Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

Abstract

New models for plane curved rods based on linear couple stress theory of elasticity have been developed. 2-D theory is developed from general 2-D equations of linear couple stress elasticity using a special curvilinear system of coordinates related to the middle line of the rod as well as special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements and rotation along with body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby, all equations of elasticity including Hooke's law have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of elasticity, a system of differential equations in terms of displacements and boundary conditions for Fourier coefficients have been obtained. Timoshenko's and EulerBernoulli theories are based on the classical hypothesis and the 2-D equations of linear couple stress theory of elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scales when taking into account couple stress and rotation effects.

Higher order couple stress theory of plates and shells

“…By considering the symmetry relations required for an isotropic linear elastic material the quadratic positive definite stored energy density function can be presented in the form where, λ and μ are Lamé constants of classical elasticity, η is an additional elastic constant that accounts for couple stress effects in an isotropic material. In the case of thin beams and rods Lamé constants should be corrected as it is shown in [] to take into account Poisson effect.…”

“…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 [].…”