Inverse Identification of the Internal Length Scale of Quasi-Brittle Materials Using Nonlocal Damage Models

Abstract

Failure in quasi-brittle materials involves a progressive damage process, which is initially driven by the growth and coalescence of micro-cracks and finally leads to the localization of deformation in narrow damage zones, such as cracks in concrete and shear bands in soils. Due to this damage process, the load carrying capacity of the material decreases at increasing deformation, which may result in a so-called strain-softening behaviour. When a constitutive law with a softening branch is applied in a standard continuum damage model, the strain softening phenomenon leads at the onset of localization to an ill-posed boundary value problem, since in quasi-static problems elipticity of the governing set of differential equations is no longer assured. As a consequence finite element analyses suffer from a severe mesh sensitivity. The deficiency of the standard continuum damage model can be circumvented by adding higher order terms in the constitutive law, which are thought to reflect phenomena at the microlevel, such as crack bridging phenomena and interactions among adjacent microcracks. Possible strategies are nonlocal and gradient damage models (Pijaudier-Cabot and Bazant 1987, Peerlings et al. 1995, de Borst et al 1996). These models introduce an internal length scale, which measures the distance over which a strong micro-structural interaction will exist. By the introduction of this internal length scale, the gradient damage model is capable of describing size effects (Carmeliet 1996).