Modelling of composite materials as microcontinua equivalent to lattice systems

The investigation of the mechanical behaviour of composite, multiphase, structured complex materials presenting different kinds of heterogeneities requires the development of appropriate models bridging several scales of observation. The ability to design such materials and to derive their macroscopic properties relies, in turn, on the ability to take into account the internal structure: size, shape, and spatial distribution of the microstructural constituents, which can span several orders of magnitude in length, starting from the submicron scale to millimetre scale or even larger scales. A basic problem in the mechanical modelling of these complex materials is the identification of suitable constitutive laws of continua with microstructure (microcontinua) able to take into account the relevant microscopic features, avoiding a direct modelling of the microstructure, whose discretization can lead to cumbersome problems with many degrees of freedom. To this aim, since many years, several homogenization or coarse-graining methods have been introduced, which require the accurate definition of the domain sizes involved in the micro-meso-macro transition process. Here, we propose a two-scale discrete-continuum equivalence procedure developed within the general framework of the principle of virtual work, which provides a guidance to the choice of the continuum approximations for such heterogeneous media, generally leading to continua with additional degrees of freedom (micromorphic, multifield, etc.). This procedure has been already applied to microcracked composites [1]; here we focus on porous metal-ceramic composites (MCC), such as tungsten or titanium carbides (WC/Co, TiC/Mo2C), or ceramic matrix composites (CMC), such as alumina/zirconia Al2O3/ZrO2. These last material, thanks to its good thermal resistance dependent on the rate of porosity, is widely used as a thermal barrier, for instance in thermal coatings of engines [2]. The architecture of the considered composites consists of a polycrystalline structure made of grains, whose shape is approximately hexagonal, interacting through micrometric interfaces (grain boundaries), generally filled by Cobalt. The pores also have an important role in the mechanics of this material, and it is assumed here that they are localized at the interface, reducing the contact area between grains. The microcontinuum identified is equivalent in terms of virtual work to a lattice description of such materials. The possibility to assume a periodic representative volume element to build–up macroscopic constitutive laws is discussed, and the different sizes and number of grains are taken into account. The effectiveness in representing the gross thermoelastic response under different states of thermal and mechanical stresses is investigated with the aid of some numerical examples.