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Material testing and hyperelastic material model curve fitting for Ogden, Polynomial and Yeoh models ScilabTEC 2015, Paris, France Michael Rackl [email protected] Technische Universit¨ at M¨ unchen (TUM) Institute for Materials Handling, Material Flow, Logistics Ostbayerische Technische Hochschule Regensburg Laboratory for Biomechanics 21 st and 22 nd May 2015
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Page 1: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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

Page 2: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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

Page 3: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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

Page 4: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 4 / 37

Page 5: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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.

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 5 / 37

Page 6: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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

Page 7: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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.

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 7 / 37

Page 8: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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 8 / 37

Page 9: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 9 / 37

Page 10: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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

Page 11: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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

Page 12: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 12 / 37

Page 13: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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 13 / 37

Page 14: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 14 / 37

Page 15: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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.

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 15 / 37

Page 16: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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

Page 17: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 17 / 37

Page 18: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 18 / 37

Page 19: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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

Page 20: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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 20 / 37

Page 21: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

Results and DiscussionOgden Model; comparison with ANSYS

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 21 / 37

Page 22: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

Results and DiscussionOgden Model; curve fitting example

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 22 / 37

Page 23: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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 23 / 37

Page 24: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

Results and DiscussionPolynomial Model; comparison with ANSYS

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 24 / 37

Page 25: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

Results and DiscussionPolynomial Model; curve fitting example

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 25 / 37

Page 26: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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 26 / 37

Page 27: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

Results and DiscussionYeoh Model; comparison with ANSYS

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 27 / 37

Page 28: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

Results and DiscussionYeoh Model; curve fitting example

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 28 / 37

Page 29: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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 29 / 37

Page 30: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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 30 / 37

Page 31: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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)

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 31 / 37

Page 32: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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 32 / 37

Page 33: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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:

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 33 / 37

Page 34: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

ConclusionOgden Model implementation II: FEA validation result

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 34 / 37

Page 35: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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.

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 35 / 37

Page 36: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

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.

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 36 / 37

Page 37: Material testing and hyperelastic material model ... - Scilab - ScilabTEC 2015.pdfScilab script (cf. previous slide) Di erences could arise from the nonlinear nature of the problem,

[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.

Michael Rackl (TUM) Material Testing and (. . . ) 21st and 22nd May 2015 37 / 37


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