The higher-order continuum theories are necessary to explain and simulate the physical behaviour of materials with microstructure. In particular, size effects evidenced at all levels of observation as well as localization and softening phenomena, which have their origin in the micro/meso-structure of materials, but are revealed in the macroscopic response, require enhanced continuum theories. The theories incorporate intrinsic length parameters and the gradient terms represent the microstructural interactions in phenomenological modelling.
A review of numerous gradient-enhanced models is performed for instance in (Askes and Sluys 2003, Rolshoven 2003, Pamin 2004}. The theories can be classified in two groups: 1) models based on Mindlin gradient continuum, involving higher order stresses in balance and evolution equations, e.g. (Fleck and Hutchinson 1997), and 2) models in which the classical equilibrium/motion equations are augmented by an additional differential equation of plastic consistency or variable averaging, e.g. (Muehlhaus and Aifantis 1991, Peerlings et al 1996).
Computational implementation of the gradient theories requires a proper linearization and discretization of two- or three-field governing equations written in a weak format. The approximation functions must satisfy increased continuity requirements. Finite element methods based on continuous or discontinuous Galerkin approximation (Amanatidou and Aravas 2002, Engel et al 2002) as well as meshless methods can be used. It is noted that discretization itself does not regularize the mathematical model.
In the research the attention is limited to linear kinematics and isothermal conditions. Finite element and element-free Galerkin implementation of the gradient-dependent plasticity theory (Muehlhaus and Aifantis 1991, de Borst and Muehlhaus 1992) is presented among others in (de Borst and Pamin 1996, Liebe and Steinmann 2001, Pamin et al 2003), while computational gradient-enhanced damage theories are covered e.g. in (Peerlings et al 1996, Geers 1997, Askes et al 2000).
Simulations of strain localization and failure in quasi-brittle and geotechnical materials (concrete, soil) under static and dynamic loading can then be performed. Benchmark examples include the simulation of static failure and standing localization waves in tensile bars, shear band formation in biaxially compressed specimens, slope stability analysis, the Brazilian split test, cracking in plain and reinforced concrete beams under monotonic and reversed loading and three-dimensional fracture tests.
The lack of pathological discretization sensitivity is verified. It is noticed that in the regularization of localization problems strain-like quantities should be averaged. The future of localized failure simulations seems to belong to gradient-enhanced models combined with extended finite elements to model displacement discontinuities (macrocracks), cf. (Simone et al 2003).