Higher order theory of micropolar plates and shells

Abstract

New higher order models of micropolar plates and shells have been developed here. The 3‐D dynamic equations of the micropolar elasticity have been presented in an orthogonal system of coordinates using a generalized variational principle. For the creation of 2‐D models of plates and shells the curvilinear system of coordinates related to the middle surface of the shell have been used along with a special hypothesis based on assumptions that take into account the fact that the rod is thin. Higher order theory is based on a generalized variational principle and the expansion of the 3‐D equations of the micropolar theory of elasticity into Fourier series in terms of Legendre polynomials. The stress and strain tensors, as well as vectors of displacements and rotation have been expanded into Fourier series in terms of Legendre polynomials with respect to thickness. Thereby, all equations of the micropolar theory of elasticity (including generalized Hooke's law) have been transformed to the corresponding equations for the Legendre polynomials coefficients. Then, in the same way as in the classical theory of elasticity, a system of differential equations in terms of displacements and rotation with initial and boundary conditions for the Legendre polynomials coefficients have been obtained. All equations for higher order theory of micropolar plates in Cartesian and polar coordinates as well as for cylindrical and spherical shells in coordinates related to the shells geometry have been developed and presented here in detail. The obtained equations can be used for calculating the stress‐strain and for modelling thin walled structures in macro, micro and nano scale when taking into account micropolar couple stress and rotation effects.

Two-scale micropolar plate model for web-core sandwich panels

“…The works of Eringen (2012), Altenbach and Eremeyev (2009) and Altenbach et al (2010) discuss the key theoretical aspects of micropolar plate theories and the literature reviews therein encompass a lengthy period of research on such plates. Of the more recent papers on micropolar plates we mention those of Kvasov and Steinberg (2013); Sargsyan and Sargsyan (2014); Abadikhah and Folkow (2015); Ansari et al (2017b) and Zozulya (2018). However, the aforementioned models are not applicable in the present context as their constitutive equations do not address any particular microstructure such as a webcore.…”

Higher order couple stress theory of plates and shells

“…The mechanics of generalized continua has a long history of development . Since the publication of the landmark book of the Cosserat brothers there have been various known generalizations and applications of the classical or Cauchy continuum, which are summarized in ma number of books and papers . More specially we have to mention remarkable books of Eringen and Nowacki as well as fundamental treatise of Truesdell and Toupin .…”

“…Another approach consists in exploring the classical physical hypothesis and a more advanced hypothesis which permits the developments of theories that are more accurate and consider the effect that classical theories ignore . The model of micropolar rods, plates and shells based on the classical Euler‐Bernoulli Timoshenko's and more advanced hypothesis has been considered in [].…”

“…In our previous publications the approach based on the use of Legendre's polynomials series expansion has been applied to the development of the high order models of rods, plates and shells. Thermoelastic contact problems of plates and shells when mechanical and thermal conditions are changed during deformation have been considered in [].…”

“…Then, the proposed approach and methodology were further developed and applied to the thermoelasticity of the laminated composite materials with the possibility of delamination along with mechanical and thermal contact in the temperature field in [], the pencil‐thin nuclear fuel rods modeling in [], the functionally graded shells in [] and the modeling of MEMS and NEMS in []. Development of the high order theories and their comparison with Euler‐Bernoulli and Timoshenko's theories for micropolar, couple stress and nonlocal curved elastic rods have been done in [], respectively. An analysis and comparison with the classical theory of elastic and thermoelastic plates and shells has been done in [].…”