Geodesic
Existence and uniqueness of weak solutions for the model representing motions of curves made of elastic materials
The Bulletin of Irkutsk State University. Series Mathematics, Tome 36 (2021), pp. 44-56 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the initial boundary value problem for the beam equation with the nonlinear strain. In our previous work this problem was proposed as a mathematical model for stretching and shrinking motions of the curve made of the elastic material on the plane. The aim of this paper is to establish uniqueness and existence of weak solutions. In particular, the uniqueness is proved by applying the approximate dual equation method.
Keywords: Beam equation, nonlinear strain, dual equation method. 
@article{IIGUM_2021_36_a3,
     author = {T. Aiki and C. Kosugi},
     title = {Existence and uniqueness of weak solutions for the model representing motions of curves made of elastic materials},
     journal = {The Bulletin of Irkutsk State University. Series Mathematics},
     pages = {44--56},
     year = {2021},
     volume = {36},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IIGUM_2021_36_a3/}
}
TY  - JOUR
AU  - T. Aiki
AU  - C. Kosugi
TI  - Existence and uniqueness of weak solutions for the model representing motions of curves made of elastic materials
JO  - The Bulletin of Irkutsk State University. Series Mathematics
PY  - 2021
SP  - 44
EP  - 56
VL  - 36
UR  - http://geodesic.mathdoc.fr/item/IIGUM_2021_36_a3/
LA  - en
ID  - IIGUM_2021_36_a3
ER  - 
%0 Journal Article
%A T. Aiki
%A C. Kosugi
%T Existence and uniqueness of weak solutions for the model representing motions of curves made of elastic materials
%J The Bulletin of Irkutsk State University. Series Mathematics
%D 2021
%P 44-56
%V 36
%U http://geodesic.mathdoc.fr/item/IIGUM_2021_36_a3/
%G en
%F IIGUM_2021_36_a3
T. Aiki; C. Kosugi. Existence and uniqueness of weak solutions for the model representing motions of curves made of elastic materials. The Bulletin of Irkutsk State University. Series Mathematics, Tome 36 (2021), pp. 44-56. http://geodesic.mathdoc.fr/item/IIGUM_2021_36_a3/

[1] Aiki T., “Weak solutions for Falk's Model of Shape Memory Alloys”, Math. Methods Appl. Sci., 23 (2000), 299–319

[2] Aiki T., Kosugi C., “Numerical schemes for ordinary differential equations describing shrinking and stretching motion of elastic materials”, Adv. Math. Sci. Appl., 29 (2020), 459–494

[3] Brocate M., Sprekels J., Hysteresis and phase transitions, Appl. Math. Sci., 121, Springer, 1996

[4] Furihata D., Matsuo T., Discrete variational derivative method: A structure preserving numerical method for partial differential equations, Chapman Hall CRC Publ, 2010

[5] Holzapfel Gerhard A., Nonlinear solid mechanics: a continuum approach for engineering, John Wiley Sons Publ, 2000

[6] Ladyzenskaja O. A., Solonnikov V. A., Ural'ceva N. N., Linear and Quasi-Linear Equations of Parabolic Type, Transl. Math. Monograph., 23, Amer. Math. Soc., Providence, R.I., 1968

[7] Lions J. L., Quelques méthods de résolution des problèmes aux limites non linéairs, Dunod, Gauthier-Villars Publ, Paris, 1969

[8] Niezgódka M., Pawlow I., “A generalized Stefan problem in several space variables”, Appl. Math. Opt., 9 (1983), 193–224

[9] Ogden Raymond William, “Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubber like solids”, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 328 (1972), 567–583

[10] J. C. Simo, C. Miehe, “Associative coupled thermoplasticity at finite strains: Formulation, numerical analysis and implementation”, Comput. Methods in App. Mech. Engi., 98 (1992), 41–104

[11] Yoshikawa S., “Weak solutions for the Falk model system of shape memory alloys in energy class”, Math. Methods Appl. Sci., 28 (2005), 1423–1443

[12] Yoshikawa S., “An error estimate for structure — preserving finite difference scheme for the Falk model system of shape memory alloys”, IMA J. Numer. Anal., 37 (2017), 477–504