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566 M.S. Bruzzi et al. / International Journal of Plasticity 17 (2001) 565±599 1. Introduction This paper is concerned with the study of a particulate reinforced metal matrix composite material (MMC). The objective of the work presented was to apply micromechanical modelling techniques to investigate microstructural deformation in the MMC and to predict the overall tensile stress±strain curve of the MMC. Of speci®c interest was a forged MMC of 2124 Al alloy, reinforced with 17% SiC particles, in the T4 condition, which was tensile tested at room temperature and 150 C. MMCs have been the subject of intense analytical and experimental investigation over the last 10±15 years to characterise and understand their mechanical properties and performance, and to assess the importance of variables such as reinforcement volume fraction and morphology. This is true for MMCs with various types of reinforcement structures including particulate, short ®bre and continuous ®bre reinforcements. As regards modelling of mechanical performance, both analytical and computational methods have been used. One method, computational micromechanics, is based on achieving representations of the material microstructure in computational models. By linking the microscale behaviour with the macroscale behaviour, achieved using assumptions on the deformation and stress ®elds within the material, computational micromechanics can be used to relate the microstructural performance, in terms of deformation mechanisms, strengthening mechanisms and failure mechanisms to the macroscopic mechanical performance of the material (McHugh et al., 1993a). Due to the fact that the complete analysis of a material system in which the matrix and reinforcement are represented separately, with realistic reinforcement geometries and distribution, is currently beyond computational capabilities, many descriptions of composites are based on periodic unit cell approaches or embedded cell methods. Excellent reviews of investigations into these aspects of MMCs are to be found in Suresh et al. (1993) and BoÈ hm (1998). In in®nite periodic microstructures stress and strain ®elds can be studied analytically by using series expansions that make use of the periodicity of the phase arrangements, e.g. Sangani and Lu (1987). In the periodic unit cell approach, however, the microstructural geometry is divided into periodically repeating unit cells, to which the investigation may be limited without greatly reducing its generality. For particle reinforced MMCs a large body of work exists in the literature showing how the periodic unit cell methodology has been very successfully used to study the room temperature, monotonic tensile loading behaviour of the materials, based on models with idealised and regular microstructural geometries (Christman et al., 1989; Bao et al., 1991; McHugh et al., 1993a). Models of this type have also been used by Llorca et al. (1992) and BoÈ hm (1991) to study cyclic loading. 3D unit cell models were used by O'Dowd et al. (1999) to study particle arrangement and loading state e€ects. Recently, unit cells describing more complex arrangements of particles have been reported in literature. In 3D, cubic shaped unit cells containing up to 64 spherical inclusions were used for describing the elastic behaviour of particle reinforced polymers (Gusev, 1997), and unit cells containing approximately 10 particles were used
M.S. Bruzzi et al. / International Journal of Plasticity 17 (2001) 565±599 567 for studying particle distribution e€ects in MMCs by Watt et al. (1996). 2D unit cell models with realistic microstructural arrangements within the cell, determined from actual composite sections, have been used (Brockenbrough and Suresh, 1990; Bruzzi and McHugh, 1998; Fischmeister and Karlsson, 1999). 3D studies of realistic particle distributions have also been published using periodic unit cell models, such as that concerned with a particle reinforced ceramic matrix composite reported by Terada and Kikuchi (1996), and an assessment of 2D versus 3D analysis by Iung and Grange (1995). As regards embedded cell methods, a powerful method for predicting tensile ¯ow stresses of two phase materials is presented in Dong and Schmauder (1996) and Farrissey et al. (1999). The emphasis in the ®rst part of the work presented here was not an exhaustive geometrical parameter study in 2D or 3D, as many previous studies have been, but the development of periodic unit cell models incorporating many realistic features of an MMC, that attempted to predict the macroscopic (and microscopic) behaviour of a speci®c material. Tensile loading was simulated at both room temperature and at 150 C. The simulations were made progressively more realistic, starting with 2D plane strain models with idealised microstructural geometries, through 2D models with ``realistic'' geometries based on actual SEM micrographs of the material, and concluding with 3D models with multiple particles. Residual stresses due to the thermal expansion mismatch of the constituents during processing were included by simulating a quenching process. The second part of the work focused on an evaluation of the performance of the models considered in the ®rst part, under variation in the following features: (1) boundary conditions and (2) matrix constitutive description. In addition, cyclic loading (low cycle fatigue) was simulated. The emphasis in this second part was not to try to reproduce experimental data but to essentially do a model parameter study. In terms of boundary conditions, a very simple cell embedding technique was used to eliminate the periodicity constraint associated with using a unit cell approach. Attention was focussed on a cell with non-periodic boundary conditions, within an embedding region, and the macroscopic stress±strain curve of the complete model was compared with that of the periodic unit cell model. Two methods of representing matrix plasticity for static loading were considered: one using isotropic hardening J 2 ¯ow theory and the other using crystal plasticity theory. The di€erences between the two theories in representing microstructural deformation were considered. This was of interest since previous comparisons of the theories, when applied to such materials, had been restricted to very idealised microstructures (McHugh et al., 1993b; Needleman and Tvergaard, 1993). This is in contrast to a large amount of work reported in the literature on the application of crystal plasticity theory to single phase polycrystals, e.g. Harren et al. (1988) and Harder (1999). For cyclic loading, the matrix plasticity was modelled using an isotropic hardening model and also a non-linear isotropic kinematic hardening model, which was calibrated using experimental low-cycle fatigue data for the matrix alloy. In this paper emphasis is given to model description and careful examination of the simulation results. Details of the descriptions of the constitutive theories used are for the most part omitted as they are presented in full detail elsewhere.
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