3D strain gradient elasticity: Variational formulations, isogeometric analysis and model peculiarities

This article investigates the theoretical and numerical analysis as well as applications of the three-dimensional theory of first strain gradient elasticity. The corresponding continuous and discrete variational formulations are established with error estimates stemming from continuity and coercivity within a Sobolev space framework. An implementation of the corresponding isogeometric Ritz–Galerkin method is provided within the open-source software package GeoPDEs. A thorough numerical convergence analysis is accomplished for confirming the theoretical error estimates and for verifying the software implementation. Lastly, a set of model comparisons is presented for revealing and demonstrating some essential model peculiarities: (1) the 1D Timoshenko beam model is essentially closer to the 3D model than the corresponding Euler–Bernoulli beam model; (2) the 3D model and the 1D beam models agree on the strong size effect typical for microstructural and microarchitectural beam structures; (3) stress singularities of reentrant corners disappear in strain gradient elasticity. The computational homogenization methodologies applied in the examples for microarchitectural beams are shown to possess disadvantages that future research should focus on.