An inverse micro-mechanical analysis toward the stochastic homogenization of nonlinear random composites

Abstract

An inverse Mean-Field Homogenization (MFH) process is developed to improve the computational efficiency of non-linear stochastic multiscale analyzes by relying on a micro-mechanics model. First full-field simulations of composite Stochastic Volume Element (SVE) realizations are performed to characterize the homogenized stochastic behavior. The uncertainties observed in the non-linear homogenized response, which result from the uncertainties of their micro-structures, are then translated to an incrementalsecant MFH formulation by defining the MFH input parameters as random effective properties. These effective input parameters, which correspond to the micro-structure geometrical information and to the material phases model parameters, are identified by conducting an inverse analysis from the full-field homogenized responses. Compared to the direct finite element analyzes on SVEs, the resulting stochastic MFH process reduces not only the computational cost, but also the order of uncertain parameters in the composite micro-structures, leading to a stochastic Mean-Field Reduced Order Model (MF-ROM). A data-driven stochastic model is then built in order to generate the random effective properties under the form of a random field used

An incremental‐secant mean‐field homogenization model enhanced with a non‐associated pressure‐dependent plasticity model

“…In future works, this MFH model will be used to construct a mean‐field stochastic reduced order model (MF‐ROM), 81 which will allow to perform macroscale simulations of composites with pressure‐dependent matrix materials accounting for the inherent stochastic properties caused by geometrical perturbations in the microstructure.…”

“…$$ This expression corresponds to a first order approximation of the normal direction in the stress space in terms of $$$ \Delta \boldsymbol{\varepsilon} $$$. As discussed by Wu et al, 81 for infinitesimal strain increments $$$ \Delta \boldsymbol{\varepsilon} \to 0 $$$ for a single phase material, this expression $$$ {\mathbf{N}}_{n+1} $$$ tends to the normal of the yield surface. At the trial state, $$$ {\mathbf{N}}_{n+1}^{\mathrm{tr}} $$$ writes: $$$$ {\mathbf{N}}_{n+1}^{\mathrm{tr}}=3{\left({\boldsymbol{C}}^{\mathrm{el}}:\Delta {\boldsymbol{\varepsilon}}_{n+1}^{\mathrm{r}}\right)}^{\mathrm{dev}}+\frac{2\beta }{3}{\left({\boldsymbol{C}}^{\mathrm{el}}:\Delta {\boldsymbol{\varepsilon}}_{n+1}^{\mathrm{r}}\right)}^{\mathrm{vol}}=3{\left({\hat{\boldsymbol{\sigma}}}_{n+1}^{\mathrm{tr}}-{\hat{\boldsymbol{\sigma}}}_n^{\mathrm{r}\mathrm{es}}\right)}^{\mathrm{dev}}+\frac{2\beta }{3}{\left({\hat{\boldsymbol{\sigma}}}_{n...$$…”

Investigation of mesh dependency issues in the simulation of crack propagation in quasi‐brittle materials by using a diffuse interface modeling approach

“…However, if crack paths are unknown in advance, this strategy becomes less efficient. Indeed, to handle crack propagation along arbitrary trajectories, one must use special adaptive procedures, involving remeshing, 19,20 nodal relocation, 21–25 and multiscale strategies 26–30 …”

Stochastic local FEM for computational homogenization of heterogeneous materials exhibiting large plastic deformations

“…Firstly, we need to design a suitable model of the microstructure and transfer macroscopic loads and deformations to the microscale. This is done using the Hill-Mandel [14,23,44,45,54] condition which states that the virtual work performed by microscopic stresses at microscopic deformation gradients must be equal to the virtual work performed by macroscopic (homogenized) stresses at macroscopic (homogenized) deformation gradients. The second step is to simulate the response of the micromodel.…”

“…This reduction results however in a highly sophisticated stochastic problem with discontinuous random fields. It is worth to notice that a similar approach towards the homogenization of elasto-plastic materials with random microstructure was recently developed in [54].…”