Vol. 15, No. 4, 2020

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A Bernoulli–Euler beam model based on the local gradient theory of elasticity

Olha Hrytsyna

Vol. 15 (2020), No. 4, 471–487
Abstract

Using the total energy balance equation and principle of the frame indifference, a fundamental set of relations of the local gradient continuum model of elastic solids is formulated. The model is based on taking account of non-convective and non-diffusive mass flux related to the changes in the material microstructure. Linear stationary governing equations of the local gradient theory and corresponding boundary conditions are also derived by variational principle. In order to investigate the size-dependent behavior of nano-scale structures, this model is combined with the Bernoulli–Euler beam theory. Deflection of the cantilever beam subjected to the end-point loading under the plane stress conditions is evaluated and compared to the corresponding ones provided by the classical theory and by the strain gradient theory. It is shown that the beam deflection within the local gradient theory is smaller than that predicted by the classical Bernoulli–Euler beam theory. This work may be of special interest for designing the devices utilizing the micro/nano-beam elements.

Keywords
gradient-type theory, continuum mechanics modeling, local gradient elasticity, Bernoulli–Euler beam, micro/nano-beams, size effect
Milestones
Received: 10 January 2020
Revised: 13 May 2020
Accepted: 27 June 2020
Published: 19 October 2020
Authors
Olha Hrytsyna
Institute of Construction and Architecture
Slovak Academy of Sciences
84503 Bratislava 45
Slovakia
Center of Mathematical Modeling
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics
National Academy of Sciences of Ukraine
79005 Lviv
Ukraine