We study a thermo-mechanical system consisting of an elastic membrane to which a shape-memory rod is glued. The slow movements of the membrane are controlled by the motions of the attached rods. A quasi-static model is used. We include the elastic feedback of the membrane on the rods. This results in investigating an elliptic boundary value problem in a domain Ω ⊂ R^2 with a cut, coupled with non-linear equations for the vertical motions of the rod and the temperature on the rod. We prove the existence of a unique global weak solution to this problem using a fixed point argument.
@article{bwmeta1.element.bwnjournal-article-amcv12i4p479bwm,
author = {Horn, Werner and Soko\l owski, Jan},
title = {An elastic membrane with an attached non-linear thermoelastic rod},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {12},
year = {2002},
pages = {479-486},
zbl = {1101.74332},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv12i4p479bwm}
}
Horn, Werner; Sokołowski, Jan. An elastic membrane with an attached non-linear thermoelastic rod. International Journal of Applied Mathematics and Computer Science, Tome 12 (2002) pp. 479-486. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv12i4p479bwm/
[000] Brokate M. and Sprekels J. (1996): Hysteresis and Phase Transitions. - Berlin: Springer Verlag. | Zbl 0951.74002
[001] Bubner N. (1995): Modellierung dehnungsgesteuerter Phasenbergnge in Formgedchtnislegierungen. - Ph.D. Thesis, Universitt Essen.
[002] Bubner N., Horn W. and Sokołowski J. (2001): Weak solutions to joined non-linear systems of PDE. - J.Appl. Math. Phys. (ZAMP), Vol. 52, No. 5, pp. 713-729. | Zbl 0994.35114
[003] Bubner N. and Sprekels J. (1998): Optimal control of martensitic phase transitions in a deformation driven experiment on shape memory alloys. - Adv. Math.Sci. Appl., Vol. 8, No. 1, pp. 299-325. | Zbl 0901.49019
[004] Hanouzet B. and Joly J.-L. (1979): Méthodes d'ordre dans l'interprétattion de certaines inéquations variationelles et applications. - J. Funct. Anal., Vol. 34, No. 2, pp. 217-249. | Zbl 0425.49009
[005] Horn W. and Sokołowski J. (2000): Models for adaptive structures using shape memory actuators. - Proc. 14-th Int. Symp. MTNS 2000, Perpignan, France, (on CD-ROM).
[006] Hoffmann K.-H. and Żochowski A. (1992): Existence of solutions to some nonlinear thermoelastic system with viscosity. - Math. Meth. Appl. Sci., Vol. 15, No.3, pp. 187-204. | Zbl 0745.35022
[007] Kondrat'ev V.A. (1967): Boundary value problems for elliptic equations in domains with conical or angular points. -Trudy Moskov. Mat. Obshch., Vol. 16, pp. 209-292, (in Russian); English transl.: Trans. Moscow Math. Soc., 1967, Vol. 16, pp. 227-313.
[008] Kondrat'ev V.A. and Oleinik O.A. (1983): Boundary value problems for partial differential equations in nonsmooth domains. -Uspekhi Mat. Nauk., Vol. 38, No. 2, pp. 3-76, (in Russian).
[009] Kozlov V.A., Maz'ya V.G. and Rossmann J. (1997): Elliptic Boundary Value Problems in Domains with Point Singularities. -Providence, R. I.: Amer. Math. Soc.
[010] Kozlov V.A. and Maz'ya V.G. (1999): Comparison principles for nonlinear operator differential equations in Banach spaces. - Amer. Math. Soc. Transl., Vol. 189, No. 2, pp. 149-157. | Zbl 0923.34057
[011] Lions J.L. and Magenes E. (1972): Non-Homogeneous Boundary Value Problems and Applications. - Berlin: Springer Verlag. | Zbl 0223.35039
[012] Nazarov S.A. and Plamenevsky B.A. (1994): Elliptic Problemsin Domains with Piecewise Smooth Boundaries. - Berlin: Walter de Gruyter. | Zbl 0806.35001
[013] Pawłow I. and Żochowski A. (2000): Existence and uniqueness of solutions for a three-dimensional thermoelastic system. - Working Paper, Systems Research Institute, Polish Academyof Sciences. | Zbl 0955.35074
[014] Sprekels J. and Zheng S. (1989): Global solutions to the equations of a Ginzburg-Landau theory for structural phase transitions in shape memory alloys. - Physica D., Vol. 39, No.1, pp. 59-76. | Zbl 0696.35145
[015] Zheng S. (1995): Nonlinear Parabolic Equations and Coupled Hyperbolic-Parabolic Systems. - Burnt Mill (UK): Longman House. | Zbl 0835.35003
[016] Żochowski A. (1992): Mathematical Problems in Shape Optimization and Shape Memory Materials. - Frankfurt Main: Verlag Peter Lang. | Zbl 0753.73006