Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models J. Niiranen *a , V. Balobanov a , J. Kiendl b , S. B. Hosseini a a Aalto University, Department of Civil Engineering P.O. Box 12100, 00076 AALTO, Finland b Norwegian University of Science and Technology, Department of Marine Technology Marinteknisk senter, F2.160, Trondheim, Otto Nielsens veg 10, Norway Abstract As the first step, variational formulations and governing equations with boundary conditions are derived for a pair of Euler–Bernoulli beam bending models following a simplified version of Mindlin’s strain gradient elasticity theory of form II. For both models, this leads to sixth-order boundary value problems with new types of boundary conditions which are given additional attributes singly and doubly ; referring to a physically relevant distinguish- ment between free and prescribed curvature, respectively. Second, the varia- tional formulations are analyzed with rigorous mathematical tools: existence and uniqueness of weak solutions are established by proving continuity and ellipticity of the associated symmetric bilinear forms. This guarantees op- timal convergence for conforming Galerkin discretization methods. Third, the variational analysis is extended to cover two other generalized beam models: another modification of the strain gradient elasticity theory and a * Corresponding author: jarkko.niiranen@aalto.fi Preprint submitted to International Journal of Solids and Structures May 11, 2017
Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models
May 17, 2023
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