Kevin Spilker

The dissertation of Mr. Kevin SPILKER is written in English and includes six chapters and 5 appendices with a total of 170 pages.
After motivating the need to develop homogenization methods in the context of multi-scale analyses of composite materials, Chapter 1 describes briefly the most commonly considered
techniques such as analytical bound estimates, mean-field homogenization (MFH) and full-field (FF) numerical simulations. Because of the accuracy and versatility of the latter, the different approaches developed in the literature in order to reduce the computation time of FF simulations are recalled. Among these methods, the micro-mechanics based clustering approaches, the reduced order models and the data-driven approaches, are reviewed through a literature study.

The context of 3-scale composite materials such as woven ones is then shortly described, motivating the need to develop multi-stage micro-mechanics-based homogenization
formalisms for non-linear elasto-plastic and anisotropic structures. The particular objectives and originalities of the thesis are then enumerated
1) To enhance micro-mechanics approaches based on the existing Transformation-Field Analysis (TFA) and Hashin-Shtrikman (HS) analysis, in which the clustering is defined
from inelastic fields instead of elastic fields as classically achieved, and to develop efficient resolution algorithms.
2) To accelerate the convergence of the TFA approach by accounting for the fluctuations of the inelastic fields inside the subdomains in the form of a correction factor and to
reformulate the Hashin-Shtrikman (HS) analysis in order to model cyclic and non-proportional loading cases.
3) To develop multi-stage micro-mechanics-based homogenization formalisms, (i) for woven structures in which a first homogenization is performed to represent the yarns
response using MFH and a second homogenization is performed with the enhanced micro-mechanics based clustering approaches, and (ii) for general composite materials
in which the enhanced TFA approach is hierarchically applied.

 

The thesis outline is then given.

In Chapter 2, Mr. Kevin SPILKER recalls the theoretical bases of the different homogenization methods considered in the thesis, i.e. computational homogenization (or FF resolution),
clustering based-micromechanics models like TFA and HS analysis, and MFH. Resolution methodologies for the TFA and HS analysis are proposed and presented in an algorithmic way.

Chapter 3 presents an extensive study of the developed TFA and HS analysis. First a clustering methodology based on inelastic fields is proposed for both the TFA and HS analysis. Then the TFA approach is enhanced by introducing a correction factor accounting for the fluctuations of the inelastic fields inside the subdomains, while the HS analysis is reformulated using an incremental secant form of the reference medium in order to be able to model cyclic and non- proportional loading cases. The TFA and HS analysis methods are then tested on several micro- structures, both isotropic and anisotropic, for elastic and elasto-plastic inclusions embedded in an elasto-plastic matrix. It is shown that, as long as the finite element mesh used for the clustering analysis is fine enough, the TFA prediction converges toward the FF results when increasing the number of clusters, with a faster convergence when considering the proposed enhancements. However, the case of high volume fraction of elastic inclusions in a perfectly plastic matrix requires a high number of clusters to achieve an admissible accuracy. The HS analysis proved to lead to a lower error for isotropic micro-structures although a convergence cannot be shown when increasing the number of clusters. The HS predictions for anisotropic micro-structures were however neither converging nor accurate.

 

A two-step homogenization process for woven composites is developed in Chapter 4. The yarns response is predicted using an incremental-secant MFH. The TFA and HS analysis are then modified in order to homogenize a woven cell response. First, in order to use as cluster constitutive material law the MFH response, the eigenstrains of the clustering based micro-
mechanical models are redefined from a virtual elastic unloading of the yarns homogenized material. Second, the yarns are clustered by considering a combination of the inelastic field
distribution and of the yarn orientation. Results predicted by the two-step homogenization are in good agreement with FF simulations of the woven unit-cell (the FF method also uses the
MFH to predict the yarns response).

Chapter 5 initiates the development of a hierarchical TFA method, which uses as material response of a cluster a TFA resulting from a subsequent decomposition of this cluster. As in

Chapter 4, in order to use as cluster constitutive material law the TFA response of the subsequent clustering, the eigenstrains of the clustering based micro-mechanical models are

redefined from a virtual elastic unloading. At this stage, the method is implemented as a two- step homogenization method and tested on unit cells of porous and elastic fiber reinforced
elasto-plastic matrix. Although good predictions can be obtained, the method does not allow to reduce the number of clusters on the different scales.

Chapter 6 presents the conclusions and outlooks of the dissertation.