bi-Axial Load:2k; y2 vs. x1
m m
mm
Line ALine BLine C
3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.44.8
5
5.2
5.4
5.6
5.8
6
6.2
6.4
6.6Sample-1;Bi-Axial Load:2k; y2 vs. x2
m m
mm
Line 1Line 2Line 3Avg. Res.
y1 vs. x1 y1 vs. x2
y2 vs. x1 y2 vs. x2
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
ØØOut of Plane StretchOut of Plane Stretch
.769
.803
.803
.934
λ3(Average)
1.1579
1.0493
0.9536
1.0259
det(F)
.0075.0075.008.0075None
.0055
.006
.006
.007
Location 4 (in.)
.0055.0065.0062
.006.0065.0061 ½
.006.0065.0061
.007.0075.007½
Location 3 (in.)
Location 2 (in.)
Location 1 (in.)
Load(kg)
•Thickness measured with modified calipers
•Results summarized below
•Appears Incompressible
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
Computing Potential•Pick a potential using “educated guess”: incompressibleMooney-Rivlin potential.
•Used 2kg case with Mathematica to solve for unknown constants using data extracted from biaxial test
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
Checking potential•Once the parameters are determined we can compare the potential with the uniaxial test data•First solve for the nominal stress in the direction of appliedload vs. lambda 1
lambda1 lambda2 lambda3 P11 (Predicted) P22(Predicted) P11(measured) P22(measured) % Error % Error1.115 0 .985 0.934 3.841 2.532 2.190 1.280 42.98% 49.46%1.168 1 .017 1.218 4.073 2.613 3.170 2.260 22.17% 13.50%1.218 1 .073 0.803 4.767 3.358 4.150 3.250 12.95% 3.22%1.284 1 .171 0.769 5.260 4.200 5.260 4.200 0.01% 0.01%
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
Uniaxial ResponseP 2 2 v s l a m2 O v e r a R a n g e o f l a m 1
-10
- 5
0
5
10
15
20
25
0 0. 5 1 1.5 2 2.5 3
l a m 2
l a m 1 = 0 . 5
l a m 1 = 1 . 0
l a m 1 = 1 . 5
l a m 1 = 2 . 0
l a m 1 = 2 . 5
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
Concerns About the MethodWhile this method sounded good to start with there are a fewissues that cause a little concern1. The set up does not self-center perfectly2. The method for measuring thickness change has limitations3. The fit parameters are very sensitive to the initial tractions
which are not easily measured(thickness and contact lengthwere not determined perfectly)
4. Some of the forces are disregarded to simplify calculations5. Incompressibility is assumed but fails at large loads
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
Conclusions:
•The material responds in an isotropic manor in the plane•At low stretch values the material appears incompressiblethis fails with higher applied tractions•A Mooney-Rivlin Potential fits the data pretty well but driftsfrom observations at low stretch values•Over all the data fits well but with more development the experiment would have more success
EN 227 Advanced ElasticityEN 227 Advanced ElasticityDivision of Engineering, Brown UniversityDivision of Engineering, Brown University
Final Project PresentationFinal Project PresentationDecember 3, 2003December 3, 2003
( ) ( ) ( ) ( )1 2 1 2, 3 1 32
W I I I Iµ
α α= − + − −
( )( )1 1 (1 )TP pF I F G Fµ α α µ α−= − + + − − −% % % %%
3.486.501.049
p MPaMPaµ
α
===
Top Related