Static analysis of the stress-gradient nanobeam by analytical and Nystrom numerical method

Document Type : Research Note

Authors

1 Department of Civil Engineering , Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

2 Department of civil engineering,Faculty of Engineering, Ferdowsi university of Mashhad, Mashhad, Iran

3 Department of civil engineering, Faculty of Engineering, Ferdowsi university of Mashhad, Mashhad, Iran

Abstract

This paper deals with the static analysis of the Euler-Bernoulli nanobeam based on the Eringen’s nonlocal theory. This theory is used for the nanoscale structures such as nanobeams which claims that the stress tensor is associated with the strain tensor by a linear integral transformation. The kernel function of the transformation contains an attenuation function. Several candidates have been proposed for the attenuation function. In this paper, the exponential attenuation function is utilized and the corresponding integral equation is solved directly. To do so, two different methods of the Nystrom numerical method and analytical method are employed, respectively. The Nystrom numerical method is one of the numerical solutions that is extensively utilized to solve different integral equations. This method builds up a linear system of equations that is conveniently solved by the computational programs. Next, the function of the answer is predicted and then examined by the analytical method. In fact, the analytical method is determination of the unknown constants in order to justify the integral equation by inserting the mentioned probable answer in the integral equation and putting both side equivalent to each other. At last, the displacement and curvature function of the nanobeam is determined according to the answer of the integral equation so that the mentioned integral equation converts to an equivalent differential equation that is newly proposed. On the other hand, the resultant displacement function is a closed form function which contains some constants that should be found by utilizing the boundary conditions of the nanobeam. For the sake of verification, the offered function is employed to determine the dimensionless displacement of a specified point of the beam and compare it with the results given in the previously proposed papers. Additionally, the mentioned function is employed to analyze several nanobeams with new boundary conditions and load functions. Then, the displacement function is plotted. Lastly, a contradiction is also determined based on the displacement graphs in the pervious section.

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Sharif Journal of Civil Engineering

Articles in Press, Accepted Manuscript
Available Online from 21 September 2024

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