Diffuse failure in geomaterials: Experiments and modelling H.D.V. Khoa a , I.O. Georgopoulos b , F. Darve a, * , F. Laouafa c a Laboratoire Sols, Solides, Structures, RNVO, Alert Geomaterials INPG, UJF, CNRS – BP 53 38041, Grenoble, France b Laboratory of Geomaterials, NTUA, Athens, Greece c INERIS – Verneuil en Halatte, France Received 4 May 2005; received in revised form 13 January 2006; accepted 19 January 2006 Available online 10 March 2006 Abstract While localized failure has been extensively studied, diffuse failure modes are still not characterized from an experimental point of view and not described by proper criteria from a theoretical point of view. The purpose of this paper is to tackle both these questions by considering, in a first part, undrained triaxial loading on very loose sands and the failure mode occurring at q-peak, and, in a second part, the bifurcation domain of sands in plane strain conditions through strain proportional paths and incrementally piece-wise linear and non-linear constitutive relations. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Diffuse failure; Bifurcation; Instability; Sand; Elasto-plasticity; Undrained triaxial 1. Introduction It is recognized since several years that some failures can appear strictly inside Mohr–Coulomb plastic limit condi- tion at least for two loading conditions: undrained triaxial tests on loosed sands (see for example [17]) and q constant tests on loose sands (see for example [3]). For loose sands, the phenomenon of shear band formation occurs very clo- sely to Mohr–Coulomb plastic limit condition and thus neither this last condition nor a localization criterion can detect these failure modes, which appear in the experiments as diffuse modes. It has been shown in the past [5] that Hill’s sufficient condition of stability [19] and the sign of second order work where good candidates as criteria for analyzing these diffuse modes. More generally a large bifur- cation domain was exhibited for loose and dense sands [9,11] and ‘‘bifurcation criteria’’ and ‘‘failure rules’’ were established. ‘‘Bifurcation’’ means here, first, that these is a rupture in the continuous evolution of the material mechanical states because of reaching limit generalized stress states and, sec- ondly, that there is a loss of constitutive uniqueness for cer- tain control loading parameters [20,22]. These bifurcation criteria characterize limit conditions related to these limit stress states reached along certain loading paths. For incre- mentally piece-wise linear constitutive relations, these bifurcation criteria are given by vanishing values of the determinants of constitutive matrices relating conjugated stress–strain variables along the loading paths (at least for the ‘‘octo-linear’’ model considered). ‘‘Failure rule’’ means here that, as for flow rules, the infinite number of incremental stress–strain solutions cor- responds to eigenvectors related to the vanishing eigen- values of the constitutive matrices just evoked before. Generally these failure rules are mixed stress–strain rela- tions, which cannot be confounded with flow rules because of this mixed character. The purpose of this paper are both, first to give exam- ples of diffuse failure modes by considering stress con- trolled undrained triaxial paths on loose sands, then to calculate analytically and to verify numerically the limits of the bifurcation domain in plane strain conditions, for loose and dense sands, for an incrementally piece-wise lin- ear constitutive relation and a thoroughly non-linear one. 0266-352X/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2006.01.002 * Corresponding author. Tel.: +33 4 76 82 52 76; fax: +33 4 76 82 70 00. E-mail address: Felix.Darve@inpg.fr (F. Darve). www.elsevier.com/locate/compgeo Computers and Geotechnics 33 (2006) 1–14