A mean-field micromechanical incremental approach for brittle damage in particulate microstructures under monotonic loading, using a piecewise uniform damage field

In this work, a mean-field micromechanical approach is proposed to determine the effective behaviour of a microstructure where damage occurs in one phase. Our approach fits into the framework of generalised standard materials proposed by Germain et al. (1983) by defining an incremental potential in the spirit of Lahellec and Suquet (2007), made up of two potentials: the free energy and the dissipation, given in this work by Lorentz and Andrieux (2003). It results in the definition of an elastic nonlinear problem at each time increment but, due to the damage evolution, this incremental potential is non convex. The proposed incremental approach also deviates from Lahellec and Suquet (2007) by considering in the damaged phase two sub-phases characterised by two damage states at the beginning of each time step. Next, this elastic nonlinear problem is regularised by introducing a linear comparison composite whose features are optimised by taking inspiration from the Full Optimised Second Order approach of Ponte-Castañeda (2016). However, the reduction of the number of degrees of freedom of the problem is realised by considering an isotropic linear comparison medium. The abilities of the proposed approach are assessed through a particular case. The composite is made up of two elastic phases : the matrix and the inclusions. The inclusions are subjected to a monotonous increasing swelling while the boundary of the RVE is stress free. For this problem, the resulting solution is given by closed-form expressions. These analytical developments are then used to obtain the effective response of the composite whose dependency on physical parameters (volume fraction of inclusions, toughness of the matrix, elastic contrast ...) is then analysed. The analysis is essentially based on an energy balance between the relaxed stored elastic energy and the dissipated energy. Explicit expressions of the critical values of the unique loading parameter (swelling of the inclusions) related to the onset of damage and the failure of the matrix phase are also given. The theoretical effective response compares well with the results of full fields calculations. •Brittle damage of particulate composites is modelled by homogenisation methods.•Existing incremental potential method is adapted to the problem of damageable phases.•The linearisation procedure uses two reference values of the fields in the matrix.•Softening response of the material depends on physical/microstructural parameters.