{"title":"非线性粘弹性中粘应力与应变率的非线性关系","authors":"Lennart Machill","doi":"10.1007/s10659-025-10118-8","DOIUrl":null,"url":null,"abstract":"<div><p>We consider a Kelvin–Voigt model for viscoelastic second-grade materials, where the elastic and the viscous stress tensor both satisfy frame-indifference. Using a rigidity estimate by <span>Ciarlet and Mardare</span> (J. Math. Pures Appl. 104:1119–1134, 2015), existence of weak solutions is shown by means of a frame-indifferent time-discretization scheme. Further, the result includes viscous stress tensors which can be calculated by nonquadratic polynomial densities. Afterwards, we investigate the long-time behavior of solutions in the case of small external loading and initial data. Our main tool is the abstract theory of metric gradient flows (Ambrosio et al. in Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser, Basel, 2005).</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"157 1","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10659-025-10118-8.pdf","citationCount":"0","resultStr":"{\"title\":\"Nonlinear Relations of Viscous Stress and Strain Rate in Nonlinear Viscoelasticity\",\"authors\":\"Lennart Machill\",\"doi\":\"10.1007/s10659-025-10118-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We consider a Kelvin–Voigt model for viscoelastic second-grade materials, where the elastic and the viscous stress tensor both satisfy frame-indifference. Using a rigidity estimate by <span>Ciarlet and Mardare</span> (J. Math. Pures Appl. 104:1119–1134, 2015), existence of weak solutions is shown by means of a frame-indifferent time-discretization scheme. Further, the result includes viscous stress tensors which can be calculated by nonquadratic polynomial densities. Afterwards, we investigate the long-time behavior of solutions in the case of small external loading and initial data. Our main tool is the abstract theory of metric gradient flows (Ambrosio et al. in Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser, Basel, 2005).</p></div>\",\"PeriodicalId\":624,\"journal\":{\"name\":\"Journal of Elasticity\",\"volume\":\"157 1\",\"pages\":\"\"},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2025-02-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s10659-025-10118-8.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Elasticity\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10659-025-10118-8\",\"RegionNum\":3,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"ENGINEERING, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Elasticity","FirstCategoryId":"5","ListUrlMain":"https://link.springer.com/article/10.1007/s10659-025-10118-8","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"ENGINEERING, MULTIDISCIPLINARY","Score":null,"Total":0}
Nonlinear Relations of Viscous Stress and Strain Rate in Nonlinear Viscoelasticity
We consider a Kelvin–Voigt model for viscoelastic second-grade materials, where the elastic and the viscous stress tensor both satisfy frame-indifference. Using a rigidity estimate by Ciarlet and Mardare (J. Math. Pures Appl. 104:1119–1134, 2015), existence of weak solutions is shown by means of a frame-indifferent time-discretization scheme. Further, the result includes viscous stress tensors which can be calculated by nonquadratic polynomial densities. Afterwards, we investigate the long-time behavior of solutions in the case of small external loading and initial data. Our main tool is the abstract theory of metric gradient flows (Ambrosio et al. in Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser, Basel, 2005).
期刊介绍：
The Journal of Elasticity was founded in 1971 by Marvin Stippes (1922-1979), with its main purpose being to report original and significant discoveries in elasticity. The Journal has broadened in scope over the years to include original contributions in the physical and mathematical science of solids. The areas of rational mechanics, mechanics of materials, including theories of soft materials, biomechanics, and engineering sciences that contribute to fundamental advancements in understanding and predicting the complex behavior of solids are particularly welcomed. The role of elasticity in all such behavior is well recognized and reporting significant discoveries in elasticity remains important to the Journal, as is its relation to thermal and mass transport, electromagnetism, and chemical reactions. Fundamental research that applies the concepts of physics and elements of applied mathematical science is of particular interest. Original research contributions will appear as either full research papers or research notes. Well-documented historical essays and reviews also are welcomed. Materials that will prove effective in teaching will appear as classroom notes. Computational and/or experimental investigations that emphasize relationships to the modeling of the novel physical behavior of solids at all scales are of interest. Guidance principles for content are to be found in the current interests of the Editorial Board.
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