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In the present paper asymptotic solution of boundary-value problem of three-dimensional micropolar theory of elasticity with free fields of displacements and rotations is constructed in thin domain of the shell. This boundary-value problem is singularly perturbed with small geometric parameter. Internal iteration process and boundary layers are constructed, problem of their jointing is studied and boundary conditions for each of them are obtained. On the basis of the results of the internal boundary-value problem the asymptotic two-dimensional model of micropolar elastic thin shells is constructed. Further, the qualitative aspects of the asymptotic solution are accepted as hypotheses and on the basis of them general applied theory of micropolar elastic thin shells is constructed. It is shown that both the constructed general applied theory of micropolar elastic thin shells and the classical theory of elastic thin shells with consideration of transverse shear deformations are asymptotically confirmed theories.

Current methods of reducing three-dimensional problem of theory of elasticity to two-dimensional problem of theory of plates and shells are the followings: 1) hypotheses method; 2) method of expansion by thickness; 3) asymptotic method [

The main problem of the general theory of micropolar or classical elastic thin plates and shells is in approximate, but adequate reduction of three-dimensional boundary-value problem of the micropolar or classical theory of elasticity to two-dimensional problem. From our point of view, for achievement of this aim [

A shell of constant thickness 2h is considered as a three-dimensional elastic body. Equations of the static problem of asymmetric (micropolar, momental) theory of elasticity with free fields of displacements and rotations are the followings [

Equilibrium equations:

Physical relations:

Geometrical relations:

Here

It should be noted that if

We’ll consider three orthogonal system of coordinates

Boundary conditions of the first boundary-value problem for front surfaces of the shell are accepted:

Boundary conditions on the edge

where

It is assumed that the thickness 2h of the shell is small compared with typical radius of curvature of the middle surface of the shell

Question of reduction of three-dimensional static problem of asymmetric theory of elasticity for thin domain of the shell to two-dimensional problem is considered on the basis of asymptotic method with boundary layer [

At first we’ll consider the construction of internal interactive process. For achievement of this aim we’ll pass to dimensionless coordinates in three-dimensional Equations (1)-(3) of asymmetric theory of elasticity:

Here quantity

On the basis of (7), (8) following system of dimensionless equations will be obtained instead of system of Equations (1)-(3).

Equilibrium equations:

Physical-geometrical relations:

Here

The case is considered when dimensionless physical parameters (8) have the following values:

Following replacements of unknown quantities will be done:

As a result following system of equations will be obtained:

where

Following to the asymptotic method, the question is the following: to reduce three-dimensional Equations (14) (with free variables

Following formulas will be obtained for displacements and rotations, force and moment stresses with asymptotic accuracy

where

The aim is to construct asymptotically strictly interactive process for averaged along the shell thickness quantities, which determine the stated problem (i.e. depending only on quantities

Keeping quantities up to

where

It must be required that averaged values along the shell thickness of quantities

Substituting (19) and (20) into conditions (22), following formulas will be obtained for

Thus for

Finally, for quantities

It should be noted that averaged along the shell thickness quantities for

Thus, taking into consideration (25), (26), we’ll have following formulas for displacements, rotations, force and moment stresses instead of (16):

The constructed asymptotics (27) for internal interaction process of the stated problem gives an opportunity to reduce three-dimensional problem to two-dimensional one (what is already done for displacements, rotations, force and moment stresses). As in the classical theory, instead of components of tensors of force and moment stresses statically equivalent to them integral characteristics are introduced in micropolar theory: forces T_{ii}, S_{ij}, N_{i}_{3}, N_{3i}, moments M_{ii}, H_{ij}, L_{ii}, L_{ij}, L_{i}_{3}, L_{33} and hypermoments

Displacements and rotations of points of the shell middle surface are introduced as follows:

Satisfying boundary conditions (4) on shell surfaces

Equilibrium equations:

Elasticity relations:

Geometric relations:

System of equations of thin shells of classical theory will be obtained from system of Equations (29)-(31) in case of

We’ll proceed from three-dimensional Equations (1)-(3) of micropolar theory of elasticity. It is assumed that the surface of the shell edge

where quantities

Solution of the obtained system of boundary-value problem must satisfy homogeneous boundary conditions on surfaces

We’ll pass to dimensionless quantities (7), (8) and introduce following notations:

As a result three-dimensional equations of micropolar theory in dimensionless form will be obtained from Equations (1)-(3) (with consideration of (7), (8)).

At level

Force plane problem:

Force non plane problem:

Momental plane problem:

Momental non plane problem:

where

The obtained equations of boundary layer in Cartesian coordinates

Requiring that solutions (35)-(38) of boundary layers have fading character when

From the above introduced relations (special for micropolar theory of elasticity) following important conclusion can be done: when force and moment stresses are balanced in boundary layer, displacements and free rotations will have the same property.

Considering problem of jointing of internal SSS and boundary layer, following symbolic formula must be introduced for the whole SSS of the shell:

Now the first variant of three-dimensional boundary conditions of micropolar theory of elasticity will be considered, when shell edge is loaded with forces and moments

At level

where

Using corresponding conditions from (39) and on the basis of (41), boundary conditions for system (29)-(31) of two-dimensional equations will be obtained:

Let us study the second variant of three-dimensional boundary conditions of micropolar theory of elasticity, when displacements and rotations are given on the shell edge

At level

where

With the help of conditions from (39) boundary conditions for two-dimensional model will be obtained:

Mixed three-dimensional boundary conditions are studied, when hinged support takes place.

Following values will be taken for quantities r and

At level

where

In this case, using conditions from (39), following boundary conditions of hinged-support will be obtained for two-dimensional model:

Thus two-dimensional theory of micropolar shells is constructed at level of initial approximation of the asymptotic method. System of equations (29)-(31) and boundary conditions (42) (or (43) or (46)) introduce the asymp- totic model of micropolar elastic thin shells with free fields of displacements and rotations.

Hypotheses method of construction of classical theory of elastic thin shells (i.e. Kirkhov-Love’s or refined hypotheses) has an advantage above the asymptotic method from point of view of engineering, because some simplifications were put in the base of theory, which have physical meaning and also visibility and clarity. Main problem of the construction of applied theory of micropolar elastic thin shells is the following: to formulate such hypotheses that let us reduce three-dimensional problem of micropolar theory of elasticity to adequate two- dimensional boundary-value problem. For achievement of this aim the use of qualitative aspects of asymptotic solution of three-dimensional boundary-value problem (1)-(5) of micropolar theory of elasticity is appropriate in thin domain of the shell.

In papers [

1) During the deformation initially straight and normal to the shell middle surface fibers rotate freely in space at an angle as a whole rigid body, without changing their length and without remaining perpendicular to the deformed middle surface.

The formulated hypothesis is mathematically written as follows: tangential displacements and normal rotation are distributed in a linear law along the shell thickness:

Normal displacement and tangential rotations do not depend on coordinate

It should be noted that from the point of view of displacements the accepted hypothesis, in essence, is Timoshenko’s kinematic hypothesis in the classical theory of elastic shells [

2) In the generalized Hook’s law (2) force stress

3) During the determination of deformations, bending-torsions, force and moment stresses, first for the force stresses

After determination of mentioned quantities, values of

4) Quantities

Now we’ll compare main equations of applied static theory of micropolar elastic thin shells from paper [

Concerning the dynamic theory of micropolar elastic thin shells, it should be noted that the corresponding asymptotic model is constructed in paper [

As in case

It should be noted that in papers [

In the present paper the question of reduction of three-dimensional boundary-value problem of micropolar and classical theories of elasticity to general applied theories of thin shells is studied. The asymptotics of singularly perturbed boundary-value problem of three-dimensional micropolar theory of elasticity is studied in thin domain of the shell. The internal iteration process and boundary-layers are constructed, jointing of these two iteration processes is studied and boundary conditions are obtained. As a result two-dimensional asymptotic model with free fields of displacements and rotations of micropolar shells is constructed. Transverse shear deformations are automatically taken into consideration in the constructed model. Particularly, classical asymptotic theory of elastic thin shells with consideration of transverse shears can be obtained from the above mentioned micropolar model.

Hypotheses are accepted for the construction of general applied theory of micropolar elastic thin shells. The hypotheses are adequate to the asymptotic behavior of the solution of three-dimensional problem. Such approach ensures the asymptotic exactness of the constructed micropolar and classical theories of thin shells with consideration of transverse shears.

SamvelSargsyan, (2015) Asymptotically Confirmed Hypotheses Method for the Construction of Micropolar and Classical Theories of Elastic Thin Shells. Advances in Pure Mathematics,05,629-642. doi: 10.4236/apm.2015.510057