Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

Composites science and technology

Predicting failure modes and ductility of dual phase steels using plastic strain localization

Generation of 3D representative volume elements for heterogeneous materials: a review

Outside of the classical microstructural detail-free estimates of effective moduli, micromechanical analyses of macroscopically uniform heterogeneous media may be grouped into two categories based on different geometric representations of material microstructure. Analysis of periodic materials is based on the repeating unit cell (RUC) concept and the associated periodic boundary conditions. This contrasts with analysis of statistically homogeneous materials based on the representative volume element (RVE) concept and the associated homogeneous boundary conditions. In this paper, using the above classification framework we provide a critical review of the various micromechanical approaches that had evolved along different paths, and outline recent emerging trends. We begin with the basic framework for the solution of micromechanics problems independent of microstructural representation, and then clarify the often confused RVE and RUC concepts. Next, we describe classical models, including the available RVE-based models, and critically examine their limitations. This is followed by discussion of models based on the concept of microstructural periodicity. In the final part, two recent unit cell-based models, which continue to evolve, are outlined. First, a homogenization technique called finite-volume direct averaging micromechanics theory is presented as a viable and easily implemented alternative to the mainstream finite-element based asymptotic homogenization of unit cells. The recent incorporation of parametric mapping into this approach has made it competitive with the finite-element method. Then, the latest work based on locally-exact solutions of unit cell problems is described. In this approach, the interior unit cell problem is solved exactly using the elasticity approach. The exterior problem is tackled with a new variational principle that successfully overcomes the non-separable nature of the overall unit cell problem.

Micromechanics of spatially uniform heterogeneous media: A critical review and emerging approaches

In this paper, we present a critical survey on homogenization theory and related techniques applied to micromechanics. The validation of homogenization results, the characterization of composite materials and the optimal design of complex structures are issues of great technological importance and are viewed here as a combination of mathematical and mechanical homogenization. The mathematical tools for modeling sequentially layered composites are explained. The influence of initial and boundary conditions on the effective properties in nonlinear problems is clarified and the notion of stability by homogenization is analyzed. Multiscale micromechanics methods are outlined and the classical as well as the emerging analytical and computational techniques are presented. Computation of effective static and dynamical properties of materials with linear or nonlinear constitutive equations is closely related to the development of generalized theories such as the strain-gradient mechanics. Selected applications of these techniques are outlined. Moreover, the extension of kinetic techniques in homogenization and the related inverse imaging problem are presented.

Homogenization Techniques and Micromechanics. A Survey and Perspectives

The scope of this chapter is to present briefly several available approaches in the literature for the study of nonlinear composites using full–field or mean–field approaches. The list of methods discussed here includes characteristic cases and should not be seen as exhaustive. To demonstrate the implementation of nonlinear behavior into a micromechanics framework, we present in more detail a simple technique based on the Mori–Tanaka and transformation field analysis approaches. We also provide numerical examples on thermoelastic, elastoplastic, and damageable media using mean–field computational strategies. 

Nonlinear composites

https://hal.univ-lorraine.fr/hal-01515204/document

Periodic homogenization for fully coupled thermomechanical modeling of dissipative generalized standard materials

Homogenization problems of a hollow cylinder made of elastic materials with discontinuous properties

In this paper, a review of papers on mathematical homogenization of dissipative composites under small strains and on the interplay between homogenization procedure and dissipation due to mechanical work is presented. Moreover, a critical survey on the links between mathematical homogenization and computational homogenization is attempted.

/pdf/mathematical-homogenization-of-inelastic-dissipative-2rae82drve.pdf

Nicolas Charalambakis

Papers

Homogenization Techniques and Micromechanics. A Survey and Perspectives

Nonlinear composites

Periodic homogenization for fully coupled thermomechanical modeling of dissipative generalized standard materials

Homogenization problems of a hollow cylinder made of elastic materials with discontinuous properties

Mathematical homogenization of inelastic dissipative materials: a survey and recent progress