Modelling of thermal residual stresses in MMC composites with the application of the homogenization method and digital image-based technique
Dariusz Golański Politechnika Warszawska, Instytut Technologii Materiałowych, ul. Narbutta 85, 02-542 Warszawa
Annals 2 No. 5, 2002 pages 354-358
DOI:
keywords:
abstract The homogenization method has been utilized with the help of digital image-based technique to analyze the local stress field in the unit cells of metal matrix composite surface layers. The matrix was a nickel-based stellite superalloy with tungsten carbide particulate inclusions. The theory of homogenization concerns statistically homogeneous or periodic composite media of domain Ωε and the representative volume element (RVE) occupying a microscopic region V with characteristic length ε. Identifying the size of the RVE with ε, we introduce two different scales: first is a macroscopic scale denoted by x, in the domain Ωε at which the heterogeneities are invisible and the other one is an microscopic one denoted by y = x/ε which enlarges the RVE region by ε such that V = εY. Thus, the superscripts introduced in variables indicate their orders as well as the dependency on both x and/or y (Fig. 1). According to the principle of minimum total potential energy for equilibrium, the displacement uε is the solution of the variational problem defined in the domain Ωε (eqn.1). With the help of two-scale asymptotic expansion method (4), the theory asserts that if the selected RVE is periodic and infinitesimally small, the actual displacement, uε , tends to the homogenized one, u0, which is the solution of the macroscopic equations whose coefficients have been homogenized (eqn. 3). Once the macroscopic displacement u0 and ΔT are obtained in the macroscopic region, these values are localized to give the micromechanical response of the unit cell. Therefore, the microscopic stress is defined by eqn. (7). Digital image-based (DIB) technique is used to catch and manipulate the image of composite microstructures (unit cells) so that they could be analyzed by the homogenization method. Also, this technique is very helpful for preparing the images of composite unit cells to some other additional processing like generation of 3D structures or changing volume fraction of inclusions. The main procedure of preparation the unit cell of a composite microstructure can be divided into the following four major stages: capture and sampling, selecting and thresholding, exporting, stacking (Fig. 2). The WC-stellite surface layers shown in Figure 3a has been used for calculation of microscopic residual stresses. After the microstructure of the WC-stellite composite has been scanned into computer memory and converted into digital unit cell (Fig. 3b) the homogenization method was used to obtain the effective composite properties dependent on the properties of its constituents and their volume fraction. This was done for the selected unit cell taken from the microstructure of composite (Fig. 3b). Next, the global structure was defined to compose the substrate material covered with a composite surface layer having uniform homogenized properties (Fig. 4). Then, the standard finite element method was applied to solve for the displacement and stress field in the macroscopic scale (global structure). The last step utilizes calculated global displacement field taken for the selected unit cell from the composite’s global structure to compute the microscale (local) stress distribution for composite microstructure represented by the selected unit cell. The results of calculations are shown in Figure 5 where principal σ 11 and effective Mises σH stress distribution in the analyzed WC-stellite composite unit cell are presented. The applied techniques presented in this paper can be seem to be the adequate tool to compare the local stress state in the metal matrix composites with hard inclusions, as well as to obtain the homogenized composite properties. We shall emphasize, that this kind of analysis is very unique as it allows the real microstructures of composite materials to be analyzed.