## Theory of Thin Elastic Micropolar Objects

#A-1183

Theory of Micropolar Thin Elastic Plates and Shells and Its Application in the Applied Mechanics of Solid Deformable Bodies with Microstructure

**Tech Area / Field**

- PHY-STM/Structural Mechanics/Physics

**Status****3** Approved without Funding

**Registration date****08.10.2004**

**Leading Institute**

Gyumri State Pedagogical Institute after M. Nalbandyan, Armenia, Gyumri

**Collaborators**

- Virginia Polytechnic Institute and State University / Department of Engineering Science and Mechanics, USA, VA, Blacksburg\nMcGill University / Department of Mechanical Engineering, Canada, QC, Montreal\nThe University of Salford, UK, Salford\nMartin-Luther Universität Halle-Wittenberg / Chair of Engineering Mechanics, Germany, Halle\nCADLM, France, Gif-sur-Yvette Cedex\nTohoku University / Graduate School of Engineering, Japan, Sendai

**Project summary**

The problem of the construction of such mathematical models of micropolar elastic thin bars, plates and shells gains actuality, as they can adequately reflect the physical essence of the corresponding three-dimensional boundary-value problems and they can also be applicable for their numerical realization.

The asymptotic methods are considered to be one of those main methods of study of the initial-boundary-value problems of Applied Mathematics. The asymptotic method is considered an especially essential method in the Theory of Thin Bars, Plates and Shells, as the latter are thin deformable bodies and the small parameter (the relative thickness) goes into the definition of the object under investigation.

In the Classical Theory of Elasticity for Thin Plates and Shells there are constructed two different asymptotic methods. The first asymptotic method is based upon the immediate integration of the three-dimensional equations of the Theory of Elasticity with the help of the interior (penetrating) and boundary - layer iterational processes by mating the mentioned asymptotic expansions [1, 10, 12,13, 34, etc.]. The second asymptotic method is mostly based upon the construction of homogeneous solutions of the Theory of Elasticity, by reducing the initial boundary-value problem to infinite systems and upon the asymptotic analysis of homogeneous solutions and of infinite systems [43].

Both of these asymptotic methods complete one another and depending on the character of the problem under consideration the first or the second asymptotic method can take precedence. These methods together make up a powerful mathematical apparatus for the construction of the General Theory of Elastic Thin Plates and Shells and for the research of a large spectrum of important classes of problems of static strength, of thermoelasticity and of dynamics of plates and shells.

Both of the asymptotic methods are the principle mathematical means of constructing the General Theories of Electromagnetic Mechanics of Thin Plates and Shells on the basis of the Classical Theory of Elasticity.

In work [32] by Sargsyan S. H. there was constructed the General Theory of Magnetoelasticity of Electroconducting Thin Plates and Shells on the basis of the first asymptotic method and there has been developed effective methods for studying the most important problems of the dynamics of plates and shells in the magnetic field.

In the scientific papers by professor Sargsyan, [33, 35-37, etc. ] on the basis of the first asymptotic method there has been developed the asymptotic approach and there has been also constructed the General Asymptotical Theory of Micropolar Elastic Thin Plates. On the basis of the interior iterational process there has been created the General Applied Two-dimensional Theory of Micropolar Elastic Thin Plates. There has also been constructed the second iterational process by which the boundary layer of micropolar elastic plates is defined. The properties and the structure of the boundary layer were exposed. The existence of the force plane and antiplane and momental plane and antiplane boundary layers has been shown. The problem of mating of the asymptotic expansion of interior problem and boundary layers has also been considered with the aim of satisfaction to boundary conditions on linear cylindrical surface of plates, as a result of which the interior (applied) problem and boundary-layer problems separated from each other as independent boundary-value problems.

This method is also developed for the micropolar bars, for the static thermoelasticity of micropolar plates and for the dynamic problems of micropolar plates by the post-graduate students and scientists of Mathematical Analysis and Differential Equations Chair at Gyumri State Pedagogical Institute under the supervision of professor Sargsyan S.H.

The aim of the present project is the creation of a powerful mathematical apparatus for the construction of the General Theory of Micropolar Elastic Thin Plates and Shells, and the research of a wide circle of applied problems of Statics, Thermoelasticity and Dynamics of the micropolar homogeneous and layered plates and shells. It’s also intended to solve a set of actual and principal problems of micropolar elastic thin plates and shells, among which we single out the following:

A. Construction of a mathematical apparatus for transition from the Three-dimensional Asymmetrical Theory of Elasticity to the Two-dimensional ones for thin plates and shells in static cases on the basis of the development of the principles of the two asymptotic methods; Realization of a qualitative asymptotic analysis of three-dimensional stress-deformed state o f micropolar plates and shells; Development of the algorithms of construction and research of interior problem and boundary layer for plates and shells on the Asymmetrical Theory of Elasticity; Construction of General Applied Two-dimensional Theories of Micropolar Plates and Shells. These include the following steps:

1. Construction of homogeneous solutions for plates on the Asymmetrical Theory of Elasticity in the static case. It’s important to consider the plates and the shells of both homogeneous and non-homogeneous thickness;

2. Development of the apparatus of homogeneous solutions for shells on the Asymmetrical Theory of Elasticity (particularly for cylindrical, conic and spherical shells);

3. Creation of the Applied Two-dimensional Theories for Micropolar Elastic and Thermoelastic Thin Plates and Shells (both for homogeneous and non-homogeneous-layered ones).

B. Development of the apparatus of the transition to two-dimensional problems for micropolar thin elastic plates and shells in the dynamic case:

1. Construction of homogeneous solutions in the dynamic case for plates and shells;

2. Construction of Applied Theories of Micropolar Plates and Shells in the dynamic case;

3. Development of the apparatus of transition from the Three-dimensional Theory of Elasticity to Two-dimensional ones for Plates and Shells in the problems of proper significance;

4. Utilization of the apparatus of homogeneous solutions of the Asymmetrical Theory of Elasticity during the analysis of forced periodical vibrations of micropolar plates and shells.

C. Thorough numerical revision of the accuracy on the asymptotic method and the solutions of concrete applied problems of strength and vibration of the micropolar plates and shells:

1. Thorough numerical revision of the accuracy of the basis of comparison between the exact solutions and their asymptotic representation and the analysis of the possibility of using the constructed Applied Two-dimensional Theories;

2. Solution of a set of applied problems of strength, of free and forced vibrations for the rectangle, for round plates and for round cylindrical shells;

3. Solution of a set of problems on concentration of stress on the Asymmetrical Theory of Elasticity around Various Holes in the Plates and Shells. Analysis of the possibility of using the constructed Applied Two-dimensional Theories for the account of the stress concentration in the general case of the stress state on the Asymmetrical Theory of Elasticity;

4. Development of effective numerical methods for the solution of definite classes of problems of strength and vibration of the micropolar elastic thin plates and shells.

The present project contains investigations which will thoroughly solve the above mentioned actual problems of Momental Theory of Elastic Thin Plates and Shells. In case of complete and successful realization of the above mentioned problems we shall obtain the complete Theory of Micropolar Elastic Thin Plates and Shells, at the same time, there’ll be developed effective methods of solution of a wide range of problems of strength, of stress concentration around the holes, of Thermoelasticity and of Dynamics of Plates and Shells. The expected long-term results will be rather weighty for the future investigations in the field of the Mechanics of the Deformable Solid Bodies, i.e. for the construction of the Theory of Electroelasticity and Magnetoelasticity of Micropolar Thin Plates and Shells and the exit to the actual and contemporary problems of Micro- and Nano-Mechanics, Seismology, etc.

The solution of the set problems is rather important from the point of view of its application, i.e. for the field of Mechanics of Thin Plates and Shells with Microstructure.

In this project there will be constructed the General Theory of Micropolar Elastic Thin Plates and Shells on the basis of the unique approach, i.e. on the basis of the asymptotic method and also on the basis of the integration of the equations of the three-dimensional problem of the Asymmetrical Theory of Elasticity (for the problems of Statics, Thermoelasticity and Dynamics) with independent fields of transition and rotation. The asymptotic method will be the basis of the construction of the General Applied Theory of Micropolar Elastic Thin Plates and Shells. There will be developed effective methods and there will be constructed algorithms with their realization on calculating machines for wide classes of applied problems of Statics, Thermoelasticity, Stress Concentration, Natural and Forced Vibrations of Micropolar Plates and Shells.

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