Material testing and hyperelastic material model curve fitting for Ogden, Polynomial and Yeoh models
By Michael Rackl, Ostbayerische Technische Hochschule Regensburg
An under-development bone fracture plating system is going to include silicone rubber as suspension material. The system’s aim is to promote bone growth by allowing for axial motion within the fracture gap. The system’s quasi-static mechanical response to loading was to be examined using finite element analysis (FEA). In order to be able to access mechanical stress and strain within the silicone, a hyperelastic material model formulation was chosen. Three phenomenological hyperelastic material model formulations (Ogden, Polynomial, Yeoh) were preselected from literature and had to be fitted to closely match previously measured stress-stretch curves from axial compression/tension as well as simple shear measurements. The stress-stretch curves had to be computed from force-displacement data, which required adjustment due to bonding effects. Despite offering to input material parameters for higher order derivatives of the selected material models, the utilised commercial FEA code ANSYS  did not feature curve fitting for each of these models. Scilab  scripts were created to be able to fit the experimental stress-stretch data to higher order hyperelastic material models than ANSYS allows for. For lower order models, the scripts were compared against the parameter sets provided by ANSYS. The comparison showed that the lower order Yeoh and Polynomial model parameters from the Scilab scripts and the commercial FEA software match perfectly. However, discrepancies were discovered when fitting the Ogden material models. Further examination suggested that these differences arose from numerical effects, e.g. the optimisation algorithm employed for curve fitting. From eleven tested hyperelastic material parameter sets, the three most suitable fits were selected for validation against the measured force/displacement curves, using FEA. Especially higher order models were subject to instability at small nominal-strains. Thus, a first order Polynomial model was chosen for finite element modelling of the plating system. The curve fitting algorithms presented in this study allow fitting of compression/tension and simple shear test data to any order Ogden models, any order Yeoh models and second as well as first order Polynomial models. As a special case of the Polynomial model, a three parameter Mooney-Rivlin model may be fitted, too. Contrary to the commercial solution, the scripts permit higher order material model parameter computation. In addition to this, implementation of auxiliary conditions and biasing of one of the loading cases can be accounted for.
hyperelasticity, material testing, curve fitting, silicone, material model, ANSYS, Scilab
 ANSYS Inc., Canonsburg, Pennsylvania, USA
 Scilab Enterprises, Versailles, France
The scripts used in the presentation are available through Scilab File Exchange here.