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

• Volume: 4, Issue: 1, page 221-236
• ISSN: 2353-7396

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## Abstract

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New models for plane curved rods based on linear nonlocal theory of elasticity have been developed. The 2-D theory is developed from general 2-D equations of linear nonlocal elasticity using a special curvilinear system of coordinates related to the middle line of the rod along with 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 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 nonlocal constitutive relations have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of local elasticity, a system of differential equations in terms of displacements for Fourier coefficients has been obtained. First and second order approximations have been considered in detail. Timoshenko’s and Euler-Bernoulli theories are based on the classical hypothesis and the 2-D equations of linear nonlocal theory of elasticity which are considered in a special curvilinear system of coordinates related to the middle line of the rod. The obtained equations can be used to calculate stress-strain and to model thin walled structures in micro- and nanoscales when taking into account size dependent and nonlocal effects.

## How to cite

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V.V. Zozulya. "Nonlocal theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models." Curved and Layered Structures 4.1 (2017): 221-236. <http://eudml.org/doc/288509>.

@article{V2017,
abstract = {New models for plane curved rods based on linear nonlocal theory of elasticity have been developed. The 2-D theory is developed from general 2-D equations of linear nonlocal elasticity using a special curvilinear system of coordinates related to the middle line of the rod along with 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 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 nonlocal constitutive relations have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of local elasticity, a system of differential equations in terms of displacements for Fourier coefficients has been obtained. First and second order approximations have been considered in detail. Timoshenko’s and Euler-Bernoulli theories are based on the classical hypothesis and the 2-D equations of linear nonlocal theory of elasticity which are considered in a special curvilinear system of coordinates related to the middle line of the rod. The obtained equations can be used to calculate stress-strain and to model thin walled structures in micro- and nanoscales when taking into account size dependent and nonlocal effects.},
author = {V.V. Zozulya},
journal = {Curved and Layered Structures},
keywords = {Curved rod; nonlocal elasticity; Legendre polynomial; high order theory; Timoshenko’s theory; Euler- Bernoulli theory},
language = {eng},
number = {1},
pages = {221-236},
title = {Nonlocal theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models},
url = {http://eudml.org/doc/288509},
volume = {4},
year = {2017},
}

TY - JOUR
AU - V.V. Zozulya
TI - Nonlocal 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 - 221
EP - 236
AB - New models for plane curved rods based on linear nonlocal theory of elasticity have been developed. The 2-D theory is developed from general 2-D equations of linear nonlocal elasticity using a special curvilinear system of coordinates related to the middle line of the rod along with 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 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 nonlocal constitutive relations have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of local elasticity, a system of differential equations in terms of displacements for Fourier coefficients has been obtained. First and second order approximations have been considered in detail. Timoshenko’s and Euler-Bernoulli theories are based on the classical hypothesis and the 2-D equations of linear nonlocal theory of elasticity which are considered in a special curvilinear system of coordinates related to the middle line of the rod. The obtained equations can be used to calculate stress-strain and to model thin walled structures in micro- and nanoscales when taking into account size dependent and nonlocal effects.
LA - eng
KW - Curved rod; nonlocal elasticity; Legendre polynomial; high order theory; Timoshenko’s theory; Euler- Bernoulli theory
UR - http://eudml.org/doc/288509
ER -

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