Mixed stabilized finite element methods in nonlinear solid mechanics Part II: Strain localization M. Cervera ⁎, M. Chiumenti, R. Codina International Center for Numerical Methods in Engineering (CIMNE), Technical University of Catalonia (UPC), Edificio C1, Campus Norte, Jordi Girona 1-3, 08034 Barcelona, Spain abstract article info Article history: Received 22 July 2009 Received in revised form 9 March 2010 Accepted 13 April 2010 Available online 21 April 2010 Keywords: Mixed finite elements Stabilization Strain softening Strain localization Local damage models Mesh dependence This paper deals with the question of strain localization associated with materials which exhibit softening due to tensile straining. A standard local isotropic Rankine damage model with strain-softening is used as exemplary constitutive model. Both the irreducible and mixed forms of the problem are examined and stability and solvability conditions are discussed. Lack of uniqueness and convergence difficulties related to the strong material nonlinearities involved are also treated. From this analysis, the issue of local discretization error in the pre-localization regime is deemed as the main difficulty to be overcome in the discrete problem. Focus is placed on low order finite elements with continuous strain and displacement fields (triangular P1P1 and quadrilateral Q1Q1), although the presented approach is very general. Numerical examples show that the resulting procedure is remarkably robust: it does not require the use of auxiliary tracking techniques and the results obtained do not suffer from spurious mesh-bias dependence. © 2010 Elsevier B.V. All rights reserved. 1. Introduction Strain localization occurs in softening materials subjected to monotonic straining. This phenomenon leads to the formation of localization bands inside the solid because, once the peak stress is reached within a band, and under further straining, strains concen- trate inside the band while the material outside the band unloads elastically. Upon continuing straining, the localization progresses, the width of the localization band diminishes and, unless there is a physical limitation, it tends to zero. The particular components of the strain tensor that localize during this process depend on the specific constitutive behavior of the material. In Rankine-type materials, only normal elongations localize, eventually forming tensile cracks; in the so-called J 2 materials, shear (or slip) strains concentrate, leading to slip surfaces (or lines). It is generally accepted that the amount of energy released during the formation of a unit area of discontinuity surface is a material property, called the fracture energy (Mode I and Mode II fracture energies in Fracture Mechanics terminology). Dimensional analysis shows that if the elastic energy stored in the solid volume is released through the area of the fracture surface, the failure process leads to what is known as structural size effect [1]. Experimental evidence shows that, for a given structural geometry, ductile behavior is observed in the small scale limit, when the energy dissipated by inelastic behavior in the formation of the failure mechanism is much larger than the total stored elastic energy; contrariwise, brittle behavior occurs in the very large scale limit, when the ratio between the dissipated inelastic and available elastic energies is close to one. The small scale limit is suitable for small laboratory specimens, and the large scale limit is appropriate for structures of very large dimensions or even for scales larger than man-made structures. Thus, it is of practical interest to develop analytical and numerical tools suitable to bridge the gap between perfectly ductile and perfectly brittle behavior. This is called quasi-brittle failure [2]. Quasi-brittle failure has been the object of intensive interest in computational solid mechanics during the last four decades. Even if the main motivation for this interest is the wide range of engineering applications connected to this field, academic concern has been sharpened by the unexpected numerical difficulties encountered. The fact is that most attempts to model strain localization in softening materials with standard, irreducible, local approaches fail and that the solutions obtained suffer from mesh-bias dependence in such a strong manner that it cannot be ignored. Consequently, many different, alternative, strategies have been devised to model strain localization and quasi-brittle fracture and the references in the bibliography are uncountable. In the last 25 years, micropolar ( [3,4]), gradient- enhanced ([5–9]) and non-local,([5,10–14], among others) models have been proposed with the common basic idea of modifying the original continuous problem to introduce an internal length that acts as a localization limiter. On a different line, viscous-regularized, strain- Computer Methods in Applied Mechanics and Engineering 199 (2010) 2571–2589 ⁎ Corresponding author. E-mail address: miguel.cervera@upc.edu (M. Cervera). 0045-7825/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2010.04.005 Contents lists available at ScienceDirect Computer Methods in Applied Mechanics and Engineering journal homepage: www.elsevier.com/locate/cma