Shota Rustaveli National Science Foundation of GeorgiaStart Date: 2016-12-09 End Date: 2019-12-08
The present scientific project belongs to the sphere of fundamental research. The project aim s to investigate general mixed boundary and initial-boundary value problem s corresponding to inhomogeneous anisotropic thermoelastic three-dimensional linear models with regard to magnetic field for solids of arbitrary shape in both the cases of the Lipschitz and non-Lipschitz boundaries, to construct and investigate two-dimensional and one-dimensional static and dynamical hierarchical models for prismatic shells of variable thickness and bars with variable rectangular cross-sections, respectively. Construction and mathematical justification of lower dimensional hierarchical models is the most important goal of the project. To this end variational formulations and spectral approximation methods will be employed. This approach does not require smallness of some geometrical parameters and in the construction of the models only mathematically justified assumptions are used.
In the case of fully coupled thermomechanical and electro-magnetic fields the model is described by formally non-self-adjoint system consisting of six partial differential equations with variable coefficients with the sought for functions being the three components of mechanical displacement vector, the temperature distribution and electric and magnetic potentials.
The duration of the project is three years and its plan consists of six stages.
On the first and second stages the project aims to investigate general three-dimensional boundary, initial-boundary and transmission problems corresponding to linear static and dynamical models for piecewise homogeneous or general inhomogeneous, in particular functionally graded, thermoelastic piezoelectric anisotropic solids by variational and special methods in appropriate function spaces, when on some part of the boundary the mechanical stress vector and the normal components of the heat flux, electric displacement, and magnetic induction vectors are prescribed, while on the remaining part of the boundary the mechanical displacement vector, temperature function, electric and magnetic potentials vanish. On the third and fourth stages the project aim s to construct and investigate new two-dimensional hierarchical static and dynamical models for layered piecewise homogeneous or general inhomogeneous, in particular functionally graded, thermoelastic piezoelectric prismatic shells with variable thickness. It will be shown that the sequences of vector-functions of three space variables restored from the solutions of two-dimensional boundary and initial-boundary value problems converge to the solutions of the corresponding three-dimensional boundary and initial-boundary value problem s in suitable spaces and under certain additional conditions the rate of convergence will be estimated. On the fifth and sixth stages, hierarchies of new one-dimensional static and dynamical models for layered piecewise homogeneous or general inhomogeneous, in particular functionally graded, thermoelastic piezoelectric bars with variable rectangular cross-sections will be constructed and investigated. The relationship between the constructed one-dimensional static and dynamical models for thermoelastic piezoelectric bars and the corresponding three-dimensional static and dynamical models will be studied. The convergence of the sequence of vector-functions of three space variables restored from the solutions of one-dimensional problem s to the solutions of the corresponding three-dimensional problem s will be shown and under certain additional conditions modeling error estimates will be obtained.
On all the six stages within the framework of the constructed two- and one-dimensional hierarchical models the peculiarities of setting boundary conditions for Lipschitz and non-Lipschitz cusped edges will be investigated. Some numerical realizations in particular cases will be carried out.