{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T12:48:44Z","timestamp":1753879724989,"version":"3.41.2"},"reference-count":30,"publisher":"ASME International","issue":"3","content-domain":{"domain":["asmedigitalcollection.asme.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2011,7,1]]},"abstract":"<jats:p>In this paper, we discuss the application of different stress\u2013optic laws for rubbers to predict the birefringence evolution and the stress\u2013stretch relationship. The main focus of this work is to propose a new formulation for the stress\u2013birefringence relationship using the Gent theory for rubber elasticity. The Gent constitutive model for the stress\u2013stretch response has been shown to provide a nearly equivalent rubber elastic behavior as that of the widely used eight-chain model. By combining the simpler stress\u2013stretch relationship from the Gent model with a Gaussian network theory for birefringence, we propose a simplified stress\u2013optic relationship. We show that our obtained results are in accord with the existing experimental results at large strains. Our proposed simplified formulation and results allow us to conclude that the Gent theory can be extended to predict optical anisotropy evolution under large strains and that these predictions are nearly equivalent to the more complex formulation based on the eight-chain model.<\/jats:p>","DOI":"10.1115\/1.4004053","type":"journal-article","created":{"date-parts":[[2011,7,4]],"date-time":"2011-07-04T11:45:13Z","timestamp":1309779913000},"update-policy":"https:\/\/doi.org\/10.1115\/crossmarkpolicy-asme","source":"Crossref","is-referenced-by-count":3,"title":["Modeling of the Stress\u2013Birefringence\u2013Stretch Behavior in Rubbers Using the Gent Model"],"prefix":"10.1115","volume":"133","author":[{"given":"A. K. Mossi","family":"Idrissa","sequence":"first","affiliation":[{"name":"University of Strasbourg, IMFS, CNRS, 2 Rue Boussingault, 67000 Strasbourg, France"}]},{"given":"S.","family":"Ahzi","sequence":"additional","affiliation":[{"name":"University of Strasbourg, IMFS, CNRS, 2 Rue Boussingault, 67000 Strasbourg, France; University of Aveiro, TEMA, 3810-193 Aveiro, Portugal e-mail:"}]},{"given":"Y.","family":"R\u00e9mond","sequence":"additional","affiliation":[{"name":"University of Strasbourg, IMFS, CNRS, 2 Rue Boussingault, 67000 Strasbourg, France"}]},{"given":"J.","family":"Gracio","sequence":"additional","affiliation":[{"name":"University of Aveiro, TEMA, 3810-193 Aveiro, Portugal"}]}],"member":"33","published-online":{"date-parts":[[2011,7,1]]},"reference":[{"key":"2019100513440807100_c1","doi-asserted-by":"publisher","first-page":"248","DOI":"10.1007\/BF01793684","article-title":"\u201cBeziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer Stoffe,\u201d","volume":"101","author":"Kuhn","journal-title":"Kolloid-Z"},{"volume-title":"The Physique of Rubber Elasticity,","author":"Treloar","key":"2019100513440807100_c2"},{"key":"2019100513440807100_c3","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1063\/1.1723785","article-title":"\u201cTheory of the Elastic Properties of Rubber,\u201d","volume":"11","author":"James","journal-title":"J. Chem. Phys."},{"key":"2019100513440807100_c4","doi-asserted-by":"publisher","first-page":"512","DOI":"10.1063\/1.1723791","article-title":"\u201cStatistical Mechanics of Cross-Linked Polymer Networks I. Rubberlike Elasticity,\u201d","volume":"11","author":"Flory","journal-title":"J. Chem. Phys."},{"key":"2019100513440807100_c5","doi-asserted-by":"publisher","first-page":"1144","DOI":"10.1063\/1.1700682","article-title":"\u201cStatistical Theory of Networks of Non-Gaussian Flexible Chains,\u201d","volume":"20","author":"Wang","journal-title":"J. Chem. Phys."},{"key":"2019100513440807100_c6","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1016\/0022-5096(93)90013-6","article-title":"\u201cA Three\u2013Dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials,\u201d","volume":"41","author":"Arruda","journal-title":"J. Mech. Phys., Solids"},{"key":"2019100513440807100_c7","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1016\/0022-5096(93)90043-F","article-title":"On Improved Network Models for Rubber Elasticity and their Applications to Orientation Hardening in Glassy Polymers","volume":"41","author":"Wu","journal-title":"J. Mech. Phys. Solids"},{"key":"2019100513440807100_c8","doi-asserted-by":"publisher","first-page":"582","DOI":"10.1063\/1.1712836","article-title":"\u201cA Theory of Large Elastic Deformation,\u201d","volume":"11","author":"Mooney","journal-title":"J. Appl. Phys."},{"issue":"835","key":"2019100513440807100_c9","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1098\/rsta.1948.0024","volume":"241","author":"Rivlin","journal-title":"Phil. Trans. R. Soc. Lond."},{"key":"2019100513440807100_c10","doi-asserted-by":"publisher","first-page":"2997","DOI":"10.1063\/1.1710039","article-title":"\u201cThe Strain\u2013Energy Function of a Hyperelastic Material in Terms of the Extension Ratios,\u201d","volume":"38","author":"Valanis","journal-title":"J. Appl. Phys."},{"key":"2019100513440807100_c11","doi-asserted-by":"publisher","first-page":"565","DOI":"10.1098\/rspa.1972.0026","article-title":"\u201cLarge Deformation Isotropic Elasticity\u2014On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids,\u201d","volume":"326","author":"Ogden","journal-title":"Proc. R. Soc. London, Ser. A"},{"key":"2019100513440807100_c12","doi-asserted-by":"publisher","first-page":"59","DOI":"10.5254\/1.3538357","article-title":"\u201cA New Constitutive Relation for Rubber,\u201d","volume":"69","author":"Gent","journal-title":"Rubber Chem. Technol."},{"key":"2019100513440807100_c13","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1039\/tf9474300277","article-title":"The Photo-Elastic Properties of Rubber. Part I: Theory of the Optical Properties of Strained Rubber,\u201d","volume":"43","author":"Treloar","journal-title":"Trans. Faraday. Soc."},{"key":"2019100513440807100_c14","doi-asserted-by":"publisher","first-page":"284","DOI":"10.1039\/tf9474300284","article-title":"The Photo-Elastic Properties of Rubber. Part II: Double Refraction and Crystallization in Stretched Vulcanized Rubber,\u201d","volume":"43","author":"Treloar","journal-title":"Trans. Faraday Soc."},{"key":"2019100513440807100_c15","doi-asserted-by":"publisher","first-page":"1191","DOI":"10.1080\/01418619508236245","article-title":"\u201cOn Network Descriptions of Mechanical and Optical Properties of Rubbers,\u201d","volume":"71","author":"Wu","journal-title":"Philos. Mag. A"},{"key":"2019100513440807100_c16","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1002\/pen.v35:5","article-title":"An Investigation Into the Three-Dimensional Stress\u2013Birefringence\u2013Strain Relationship in Elastomers,\u201d","volume":"35","author":"Arruda","journal-title":"Polym. Eng. Sci."},{"key":"2019100513440807100_c17","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1007\/BF01170305","article-title":"\u201cA Network Description of the Non\u2013Gaussian Stress\u2013Optic and Raman Scattering Responses of Elastomer Networks,\u201d","volume":"134","author":"Von Lockette","journal-title":"Acta Mech."},{"key":"2019100513440807100_c18","doi-asserted-by":"publisher","first-page":"1716","DOI":"10.1002\/pen.v40:7","article-title":"\u201cModeling of the Mechanical Response and Evolution of Optical Anisotropy in Deformed Polyaniline,\u201d","volume":"40","author":"Makradi","journal-title":"Polym. Eng. Sci."},{"key":"2019100513440807100_c19","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1098\/rspa.1979.0163","article-title":"\u201cA Non-Gaussian Theory for Rubber in Biaxial Strain. I. Mechanical Properties,\u201d","volume":"369","author":"Treloar","journal-title":"Proc. R. Soc. London Ser. A"},{"key":"2019100513440807100_c20","doi-asserted-by":"publisher","first-page":"781","DOI":"10.5254\/1.3538401","article-title":"\u201cDirect Comparison of the Gent and the Arruda\u2013Boyce Constitutive Models of Rubber Elasticity,\u201d","volume":"69","author":"Boyce","journal-title":"Rubber Chem. Technol."},{"key":"2019100513440807100_c21","doi-asserted-by":"publisher","first-page":"724","DOI":"10.5254\/1.3547847","article-title":"\u201cA Comparison of the Hart\u2013Smith Model With Arruda\u2013Boyce and Gent Formulations for Rubber Elasticity,\u201d","volume":"77","author":"Chagnon","journal-title":"Rubber Chem. Technol."},{"key":"2019100513440807100_c22","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1023\/A:1026029111723","article-title":"\u201cA Molecular\u2013Statistical Basis for the Gent Constitutive Model of Rubber Elasticity,\u201d","volume":"68","author":"Horgan","journal-title":"J. Elasticity"},{"key":"2019100513440807100_c23","doi-asserted-by":"publisher","first-page":"270","DOI":"10.1007\/BF00366640","article-title":"\u201cA Pad\u00e9 Approximant to the Inverse Langevin Function,\u201d","volume":"30","author":"Cohen","journal-title":"Rheol. Acta"},{"key":"2019100513440807100_c24","doi-asserted-by":"publisher","first-page":"7938","DOI":"10.1016\/j.ijsolstr.2007.05.018","article-title":"\u201cModeling and Validation of the Large Deformation Inelastic Response of Amorphous Polymers Over a Wide Range of Temperatures and Strain Rates,\u201d","volume":"44","author":"Richeton","journal-title":"Int. J. Solids Struct."},{"key":"2019100513440807100_c25","doi-asserted-by":"publisher","first-page":"800","DOI":"10.1021\/ma00231a022","article-title":"\u201cTheory of Elasticity of Polymer Networks.3,\u201d","volume":"15","author":"Flory","journal-title":"Macromolecules"},{"key":"2019100513440807100_c26","doi-asserted-by":"publisher","first-page":"806","DOI":"10.1021\/ma00231a023","article-title":"\u201cRelationships Between Stress, Strain and Molecular Constitution of Polymer Networks. Comparison of Theory With Experiments,\u201d","volume":"15","author":"Erman","journal-title":"Macromolecules"},{"key":"2019100513440807100_c27","doi-asserted-by":"publisher","first-page":"1601","DOI":"10.1021\/ma00244a010","article-title":"\u201cTheory of Strain Birefringence of Amorphous Polymer Networks,\u201d","volume":"16","author":"Erman","journal-title":"Macromolecules"},{"key":"2019100513440807100_c28","doi-asserted-by":"publisher","first-page":"1607","DOI":"10.1021\/ma00244a011","article-title":"\u201cExperimental Results Relating Stress and Birefringence to Strain in Poly(Dimethylsiloxane) Networks. Comparisons With Theory,\u201d","volume":"16","author":"Erman","journal-title":"Macromolecules"},{"volume-title":"Advances in Elastomers and Rubber Elasticity","author":"Stepto","key":"2019100513440807100_c29"},{"volume-title":"Advances in Elastomers and Rubber Elasticity","author":"Ullman","key":"2019100513440807100_c30"}],"container-title":["Journal of Engineering Materials and Technology"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/asmedigitalcollection.asme.org\/materialstechnology\/article-pdf\/doi\/10.1115\/1.4004053\/5679500\/030905_1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/asmedigitalcollection.asme.org\/materialstechnology\/article-pdf\/doi\/10.1115\/1.4004053\/5679500\/030905_1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,5]],"date-time":"2019-10-05T17:44:11Z","timestamp":1570297451000},"score":1,"resource":{"primary":{"URL":"https:\/\/asmedigitalcollection.asme.org\/materialstechnology\/article\/doi\/10.1115\/1.4004053\/471475\/Modeling-of-the-StressBirefringenceStretch"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,7,1]]},"references-count":30,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2011,7,1]]}},"URL":"https:\/\/doi.org\/10.1115\/1.4004053","relation":{},"ISSN":["0094-4289","1528-8889"],"issn-type":[{"type":"print","value":"0094-4289"},{"type":"electronic","value":"1528-8889"}],"subject":[],"published":{"date-parts":[[2011,7,1]]},"article-number":"030905"}}