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The present work is devoted to the modelling of strongly size-dependent bending, buckling and vibration phenomena of 2D triangular lattices with the aid of a simplified first strain gradient elasticity continuum theory. As a start, the corresponding generalized Bernoulli-Euler and Timoshenko sandwich beam models are derived. The effective elastic moduli corresponding to the classical theory of elasticity are defined by means of a computational homogenization technique. The two additional length scale parameters involved in the models, in turn, are validated by matching the lattice response in benchmark problems for static bending and free vibrations calibrating the strain energy and inertia gradient parameters, respectively. It is demonstrated as well that the higher-order material parameters do not depend on the problem type, boundary conditions or the specific beam formulation. From the application point of view, it is first shown that the bending rigidity, critical buckling load and eigenfrequencies strongly depend on the lattice microstructure and these dependencies are captured by the generalized Bernoulli-Euler beam model. The relevance of the Timoshenko beam model is then addressed in the context of thick beams and sandwich beams. Applications to auxetic strut lattices demonstrate a significant increase in the stiffness of the metamaterial combined with a clear decrease in mass. Furthermore, with the introduced generalized beam finite elements, essential savings in the computational costs in computational structural analysis can be achieved. For engineering applications of architectured materials or structures with a microstructure utilizing triangular lattices, generalized mechanical properties are finally provided in a form of a design table for a wide range of mass densities.

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 distinguishment between free and prescribed curvature, respectively. Second, the variational 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 optimal 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

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The Timoshenko beam bending problem is formulated in the context of strain gradient elasticity for both static and dynamic analysis. Two non-standard variational formulations in the Sobolev space framework are presented in order to avoid the numerical shear locking effect pronounced in the strain gradient context. Both formulations are shown to be reducible to their locking-free counterparts of classical elasticity. Conforming Galerkin discretizations for numerical results are obtained by an isogeometric C p−1-continuous approach with B-spline basis functions of order p ≥ 2. Convergence analyses cover both statics and free vibrations as well as both strain gradient and classical elasticity. Parameter studies for the thickness and gradient parameters, including micro-inertia terms, demonstrate the capability of the beam model in capturing size effects. Finally, a model comparison between the gradient-elastic Timoshenko and Euler-Bernoulli beam models justifies the relevance of the former, confirmed by experimental results on nano-beams from literature.