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SOME ISSUES OF GRADIENT THEORIES OF THIN BODIESтезисы доклада
- Авторы: Nikabadze M.U., Ulukhanyan A.R., Mardaleishvili N.V.
- Сборник: Book of Abstrakts, XV Annual International Meeting of the Georgian Mechanical Union
- Тезисы
- Год издания: 2024
- Место издания: Kutaisi, Youth Avenue 25-A
- Первая страница: 153
- Последняя страница: 156
- Аннотация: The expressions for specific energy, constitutive relations, equations of motion and equilibrium are given (similar relations can be found in [1-4]), and the initial-boundary value problems of the second strain tensor and velocity vector gradient theory for an arbitrary anisotropic bodies are formulated. From the latter, 3D formulations of the initial-boundary value problems of the second strain tensor and velocity vector gradient 3D theory of elastic thin bodies are obtained, on the basis of which, using the method of orthogonal polynomials similarly to [8] (see also [9-11]), the corresponding formulations of the initial-boundary value problems in moments are derived. Constitutive relations for gradient bodies are written using tensor-block matrices. In addition, static boundary conditions and equations of motion and equilibrium in the case of gradient theories are represented by differential tensor-block matrix operators, for which, in the case of homogeneous bodies, differential tensor-block matrix operators of cofactors are constructed, which make it possible to split the initial-boundary value problems of some gradient theories (differential tensor -block matrix operator of static boundary conditions is called differential tensor-block matrix stress operator). Herewith static boundary conditions are not always split. The differential tensor-block matrix operator of cofactors for the differential tensor-block matrix operator of stress can be constructed only for a homogeneous body having a piecewise flat boundary, i.e. for each flat section of the boundary it is possible to construct a differential tensor-block matrix operator of cofactors and, therefore, for this flat section, the static boundary conditions can be split. Further, from 3D formulations of initial-boundary value problems for some gradient theories of elasticity, the formulations of initial-boundary value problems for gradient theories of elastic thin bodies are derived under the new parameterization [7-8] of the regions of these bodies, from which, in turn, similarly to [8] (see also [9-11]) the initial-boundary value problems in moments of the displacement vector with respect to arbitrary system of orthogonal polynomials and also Legendre polynomials are obtained. Several first approximations of formulations of boundary value problems for prismatic bodies are considered as special cases. A system of equations in moments of the displacement vector of the 20th order approximation is obtained. The system of equations in moments of the displacement vector of the 10th order approximation is considered in more detail. In particular, it, as any system with any order of approximation, splits into two systems, one of which contains moments of the displacement vector of even order, and the other contains moments of odd order. Each of these systems, in turn, is split and for each moment included in these systems of equations, an equation of higher order elliptic type is obtained separately, for which an analytical solution is written using the Vekua method. As a special case, an elastic and viscoelastic rectangular plate, as well as an elastic and viscoelastic rectangle, were considered (the relations needed to solve these problems were taken from [5,6,12] works). It is noteworthy that the methodologies outlined above can be readily extended to accommodate other rheological bodies, including higher-order gradient theories, as well as both linear and nonlinear classical [6,12] and micropolar [5] theories of viscoelastic bodies. Acknowledgement: This work was supported by the financial support of the Shota Rustaveli National Science Foundation (project No FR-21-3926). References 1.Askes H., Aifantis E.C. (2011) Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct. 48, 1962–1990. 2.Castrenze Polizzotto (2013) A second strain gradient elasticity theory with second velocity gradient inertia – Part I: Constitutive equations and quasi-static behavior. Int. J. Solids Struct. 50, 3749–3765 http://dx.doi.org/10.1016/j.ijsolstr.2013.06.024 3. Castrenze Polizzotto (2013) A second strain gradient elasticity theory with second velocity gradient inertia – Part II: Dynamic behavior. Int. J. Solids Struct. 50, 3766–3777 http://dx.doi.org/10.1016/j.ijsolstr.2013.07.026 4.Dell’Isola F., Sciarra G., Vidoli S. (2009) Generalized Hooke’s law for isotropic second gradient materials. Royal Society of London, vol. 465, pp. 2177–2196. 5.Eringen A.C. (1999) Microcontinuum Field Theories. 1. Foundation and solids. Springer-Verlag. N.Y.: 341 p. 6.Ilyushin A.A., Pobedrya B.E. (1970) Fundamentals of the mathematical theory of thermo-viscoelasticity. М.: Nauka. 280 p. 7.Nikabadze M.U. (2017) Topics on tensor calculus with applications to mechanics. Journal of Mathematical Sciences, Vol. 225, No. 1. 194 p. https://doi.org/10.1007/s10958-017-3467-4 8.Nikabadze M.U. (2023) Development of the method of orthogonal polynomials in the classical and micropolar mechanics of elastic thin bodies. Moscow University Publishing House, Moscow. 665 p. 9.Nikabadze M., Ulukhanyan A. (2020) Modeling of multilayer thin bodies // Continuum Mechanics and Thermodynamics. Vol. 32, No 4. https://doi.org/10.1007/s00161-019-00762-6 10.Nikabadze M., Ulukhanyan A. (2022) On some variational principles in micropolar theories of single-layer thin bodies // Continuum Mechanics and Thermodynamics. Vol. 34, No 2. https://doi.org/10.1007/s00161-022-01089-5 11.Nikabadze M., Ulukhanyan A. (2022) Generalized Reissner-type variational principles in the micropolar theories of multilayer thin bodies with one small size // Continuum Mechanics and Thermodynamics. Vol. 34, No 2. https://doi.org/10.1007/s00161-022-01091-x 12.Pobedrya B.E. (1995) Numerical methods in the theory of elasticity and plasticity. Moscow University Publishing House, Moscow.
- Добавил в систему: Никабадзе Михаил Ушангиевич