Start Date: 2012-04-02 End Date: 2014-04-01
The bilateral project is devoted to investigation of some partial differential equations (PDEs) and systems arising in mechanics and biology. Namely, Georgian team will construct and investigate two-dimensional (2D) hierarchical models for prismatic shells on the bases of thermal elasticity with microtemperatures. The dynamical governing system of the linear theory of elastic materials with inner structure whose particles in addition to the classical displacement and temperature fields possess microtemperature was constructed by Iesan and Quintanilla [1] in 2000. The works of Scalia, Svanadze, and Tracina (see, e.g. [2,3]) are devoted to the investigation of this three-dimensional (3D) model. Each of the hierarchical models which will be constructed for prismatic thermoelastic shells with microtemperatures can be considered as an independent physical model. The researchers of the Georgian team have an experience of construction and investigation of hierarchical models of elastic prismatic and standard shells [4-6], in particular, with the thickness vanishing (so called cusped shells) on the boundary (see [7] and references therein). The outcomes will be: existence and uniqueness theorems for boundary value problems (BVPs) and initial-boundary value problems (IBVPs) within the framework of 2D hierarchical models of the prismatic thermoelastic shells with microtemÂ¬peratures; proof of the convergence of the sequence of approximate solutions of three space variables restored from the solutions of BVPs and IBVPs corresponding to the constructed 2D hierarchical models to the exact solutions of the original 3D BVPs and IBVPs; in the lower order approximations (hierarchical models) the peculiarities of setting boundary conditions (BCs) depending on the geometry of the sharpening of the cusped edges of the prismatic thermoelastic shells with microtemÂ¬peratures will be studied. During the cooperation of the Italian and Georgian teams the background of outlooks for further cooperation in the study of the geometrically and physically nonlinear prismatic thermoelastic shells with microtemÂ¬peratures will be prepared. That will be achieved by the systematic joint discussions of tasks by both the Italian and Georgian teams by electronic means and during mutual visits. Mathematically the above nonlinear problems will lead to the analysis of BVPs and IBVPs of nonlinear PDEs and systems. It is remarkable that some members of both the teams have an experience of the successful cooperation within the framework of the INTAS- South Caucasus project (see [8]).
References
1. Iesan D. and Quintanilla R. (2000) On a theory of thermoelasticity with microtemperatures. J. of Thermal Stresses: vol. 23, pp. 199-215.
2. Svanadze M. (2004) Fundamental Solutions of the Equations of the Theory of Thermoelasticity with Microtemperatures. J. of Thermal Stresses: vol. 27, pp. 151-170.
3. Scalia A., Svanadze M., and Tracina R. (2010) Basic theorems in the equilibrium theory of thermoelasticity with microtemperatures. J. of Thermal Stresses: vol. 33, 721-753.
4. Gordeziani, D.G. (1974) To the exactness of one variant of the theory of thin shells. Soviet. Math. Dokl.: vol. 215 (4), pp. 751-754.
5. Gordeziani, D.G. (1974) On the solvability of some boundary value problems for a variant of the theory of thin shells (Russian). Dokl. Akad. Nauk SSSR: vol. 215 (6), pp. 1289-1292.
6. Avalishvili, G., Avalishvili, M., Gordeziani, D., and Miara, B. (2010) Hierarchical modeling of thermoelastic plates with variable thickness. Anal. Appl.: vol. 8 (2), pp. 125-159.
7. Jaiani G. Cusped Shell-like Structures, SpringerBriefs in Applied Science and Technology, Springer-Heidelberg-Dordrecht-London-New York, 2011.
8. Chinchaladze, N., Jaiani, G., Maistrenko, B., and Podio-Guidugli, P. (2011) Concentrated contact interactions in cuspidate prismatic shell-like bodies, Archive of Applied Mechanics, vol 81 (10), pp. 1487-1505
The modern period is characterized by mathematization of different scientific fields. Relatively new ones, such as there are mathematical biology and bio-mechanics, are arisen and intensively developed. On the other, hand the fields of mechanics such as theory of elasticity, structural mechanics, and theories of shells, plates, and bars were mathematicized from the very beginning. In the last period there is considered very important to study stress-strain state of different structures under influence of temperature and biological processes (e.g., arising of biofilms on the boundaries of structures). Besides, investigation of concrete physical problems and corresponding differential equations and systems, and embracing them differential systems obtained by their combining are very topical. The present join project was aimed to carry out investigations of such a type. In fact, members of both the teams are participating in investigation of all the problems of both the teams but the responsibility to carry out investigation of tasks of each team will fall on the corresponding team.