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Open AccessArticle

Monte Carlo Study of Rubber Elasticity on the Basis of Finsler Geometry Modeling

1
National Institute of Technology (KOSEN), Sendai College, 48 Nodayama, Medeshima-Shiote, Natori-shi, Miyagi 981-1239, Japan
2
ELyTMaX UMI 3757, CNRS-Universite de Lyon, Tohoku University, International Joint Unit, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
3
Frontier Research Institute for Interdisciplinary Sciences (FRIS), Tohoku University, 6-3 Aoba Aramaki, Aoba-ku, Sendai 980-8578, Japan
4
Materials Engineering and Science (MATEIS), CNRS, INSA Lyon UMR 5510, Université de Lyon Batiment B. Pascal, Avenue Jean Capelle, 69621 Villeurbanne, CEDEX, France
5
Institute of Fluid Science (IFS), Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-0812, Japan
*
Author to whom correspondence should be addressed.
This work is supported in part by JSPS KAKENHI Grant No. JP17K05149 and the Collaborative Research Project of the Institute of Fluid Science, Tohoku University.
Symmetry 2019, 11(9), 1124; https://doi.org/10.3390/sym11091124
Received: 30 June 2019 / Revised: 12 August 2019 / Accepted: 17 August 2019 / Published: 4 September 2019
Configurations of the polymer state in rubbers, such as so-called isotropic (random) and anisotropic (almost aligned) states, are symmetric/asymmetric under space rotations. In this paper, we present numerical data obtained by Monte Carlo simulations of a model for rubber formulations to compare these predictions with the reported experimental stress–strain curves. The model is defined by extending the two-dimensional surface model of Helfrich–Polyakov based on the Finsler geometry description. In the Finsler geometry model, the directional degree of freedom σ of the polymers and the polymer position r are assumed to be the dynamical variables, and these two variables play an important role in the modeling of rubber elasticity. We find that the simulated stresses τ sim are in good agreement with the reported experimental stresses τ exp for large strains of up to 1200 % . It should be emphasized that the stress–strain curves are directly calculated from the Finsler geometry model Hamiltonian and its partition function, and this technique is in sharp contrast to the standard technique in which affine deformation is assumed. It is also shown that the obtained results are qualitatively consistent with the experimental data as influenced by strain-induced crystallization and the presence of fillers, though the real strain-induced crystallization is a time-dependent phenomenon in general. View Full-Text
Keywords: rubber elasticity; mathematical modeling; Finsler geometry; strain induced crystallization; Monte Carlo; stress strain curves; statistical mechanics rubber elasticity; mathematical modeling; Finsler geometry; strain induced crystallization; Monte Carlo; stress strain curves; statistical mechanics
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MDPI and ACS Style

Koibuchi, H.; Bernard, C.; Chenal, J.-M.; Diguet, G.; Sebald, G.; Cavaille, J.-Y.; Takagi, T.; Chazeau, L. Monte Carlo Study of Rubber Elasticity on the Basis of Finsler Geometry Modeling. Symmetry 2019, 11, 1124.

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