This contribution presents a two-scale formulation devised to simulate failure in materials with heterogeneous micro-structure. The mechanical model accounts for the nucleation of cohesive cracks in the micro-scale domain. The evolution and propagation of cohesive micro-cracks can induce material instability at the macro-scale level. Then, a cohesive crack is nucleated in the macro-scale model which considers, in a homogenized sense, the constitutive response of the intricate failure mode taking place at the smaller length scale.The two-scale semi-concurrent model is based on the concept of Representative Volume Element (RVE). It is developed following an axiomatic variational structure. Two hypotheses are introduced in order to build the foundations of the entire theory, namely: (i) a mechanism for transferring kinematical information from macro-to-micro scale along with the concept of "Kinematical Admissibility", and (ii) a Multiscale Variational Principle of internal virtual power equivalence between the involved scales of analysis. The homogenization formulae for the generalized stresses, as well as the equilibrium equations at the micro-scale, are consequences of the variational statement of the problem.The present multiscale technique is a generalization of a previous model proposed by the authors [1, 2] and could be viewed as an application of a recent contribution [3]. The main novelty in this article lies on the fact that failure modes in the micro-structure involve a set of multiple cohesive cracks, connected or disconnected, with arbitrary orientation, conforming a complex tortuous failure path. Following the present multiscale modeling approach, the tortuosity effect is introduced as a kinematical concept and has a direct consequence in the homogenized mechanical response.Numerical examples are presented showing the potentialities of the model to simulate complex and realistic fracture problems in heterogeneous materials. In order to validate the multiscale technique in a rigorous manner, comparisons with the so-called DNS (Direct Numerical Solution) approach are also presented.

In the first part of this contribution, a brief theoretical revision of the mechanical and variational foundations of a Failure-Oriented Multiscale Formulation (FOMF) devised for modeling failure in heterogeneous materials is described.The proposed model considers two well separated physical length scales, namely: (i) the "macro" scale where nucleation and evolution of a cohesive surface is considered as a medium to characterize the degradation phenomenon occurring at the lower length scale, and (ii) the "micro" scale where some mechanical processes that lead to the material failure are taking place, such as strain localization, damage, shear band formation, etc. These processes are modeled using the concept of Representative Volume Element (RVE). On the macro scale, the traction separation response, characterizing the mechanical behavior of the cohesive interface, is a result of the failure processes simulated in the micro scale. The traction separation response is obtained by a particular homogenization technique applied on specific RVE subdomains. Standard, as well as, Non-Standard boundary conditions are consistently derived in order to preserve "objectivity" of the homogenized response with respect to the micro-cell size.In the second part of the paper, and as an original contribution, the detailed numerical implementation of the two-scale model based on the Finite Element Method is presented. Special attention is devoted to the topics which are distinctive of the FOMF, such as: (i) the finite element technologies adopted in each scale along with their corresponding algorithmic expressions, (ii) the generalized treatment given to the kinematical boundary conditions in the RVE and (iii) how these kinematical restrictions affect the capturing of macroscopic material instability modes and the posterior evolution of failure at the RVE level.Finally, a set of numerical simulations is performed in order to show the potentialities of the proposed methodology, as well as, to compare and validate the numerical solution furnished by the two-scale model with respect to a mono-scale Direct Numerical Simulation (DNS) approach.