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The objective of this chapter is to provide a comprehensive and systematic account on the subject of representative volume elements (RVEs) and unit cells (UCs). To construct an RVE or UC, intuition has been often perceived sufficient to facilitate the analysis, but down to the details, approaches taken turn out to be rather mythological. It will be demonstrated in this chapter that there is absolutely no room for any myth on the subject if the basic concepts of mathematics and mechanics, viz. symmetry and free body diagrams, have been applied correctly and consistently. Only then, effective and reliable means of material characterisation based on the use of RVEs and UCs can be established, in particular, for composites where micro/mesostructures often dictate their behaviours. The logic employed defines the boundary of applicability of the methodology.

The explanation of transition between ductile and brittle modes of fracture based on the “incubation time” concept is proposed. This approach allows us to establish relation between the influences of different loading parameters (strain-rate, temperature) on the mode of fracture. Proposed criteria give explanation to anomalous behavior of yield limit and possible high-temperature embrittlement.

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