A Short Introduction to Continuum Micromechanics
Basic issues in continuum mechanical modeling of microstructured materials are discussed and a number of physically based modeling approaches are presented, among them mean field and bounding methods as well as unit cell and embedding models. In addition,
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1 Basic Considerations In the following some basic issues in continuum mechanical modeling of microstructured materials are discussed. The emphasis is put on application related aspects of the field of continuum micromechanics of inhomogeneous materials, the topics being presented mainly from an "engineering point of view" . For more formal treatments the reader is referred to the books by Mura (1987), Aboudi (1991), Nemat-Nasser and Hori (1993), Suquet (1997a), Markov and Preziosi (2000), Bornert et al. (2001), Jeulin and OstojaStarzewski (2001), Torquato (2001), and Milton (2002). Short overviews were given e.g. by Hashin (1983) and Zaoui (2002). For simplicity modeling approaches are presented within the framework of two-phase materials consisting of reinforcements embedded in a contiguous matrix, i.e. composites in the strict sense. The concepts involved, however, can be extended to microstructured materials containing a higher number of constituents or showing other microtopologies (e.g. polycrystals) as well as to porous and, to a considerable extent, to cellular materials. Materially and geometrically linear behavior is assumed unless explicitly stated otherwise.
1.1
Length Scales
As implied by their name, microstructured materials are inhomogeneous at some small length scale, where different constituents (or phases) can be distinguished. The phase arrangements at the microscale may show matrix-inclusion (composites), interwoven (e.g. some duplex steels), or granular (polycrystals) microtopologies. The microgeometries of most practically relevant materials are too complex for a deterministic description in full detail, so that the geometrical arrangement of phases has to be treated statistically and described by appropriate distribution or correlation functions, H. J. Böhm (ed.), Mechanics of Microstructured Materials © Springer-Verlag Wien 2004
H. J. Bohm
2
compare Torquato (2001), Jeulin and Ostoja-Starzewski (2001) and Pyrz (2004). Accordingly, statistically based microstructure-property relations for inhomogeneous materials are an important issue in micromechanical studies (Pyrz, 2004; Siegmund et al., 2004). In many materials of interest the microstructure is statistically homogeneous (or stationary), i.e. the statistical descriptors of the geometrical arrangement do not depend on the position they are evaluated at. For such systems it makes sense to define volume averaged properties, which are then independent of the size and position of the volume element considered, provided it is sufficiently large. A volume element that contains all the necessary information for the statistical description of a given microstructure is called a reference volume element (RVE). Statistically homogeneous microstructures may be statistically isotropic, the statistical descriptors being rotationally invariant, or statistically anisotropic, in which case there are preferred directions in the shapes, orientations or positions of the constituents. In the simplest case a microstructured material is associated with two
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