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Integral and differential nonlocal micromorphic theory

Integral and differential nonlocal micromorphic theory 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/approachTo 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.FindingsIt 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/valueDeveloping 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Engineering Computations: International Journal for Computer-Aided Engineering and Software Emerald Publishing

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References (96)

Publisher
Emerald Publishing
Copyright
© Emerald Publishing Limited
ISSN
0264-4401
DOI
10.1108/ec-01-2019-0008
Publisher site
See Article on Publisher Site

Abstract

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/approachTo 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.FindingsIt 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/valueDeveloping 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.

Journal

Engineering Computations: International Journal for Computer-Aided Engineering and SoftwareEmerald Publishing

Published: Aug 19, 2019

Keywords: Finite element analysis; Timoshenko beam; Bending; Nonlocal continuum; Integral nonlocal formulation; Micromorphic theory

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