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

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 Cp-1-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.