{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T05:06:23Z","timestamp":1773810383415,"version":"3.50.1"},"reference-count":55,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2019,9,4]],"date-time":"2019-09-04T00:00:00Z","timestamp":1567555200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["JP17K05149"],"award-info":[{"award-number":["JP17K05149"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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\u2013strain curves. The model is defined by extending the two-dimensional surface model of Helfrich\u2013Polyakov based on the Finsler geometry description. In the Finsler geometry model, the directional degree of freedom     \u03c3 \u2192     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     \u03c4 sim     are in good agreement with the reported experimental stresses     \u03c4 exp     for large strains of up to     1200 %    . It should be emphasized that the stress\u2013strain 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.<\/jats:p>","DOI":"10.3390\/sym11091124","type":"journal-article","created":{"date-parts":[[2019,9,5]],"date-time":"2019-09-05T03:22:36Z","timestamp":1567653756000},"page":"1124","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Monte Carlo Study of Rubber Elasticity on the Basis of Finsler Geometry Modeling"],"prefix":"10.3390","volume":"11","author":[{"given":"Hiroshi","family":"Koibuchi","sequence":"first","affiliation":[{"name":"National Institute of Technology (KOSEN), Sendai College, 48 Nodayama, Medeshima-Shiote, Natori-shi, Miyagi 981-1239, Japan"}]},{"given":"Chrystelle","family":"Bernard","sequence":"additional","affiliation":[{"name":"ELyTMaX UMI 3757, CNRS-Universite de Lyon, Tohoku University, International Joint Unit, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan"},{"name":"Frontier Research Institute for Interdisciplinary Sciences (FRIS), Tohoku University, 6-3 Aoba Aramaki, Aoba-ku, Sendai 980-8578, Japan"}]},{"given":"Jean-Marc","family":"Chenal","sequence":"additional","affiliation":[{"name":"Materials Engineering and Science (MATEIS), CNRS, INSA Lyon UMR 5510, Universit\u00e9 de Lyon Batiment B. Pascal, Avenue Jean Capelle, 69621 Villeurbanne, CEDEX, France"}]},{"given":"Gildas","family":"Diguet","sequence":"additional","affiliation":[{"name":"ELyTMaX UMI 3757, CNRS-Universite de Lyon, Tohoku University, International Joint Unit, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4725-3489","authenticated-orcid":false,"given":"Gael","family":"Sebald","sequence":"additional","affiliation":[{"name":"ELyTMaX UMI 3757, CNRS-Universite de Lyon, Tohoku University, International Joint Unit, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan"}]},{"given":"Jean-Yves","family":"Cavaille","sequence":"additional","affiliation":[{"name":"ELyTMaX UMI 3757, CNRS-Universite de Lyon, Tohoku University, International Joint Unit, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan"}]},{"given":"Toshiyuki","family":"Takagi","sequence":"additional","affiliation":[{"name":"ELyTMaX UMI 3757, CNRS-Universite de Lyon, Tohoku University, International Joint Unit, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan"},{"name":"Institute of Fluid Science (IFS), Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-0812, Japan"}]},{"given":"Laurent","family":"Chazeau","sequence":"additional","affiliation":[{"name":"Materials Engineering and Science (MATEIS), CNRS, INSA Lyon UMR 5510, Universit\u00e9 de Lyon Batiment B. Pascal, Avenue Jean Capelle, 69621 Villeurbanne, CEDEX, France"}]}],"member":"1968","published-online":{"date-parts":[[2019,9,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3083","DOI":"10.1016\/S0032-3861(99)00664-3","article-title":"The conformation of poly (dimethylsiloxane) in the crystalline state","volume":"41","author":"Albouy","year":"2000","journal-title":"Polymer"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/j.polymer.2016.04.023","article-title":"Strain-induced crystallization in sustainably crosslinked epoxidizednatural rubber","volume":"93","author":"Imbernon","year":"2016","journal-title":"Polymer"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2540","DOI":"10.1016\/j.polymer.2012.04.027","article-title":"Characteristic time of strain induced crystallization of crosslinked natural rubber","volume":"53","author":"Candau","year":"2012","journal-title":"Polymer"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"956","DOI":"10.1002\/polb.10679","article-title":"Structural developments in synthetic rubbers during uniaxial deformation by in situ synchrotron X-ray diffraction","volume":"42","author":"Toki","year":"2004","journal-title":"J. Polym. Sci. B"},{"key":"ref_5","unstructured":"Treloar, L.R.G. (1975). The Physics of Rubber Elasticity, Claredon Press. [3rd ed.]."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1080\/00107517108205104","article-title":"Rubber elasticity","volume":"12","author":"Treloar","year":"1971","journal-title":"Contemp. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1002\/pol.1946.120010505","article-title":"Dependence of the average transversal on the longitudinal dimensions of statistical coils formed by chain molecules","volume":"1","author":"Kuhn","year":"1946","journal-title":"J. Polym. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"258","DOI":"10.1007\/BF01451143","article-title":"Beziehungen zwischen Molek\u00fclgr\u00f6se, statistischer Molek\u00fclgestalt und elastischen Eigenschaften hochpolymerer Stoffe","volume":"76","author":"Kuhn","year":"1936","journal-title":"Kolloid Z."},{"key":"ref_9","first-page":"96","article-title":"Network topology and the theory of rubber elasticity","volume":"17","author":"Flory","year":"1985","journal-title":"Polym. Int."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Tanaka, F. (2011). Polymer Physics: Applications to Molecular Association and Thermoreversible Gelation, Cambridge Univ. Press Ithaca.","DOI":"10.1017\/CBO9780511975691"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1039","DOI":"10.1063\/1.1699106","article-title":"Statistical Thermodynamics of Rubber Elasticity","volume":"21","author":"James","year":"1953","journal-title":"J. Chem. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1063\/1.1723793","article-title":"Statistical Thermodynamics of Rubber. III","volume":"11","author":"Wall","year":"1943","journal-title":"J. Chem. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1098\/rspa.1976.0146","article-title":"Statistical thermodynamics of random networks","volume":"351","author":"Flory","year":"1976","journal-title":"Proc. Roy. Soc. Lond. A"},{"key":"ref_14","unstructured":"Flory, P.J. (1953). Principles of Polymer Chemistry, Cornell Univ. Press."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1435","DOI":"10.1063\/1.1748098","article-title":"Statistical Thermodynamics of Rubber Elasticity","volume":"19","author":"Wall","year":"1951","journal-title":"J. Chem. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1063\/1.1747424","article-title":"Statistical Mechanics of Swelling of Network Structures","volume":"18","author":"Flory","year":"1950","journal-title":"J. Chem. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"582","DOI":"10.1063\/1.1712836","article-title":"A Theory of Large Elastic Deformation","volume":"11","author":"Mooney","year":"1940","journal-title":"J. Appl. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1098\/rsta.1948.0002","article-title":"Large elastic deformations of isotropic materials I. Fundamental concepts","volume":"240","author":"Rivlin","year":"1948","journal-title":"Phil. Trans. R. Soc. Lond. Soc. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1098\/rsta.1948.0024","article-title":"Large elastic deformations of isotropic materials IV. Further depelopements of the general theory","volume":"241","author":"Rivlin","year":"1948","journal-title":"Phil. Trans. R. Soc. Lond. Soc. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1098\/rsta.1976.0001","article-title":"The theory of rubber elasticity","volume":"280","author":"Deam","year":"1976","journal-title":"Phil. Trans. Roy. Soc. Lond."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3440","DOI":"10.1002\/polb.21010","article-title":"An experimentalist\u2019s view of the physics of rubber elasticity","volume":"44","author":"Urayama","year":"2006","journal-title":"J. Polym. Sci. B"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.physa.2013.08.006","article-title":"Monte Carlo studies of a Finsler geometric surface model","volume":"393","author":"Koibuchi","year":"2014","journal-title":"Phys. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1016\/j.polymer.2017.02.065","article-title":"Finsler geometry modeling and Monte Carlo study of 3D liquid crystal elastomer","volume":"114","author":"Osari","year":"2017","journal-title":"Polymer"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"042411","DOI":"10.1103\/PhysRevE.95.042411","article-title":"J-shaped stress\u2013strain diagram of collagen fibers: Frame tension of triangulated surfaces with fixed boundaries","volume":"95","author":"Takano","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Mitsuhashi, K., Ghosh, S., and Koibuchi, H. (2018). Mathematical modeling and simulations for large-strain J-shaped diagrams of soft biological tissues. Polymers, 10.","DOI":"10.1101\/275206"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"012081","DOI":"10.1088\/1742-6596\/1141\/1\/012081","article-title":"Mathematical Modeling of Rubber Elasticity","volume":"1142","author":"Koibuchi","year":"2018","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1103\/PhysRevA.6.426","article-title":"Nematic-Liquid-Crystal Order?A Monte Carlo Calculation","volume":"6","author":"Lebwohl","year":"1972","journal-title":"Phys. Rev. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3323","DOI":"10.1088\/0305-4470\/27\/10\/009","article-title":"Random surfaces: From polymer membranes to strings","volume":"27","author":"Wheater","year":"1994","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"078103","DOI":"10.1103\/PhysRevLett.100.078103","article-title":"Direct Calculation from the Stress Tensor of the Lateral Surface Tension of Fluctuating Fluid Membranes","volume":"100","author":"Fournier","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2298","DOI":"10.1103\/PhysRevLett.72.2298","article-title":"Liquid properties of embryonic tissues: Measurement of interfacial tensions","volume":"72","author":"Foty","year":"1994","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","unstructured":"Creutz, M. (1983). Quarks, Gluons and Lattices, Cambridge University Press."},{"key":"ref_32","unstructured":"Schoeberl, J. (2016, January 01). Netgen Mesh Generator, a Free Software for 3D Meshing. Available online: https:\/\/ngsolve.org\/."},{"key":"ref_33","unstructured":"Chern, S.-S. (1996). Finsler Geometry Is Just Riemannian Geometry without the Quadratic Restriction. Notices of the AMS, American Mathematical Society."},{"key":"ref_34","unstructured":"Matsumoto, M. (1986). Foundations of Finsler Geometry and Special Finsler Spaces, Kaiseisya."},{"key":"ref_35","unstructured":"Matsumoto, M. (1975). Keiryou Bibun Kikagaku, Shokabo. (In Japanese)."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Bao, D., Chern, S.-S., and Shen, Z. (2000). An Introduction to Riemann-Finsler Geometry, GTM 200, Springer.","DOI":"10.1007\/978-1-4612-1268-3"},{"key":"ref_37","unstructured":"The variable \u03c3\u2192 represents the \u201cmean value\u201d of the direction of the polymer chains at a given point in the material, and this mean direction is identical to that expressed by \u2212\u03c3\u2192. For this reason, the directions are represented by the left-right arrows in Figure 4c."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1515\/znc-1973-11-1209","article-title":"Elastic Properties of Lipid Bilayers: Theory and Possible Experiments","volume":"28","author":"Helfrich","year":"1973","journal-title":"Z. Naturforsch C"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1016\/0550-3213(86)90162-8","article-title":"Fine structure of strings","volume":"268","author":"Polyakov","year":"1986","journal-title":"Nucl. Phys. B"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"3056","DOI":"10.1103\/PhysRevA.35.3056","article-title":"Tethered surfaces: Statics and dynamics","volume":"35","author":"Kantor","year":"1987","journal-title":"Phys. Rev. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"4020","DOI":"10.1103\/PhysRevA.36.4020","article-title":"Phase Transitions in Flexible Polymeric Surfaces","volume":"36","author":"Kantor","year":"1987","journal-title":"Phys. Rev. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/S0370-1573(00)00128-9","article-title":"The statistical mechanics of membranes","volume":"344","author":"Bowick","year":"2001","journal-title":"Phys. Rep."},{"key":"ref_43","unstructured":"Domb, C., and Lebowitz, J.L. (2000). Polymerized Membranes, a Review. Phase Transitions and Critical Phenomena 19, Academic Press."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Nelson, D., Piran, T., and Weinberg, S. (2004). The Statistical Mechanics of Membranes and Interfaces. Statistical Mechanics of Membranes and Surfaces, World Scientific. [2nd ed.].","DOI":"10.1142\/5473"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Nelson, D., Piran, T., and Weinberg, S. (2004). Triangulated-surface Models of Fluctuating Membranes. Statistical Mechanics of Membranes and Surfaces, World Scientific. [2nd ed.].","DOI":"10.1142\/5473"},{"key":"ref_46","unstructured":"Doi, M., and Edwards, S.F. (1986). The Theory of Polymer Dynamics, Oxford University Press."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Nelson, D., Piran, T., and Weinberg, S. (2004). Geometry and Field Theory of Random Surfaces and Membranes. Statistical Mechanics of Membranes and Surfaces, World Scientific. [2nd ed.].","DOI":"10.1142\/5473"},{"key":"ref_48","unstructured":"Stress is normally expressed by the symbol \u03c3, however, this symbol \u03c3 is already used for the directional degree of freedom of polymers. For this reason, the symbol \u03c4 is used instead of \u03c3 for stress in this paper."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1063\/1.1699114","article-title":"Equation of State Calculations by Fast Computing Machines","volume":"21","author":"Metropolis","year":"1953","journal-title":"J. Chem. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"2997","DOI":"10.1103\/PhysRevB.13.2997","article-title":"Finite-size behavior of the Ising square lattice","volume":"13","author":"Landau","year":"1976","journal-title":"Phys. Rev. B"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"3122","DOI":"10.1016\/j.eurpolymj.2008.07.025","article-title":"Elastomer\/LDH nanocomposites: Synthesis and studies on nanoparticle dispersion, mechanical properties and interfacial adhesion","volume":"44","author":"Pradhan","year":"2008","journal-title":"Eur. Polym. J."},{"key":"ref_52","unstructured":"The high-precision number assumed for c does not always imply that the results are very sensitive to c. In the simulation program, the parameter c\u2032(=(2\/3)c) was used instead of c. As a result, the simple input numbers c\u2032 = 0.7 and c\u2032 = 0.03 turn to be c = 1.05 and c = 0.045, respectively."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1177\/0731684413475911","article-title":"Thermal and mechanical behavior of cotton\/vinyl ester composites: Effects of some flame retardants and fiber treatment","volume":"32","author":"Shahedifar","year":"2013","journal-title":"J. Reinf. Plast. Compos."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1590\/S0104-14282002000200008","article-title":"Stress-Strain Curves of Nafion Membranes in Acid and Salt Forms","volume":"12","author":"Kawano","year":"2002","journal-title":"Polim. Cienc. Tecnol."},{"key":"ref_55","unstructured":"In the hysteresis simulation for the variation of the strain \u03f5, the final configuration of the previous simulation is used as an input of the following simulation, where the total number of Monte Carlo sweep is always kept small so that the configuration may not be convergent."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/9\/1124\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:16:51Z","timestamp":1760188611000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/9\/1124"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9,4]]},"references-count":55,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2019,9]]}},"alternative-id":["sym11091124"],"URL":"https:\/\/doi.org\/10.3390\/sym11091124","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,9,4]]}}}