Computational structural mechanics within strain gradient elasticity: mathematical formulations and isogeometric analysis for metamaterial design

The dissertation studies the majority of the most relevant and widespread physico-mathematical models of structural mechanics within the theory of strain gradient elasticity: gradient-elastic bars, beams, two- and three-dimensional solids and shells. Hamilton's principle, a variational energy approach, is utilized for deriving the strong, weak and finite element formulations of the related problems of statics and dynamics. For the most fundamental problems of the work, existence and uniqueness of weak solutions as well as error estimates for the corresponding conforming Galerkin discretizations are proved within the framework of Sobolev spaces. This theoretical foundation serves as a basis for the development and implementation of isogeometric conforming Galerkin methods within both open source and commercial finite element software packages. A set of benchmark problems for statics and free vibrations is solved for verification purposes and, in particular, for confirming the optimal convergence properties of the methods provided by the theoretical analysis. The numerical shear locking phenomenon for the Timoshenko beam model is studied and, furthermore, two different locking-free formulations are proposed and shown to guarantee optimal convergence results. Various generalized beam models are compared to each other and the most crucial differences between these models, related to the so-called stiffening size effect, are demonstrated by analytical and numerical solutions. The importance of higher-order rotatory inertia terms is highlighted in the context of gradient elasticity. Boundary layers arising due to the presence of the parameter-dependent higher-order terms and non-standard boundary conditions of the gradient-elastic Euler–Bernoulli beams are addressed. All the considered beam models, the Euler–Bernoulli, Timoshenko and the higher-order shear deformable ones, are extended for a case of anisotropic materials. Another advantage of the strain and velocity gradient elasticity theory, regularization of stress singularities, is demonstrated in the context of shell structures, in particular. The ability of the generalized beam models to capture size effects of microstructured continua at different length scales from nano- to macro-scale is demonstrated by comparisons to experimental results for nano- and micro-beams and by comparisons to computational results obtained from fine-scale models for lattice structures and auxetic metamaterials. The computational results cover engineering sandwich lattice beams as well. Extending these results to plates and shells, especially, unlocks a door for utilizing the theoretical results and computational methods of the dissertation for designing microarchitectured materials or mechanical metamaterials with predefined properties.