Vol. 8, No. 4, 2020

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A geometrically nonlinear Euler–Bernoulli beam model within strain gradient elasticity with isogeometric analysis and lattice structure applications

Loc V. Tran and Jarkko Niiranen

Vol. 8 (2020), No. 4, 345–371
DOI: 10.2140/memocs.2020.8.345
Abstract

The nonlinear governing differential equation and variational formulation of the Euler–Bernoulli beam model are formulated within Mindlin’s strain gradient elasticity theory of form II by adopting the von Kármán strain assumption. The formulation can retrieve some simplified beam models of generalized elasticity such as the models of simplified strain gradient theory (SSGT), modified strain gradient theory (MSGT), and modified couple stress theory (MCST). Without the presence of nonlinear terms, the resulting linear differential equation is solvable by analytical means, whereas the mathematical complexity of the nonlinear problem is treated with the Newton–Raphson iteration and a conforming isogeometric Galerkin method with Cp1-continuous B-spline basis functions of order p 3. Through a set of numerical examples, the accuracy and validity of the present theoretical formulation at linear and nonlinear regimes are confirmed. Finally, an application to lattice frame structures illustrates the benefits of the present beam model in saving computational costs, while maintaining high accuracy as compared to standard 2D finite element simulations.

Keywords
strain gradient elasticity, geometric nonlinearity, beam model, isogeometric analysis, lattice structure
Mathematical Subject Classification 2010
Primary: 65M60, 74A60, 74B20, 74K10, 74Q15
Milestones
Received: 21 November 2019
Revised: 18 May 2020
Accepted: 23 June 2020
Published: 11 December 2020

Communicated by Victor A. Eremeyev
Authors
Loc V. Tran
Faculty of Civil Engineering
Ton Duc Thang University
Ho Chi Minh City
Vietnam
Jarkko Niiranen
Department of Civil Engineering
Aalto University
Espoo
Finland