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

• Volume: 4, Issue: 1, page 119-133
• ISSN: 2353-7396

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## Abstract

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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 Euler-Bernoulli 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.

## How to cite

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V.V. Zozulya. "Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models." Curved and Layered Structures 4.1 (2017): 119-133. <http://eudml.org/doc/288439>.

@article{V2017,
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 Euler-Bernoulli 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.},
author = {V.V. Zozulya},
journal = {Curved and Layered Structures},
keywords = {Curved rod; couple stress; Legendre polynomial;Timoshenko theory; Euler-Bernoulli theory; high order theory},
language = {eng},
number = {1},
pages = {119-133},
title = {Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models},
url = {http://eudml.org/doc/288439},
volume = {4},
year = {2017},
}

TY - JOUR
AU - V.V. Zozulya
TI - Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models
JO - Curved and Layered Structures
PY - 2017
VL - 4
IS - 1
SP - 119
EP - 133
AB - 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 Euler-Bernoulli 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.
LA - eng
KW - Curved rod; couple stress; Legendre polynomial;Timoshenko theory; Euler-Bernoulli theory; high order theory
UR - http://eudml.org/doc/288439
ER -

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