Micropolar curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

Abstract

New models for micropolar plane curved rods have been developed. 2-D theory is developed from general 2-D equations of linear micropolar elasticity using a special curvilinear system of coordinates related to the middle line of the rod and 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 and 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, system of differential equations in term of displacements and boundary conditions for Fourier coefficients have been obtained. The Timoshenko's and EulerBernoulli theories are based on the classical hypothesis and 2-D equations of linear micropolar 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 scale when taking into account micropolar couple stress and rotation effects.

Micropolar modeling approach for periodic sandwich beams

“…If we consider only the homogeneous case (q = 0), the solution to the uncoupled ordinary differential equation (A.4) is given by Eq. (33). To continue, we introduce two additional variables that have also been used for couple-stress beams [18,41] γ ≡ u y + φ and ω ≡ u y − φ.…”

“…We show that a couple-stress Timoshenko beam may provide too stiff results for sandwich beams due to the inherent rotational constraint. We note that micropolar Timoshenko beam theories have been developed by several authors in recent years [28][29][30][31][32][33]. In light of this, the main novel features of the current study are that we derive the explicit general solution to the equilibrium equations of the micropolar Timoshenko beam; use the solution to develop a nodally-exact micropolar Timoshenko beam finite element (FE) and, finally, we apply the beam model and the finite elements to practical sandwich beam problems with the micropolar ESL stiffness parameters determined through the unit cell analysis of a web-core sandwich beam.…”

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

“…Analysis of the systems of partial differential equations (54), (55) for first order theory, (66), (67) for Timoshenko's theory and (81), (82) for Euler-Bernoulli theory show that all of them are coupled and related longitudinal, flexural and rotational deformation modes. The first order approximation theory is more complete and all quantities are approximated by linear functions.…”

“…In most publications the considered models are based on Euler-Bernoulli and Timoshenko's hypothesis. Among the many articles on couple stress theories of beams based on Euler-Bernoulli hypothesis we mention here [43][44][45][46] and on Timoshenko's hypothesis we mention here [46][47][48][49][50][51][52], curved rods along with others have been considered in [4,16,42,[53][54][55], for more information see the extensive review paper [56].…”

“…Thermoelastic contact problems of plates and shells when mechanical and thermal conditions are changed during deformation have been considered in [66,67]. Then, the proposed approach and methodology were further developed and extended to thermoelasticity of the laminated composite materials with the possibility of delamination along with mechanical and thermal contact in the temperature field in [68], the pencil-thin nuclear fuel rods modeling in [69], modeling of MEMS and NEMS in [73,74], the functionally graded shells in [70,75] as well as micropolar curved elastic rods in [55]. Analysis and comparison with classical theory of elastic and thermoelastic plates and shells has been done in [71,72].…”