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Adhesive connections in glass structures—part II: material parameter identification on thin structural silicone

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Abstract

The present paper proposes two methodologies of identifying hyperelastic material parameters of thin structural silicones based on so-called direct and inverse methods. Based on part I of this paper, analytical investigations were performed to conduct homogeneous experiments with structural silicones. To obtain more insight wether or not an experiment provides a homogeneous stress state, the so-called triaxiality was introduced, which allows one to illustrate differences between homogeneous and inhomogeneous experiments. With the help of this scalar, it was possible to design experimental test setups, which ensure a homogeneous stress and strain distribution within the tested rubber-like material. Furthermore an engineering approach to determine the testing speed of arbitrary experiments dependent on one reference testing speed and experiment was presented. This approach ensured equivalent strain energies between arbitrary and reference test specimens during testing, by which expensive strain rate controlled experiments can be relinquished. Based on these analytical studies, experimental data could be provided for the material parameter identification, which exhibits firstly a nearly homogeneous stress state in accordance to the desired stress and strain field of the applied mathematical model and secondly providing nearly equivalent strain energies within different experimental test set-ups and geometries of test specimens. Returning to the present paper, the first methodology identifies simultaneously hyperelastic material parameters based on a set of conventional and homogeneous experimental tests, like uniaxial tension and uniaxial compression, biaxial tension as well as shear-pancake tests. The second methodology determines inversely hyperelastic material parameters utilizing the inverse Finite Element Method based on one single unconventional and inhomogeneous experimental test, here a microindentation test. The main idea is to obtain reliable hyperelastic material parameters based on a single, inhomogeneous experiment to avoid many, time-consuming homogeneous experiments. To validate the inversely determined hyperelastic material parameters, simultaneous multi-experiment data fits are performed to relate the obtained material parameters to those of the microindentation tests. Considering the set of homogeneous experiments, two classical hyperelastic constitutive equations (Neo-Hooke and Mooney–Rivlin) were utilized to determine constitutive parameters. Due to the simplicity of the classical material laws, a more sophisticated, novel phenomenological hyperelastic material law will be proposed and compared with the results of the classical models respectively the results obtained by a modern hyperelastic material model after Kaliske & Heinrich, which generally delivers outstanding results for the material parameter identification.

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Change history

  • 13 December 2017

    The name of the second author contained a ty** error. The original article has been corrected.

Abbreviations

UT:

Uniaxial tension test

UC:

Uniaxial compression test

BT:

Biaxial tension test

SPC:

Shear pancake test

MI:

Microindentation test

MPI:

Material parameter identification

iFEM:

Inverse finite element method

FEMU:

Finite element model updating

TRM:

Trust region method

LAR:

Least absolute residuals

MOP:

Meta model of optimal prognosis

RMSE:

Root mean squared error

\({\left( \bullet \right) _{\mathrm{{iso}}}}\) :

Isochoric/volume-preserving

tr\(\left( \bullet \right) \) :

Trace of argument

Grad\(\left( \bullet \right) \) :

Gradient of argument

F :

Deformation gradient

J :

Relative volume

C :

Right Cauchy-Green tensor

b :

Left Cauchy-Green tensor

\({\bar{\mathbf{b}}}\) :

Isochoric left Cauchy-Green tensor

\(\lambda _i\) :

Principal stretches

\(\varepsilon _i^{\mathrm{{eng}}}\) :

Engineering strain

\({I_{\mathbf{{ b}}}}\) :

First principal strain invariant of \(\mathbf b \)

\({II_{\mathbf{{ b}}}}\) :

Second principal strain invariant of \(\mathbf b \)

\({III_{\mathbf{{ b}}}}\) :

Third principal strain invariant of \(\mathbf b \)

\(t_i\) :

Principal engineering stress

\(\sigma _i\) :

Principal Cauchy stress

p :

Hydrostatic stress

\(\varPsi ( \bullet )\) :

Helmholtz free energy

\(\mathcal {S}\) :

Objective function

\(\varPhi _{{k}}\) :

Trigger function

p :

Vector of material parameters

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Acknowledgements

We would like to thank Dow Corning Inc. and Interpane Glas Industrie AG gratefully for their support during our studies by providing us testing material.

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Authors and Affiliations

  1. Institute of Structural Mechanics and Design, Technische Universität Darmstadt, Darmstadt, Germany

    Michael Drass, Gregor Schwind & Jens Schneider

  2. Institute of Mechanics and Materials, Technische Hochschule Mittelhessen, Giessen, Germany

    Stefan Kolling

Authors
  1. Michael Drass
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  2. Gregor Schwind
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  3. Jens Schneider
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  4. Stefan Kolling
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Corresponding author

Correspondence to Michael Drass.

Additional information

The original article has been revised: the name of the second author has been corrected.

A correction to this article is available online at https://doi.org/10.1007/s40940-017-0054-5.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (xlsx 92 KB)

Appendices

Appendix-A

A data package of the presented experimental raw data including the engineering stress strain responses for UT, UC, BT, SPC experiments based on mean values can be granted upon e-mail request to the authors. Table 2 gives an overview of the testing data and the utilized fitting algorithm.

Table 2 Summary of experiments (see part I)
Full size table

Appendix-B

In this appendix, the determined constitutive material parameters for thin structural silicone will be provided based on the applied two methodologies - direct/inverse method. The constitutive parameters are listed with respect to the applied constitutive model (Table 3).

Table 3 Material parameters of MPI utilizing the direct and inverse method
Full size table
Fig. 10
figure 10figure 10

Illustration of the fitting and simulation results for the direct and inverse method utilizing different constitutive models: a–d NH model; e–h ExtTube model and i–l MD model

Full size image

Appendix-C

For the sake of completeness, in the following graphs the comparison for the conventional and unconventional MPI will be illustrated for the material models NH, ExtTube and MD (see Fig. 10) in accordance to Fig. 8.

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Drass, M., Schwind, G., Schneider, J. et al. Adhesive connections in glass structures—part II: material parameter identification on thin structural silicone. Glass Struct Eng 3, 55–74 (2018). https://doi.org/10.1007/s40940-017-0048-3

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