On the Neuber theory of micropolar elasticity. formulation

The present paper deals with a pseudotensor formulation of the Neuber theory of micropolar elasticity. The dynamic equations of the micropolar continuum in terms of relative tensors (pseudotensors) are presented and discussed. The constitutive equations for a linear isotropic micropolar solid is given in the pseudotensor form. The final forms of the dynamic equations for the isotropic micropolar continuum in terms of displacements and microrotations are obtained in terms of relative tensors. The refinements of Neuber’s dynamic equations are discussed. Those are also considered in the cylindrical coordinate net.


Preliminary remarks
The classical theories of continuum mechanics often is not acceptable for mathematical modeling of the modern materials behavior (for example, elastic metamaterials [1,2] or biomaterials [3]: among them sands, soils and other granular elastic media, even perfectly plastic media exhibiting irreversible volume transformation (for instance, the Coulomb-Mohr media), fibrous media, honeycomb structures, reinforced composite materials, bones, vessels, muscles, and other tissues). In those materials the waves of microtations coupled to the displacements waves are observed due to the microstructure effects. Moreover, mirror modes of propagating waves in hemitropic media are caused by a physical mechanism manifested in the hemitropic elastic equations as their sensitivity to mirror reflections and 3D-space inversions.
A first variant of asymmetric elasticity theory was developed by the Cosserats brothers (1909) in the pioneering work [4]. Further consideration of the micropolar theory for finite deformation have been carried out by Truesdell and Toupin [5]. The Aero and Kuvshinskii derived linear constitutive equation of micropolar continuum in [6]. The material anisotropy of micropolar media has been considered and discussed in the Mindlin studies (see for example [7,8]). The problem of stress concentration is the subject of the Neuber papers [9][10][11]. An extension of micropolar theory to the hemitropic case can be found in [12,13].
In the general case of micropolar anisotropy the elastic material is specified by the 171 constitutive constant, which extremely complicates the equations analysis while solving applied problems. A semi-isotropic (hemitropic) solid is determining by nine constitutive constants of which only three new dimensionless ones if compared to the isotropic case. Literary search shows that papers devoted to micropolar theory often contain errors and misprints in the fundamental equations (see for example [10,14]) making them difficult to understand.
Another important issue in mathematical modelling of micropolar material behaviour is a deficiency of relative tensors technique [5,[15][16][17][18][19][20][21][22][23][24][25], since micropolar characteristics actually are relative tensors. The relative tensors notation provides a deep insight to the physical and geometric nature of considered physical fields. Nonetheless, relative tensors notation in the continuum mechanics is not widespread. The most recent relative tensor formulation of hemitropic micropolar continuum in application to growing solid mechanics is discussed in [26].
The present paper is arranged as follows. The second section is devoted to a number of fundamental definitions from relative tensors algebra. The covariant derivative of an arbitrary relative tensor is considered.
In Sec. 3 Neuber's dynamic equations are derived in terms of relative tensors. The constitutive equations for linear isotropic micropolar continuum are furnished by pseudotensors notation. The weights of relative tensors of linear micropolar elastic medium and constitutive scalars are verified and presented by tables 1 and 2. The final form of Neuber's dynamic equations in an arbitrary curvilinear coordinate system is obtained. The misprints in Neuber's dynamic equations known from [10] are corrected.
The Sec. deals with a formulation of dynamic equations in cylindrical coordinate net. The obtained equations are of crucial importance for investigating wave propagation in long cylindrical waveguides.
The final section contains concluding remarks.

Relative tensors algebra and covariant differentiation
The permutation symbols and the fundamental orienting pseudoscalar are fundamental objects of relative tensor theory. This theory is a subject of many discussions found in multidimensional geometry tutorials and tensor analysis books [5,[15][16][17][18][19][20][21][22][23][24][25]27]. A re-orientation of a coordinate frame (left-handed into into right-handed or vice versa) can be afforded by re-enumeration of coordinate axes, thus allowing to introduce the fundamental object of relative tensor algebra and multidimensional geometry -the Levi-Civita permutation symbols [18]. It is well known, that the permutation symbols determined according to 0 in all other cases; are not absolute tensors. In fact, permutation symbols and [+1] are the relative tensors (pseudotensors) of the weight −1 (w.g.t. = −1) and at the same time -relative contravariant tensors of the weight +1 (w.g.t. = +1). Hereinafter, position above a root symbol is reserved for weight of a relative tensor which is additionally embraced by square brackets.
We proceed to discussion of an orienting pseudoscalar (relative scalar of weight +1 (w.g.t. = +1)), defined by the sequential application of inner and cross products to the covariant basis vectors: (1) and the relative scalar of the negative weight −1 (w.g.t. = −1): Note that the pseudoscalar (1) is related to the parallelepiped volume built on the vectors . In further considerations, we will omit the weight indication for fundamental symbols such as , , and also true for zero weight relative tensors. Here once again, we emphasize that > 0 for a right-handed coordinate system, < 0 for a left-handed coordinate system.

Reminder and refinement of Neuber's micropolar elasticity theory
The applying relative tensors formalism to the Neuber's theory allows to clarify its physical sense. The dynamic equations in terms of relative tensors can be presented in contrary to [9] ∇ = · · , In the latter equation contrary to the Neuber theory we use despite of weights unbalance.
The equations (3) in a curvilinear coordinate net can be rearranged due to (2) as follows Linear isotropic micropolar elastic constitutive equations [9] in terms of rela- ].

(5)
In the above formulae the constitutive scalars and pseudoscalars are denoted by: is the shear modulus of elasticity; -is the Poisson ratio; is the micropolar characteristic length; , , are dimensionless constitutive scalars. Upon substituting constitutive equations (5) in equations (3) the Neuber dynamic equations in terms of relative tenors read by Note that in the original paper [9] the multiplier 2 is omitted in the second term of the second equation of the system (6).
Dynamic equations of linear micropolar elasticity by notation introduced in [13,26] in relative tensors are represented by Comparison of equations (6) and (7) leads to the relations between the micropolar constitutive constants in the form (1 + ), The weights of the Neuber constitutive scalars and pseudoscalars , , , are given by table 2. The weight of constitutive scalar in table 2 is verified by formula On the Neuber theory of micropolar elasticity. A pseudotensor formulation

Neuber's dynamic equations in cylindrical coordinates
We proceed the paper to consideration of Neuber's dynamic equations in cylindrical coordinate net ( , , ). It is convenient to assume that the reference plane of the former is the Cartesian -plane (with equation = 0), and the cylindrical axis is the Cartesian -axis. Then the -coordinate is the same in both systems. The transformation formulae between cylindrical ( , , ) and Cartesian coordinates ( , , ) can be furnished by The nonzero components of metric tensor and Christoffel symbols are determined by Thus, the dynamic equations (4) can be presented in following form , .
The obtained equations (8) are of crucial importance for investigating waves propagation in long cylindrical waveguides. Competing interests. We declare that we have no competing interests.

Conclusions
Author's Responsibilities. We take full responsibility for submitting the final manuscript in print. We approved the final version of the manuscript.
Funding. This study was in part financially supported by the Ministry of Science and Higher Education of the Russian Federation (State Registration Number AAAA-A20-120011690132-4) and by the Russian Foundation for Basic Research (projects nos. 18-01-00844, 20-01-00666).