Conditions for discontinuous bifurcation in limit states of selective non-local thermodynamically consistent gradient theory for quasi-brittle materials like concrete are evaluated by means of both geometrical and analytical procedures. This constitutive formulation includes two internal lengths, one related to the strain gradient field that considers the degradation of the continuum in the vicinity of the considered material point. The other characteristic length takes into account the material degradation in the form of energy release in the cracks during failure process evolution. The variation from ductile to brittle failure in quasi-brittle materials is accomplished by means of the pressure dependent formulation of both characteristic lengths as described by Vrech and Etse (2009). In this paper the formulation of the localization ellipse for constitutive theories based on gradient plasticity and fracture energy plasticity is proposed as well as the explicit solutions for brittle failure conditions in the form of discontinuous bifurcation. The geometrical, analytical and numerical analysis of discontinuous bifurcation condition in this paper are comparatively evaluated in different stress states and loading conditions. The included results illustrate the capabilities of the thermodynamically consistent selective non-local gradient constitutive theory to reproduce the transition from ductile to brittle and localized failure modes in the low confinement regime of concrete and quasi-brittle materials. © 2012 Elsevier Ltd. All rights reserved.
Vrech, S. M., & Etse, G. (2012). Discontinuous bifurcation analysis in fracture energy-based gradient plasticity for concrete. International Journal of Solids and Structures, 49(10), 1294–1303. https://doi.org/10.1016/j.ijsolstr.2012.02.009