Higher-order continuum theories are often necessary to explain and convenient to simulate the physical behaviour of heterogeneous materials (materials with microstructure). This is especially the case for localization and softening phenomena which have their origin at the lower level of observation, but are observed in the macroscopic response. The theories incorporate internal length parameters and the gradient terms represent the microstructural interactions in phenomenological modelling.
After the introductory Chapter 1, Chapter 2 contains an overview of gradient-enhanced models developed in the last four decades and a comparison of those based on a Laplacian of an (inelastic) strain measure. Chapter 3 discusses the discretization of the selected gradient damage and plasticity models which lead to two-field formulations: the nonlocal (plastic) strain measure is a primary unknown in addition to the displacements. The finite element and element-free Galerkin methods are employed. The latter discretization has the advantage that increased continuity requirements typical for higher-order continua are easily satisfied.
Chapter 4 deals with the issue of material instabilities and shows an application of the non-associated Burzy\'nski-Drucker-Prager gradient plasticity with cohesion hardening in the simulation of shear banding. Chapter 5 presents a theory of scalar gradient damage coupled to plasticity with isotropic hardening. The theory reproduces the strength and stiffness degradation observed experimentally for quasi-brittle materials and is equipped with a crack-closing projection.
Chapter 6 addresses the more practical issue of reinforced concrete modelling. The analysis is focused on the numerical simulation of crack patterns and crack spacing in reinforced bars and beams. Some comparisons and general remarks on concrete modelling are included. Chapter 7 presents the strain-space theory of plasticity with an averaged strain measure, which is an alternative to the earlier discussed stress-space formulations.
In the reported work consistently linearized Newton algorithms are used to solve multiple one- and two-dimensional examples of localized deformation, including bars in direct tension, specimens in biaxial compression and beams in four-point bending. The final Chapter 8 contains several conclusions about the employed formulations and their efficiency in the numerical simulation of instabilities, localization and failure. Some prospects of future developments are also given.