Please cite this article as: R. Alessi, F. Freddi and L. Mingazzi, Phase-field numerical strategies for deviatoric driven fractures, Computer Methods in Applied Mechanics and Engineering (2019) 112651, https://doi.org/10.1016/j.cma.2019.112651. Available online at www.sciencedirect.com ScienceDirect Comput. Methods Appl. Mech. Engrg. xxx (xxxx) xxx www.elsevier.com/locate/cma Phase-field numerical strategies for deviatoric driven fractures R. Alessi a , F. Freddi b,∗ , L. Mingazzi b a Department of Civil and Industrial Engineering, Università di Pisa, Largo Lucio Lazzarino, 56122 Pisa, Italy b Department of Engineering and Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy Received 10 May 2019; received in revised form 3 September 2019; accepted 24 September 2019 Available online xxxx Abstract The phase-field approach regularizes the variational theory of fracture by approximating cracks with a smeared damage field. In this work, the attention is focused on those formulations approximating mode II fractures (shear fractures). In these models, only the deviatoric part of the strain elastic energy, penalized by the phase-field, drives the crack onset and evolution, whereas the elastic hydrostatic energetic contribution has no influence on the failure process. Consequently, cracks evolves according to the von Mises–Hencky–Hüber, also known as J2, failure criterion. Unfortunately, volumetric locking problem arises in the damaged zones if classical numerical solution strategies are adopted. As a consequence, damage localization bands appear with an excessive thickness, thus overestimating the fracture energy. In addition, the crack path geometry may be erroneously described because of the loss of precision of the displacement field in damaged zones. To circumvent these drawbacks, two numerical techniques are proposed, namely selective reduced integration and mixed displacement/pressure formulation, and their effectiveness evidenced by a numerical investigation. c ⃝ 2019 Elsevier B.V. All rights reserved. Keywords: Variational fracture mechanics; Phase-field; Shear fracture; Gamma-convergence 1. Introduction In the last decade, the so called phase-field approach to fracture has gained widespread popularity becoming a powerful computational tool for the analysis of crack nucleation and propagation in solids and structures. Derived from the variational revisitation of Griffith’s brittle fracture model [1], the fracture problem solution is obtained by minimizing an energy functional where sharp cracks are captured and described by a smeared field, the phase-field. This functional is composed by the sum of two terms, one associated with the elastic strain energy of the damageable material and the other with the fracture dissipation. This phase-field approach can recover the Griffith’s theory as a limit case in the Γ -convergence sense [2]. The original model proposed in [3] has been reformulated to reproduce more complex failure modes and to prevent material interpenetration [4–9]. In particular, in order to incorporate the idea of less brittle “deviatoric- type fracture”, referenced as shear fracture throughout this work, Lancioni and Royer-Carfagni in [5] proposed a formulation according to the von Mises–Hencky–H¨ uber criterion of local rupture. In these models, it is the deviatoric part of the elastic strain energy which is assumed to be responsible of the material degradation and therefore driving ∗ Corresponding author. E-mail addresses: roberto.alessi@unipi.it (R. Alessi), francesco.freddi@unipr.it (F. Freddi), lorenzo.mingazzi@unipr.it (L. Mingazzi). https://doi.org/10.1016/j.cma.2019.112651 0045-7825/ c ⃝ 2019 Elsevier B.V. All rights reserved.