1. Introduction
For isotropic materials, eigenmodules (eigenvalues) and eigenstates (eigentensors) are known since monograph [
1]. For anisotropic ones, the said notions were introduced by Kelvin in the middle of the 19th century (other terms were used). However, the investigation in this area was continued only about 40 years ago (see, e.g., [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15] and the bibliography of [
10]). In [
10], Ostrosablin has studied the inner structure rather well, classified anisotropic classically linearly elastic materials, and studied many other important problems (see also [
16]). The notion of elastic eigenstates is applied in the plasticity theory (see [
17]) and the flow theory (see [
18]).
Representations of general solutions of the Lamé’s equations were made by many scientists (see, for example, [
19,
20,
21,
22,
23]), and representations of general solutions of equations in displacements and rotations of the micropolar theory of elasticity can be found, for example, in [
22,
24,
25,
26]. Note that in this case the equations of the classical and micropolar theory of elasticity are decomposed, but for decomposing the equations, as well as static boundary conditions, the algebraic method turned out to be more efficient. If this method is used, it is advisable to present the equations and static boundary conditions in tensors–operators (tensor–block operators) in the case of classical (micropolar) medium, and then find the tensors–operators (tensor–block operators) of algebraic cofactors for these operators. Of course, the algebraic method can be used for decomposing the static boundary conditions only for bodies with piecewise plane boundaries. These questions are described in some detail in [
27,
28], and in this work, special attention is paid to the canonical representations of equations and boundary conditions.
Note that in [
28] some questions from monograph [
27], which was published in Russian in the Mechanics and Mathematics faculty of Lomonosov Moscow State University, were presented in English with some clarifications and changes. In particular, the authors presented some questions of tensor calculus; constructed new versions of theories of single-layer and multi-layer elastic thin bodies via the developed method of orthogonal polynomials and also obtained the corresponding decomposed equations of quasistatic problems of classical (micropolar) theory of prismatic bodies with constant thickness in displacements (in displacements and rotations) from the decomposed equations of classical (micropolar) theory of elasticity. The above results and eigenvalue problems for tensor and tensor–block matrices (see [
29]) are used in this work for mathematical modeling of the micropolar thin bodies.
3. Equations of Motion Relative to the Displacement and Rotation Vectors for an Elastic Material without a Center of Symmetry
The constitutive relations (CR) given in [
25,
30,
31] for a linearly elastic inhomogeneous anisotropic material without a center of symmetry for small displacements and rotations and isothermal processes can be written as
where
and
are the stress and couple-stress tensors,
and
are the tensors of deformation and bending-torsion,
and
are the displacement and rotation vectors,
,
and
are the material tensors of the fourth rank,
is the discriminant tensor of third rank,
is the inner two-product [
27,
29,
32,
33,
34,
35], the superscript
T in the upper right corner of the quantities denotes transposition.
Introducing the tensor columns of the deformation and bending-torsion tensors and stress and couple-stress tensors, as well as the fourth rank tensor–block matrix (TBM) of the elastic modulus tensors
the specific strain energy and the CR can be written in the form
If the material has a center of symmetry in the sense of elastic properties, then
, where
is the zero tensor of the fourth rank and the tensor–block matrix of the elastic modulus tensors (
3) will take the form of a tensor–block-diagonal matrix.
Substituting (
2) in the equations of motion for small displacements and rotations
and introducing the 2nd rank tensor–block matrix operator of the equations of motion and the vector columns of the displacement and rotation vectors and vectors of volume forces and moments
we obtain the equations of motion in displacements and rotations in the form
where the differential tensors–operators
,
,
and
have expressions
Here and below, is the unit tensor of the second rank, t is the time, and is the partial derivative operator with respect to time.
If we solve the eigenvalue problem for the tensor–block matrix (
3), then we obtain its canonical presentation [
29,
34]
where
,
, are eigenvalues for positive-definite TBM from Equation (
3). Moreover,
, and
,
, is a complete orthonormal system of eigentensor columns satisfying the orthonormality conditions [
29,
34]
Taking into account the expressions for TBM in Equation (
3), from Equation (
8) we get
It is not difficult to prove that the inverse TBM
to (
8) has the form
Generally speaking, for any integer
we have
In particular, from (
11) for
we have
where
is the unit TBM of the fourth rank relative to the operations of the inner 2-product. Given (
8), from (
4) we get
Multiplying both sides of the second relation of (
13) scalarly by
and taking into account (
9), the CR can be written in the form
Note that the formulas in Equation (
14) give equivalent records of the CR. Introducing the notations
the specific deformation energy (the first equality of (
13)) and the CR (
14) are represented in the form
Multiplying both sides of the first equality of (
15) from the left tensorly by
, and then if we sum the resulting ratio from 1 to 18, by (
12) we get
where the second formula of (
16) is obtained by analogy with the first. It should be noted that the formulas in Equation (
16) are decompositions of the tensor columns
and
with respect to the orthonormal basis
,
, and
and
are projections
and
respectively on
.
It is not difficult to see that, taking into account the first equality of (
15), from the second relation of (
13) we obtain the following presentations of stress and couple-stress tensors:
It is not difficult to write the reverse CR. In fact, for example, from the second equality of (
4) by virtue of (
10) and the second equality of (
15) we find
Taking into account the second equality of (
15), from the first equality of (
18) similarly to (
17) we have
It is not difficult to see that
can be represented as
By virtue of Equation (
19), the CR in Equation (
4) can also be written in the form
5. Formulation of Initial-Boundary Value Problems
Let us introduce the definitions.
Definition 1. If the displacement and rotation vectors (kinematic boundary conditions) are given on the body boundary S, then such conditions are called boundary conditions of the first kind, and the problem of the micropolar solids mechanics (SM), using these conditions, and also the initial conditions is called the first initial-boundary value problem.
In the case under consideration, the first initial-boundary value problem includes: the equations of motion (
6), the kinematic boundary conditions (
26) and the initial conditions (
28).
Definition 2. If static boundary conditions (stress and couple-stress vectors) are given on the body boundary S, then such boundary conditions are called boundary conditions of the second kind, and the problem of micropolar SM using them and initial conditions is called the second initial-boundary value problem.
In the case under consideration, the second initial-boundary value problem includes: the equations of motion (
6), the static boundary conditions (
25) and the initial conditions (
28).
Definition 3. If kinematic boundary conditions are given on one part of the body boundary , and on the remaining part of it are given the static boundary conditions, where , , then such boundary conditions are called mixed boundary conditions, and the problem of micropolar SM, using them and initial conditions is called the mixed initial-boundary value problem.
In this case, the mixed (third) initial-boundary value problem includes: the equations of motion (
6), the kinematic boundary conditions (
26) on one part of the body boundary and the static boundary conditions (
25) on the rest of the body boundary and the initial conditions (
28) (see also (
27)).
Note that, excluding the characteristics of the micropolar theory from the above definitions, we obtain the corresponding definitions for classical SM.
It should be noted that the kinematic boundary conditions and the initial conditions do not need to be split, since they were set in a split form. Hence, for the splitting of the first initial-boundary value problem, it is sufficient to split only the equations of motion, since, as already mentioned in the previous proposition, the kinematic boundary conditions and the initial conditions are split. In this connection, the splitting of the static boundary conditions is of great interest. If the equations of motion (
6) and the static boundary conditions (
25) can be split under some conditions, then under the same conditions all the initial-boundary value problems formulated above can be split. Hence, it is necessary to establish the conditions under which the equations of motion (
6) and the static boundary conditions (
25) are split.
6. Decomposition of the Equation of Motion in the Case of a Homogeneous Isotropic Micropolar Medium
In this case, as many authors (see, for example, [
25,
30,
31]) consider
and differential tensors–operators
,
,
and
(see (
7)) have the form
where
,
,
and
are wave operators, and the elasticity tensors have expressions
Here , and are the basic isotropic tensors of the fourth rank, and for material constants we use the notation , , , , and .
Denoting by
the tensor–block matrix operator of the cofactors for the tensor–block matrix operator
of Equation (
6), after cumbersome calculations we obtain [
27,
28,
32]
It is easy to see that by virtue of Equation (
31) the tensor–block matrix operator of the cofactors (
30) can be represented as follows:
where tensor–block matrix operators are introduced
It is easy to prove the relations
Then if we shall seek the solution of Equation (
6) in the form (similar to Galerkin)
then, by virtue of the appropriate relation (
32), we obtain the following split equations:
Note that similar equations were obtained by [
26]. Similar equations in another way were obtained by [
22]. Finally, Sandru and Nowacki gave the same representations of the displacement and rotation vectors and they are reduced to (
33). The paper [
24] deserve great attention because the system of equilibrium equations for the isotropic elastic body without a center of symmetry and in the absence of mass loads is decomposed into two independent systems of equations.
Applying the operator
to the left of (
6), by the first relation of (
32) we will have
For
(the case of a reduced medium), the classical equation follows from the first equation of (
34), and the second equation has a similar form.
7. Decomposition of Static Boundary Conditions
In the case of an isotropic micropolar material without a center of symmetry, by virtue of Equation (
29) and
, from Equation (
23) we have
It should be noted that some authors (see, for example, [
25,
30,
31]) consider that
is an asymmetric tensor, therefore in the case of an isotropic medium it is zero, as was done above. However, some authors prove that
is a symmetric tensor, therefore, in the case of an isotropic medium, it is not equal to zero, and how any isotropic fourth-rank tensor is generally determined by three parameters, as is customary in this case. Further it is easy to see that
Assuming that the body has a piecewise-plane boundary and denoting by
and
,
and
,
and
the differential tensors–operators of the cofactors and determinants for the tensor–operators
,
and
respectively, after simple calculations we obtain
Note that we want to obtain boundary conditions separately for
and
. In order to shorten the letter, we consider the case when
,
. Then
,
and the boundary conditions from Equation (
25) can be written in the form
In this case, it is easy to obtain the boundary conditions separately for
and
Note that the eigenvalue problem can also be considered for differential tensor–operator and tensor–block matrix operator. Consequently, in this case the eigenvalues are differential operators whose product, taking into account their multiplicities, will be equal to the determinant of the object for which the eigenvalue problem is considered. Of course, we can find our eigenoperators from the solution of the characteristic equation. For example, the characteristic equation for an arbitrary tensor–block matrix
of the second rank consisting of four second-rank tensors of three-dimensional space has the form [
29,
34]
where
,
, denote the invariants of the tensor–block matrix
. In this case, the inverse relations to Equation (
36) are represented in the form
Replacing in Equations (
35)–(
37)
by the tensor–block matrix operators of the equations of motion in displacements and rotations under various anisotropic media, we obtain the characteristic equations for them. If we find the roots (eigenoperators) of the obtained characteristic equations for the above tensor–block matrix operators, then their determinants can be represented as a product of simple eigenoperators. Thus, the equations are split.
8. Decomposition of the Canonical Equations of the Classical Elasticity Theory for the Transversely Isotropic Body
Under the canonical equations we call equations that are obtained by the canonical presentations of the material tensors. In a similar sense, the term canonical mechanics can also be used. Thus, in this case the elastic tensor
is represented in the canonical form [
29,
34], that is
where
and
,
, are the eigenvalues and eigentensors for
. Then by Equation (
38) the vector equation with respect to the displacement vector can be written in the form
It is seen that for the splitting of the last equation it is necessary to find the cofactors
for
(see the second relation of (
39)), and for this, in turn it is necessary to find expressions for the determinant of a linear combination of several tensors of the second rank (from two to six). We give below the expressions for the determinant and the cofactors of the sum of six second-rank tensors without proofs, from which, in turn, it is easy to obtain analogous expressions for the sum of a smaller number of tensors. The expressions for the determinant and the cofactor tensor of the sum of six tensors of the second rank are presented as follows:
Given
, we obtain from the previous formula
Here and , , are tensors of cofactors of second rank and second order and fourth rank and first order, respectively.
Further, before we consider the canonical presentations of tensors, we introduce the following definition.
Definition 4. The symbol , where k is the number of different eigenvalues of the tensor, and is the multiplicity of the eigenvalue , is called the anisotropy symbol (structure symbol) of the tensor.
We note that on the basis of this definition, the classification of classical and microcontinuum anisotropic materials is given in [
29,
34]. By virtue of this classification, classical (micropolar) isotropic materials are special cases of materials in which the anisotropy symbol consists of not more than two (three) elements. A similar situation occurs for other anisotropies [
29,
34]. In particular, classical transversely isotropic materials are special cases of anisotropic materials whose anisotropy symbols consist of four elements, and orthotropic materials are special cases of anisotropic media whose structure symbols consist of no more than 6 elements. The anisotropy symbol of an orthotropic micropolar material with a symmetry center consists of not more than nine elements [
29,
34]. It should also be noted that the mathematical structure of elastic media was investigated in [
13], and the classification of classical elastic media is given in [
9].
We now consider the canonical representation of the transversely isotropic elastic modulus tensor with the anisotropy symbol {1,1,2,2} [
29,
34]:
(by force of the classification of materials adopted in [
29,
34], transversely isotropic materials can be of the following types: {1,1,2,2}, {1,2,1,2}, {1,2,2,1}, {2,1,1,2}, {2,1,2,1}, {2,2,1,1}). For the material under consideration, the eigenvalues are determined by the formulas [
29,
34]
and the eigentensors are represented in the form
The canonical representations of the tensor–operator of the equations and its determinant and the tensor of cofactors and its components, as well as the stress tensor–operator, its determinant and the components of the tensor–operator of cofactors by virtue of Equations (
40)–(
42) have the form
where the following notations are introduced:
If we apply the operator
from the left with a single multiplication to the equation (the first relation in Equation (
39)), and the operator
(see the corresponding relations from Equation (
43)) to the boundary conditions
, then we obtain the following split equations and boundary conditions:
and if we look for the solution
in the form
, then we have the following split equations and boundary conditions:
When obtaining Equations (
44) and (
45), we took into account the relation
Further, applying, for example, to the split equations from (
45), the
k-th moment operator [
27,
32], the equations for prismatic bodies in moments with respect to any system of orthogonal polynomials can be represented in the form
where in the application of the system of Legendre polynomials, the expressions for
,
and
are defined using the following relationship:
Note that analogous to Equation (
46) equations with respect to
can be easily obtained from Equation (
44). In order to shorten the letter, we will not write them out. By virtue of the first relation of (
43), it is not difficult to calculate that
Note that the dynamic equations (equilibrium equations) in displacements for any homogeneous anisotropic body can be written in the form
Consequently, the decomposed equations will have the form
Next, we represent
(see the first equality of Equation (
49))) as a product of simple operators
From this we see that , and are eigenoperators for , and , and are the eigenoperators for .
Note that, knowing the expressions for
,
(the case of a transversally isotropic medium, see Equation (
48)), the simple (linear) operators
,
can be found using Equation (
50).
Based on the above, it is not difficult to consider other cases of anisotropic media, as well as cases of classical and micropolar isotropic materials, but in order to reduce the writing we will not dwell on them.
10. The Quasistatic Canonical Problem of the Micropolar Theory of Prismatic Bodies of Constant Thickness in Displacements and Rotations and in the Moments of Displacement and Rotation Vectors
Let us consider a prismatic body of constant thickness
. As the base plane, we take the middle plane. Then in this case
,
,
and the nabla-operator
, the Laplacian
,
and
are represented in the form
By the corresponding formula (
55), Equation (
52) for the theory of prismatic bodies of constant thickness in displacements and rotations can be written in the form
Applying the
k-th moment operator of some system of orthogonal polynomials (Legendre, Tchebyshev) to the equations in (
56), we find the following equations for the micropolar theory of prismatic bodies of constant thickness in the moments of the displacement and rotation vectors:
Having Equation (
57), by the formula (
47) it is easy to obtain systems of equations of any approximation in moments with respect to the system of Legendre polynomials. We note that the equations of the fifth (in the classical case) and the 8th (in the micropolar case) approximations in moments were obtained in the papers [
27,
28,
32] for isotropic material in the traditional form, as well as the similarly to Equations (
52), (
54), (
56) and (
57) in the traditional form are given in [
27,
28,
32]. Equations in moments for thin bodies with two small dimensions and thin multi-layered structures are also given there.
Adding the corresponding canonical boundary conditions to Equation (
57), we obtain a canonical statement of quasistatic boundary value problems for prismatic bodies. In order to shorten the letter, we shall not dwell on this in this paper, but refer to the interested reader in the papers [
27,
32], in which the formulations of boundary-value problems in the traditional form are given in detail, and they easily extend to canonical statements.
It should be noted that for the theory of thin bodies, the decomposed equations at equilibrium, depending on the order of approximation, are equations of elliptical type of high order [
27,
28,
32] and using the Vekua method [
36], for them it is possible to write out analytical solutions. Note also that some questions about the application of problems on eigenvalues of tensor objects are set out in the works [
37,
38,
39,
40].
Note that using canonical representations of material objects, based on differential statements of initial-boundary value problems, it is easy to obtain variational statements, and then the application of this method can be generalized to the case of media considered, for example, in [
41,
42,
43,
44].