Nonlocal theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models
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 hypothes...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2017-09-01
|
| Series: | Curved and Layered Structures |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/cls-2017-0015 |
| _version_ | 1818720526085914624 |
|---|---|
| author | Zozulya V.V. |
| author_facet | Zozulya V.V. |
| author_sort | Zozulya V.V. |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-17T20:24:14Z |
| format | Article |
| id | doaj.art-b5bc3023cecc42fba6a8fbd316d48f08 |
| institution | Directory Open Access Journal |
| issn | 2353-7396 |
| language | English |
| last_indexed | 2024-12-17T20:24:14Z |
| publishDate | 2017-09-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Curved and Layered Structures |
| spelling | doaj.art-b5bc3023cecc42fba6a8fbd316d48f082022-12-21T21:33:51ZengDe GruyterCurved and Layered Structures2353-73962017-09-014122123610.1515/cls-2017-0015cls-2017-0015Nonlocal theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli modelsZozulya V.V.0Centro de Investigacion Cientifica de Yucatan, A.C., Calle 43, No 130, Colonia: Chuburna de Hidalgo, C.P. 97200, Merida, Yucatan, MexicoNew 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.https://doi.org/10.1515/cls-2017-0015curved rodnonlocal elasticitylegendre polynomialhigh order theorytimoshenko’s theoryeuler- bernoulli theory |
| spellingShingle | Zozulya V.V. Nonlocal theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models Curved and Layered Structures curved rod nonlocal elasticity legendre polynomial high order theory timoshenko’s theory euler- bernoulli theory |
| title | Nonlocal theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models |
| title_full | Nonlocal theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models |
| title_fullStr | Nonlocal theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models |
| title_full_unstemmed | Nonlocal theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models |
| title_short | Nonlocal theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models |
| title_sort | nonlocal theory of curved rods 2 d high order timoshenko s and euler bernoulli models |
| topic | curved rod nonlocal elasticity legendre polynomial high order theory timoshenko’s theory euler- bernoulli theory |
| url | https://doi.org/10.1515/cls-2017-0015 |
| work_keys_str_mv | AT zozulyavv nonlocaltheoryofcurvedrods2dhighordertimoshenkosandeulerbernoullimodels |