Integral and differential nonlocal micromorphic theory: Finite element bending analysis of Timoshenko micro-/nano-beams

Amir Norouzzadeh (Department of Mechanical Engineering, University of Guilan, Rasht, Islamic Republic of Iran)
Mohammad Faraji Oskouie (Department of Mechanical Engineering, University of Guilan, Rasht, Islamic Republic of Iran)
Reza Ansari (Department of Mechanical Engineering, University of Guilan, Rasht, Islamic Republic of Iran)
Hessam Rouhi (Department of Engineering Science, University of Guilan, Rudsar-Vajargah, Islamic Republic of Iran)

Engineering Computations

ISSN: 0264-4401

Publication date: 19 August 2019

Abstract

Purpose

This paper aims to combine Eringen’s micromorphic and nonlocal theories and thus develop a comprehensive size-dependent beam model capable of capturing the effects of micro-rotational/stretch/shear degrees of freedom of material particles and nonlocality simultaneously.

Design/methodology/approach

To consider nonlocal influences, both integral (original) and differential versions of Eringen’s nonlocal theory are used. Accordingly, integral nonlocal-micromorphic and differential nonlocal-micromorphic beam models are formulated using matrix-vector relations, which are suitable for implementing in numerical approaches. A finite element (FE) formulation is also provided to solve the obtained equilibrium equations in the variational form. Timoshenko micro-/nano-beams with different boundary conditions are selected as the problem under study whose static bending is addressed.

Findings

It was shown that the paradox related to the clamped-free beam is resolved by the present integral nonlocal-micromorphic model. It was also indicated that the nonlocal effect captured by the integral model is more pronounced than that by its differential counterpart. Moreover, it was revealed that by the present approach, the softening and hardening effects, respectively, originated from the nonlocal and micromorphic theories can be considered simultaneously.

Originality/value

Developing a hybrid size-dependent Timoshenko beam model including micromorphic and nonlocal effects. Considering the nonlocal effect based on both Eringen’s integral and differential models proposing an FE approach to solve the bending problem, and resolving the paradox related to nanocantilever.

Keywords

Citation

Norouzzadeh, A., Faraji Oskouie, M., Ansari, R. and Rouhi, H. (2019), "Integral and differential nonlocal micromorphic theory: Finite element bending analysis of Timoshenko micro-/nano-beams", Engineering Computations, Vol. 37 No. 2, pp. 566-590. https://doi.org/10.1108/EC-01-2019-0008

Publisher

:

Emerald Publishing Limited

Copyright © 2019, Emerald Publishing Limited

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