A phase-field model for strain localization

Strain localization in quasi-brittle materials occurs when a material is subjected to a high level of mechanical solicitations and inelastic strains develop in relatively narrow zone where micro-cracks appear. The gradual evolution of the micro-cracks results in the formation of localized bands up to the development of stress-free cracks. The localized zone or plastic band is generally associated to a faster growth of strain and is characterized by inelastic phenomena such as opening and propaga- tion of cracks, initiation and growth of voids. Conversely, outside of this zone, the material unloads elastically. Extensive research has been carried out to address issues related to the modeling of localized defor- mation. A crucial point is the kinematic description of the band that has been addressed by three main models [1]. The first category of models considers the presence of a strong discontinuity in the displacement field and is typical of elastic fracture mechanics. The second approach considers the plastic band with finite thickness separated from the remaining part of the body by two weak discontinuities, as a zero-thickness interface characterized by its own tractions-displacement jumps law. Lastly, the third group considers constitutive enrichments with an internal length scale related to the width of the localization zone. Nonlocal and gradient theories that relate the constitutive be- havior of a material point with those in the neighboring region, fall into this category. Recently, phase-field models, usually adopted to describe gradual chemical changes from one phase to another, have been applied to model the transition between the fully broken and the sound material in a diffusive way. These models are characterized by the evolution of an auxiliary field (the phase field) that takes the role of an order parameter. They have been used to model brittle fracture [2] and ductile fracture [3]. The present paper presents a thermodynamically consistent formulation of the localization problem in quasi-static regime adopting the phase field approach. The introduction of the phase field vari- able enriches the solid kinematics, in this sense the proposed formulation can be categorized in the class of the regularized models where the plastic band is smeared on its neighboring volume with the order parameter assuming the unit value in middle surface of the band and zero value far from the same one. Several examples demonstrate the ability of the model to reproduce some important phenomenological features of brittle fracture as reported in the experimental literature.