A two‐scale failure model for heterogeneous materials: numerical implementation based on the finite element method

Abstract

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.

Cohesive surface model for fracture based on a two-scale formulation: computational implementation aspects

“…For example, in Sánchez et al [22] and Toro et al [43], a semi-concurrent two-scale approach for material failure analysis has been proposed. The failure phenomena at the micro-scale are represented by the existence of strain localization bands, modeled with a smeared crack technique.…”

“…Also, based on the MMVP concept, Toro et al [44] have subsequently generalized and improved the methodology proposed in [22] and [43] by introducing cohesive surfaces at both scales of analysis. New equations for transferring information between scales (scale-bridging equations) have been developed and presented.…”

“…Then, the linear operator A(y) can be determined by solving three linear micro-scale equilibrium problems (in R 2 ). See additional details in [22] and [43].…”

“…Following the denomination introduced in our previous work (see [43]), we call it the Standard Boundary Conditions (SBC) and is the typical kinematical constraint defined in conventional semi-concurrent multi-scale models. The last constraint imposed along S L µ , in expression (82), is called the Non-Standard Boundary Conditions (NSBC).…”

“…However, kinematically more constrained sub-models can be defined, such as: Taylor, Linear or Periodic (see Toro et al [43]). The kinematical constraints introduced by different sub-models play a very important role for adequately capturing the failure mechanism of the RVE leading to complete degradation of the material.…”