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.

Toyohiko Aiki, Dr. Sci. (Phys.–Math.), Prof., Department of Mathematics, Faculty of Science, Japan Women’s University, 2-8-1 Mejirodai, Bunkyo-ku, Tokyo, 112-8681, Japan, email: aikit@fc.jwu.ac.jp

Chiharu Kosugi, Postgraduate Student, Department of graduate School of Science, Japan Women’s University, 2-8-1 Mejirodai, Bunkyo-ku, Tokyo,112-8681, Japan, email: m1416034kc@ug.jwu.ac

Aiki T., Kosugi C. 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, 2021, vol. 36, pp. 44-56. https://doi.org/10.26516/1997-7670.2021.36.44

1. Aiki T. Weak solutions for Falk’s Model of Shape Memory Alloys. Math. Methods Appl. Sci., 2000, vol. 23, pp. 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., 2020, vol. 29, pp. 459-494.
3. Brocate M., Sprekels J. Hysteresis and phase transitions. Springer, Appl. Math. Sci., 1996, vol. 121.
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, Vol. 23, Amer. Math. Soc., Providence, R. I. 1968.
7. Lions J.L. Quelques m´ethods de r´esolution des probl`emes aux limites non lin´eairs. Paris, Dunod, Gauthier - Villars Publ., 1969.
8. Niezg´odka M., Pawlow I. A generalized Stefan problem in several space variables. Appl. Math. Opt., 1983, vol. 9, pp. 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., 1972, vol. 328, pp. 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., 1992, vol. 98, pp. 41-104.
11. Yoshikawa S. Weak solutions for the Falk model system of shape memory alloys in energy class. Math. Methods Appl. Sci., 2005, vol. 28, pp. 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., 2017, vol. 37, pp. 477-504.