On the Neuber theory of micropolar elasticity. A pseudotensor formulation

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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2020, vol. 24, no. 4, pp. 752-761 ISSN: 2310-7081 (online), 1991-8615 (print) https://doi.org/10.14498/vsgtu1799 Short Communications MSC: 74A20, 74A35, 74A60 On the Neuber theory of micropolar elasticity. A pseudotensor formulation © V. A. Kovalev1 , E. V. Murashkin2 , Y. N. Radayev2 1 Moscow City Government University of Management, 28, Sretenka st., Moscow, 107045, Russian Federation. 2 A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences, 101, pr. Vernadskogo, Moscow, 119526, Russian Federation. Abstract 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 mi- crorotations 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. Keywords: micropolarity, elasticity, continuum, microrotation, pseudoscalar, relative tensor, weight, constitutive equation. Received: 16th July, 2020 / Revised: 17th October, 2020 / Accepted: 16th November, 2020 / First online: 22nd November, 2020 Short Communication cb The content is published under the terms of the Creative Commons Attribution 4.0 In- ternational License (http://creativecommons.org/licenses/by/4.0/) Please cite this paper in press as: K o v a l e v V. A., M u r a s h k i n E. V. , R a d a y e v Y. N. On the Neuber theory of micropo- lar elasticity. A pseudotensor formulation, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2020, vol. 24, no. 4, pp. 752-761. https://doi.org/10.14498/vsgtu1799. Authors' Details: Vladimir A. Kovalev https://orcid.org/0000-0003-2991-9531 D.Sc. (Phys. & Math. Sci.), Ph.D., M.Sc., Professor; Professor; Dept. of Financial Management and Financial Law; e-mail: vlad_koval@mail.ru Evgenii V. Murashkin https://orcid.org/0000-0002-3267-4742 Cand. Phys. & Math. Sci., PhD, MD; Senior Researcher; Lab. of Modeling in Solid Mechanics; e-mail: evmurashkin@google.com Yuri N. Radayev https://orcid.org/0000-0002-0866-2151 D.Sc. (Phys. & Math. Sci.), Ph.D., M.Sc., Professor; Leading Researcher; Lab. of Modeling in Solid Mechanics; e-mail: radayev@ipmnet.ru, y.radayev@gmail.com 752 © Samara State Technical University On the Neuber theory of micropolar elasticity. A pseudotensor formulation 1. Preliminary remarks The classical theories of continuum mechanics often is not acceptable for math- ematical modeling of the modern materials behavior (for example, elastic meta- materials [1, 2] or biomaterials [3]: among them sands, soils and other granular elastic media, even perfectly plastic media exhibiting irreversible volume trans- formation (for instance, the Coulomb-Mohr media), fibrous media, honeycomb structures, reinforced composite materials, bones, vessels, muscles, and other tis- sues). 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 man- ifested 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 con- tinuum 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 [9-11]. 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 determin- ing by nine constitutive constants of which only three new dimensionless ones if compared to the isotropic case. Literary search shows that papers devoted to mi- cropolar 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, 15-25], since micropolar characteristics actually are relative tensors. The relative tensors notation pro- vides 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 curvilin- ear 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 coordi- nate net. The obtained equations are of crucial importance for investigating wave propagation in long cylindrical waveguides. The final section contains concluding remarks. 753 K o v a l e v V. A., M u r a s h k i n E. V. , R a d a y e v Y. N. 2. 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, 15-25, 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 [-1] = [+1] = +1 for triplets (1, 2, 3), (2, 3, 1), (3, 1, 2); -1 for triplets (3, 2, 1), (1, 3, 2), (2, 1, 3); 0 in all other cases; are not absolute tensors. In fact, permutation symbols [-1] 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 prod- ucts to the covariant basis vectors: [+1] = 1 · ( 2 3 ) (1) and the relative scalar of the negative weight -1 (w.g.t. = -1): [-1] -1 = 1 · ( 2 3 ). 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. In general, the transformation formula for a relative tensor of weight reads by [15-17] ··· ··· = ( )( ) · · · ( )( )( ) · · · ( )··· ··· , where = det( ), = , = . Here, an overlined symbol should be considered as related to new coordinates ( = 1, 2, 3), denotes the transformation Jacobian. Covariant derivative of the relative tensor ··· ··· of a given weight is sim- ilarly defined by the corresponding derivative for an absolute tensor [15, 17, 21]: 754 On the Neuber theory of micropolar elasticity. A pseudotensor formulation [] ··· ··· = [] ··· ··· + [] ··· ··· + · · · + [] ··· ··· - - [] ··· ··· - · · · - [] ··· ··· - [] ··· ··· . (2) 3. 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] = ·· , [-1] + = - [-2] ·· [+1] . (3) 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 + + = ·· , [-1] + [-1] + [-1] + [-1] + = - [-2] ·· [+1] . (4) Linear isotropic micropolar elastic constitutive equations [9] in terms of rela- Table 1 Relative tensors of the isotropic micropolar elasticity Standard terminology Notation adopted in [26] Neuber's notation Weight Transformation to absolute tensor displacements vector 0 asymmetric strain tensor 0 force stress tensor 0 mass density 0 couple stress tensor · · -1 = [-1] microinertia I -2 = 2 [-2] microrotation vector +1 = 1 [+1] wryness tensor · · +1 = 1 [+1] 755 K o v a l e v V. A., M u r a s h k i n E. V. , R a d a y e v Y. N. tive tensors are furnished by = [(1 + 2[-2] ) + (1 - 2[-2] ) + 2 [-2] [+1] + + 2(1 - 2)-1 ], [-1] = 4 [-1] [-1] [[+1] + [+1] + [+1] ]. (5) In the above formulae the constitutive scalars and pseudoscalars are denoted by: - is the shear modulus of elasticity; - is the Poisson ratio; - [-1] is the micropolar characteristic length; - [-2] , , are dimensionless constitutive scalars. Upon substituting constitutive equations (5) in equations (3) the Neuber dy- namic equations in terms of relative tenors read by (1 + 2[-2] ) + ((1 - 2)-1 - 2[-2] ) + + 2 [-2] [+1] = -1 ·· , ( [-2] - [-1] [-1] ) [+1] - [-1] [-1] ( + ) [+1] - - [-2] 2 = - [-2] (4)-1 ·· [+1] . (6) 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 (1 + 2[-2] 1) + (1 - 2[-2] 1 + 2(1 - 2)-1 ) + + 2 [-2] 1 [+1] = -1 ·· , [-1] [-1] [(1 + -2[+2] 2) [+1] + (1 - -2[+2] 2 + 23) [+1] ]- - 2 [-2] 1(2 [+1] - ) = [-2] I -1 ·· [+1] . (7) Comparison of equations (6) and (7) leads to the relations between the mi- cropolar constitutive constants in the form [-1] [-1] = 2 [-1] [-1] (1 + ), [-2] 1 = [-2] , [+2] 2 = 1 - 1 + , 3 = 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 = 2 - [+2] 2 2 + [+2] 2 . 756 On the Neuber theory of micropolar elasticity. A pseudotensor formulation Table 2 Micropolar constitutive scalars of Neuber's theory Standard terminology Root notation Weight Transformation to absolute tensor shear modulus of elasticity 0 the Poisson ratio 0 micropolar characteristic length -1 = [-1] dimensionless micropolar modulus i -2 = 2 [-2] 1 dimensionless micropolar modulus ii 0 dimensionless micropolar modulus iii 0 4. Neuber's dynamic equations in cylindrical coordinates We proceed the paper to consideration of Neuber's dynamic equations in cylin- drical coordinate net (, , ). It is convenient to assume that the reference plane of the former is the Cartesian -plane (with equation = 0), and the cylin- drical 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 = cos , = sin , = . The nonzero components of metric tensor and Christoffel symbols are deter- mined by 11 = 1, 22 = 2 , 33 = 1, 1 22 = -, 2 12 = 2 21 = -1 . Thus, the dynamic equations (4) can be presented in following form () + + - = ·· , () + () + + = ·· , () + + () = ·· , ( [-1] ) - [-1] + [-1] + [-1] + [-1] + ( - ) = [-2] ·· [+1] , [-1] + ( [-1] )+ [-1] + [-1] + -1[-1] + ( - )= [-2] ·· [+1] , ( [-1] ) + [-1] + [-1] + [-1] + ( - ) = [-2] ·· [+1] . (8) The obtained equations (8) are of crucial importance for investigating waves propagation in long cylindrical waveguides. 757 K o v a l e v V. A., M u r a s h k i n E. V. , R a d a y e v Y. N. 5. Conclusions (i) The Neuber dynamic equations of the linear micropolar continuum in terms of relative tensors (pseudotensors) in an arbitrary curvilinear coordinate system are presented and discussed. The misprints in original Neuber's dy- namic equations are eliminated. (ii) The constitutive equations for linear isotropic micropolar continuum are furnished by pseudotensors notation. (iii) The weights of relative tensors of linear micropolar elastic medium and the Neuber constitutive scalars are verified and given by tables 1 and 2. (iv) The final forms of the dynamic equations for the isotropic micropolar con- tinuum in terms of displacements and microrotations are obtained in terms of relative tensors. (v) The refinements of the final form of Neuber's dynamic equations are dis- cussed. Competing interests. We declare that we have no competing interests. Author's Responsibilities. We take full responsibility for submitting the final manu- script in print. We approved the final version of the manuscript. Funding. 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Аннотация Рассматривается псевдотензорная формулировка теории микропо- лярной упругости Нейбера. Приведены и обсуждаются динамические уравнения микрополярного континуума в терминах относительных тен- зоров (псевдотензоров). Даны определяющие уравнения для линейно- го изотропного микрополярного твердого тела. Окончательные формы динамических уравнений для изотропного микрополярного континуума в терминах смещений и микровращений получены в терминах относи- тельных тензоров. Устранены недочеты в окончательной форме динами- ческих уравнений Нейбера. Получены динамические уравнения Нейбера в цилиндрической системе координат. Ключевые слова: микрополярность, упругость, континуум, микровра- щение, псевдоскаляр, относительный тензор, вес, определяющее уравне- ние. Получение: 16 июля 2020 г. / Исправление: 17 октября 2020 г. / Принятие: 16 ноября 2020 г. / Публикация онлайн: 22 ноября 2020 г. Краткое сообщение cb Контент публикуется на условиях лицензии Creative Commons Attribution 4.0 International (https://creativecommons.org/licenses/by/4.0/deed.ru) Образец для цитирования K o v a l e v V. A., M u r a s h k i n E. V. , R a d a y e v Y. N. On the Neuber theory of micropo- lar elasticity. A pseudotensor formulation, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2020, vol. 24, no. 4, pp. 752-761. https://doi.org/10.14498/vsgtu1799. Сведения об авторах Владимир Александрович Ковалев https://orcid.org/0000-0003-2991-9531 д.ф.-м.н.; профессор; каф. финансового менеджмента и финансового права; e-mail: vlad_koval@mail.ru Евгений Валерьевич Мурашкин https://orcid.org/0000-0002-3267-4742 к.ф.-м.н.; старший научный сотрудник; лаб. моделирования в механике деформируемого твердого тела; e-mail: evmurashkin@google.com Юрий Николаевич Радаев https://orcid.org/0000-0002-0866-2151 доктор физико-математических наук, профессор; ведущий научный сотрудник; лаб. моделирования в механике деформируемого твердого тела; e-mail: radayev@ipmnet.ru, y.radayev@gmail.com 760 © Самарский государственный технический университет К теории микрополярной упругости Нейбера. Псевдотензорная формулировка Конкурирующие интересы. Заявляем, что в отношении авторства и публикации этой статьи конфликта интересов не имеем. Авторский вклад и ответственность. Все авторы принимали участие в разра- ботке концепции статьи и в написании рукописи. Авторы несут полную ответствен- ность за предоставление окончательной рукописи в печать. Окончательная версия рукописи была одобрена всеми авторами. Финансирование. Работа выполнена в рамках государственного задания ( госре- гистрации АААА-А20-120011690132-4) и при поддержке Российского фонда фун- даментальных исследований (проекты 18-51-00844, 20-01-00666). Благодарности. Авторы благодарят рецензента за внимательное прочтение статьи, ценные предложения и комментарии. 761

About the authors

Vladimir Aleksandrovich Kovalev

Moscow City Government University of Management Moscow

Email: vlad_koval@mail.ru, kovalev.kam@gmail.com
Doctor of physico-mathematical sciences, Professor

Eugenii Valeryevich Murashkin

Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences

Email: murashkin@dvo.ru, murashkin@ipmnet.ru, evmurashkin@gmail.com
Candidate of physico-mathematical sciences, no status

Yuri Nikolaevich Radayev

Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences

Email: y.radayev@gmail.com, radayev@ipmnet.ru
Doctor of physico-mathematical sciences, Professor

References

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