Strain gradient continuum mechanics: simplified models, variational formulations and isogeometric analysis with applications

This dissertation is devoted to two families of generalized continuum theories: the first and second strain gradient elasticity theories including the first and second velocity gradient inertia, respectively. First of all, a number of model problems is studied by analytical means revealing the key characters and potential of generalized continuum theories. In particular, the classical Kirsch problem is extended to the case of a simplified first strain gradient elasticity model demonstrating the size dependency of stresses and strains in the vicinity of a round hole in a plate in tension. Within linearly isotropic second strain gradient elasticity theory, instead, a simplified model is proposed, still capable of capturing free surface effects and surface tension, in particular, arising in solids of both nano- and macro-scales. With a series of benchmark problems, including a comprehensive set of stability analyses, the role of higher-order material parameters is revealed. On the way towards computational analysis, the boundary value problems of the fourth- and sixth-order partial differential equations arising in the first and second strain gradient models, respectively, are formulated and analysed in a mathematical variational form within appropriate Sobolev space settings. For numerical simulations, isogeometric Galerkin methods meeting higher-order continuity requirements are implemented in a user element framework of a commercial finite element software. Various benchmarks for statics and free vibrations confirm the optimal convergence properties of the numerical methods, verify the implementation and demonstrate the key properties of the underlying higher-order continuum models. Regarding model validation and applications, thorough analyses of stretching, shearing and vibration phenomena of complex triangular lattices homogenized by the simplified second strain gradient elasticity model reveal the strong size dependency of lattice structures and hence provide pivotal information for practical applications of materials and structures with a microstructure or microarchitecture.