| Title:
|
One-dimensional model describing the non-linear viscoelastic response of materials (English) |
| Author:
|
Bárta, Tomáš |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
55 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
227-246 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we consider a model of a one-dimensional body where strain depends on the history of stress. We show local existence for large data and global existence for small data of classical solutions and convergence of the displacement, strain and stress to zero for time going to infinity. (English) |
| Keyword:
|
viscoelasticity |
| Keyword:
|
integrodifferential equation |
| Keyword:
|
classical solution |
| Keyword:
|
global existence |
| Keyword:
|
implicit constitutive relations |
| MSC:
|
35A09 |
| MSC:
|
35M33 |
| MSC:
|
45G10 |
| MSC:
|
45K05 |
| MSC:
|
74D10 |
| MSC:
|
74H20 |
| MSC:
|
74H40 |
| idZBL:
|
Zbl 06391540 |
| idMR:
|
MR3193928 |
| . |
| Date available:
|
2014-06-07T15:39:30Z |
| Last updated:
|
2016-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143804 |
| . |
| Reference:
|
[1] Bulíček M., Gwiazda P., Málek J., Świerczewska-Gwiazda A.: On unsteady flows of implicitly constituted incompressible fluids.SIAM J. Math. Anal. 44 (2012), no. 4, 2756–2801. Zbl 1256.35074, MR 3023393, 10.1137/110830289 |
| Reference:
|
[2] Dafermos C.M., Nohel J.A.: A nonlinear hyperbolic Volterra equation in viscoelasticity.. Contributions to analysis and geometry (Baltimore, Md., 1980), pp. 87–116, Johns Hopkins Univ. Press, Baltimore, Md., 1981. Zbl 0588.35016, MR 0648457 |
| Reference:
|
[3] Gripenberg G., Londen S.O., Staffans O.: Volterra integral and functional equations.Encyclopedia of Mathematics and its Applications, 34, Cambridge University Press, Cambridge, 1990. Zbl 1159.45001, MR 1050319 |
| Reference:
|
[4] Hrusa W.J.: A nonlinear functional-differential equation in Banach space with applications to materials with fading memory.Arch. Rational Mech. Anal. 84 (1984), no. 2, 99–137. Zbl 0544.73056, MR 0713121, 10.1007/BF00252129 |
| Reference:
|
[5] Hrusa W.J., Nohel J.A.: The Cauchy problem in one-dimensional nonlinear viscoelasticity.J. Differential Equations 59 (1985), no. 3, 388–412. Zbl 0535.35057, MR 0807854, 10.1016/0022-0396(85)90147-0 |
| Reference:
|
[6] Hrusa W.J., Renardy M.: A model equation for viscoelasticity with a strongly singular kernel.SIAM J. Math. Anal. 19 (1988), no. 2, 257–269. Zbl 0644.73041, MR 0930025, 10.1137/0519019 |
| Reference:
|
[7] MacCamy R.C.: A model for one-dimensional nonlinear viscoelasticity.Quart. Appl. Math. 37 (1977), 21–33. Zbl 0355.73041, MR 0478939 |
| Reference:
|
[8] Málek J.: Mathematical properties of flows of incompressible power-law-like fluids that are described by implicit constitutive relations.Electron. Trans. Numer. Anal. 31 (2008), 110–125. Zbl 1182.35182, MR 2569596 |
| Reference:
|
[9] Málek J., Průša P., Rajagopal K.R.: Generalizations of the Navier–Stokes fluid from a new perspective.Internat. J. Engrg. Sci. 48 (2010), no. 12, 1907–1924. Zbl 1231.76073, MR 2778752, 10.1016/j.ijengsci.2010.06.013 |
| Reference:
|
[10] Muliana A., Rajagopal K.R., Wineman A.S.: A new class of quasi-linear models for describing the nonlinear viscoelastic response of materials.Acta Mechanica (2013), 1–15. |
| Reference:
|
[11] Průša V., Rajagopal K.R.: On implicit constitutive relations for materials with fading memory.Journal of Non-Newtonian Fluid Mechanics 181-182 (2012), 22–29. 10.1016/j.jnnfm.2012.06.004 |
| Reference:
|
[12] Rajagopal K.R.: On implicit constitutive theories.Appl. Math. 48 (2003), 279–319. Zbl 1097.76009, MR 1994378, 10.1023/A:1026062615145 |
| Reference:
|
[13] Renardy M., Hrusa W.J., Nohel J.A.: Mathematical Problems in Viscoelasticity.Pitman Monographs and Surveys in Pure and Applied Mathematics, 35, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. Zbl 0719.73013, MR 0919738 |
| Reference:
|
[14] Staffans O.J.: On a nonlinear hyperbolic Volterra equation.SIAM J. Math. Anal. 11 (1980), no. 5, 793–812. Zbl 0464.45010, MR 0586908, 10.1137/0511071 |
| . |
