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This paper presents an extension of mathematical static model to dynamic problems of micropolar elastic plates, recently developed by the authors. The dynamic model is based on the generalization of Hellinger-Prange-Reissner (HPR) variational principle for the linearized micropolar (Cosserat) elastodynamics. The vibration model incorporates high accuracy assumptions of the micropolar plate deformation. The computations predict additional natural frequencies, related with the material microstructure. These results are consistent with the size-effect principle known from the micropolar plate deformation. The classic Mindlin-Reissner plate resonance frequencies appear as a limiting case for homogeneous materials with no microstructure.

Classical theory of elasticity ignores the size effects of the particles and their mutual rotational interactions, thus considering the material particles to have only three degrees of freedom that represent their macrodisplacements. The stress tensor is symmetric and the surface loads are assumed to be solely determined by the force vector. Classical theory of elasticity is widely used in engineering and is successfully applied under small deformations to such linear elastic materials as stainless steel, concrete, plastic, aluminium, etc. Many modern engineering materials, however, contain fibers, grains, pores or macromolecules, which in turn make them exhibit the defor- mation that cannot be adequately described by the classical elasticity (see, for example, the studies of a low- density polymeric foam in [

The first theory of elasticity that took into account the microstructure of the material was developed in 1909 by Cosserat brothers. They presented the equations of local balance of momenta for stress and couple stress, and the expressions for surface tractions and couples [

In this paper, we present an extension of our static approach to the dynamics of micropolar elastic plates. We reformulate a generalization of Hellinger-Prange-Reissner (HPR) variational principle [

Throughout this paper we will use the Einstein summation notation. The Latin subindices take values in the set

The Cosserat linear elasticity balance equations without body forces represent the balance of linear and angular momentums of micropolar elastodynamics and have the following form:

where the quantity

ment and rotation vectors,

material density and the rotatory inertia characteristics,

The linearized constitutive equations are given in the form [

and the strain-displacement and torsion-rotation relations

where

The constitutive equations in the reverse form can be written as

where

We consider a Cosserat elastic body

and initial conditions

where

The strain stored energy

where non-negative

then the constitutive relations (3)-(4) can be written in the form:

For future convenience, we present the stress energy

where

The constitutive relations in the reverse form (6)-(7) can be also written in form:

The total internal work done by the stresses

and

provided the constitutive relations (3)-(4) hold.

We also consider the stored kinetic energy of the body

We also present the kinetic energy as

where

or

and

The internal work done by the inertia forces over displacement and microrotation is

Using the integration by parts

and taking into account the zero variation of

Note that since the variations

We modify the HPR principle [

of the functional

at

Proof of the Principle

Let us consider the variation of the functional

Taking into account (5), we can perform the integration by parts

and based on (16)-(19)

Then, keeping in mind

The latter expression provides the proof of the principle.

In this section we review our stress, couple stress and kinematic assumptions of the Cosserat plate [

The set of points

lateral part of the boundary where displacements and microrotations are prescribed. The notation

of the remainder we use to describe the lateral part of the boundary edge

couple stress are prescribed. We also use notation

In our case we consider the vertical load and pure twisting momentum boundary conditions at the top and bottom of the plate, which can be written in the form:

where

Some basic stress and kinematic assumptions are similar to the Reissner plate theory [

We reproduce the main micropolar plate assumptions presented in [

where

functions

and

We also will use the notation of the normalized components of the micropolar plate stress set

Here,

where

The terms

components, and

The components of the corresponding micropolar plate strain set

The components of Cosserat plate strain can also be represented in terms of the components of set

The formulas (54) are called the Cosserat plate strain-displacement relation.

We also assume that the initial condition can be presented in the similar form:

The HPR variational principle for a Cosserat plate dynamics is most appropriately expressed in terms of corres- ponding integrands calculated across the whole thickness. We also introduce the weighted characteristics of dis- placements, microrotations, strains and stresses of the plate, which will be used to produce the explicit forms of these integrands.

We define the plate stress energy density by the formula [

Then the stress energy of the plate P

where

We define the plate stress energy density by the formula;

Taking into account the kinematics assumptions and integrating

Then the kinetic energy of the plate can be written

In the following consideration we also assume that the proposed stress, couple stress, and kinematic assumptions are valid for the lateral boundary of the plate P as well.

We evaluate the density of the work over the boundary

Taking into account the stress and couple stress assumptions (26)-(34) and kinematic assumptions (42)-(45) we are able to represent

where the sets

and

In the above

The density of the work over the boundary

can be presented in the form

where

Now

We are able to evaluate the work done at the top and bottom of the Cosserat plate by using boundary con- ditions (22) and (24)

Here we define the density of the work done by the stress and couple stress over the Cosserat strain field:

Substituting stress and couple stress assumptions and integrating the expression (64) we obtain the following expression:

where

Here we define the density of the kinetic energy:

which can be presented in the form

where

Let

for every

Then

is equivalent to the plate bending system of equations (A) and constitutive formulas (B) mixed problems.

A. The bending equilibrium system of equations:

where

at the part

at the part

The constitutive formulas have the following reverse form^{1}:

Proof of the principle. The variation of

where

where we call

We apply Green’s theorem and integration by parts for

Then based on the fact that

If s is a solution of the mixed problem, then

On the other hand, some extensions of the fundamental lemma of calculus of variations [

Remark. The above equilibrium equations and boundary conditions for the Cosserat plate can also be obtained by substituting polynomial approximations of stress and couple stress directly to the elastic equilibrium (1)-(2) and the boundary conditions (22)-(25) and collecting and equating to zero all coefficients of the resulting poly- nomials with respect to variable

In order to obtain the micropolar plate bending field equations in terms of the kinematic variables, we substitute the constitutive formulas in the reverse form (76)-(88) into the bending system of Equations (67)-(72). The micropolar plate bending field equations can be written in the following form:

where

The operators

where

The right-hand side, and therefore the solution

Let us consider a square plate _{t} = 0.62 mm,

The boundary

and the hard simply supported boundary conditions can be represented in the following mixed Dirichlet- Neumann form [

By applying the method of separation of variables for the two-dimensional eigenvalue problem (89) with the hard simply supported boundary conditions we obtain the kinematic variables in the following form:

and a standard eigenvalue problem for a system of 9 algebraic equations. Thus the model produces a spectrum of 9 infinite sequences

Preliminary computations show that only

We perform computations for different levels of the asymmetric microstructure by reducing the values of the elastic asymmetric parameters.

torsion and bending of cylindrical rods of a Cosserat solid.

We also check how the total energy of the free oscillation depends of the value of the shear correction factor

We consider a rectangular plate

plates. This result is consistent with the changes of this factor used in the Mindlin-Reissner plate theory.

This paper presents a mathematical model for the vibration of micropolar elastic plates. This model is based on the proposed generalization of Hellinger-Prange-Reissner (HPR) variational principle for the linearized micro- polar (Cosserat) elastodynamics. The modeling of the plate vibration is based on the HPR variational principle for the dynamics of Cosserat plates, which incorporates most of assumptions of the authors’ enhanced mathe- matical model for Cosserat plate deformation. The dynamic theory of the plates obtained from the dynamic

variational principle includes a system of dynamics equations and the constitutive relations. The preliminary computations of the rectangular plate vibration predict additional natural frequencies, which are related with the material microstructure and obey the size-effect principle similar to the known from the micropolar plate de- formation. The computations also show how natural frequencies of micropolar plate converge to classic Mindlin-Reissner plates and the total vibration energy can get 4% - 5% smaller depending on a parameter in the constitutive formulas and the geometry of the plates. These result are consistent with the modification (from