Homogenization-based multiscale crack modelling: From micro-diffusive damage to macro-cracks

Abstract

The existence of a representative volume element (RVE) for a class of quasi-brittle materials having a random heterogeneous microstructure in tensile, shear and mixed mode loading is demonstrated by deriving traction-separation relations, which are objective with respect to RVE size. A computational homogenization based multiscale crack modelling framework, implemented in an FE 2 setting, for quasi-brittle solids with complex random microstructure is presented. The objectivity of the macroscopic response to the micro sample size is shown by numerical simulations. Therefore, a homogenization scheme, which is objective with respect to macroscopic discretization and microscopic sample size, is devised. Numerical examples including a comparison with direct numerical simulation are given to demonstrate the performance of the proposed method.

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

“…Finally, the third term in the right part of (27) corresponds to the micro-strain fluctuation increments.…”

“…As shown in Box 1, see also Figure 2, the kinematics of the points x S , in S, is characterized by the terms ε R , β and the normal vector to the discontinuity surface: n. These kinematical variables are injected into the RVE using a specific strategy, such that the micro-strain increments are expressed by means of the equation (27) in Box 3. The first two terms in the right part of (27) represent the macro strain that is injected into the RVE:…”

“…The first two terms in the right part of (27) represent the macro strain that is injected into the RVE:…”

“…Also, when the bifurcation condition is fulfilled, the Cohesive Multiscale Model defined in Sub-Section 2.3.2 and Box 3 is used for homogenizing the traction increment in the point PGL. The incremental generalized kinematics, characterized by dε R , dβ and n e , is inserted into the RVE using the equation (27) of Box 3. Note that both, the Standard as well as the Non-Standard Boundary Conditions (SBC and NSBC) are prescribed in this RVE.…”

“…This methodological approach has been the subject of recent intensive research in the scientific community. A number of novel contributions have been made in this direction, see for example the approaches of Matous et al [21], Verhoosel et al [44], Nguyen et al [27,29] Belytschko and coworkers [2,41], Geers and coworkers [6], Unger [43] and Souza et al [42]. However, the development of a consistent method has remained full of major theoretical challenges.…”