Closed-form solution for the micropolar plates: Carrera unified formulation (CUF) approach
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O R I G I NA L
E. Carrera · V. V. Zozulya
Closed-form solution for the micropolar plates: Carrera unified formulation (CUF) approach
Received: 14 July 2020 / Accepted: 18 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Navier’s closed-form solution for the higher order theory of micropolar plates based on the CUF approach has been developed here. Obtained using a principle of virtual displacements the 2D system of the differential equations for the higher order theory of micropolar elastic plates is solved here for the case of simply supported plates using Navier’s method of the variables separation. For the higher order theory of micropolar plates developed here, which is based on CUF, some numerical examples have been done and the influence the rotation field on the stress–strain fields has been analyzed. Methods of determination of the classical and micropolar elastic moduli of different materials have been analyzed, and available experimental data have been presented in Nowacki’s notations. The obtained equations can be used for calculating the stress–strain and for modeling thin walled structures in macro-, micro- and nanoscale when taking into account micropolar couple stress and rotation effects. Keywords Plates · CUF · Micropolar · Series expansion · Higher order theory
1 Introduction In contrast to the classical theory of elasticity, micropolar theory considers additional rotational degrees of freedom of the material particles, which are considered as small rigid bodies. Interactions between adjoined particles occur in terms of the classical force stress tensor and the micropolar stress tensor which originate due to the rotation of particles. As a result, the micropolar theory of elasticity considers the length-scale effect, which is thought to be microstructure-dependent. Considering the microstructure of a material is very important in modeling devices and structures made of heterogeneous material especially at micro- and nanoscale, for example MEMS and NEMS, see Altenbach and Eremeyev [1], McFarland and Colton [51], Waseem et al. [63] and Yoder et al. [66] and in biomechanics for bone modeling, see Eremeyev et al. [28,29] and Fatemi et al. [33]. Since the publication of the Cosserat brother’s landmark book [22], there is a lot more books and papers dedicated to various aspects of the theory of micropolar continua and its applications. Among many other, the noteworthy books of Eringen [32] and Nowacki [52] as well as the recently published book of Eremeyev et al. [27] are mentioned here. For more information between others, one can also refer to our previous publications Carrera and Zozulya [18,19]. E. Carrera Department of Aeronautics and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24 10129 Turin, Italy E-mail: [email protected] V. V. Zozulya (B) Centro de Investigacion Cientifica de Yucatan, A.C., Calle 43, No 130,Colonia: Chuburna de Hidalgo, C.P. 97200 Mérida, Yucatan, Mexico E-mail: [email protected]
E. Carrera, V. V. Zozulya
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