Questioning numerical integration methods for microsphere (and microplane) constitutive equations |
| |
| Affiliation: | 1. Sorbonne Universités, Université de Technologie de Compiègne, Roberval UMR-CNRS 7337, Compiègne, France;2. Center of Molecular and Macromolecular Studies of Polish Academy of Sciences, Łódź, Poland;3. Northwestern University, member of the “DND-CAT Synchrotron Research Center”, ANL Argonne, IL, USA;1. 3SR Laboratory, Grenoble Alpes University, 38041 Grenoble, France;2. LEM3 Laboratory, Lorraine University, Ile du Saulcy, 57045 Metz, France;3. CEA, DAM, GRAMAT, F-46500 Gramat, France;1. TU Dresden, Fakultät Maschinenwesen, Institut für Verarbeitungsmaschinen und Mobile Arbeitsmaschinen, Bergstraße 120, D-01069 Dresden, Germany;2. KTH Royal Institute of Technology, Department of Solid Mechanics, SE-100 44 Stockholm, Sweden;1. Institute of Biomechanics, Graz University of Technology, Stremayrgasse 16-II, 8010 Graz, Austria;2. Norwegian University of Science and Technology (NTNU), Faculty of Engineering Science and Technology, 7491 Trondheim, Norway;3. School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8SQ, Scotland, UK |
| |
| Abstract: | In the last few years, more and more complex microsphere models have been proposed to predict the mechanical response of various polymers. Similarly than for microplane models, they consist in deriving a one-dimensional force vs. stretch equation and to integrate it over the unit sphere to obtain a three-dimensional constitutive equation. In this context, the focus of authors is laid on the physics of the one-dimensional relationship, but in most of the case the influence of the integration method on the prediction is not investigated.Here we compare three numerical integration schemes: a classical Gaussian scheme, a method based on a regular geometric meshing of the sphere, and an approach based on spherical harmonics. Depending on the method, the number of integration points may vary from 4 to 983,040! Considering simple quantities, i.e. principal (large) strain invariants, it is shown that the integration method must be carefully chosen. Depending on the quantities retained to described the one-dimensional equation and the required error, the performances of the three methods are discussed. Consequences on stress–strain prediction are illustrated with a directional version of the classical Mooney–Rivlin hyperelastic model. Finally, the paper closes with some advices for the development of new microsphere constitutive equations. |
| |
| Keywords: | Constitutive equations Microsphere models Numerical integration algorithms |
| 本文献已被 ScienceDirect 等数据库收录！ |
|
