material testing and hyperelastic material model ... - scilab - scilabtec 2015.pdfscilab script (cf....
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Material testing and hyperelastic materialmodel curve fitting for Ogden, Polynomial
and Yeoh modelsScilabTEC 2015, Paris, France
Michael [email protected]
Technische Universitat Munchen (TUM)Institute for Materials Handling, Material Flow, Logistics
Ostbayerische Technische Hochschule RegensburgLaboratory for Biomechanics
21st and 22nd May 2015
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Outline1 Introduction
Background informationFitting hyperelastic material constantsProblem description and approach
2 Materials and ModelsMaterial testsAnalytical equation for the Yeoh Model in tension/compression(example)Optimisation problem
3 Results and DiscussionOgden ModelPolynomial ModelYeoh Model
4 ConclusionGeneralDiscrepancies with the Ogden Model implementation
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1 IntroductionBackground informationFitting hyperelastic material constantsProblem description and approach
2 Materials and ModelsMaterial testsAnalytical equation for the Yeoh Model in tension/compression(example)Optimisation problem
3 Results and DiscussionOgden ModelPolynomial ModelYeoh Model
4 ConclusionGeneralDiscrepancies with the Ogden Model implementation
Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 3 / 37
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IntroductionBackground
Static structural investigation on a silicone-suspended fractureplating system
Modelling by means of finite element method (FEM)
⇒ How to model silicone in FEM?
Answer from literature: hyperelastic material models
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IntroductionHyperelastic material models
Two types of hyperelastic material models1
Phenomenological models
material constants generatedby curve fitting
Micro-mechanical models
material constants generatedby specific material tests
The use of phenomenological model is suggested, as the physicalsignificance of micro-mechanical material constants is often unclear2.
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1 IntroductionBackground informationFitting hyperelastic material constantsProblem description and approach
2 Materials and ModelsMaterial testsAnalytical equation for the Yeoh Model in tension/compression(example)Optimisation problem
3 Results and DiscussionOgden ModelPolynomial ModelYeoh Model
4 ConclusionGeneralDiscrepancies with the Ogden Model implementation
Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 6 / 37
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Fitting hyperelastic material constantsworkflow and material model selection
1 Material tests: stress-stretch curves from e. g. uni- or bi-axialtensions tests, shear tests.
2 Curve fitting: adapting the material constants based onanalytical models, to closely reproduce the measured curves.
3 Validation by modelling the material tests with FEM andcomparison of the results.
Material models selected for this work:
Ogden Model3,
Polynomial Model4 (includes the Mooney-Rivlin Model5;6),
Yeoh Model7.
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1 IntroductionBackground informationFitting hyperelastic material constantsProblem description and approach
2 Materials and ModelsMaterial testsAnalytical equation for the Yeoh Model in tension/compression(example)Optimisation problem
3 Results and DiscussionOgden ModelPolynomial ModelYeoh Model
4 ConclusionGeneralDiscrepancies with the Ogden Model implementation
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Problem description and approach
FEM software Ansysa offers higher-order hyperelastic material modelinput, but does not feature curve fitting for each of these models.
Approach:Create a Scilabb script and corresponding functions to conductnon-restrictive curve fitting.
aANSYS Inc., Canonsburg, Pennsylvania, USA
bScilab Enterprises, Versailles, France
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1 IntroductionBackground informationFitting hyperelastic material constantsProblem description and approach
2 Materials and ModelsMaterial testsAnalytical equation for the Yeoh Model in tension/compression(example)Optimisation problem
3 Results and DiscussionOgden ModelPolynomial ModelYeoh Model
4 ConclusionGeneralDiscrepancies with the Ogden Model implementation
Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 10 / 37
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1 IntroductionBackground informationFitting hyperelastic material constantsProblem description and approach
2 Materials and ModelsMaterial testsAnalytical equation for the Yeoh Model in tension/compression(example)Optimisation problem
3 Results and DiscussionOgden ModelPolynomial ModelYeoh Model
4 ConclusionGeneralDiscrepancies with the Ogden Model implementation
Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 11 / 37
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Materials and MethodsMaterial tests
Bonded compression/tensiontest (stress results to becorrected8)
12,7
rod
rod
silicone
3
Simple shear test
strip
strip
silicone
A
A
10
10
25,4
0,25
all dimensions inMillimetres;“strip” refers toa steel strip
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1 IntroductionBackground informationFitting hyperelastic material constantsProblem description and approach
2 Materials and ModelsMaterial testsAnalytical equation for the Yeoh Model in tension/compression(example)Optimisation problem
3 Results and DiscussionOgden ModelPolynomial ModelYeoh Model
4 ConclusionGeneralDiscrepancies with the Ogden Model implementation
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Analytical equation of the Yeoh Model intension/compression (example) I
1. Base equation of the Yeoh Model: W =∑n
i=1 Ci ·(I1 − 3
)iW : strain energy,
Ci : material constant,
I1: strain invariant,
n: model order.
2. strain invariants I1 for compression/tension9;10:I1 = λ2 + 2λ−1,I2 = λ−2 + 2λ.
λ: stretch due to tensile/compressive stress
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Analytical equation of the Yeoh Model intension/compression (example) II
3. relation between engineering stress σe and stretch λ for anincompressible material under tension/compression6;11:
σe = 2 ·(λ− λ−2
)·(∂W∂I1
+ 1λ· ∂W∂I2
).
4. Combining the aforementioned equations yields
σe(λ)
=n∑
i=1
2 · Ci · i ·(λ− λ−2
)·(λ2 + 2λ−1 − 3
)i−1
for tension/compression of an nth order Yeoh Model.
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1 IntroductionBackground informationFitting hyperelastic material constantsProblem description and approach
2 Materials and ModelsMaterial testsAnalytical equation for the Yeoh Model in tension/compression(example)Optimisation problem
3 Results and DiscussionOgden ModelPolynomial ModelYeoh Model
4 ConclusionGeneralDiscrepancies with the Ogden Model implementation
Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 16 / 37
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Optimisation problemScilab script overview
index “M” denotes data from measurement
Material modelfunctions forYeoh Model
tension/compression
simple shear
σ(λM,C)
τ(γM,C)
εσ
ετ
σM
τM
measurementdata
ε=εσ+ετ
minimise ε!fminsearch
Materialconstantsvector C
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Optimisation problemnotes
Error criteria εσ and ετ may be normalised or unnormalised(absolute)
εnorm =∑(
1−σ(λ, ~C
)σM(λ) )2
εabs =∑(
σM(λ)− σ
(λ, ~C
))2
Adjusted coefficient of determination12 was computed forgoodness of fit evaluation
Results for lower-order material models were comparedto those from Ansys
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1 IntroductionBackground informationFitting hyperelastic material constantsProblem description and approach
2 Materials and ModelsMaterial testsAnalytical equation for the Yeoh Model in tension/compression(example)Optimisation problem
3 Results and DiscussionOgden ModelPolynomial ModelYeoh Model
4 ConclusionGeneralDiscrepancies with the Ogden Model implementation
Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 19 / 37
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1 IntroductionBackground informationFitting hyperelastic material constantsProblem description and approach
2 Materials and ModelsMaterial testsAnalytical equation for the Yeoh Model in tension/compression(example)Optimisation problem
3 Results and DiscussionOgden ModelPolynomial ModelYeoh Model
4 ConclusionGeneralDiscrepancies with the Ogden Model implementation
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Results and DiscussionOgden Model; comparison with ANSYS
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Results and DiscussionOgden Model; curve fitting example
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1 IntroductionBackground informationFitting hyperelastic material constantsProblem description and approach
2 Materials and ModelsMaterial testsAnalytical equation for the Yeoh Model in tension/compression(example)Optimisation problem
3 Results and DiscussionOgden ModelPolynomial ModelYeoh Model
4 ConclusionGeneralDiscrepancies with the Ogden Model implementation
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Results and DiscussionPolynomial Model; comparison with ANSYS
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Results and DiscussionPolynomial Model; curve fitting example
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1 IntroductionBackground informationFitting hyperelastic material constantsProblem description and approach
2 Materials and ModelsMaterial testsAnalytical equation for the Yeoh Model in tension/compression(example)Optimisation problem
3 Results and DiscussionOgden ModelPolynomial ModelYeoh Model
4 ConclusionGeneralDiscrepancies with the Ogden Model implementation
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Results and DiscussionYeoh Model; comparison with ANSYS
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Results and DiscussionYeoh Model; curve fitting example
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1 IntroductionBackground informationFitting hyperelastic material constantsProblem description and approach
2 Materials and ModelsMaterial testsAnalytical equation for the Yeoh Model in tension/compression(example)Optimisation problem
3 Results and DiscussionOgden ModelPolynomial ModelYeoh Model
4 ConclusionGeneralDiscrepancies with the Ogden Model implementation
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1 IntroductionBackground informationFitting hyperelastic material constantsProblem description and approach
2 Materials and ModelsMaterial testsAnalytical equation for the Yeoh Model in tension/compression(example)Optimisation problem
3 Results and DiscussionOgden ModelPolynomial ModelYeoh Model
4 ConclusionGeneralDiscrepancies with the Ogden Model implementation
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ConclusionGeneral
Curve fitting script for fitting any order Ogden, any order Yeohand second as well as first order Polynomial models, plus a threeparameter Mooney-Rivlin Model.
Yeoh and Polynomial Model curve fitting for lower-order modelssuccessfully validated against results from Ansys
Scilab script allows for
curve fitting of higher order models than Ansys,biasing of one of the material test results possible.
higher order models tend to instabilities at small nominal-strains(Ansys warning message)
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1 IntroductionBackground informationFitting hyperelastic material constantsProblem description and approach
2 Materials and ModelsMaterial testsAnalytical equation for the Yeoh Model in tension/compression(example)Optimisation problem
3 Results and DiscussionOgden ModelPolynomial ModelYeoh Model
4 ConclusionGeneralDiscrepancies with the Ogden Model implementation
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ConclusionDiscrepancies with the Ogden Model implementation I
Ogden Model implementation ⇒ difference in material constantsbetween Ansys and the Scilab script
Results for lower-order models from verification process:
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ConclusionOgden Model implementation II: FEA validation result
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ConclusionDiscrepancies with the Ogden Model implementation III
Ogden Model appears to be implemented correctly within theScilab script (cf. previous slide)
Differences could arise from
the nonlinear nature of the problem,
the fact that more than one minimum exists.
Besides: Even identical start values may lead to completely differentmaterial parameter sets13.
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Thank you!Do you have questions or feedback?
References (two slides)
[1] P. Steinmann, M. Hossain, and G. Possart. Hyperelastic models for rubber-like materials: consistent tangent operatorsand suitability for Treloar’s data. Archive of Applied Mechanics, 82(9):1183–1217, 2012.
[2] R. W. Ogden, G. Saccomandi, and I. Sgura. Fitting hyperelastic models to experimental data. Computational Mechanics,34(6):484–502, 2004.
[3] R. W. Ogden. Large Deformation Isotropic Elasticity - On the Correlation of Theory and Experiment for IncompressibleRubberlike Solids. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,326(1567):565–584, 1972.
[4] R. S. Rivlin and D. W. Saunders. Large Elastic Deformations of Isotropic Materials. VII. Experiments on the Deformationof Rubber. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences,243(865):251–288, 1951.
[5] M. Mooney. A Theory of Large Elastic Deformation. Journal of Applied Physics, 11(9):582–592, 1940.
[6] R. S. Rivlin. Large Elastic Deformations of Isotropic Materials. IV. Further Developments of the General Theory.Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 241(835):379–397,1948.
[7] O. H. Yeoh. Some Forms of the Strain Energy Function for Rubber. Rubber Chemistry and Technology, 66(5):754–771,1993.
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[8] J. M. Horton, G. E. Tupholme, and M. J. C. Gover. Axial Loading of Bonded Rubber Blocks. Journal of AppliedMechanics, 69(6):836, 2002.
[9] L. R. G. Treloar. The elasticity and related properties of rubbers. Reports on Progress in Physics, 36(7):755–826, 1973.
[10] F. R. Schwarzl. Polymermechanik: Struktur und mechanisches Verhalten von Polymeren. Springer, Berlin, 1990.p. 298-299.
[11] R. S. Rivlin. Large Elastic Deformations. In F. R. Eirich, editor, Rheology, pages 351–385. Academic Press, New York,1956.
[12] L. Fahrmeir, T. Kneib, and S. Lang. Regression: Modelle, Methoden und Anwendungen. Statistik und ihre Anwendungen.
Springer, Berlin and Heidelberg, 2nd edition, 2009. p. 161.
[13] A. Zielesny. From curve fitting to machine learning: An illustrative guide to scientific data analysis and computationalintelligence, volume 18 of Intelligent systems reference library. Springer, Berlin, 2011. p. 146.
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