Please cite this article in press as: E. Kishta, et al., Strong discontinuity analysis of a class of anisotropic continuum damage constitutive models – Part II: Concrete material application, Mech. Re. Commun. (2017), http://dx.doi.org/10.1016/j.mechrescom.2017.03.004 ARTICLE IN PRESS G Model MRC-3155; No. of Pages 5 Mechanics Research Communications xxx (2017) xxx–xxx Contents lists available at ScienceDirect Mechanics Research Communications journal h om epa ge: www.elsevier.com/locate/mechrescom Strong discontinuity analysis of a class of anisotropic continuum damage constitutive models – Part II: Concrete material application Ejona Kishta a,b , Benjamin Richard a,∗ , Cédric Giry b , Frédéric Ragueneau b a DEN-Service d’études mécaniques et thermiques (SEMT), CEA, Université Paris-Saclay, F-91191, Gif-sur-Yvette, France b LMT, ENS Cachan, CNRS, Université Paris-Saclay, 61 avenue du Président Wilson, F-94230 Cachan, France a r t i c l e i n f o Article history: Received 24 October 2016 Accepted 26 March 2017 Available online xxx MSC: 00-01 99-00 Keywords: Discontinuity Failure Damage Anisotropy Willam’s test a b s t r a c t On the basis of the strong discontinuity analysis, a discrete model expressed in terms of traction vector- displacement jump has been constructed from a continuous model expressed in terms of stress–strain law. In the first part of the paper, this approach has been extended to a class of anisotropic continuum damage constitutive models [1]. In this second part of the paper, the proposed class of discrete anisotropic damage constitutive models is particularized. More precisely, a micromechanical-based anisotropic dam- age constitutive model is derived. This model accounts in a natural manner for particular crack families orientation. The aims of this paper are (i) to illustrate the capabilities of the proposed discrete enhanced model in reproducing the induced anisotropy appearing in quasi-brittle materials when cracking and (ii) to assess the numerical robustness of the time integration scheme. For this purpose, two numerical examples at the material point level are exposed after a brief presentation of the time integration scheme. The correspondence between the continuous and the discrete model as well as the induced anisotropy features are emphasized. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction In the first part of this paper, it was shown that the strong discon- tinuity kinematics enhancement leads naturally to the definition of a discrete constitutive model, expressed in terms of traction vector-displacement jump, for a class of anisotropic continuum damage based constitutive models [1]. This framework was already well-established for isotropic damage and plasticity model [2,3]. The discrete framework is built considering the traction continu- ity condition and regularization of the hardening parameter. These considerations allow for the kinematic compatibility between the continuum model, expressed in terms of stress–strain relation, and the strong discontinuity kinematics. For an easier reading, the dis- crete constitutive equations of the proposed class of anisotropic damage models are summarized hereafter (see Eqs. (1a)–(1f)). State potential ¯  = 1 2 t 0 · [u ] − n  i=1 ¯  i g i (([u ] ⊗ n ) s ) (1a) ∗ Corresponding author. E-mail address: Benjamin.Richard@cea.fr (B. Richard). Traction vector t = ∂ ¯  ∂[u ] = t 0 − n  i=1 ¯  i dg i d[u ] (([u ] ⊗ n ) s ) (1b) Thermodynamic forces ¯ F ¯  i = − ∂ ¯  ∂ ¯  i = g i (([u ] ⊗ n ) s ) (1c) Threshold surface ¯   i ( ¯ F  i , ¯ Z i ) = ¯ F  i − ¯ Z i (1d) Flow rules ˙ ¯  i = ˙ ¯  i ∂ ¯   i ∂ ¯ F ¯  i = ˙ ¯  i ; ˙ ¯ z i = ˙ ¯  i ∂ ¯   i ∂ ¯ Z i = − ˙ ¯  i (1e) Loading/unloading conditions ˙ ¯  i ≥ 0; ¯  ¯  i ≤ 0; ˙ ¯  i ¯  ¯  i = 0 (1f) In this second part, the discrete framework is applied to a specific micromechanical-based anisotropic damage model. This model is chosen because it describes the anisotropic character of concrete-like materials illustrated by different oriented crack families. Furthermore, this framework is useful for future develop- ments on reinforced concrete structures characterized by normal http://dx.doi.org/10.1016/j.mechrescom.2017.03.004 0093-6413/© 2017 Elsevier Ltd. All rights reserved.