Doubly Periodic Problems

We are interested in doubly periodic problems for two reasons. First, these problems arise in practice when dealing with perforated plates, regular composites and regular structures in solids and constructions. Secondly, as is especially important today, they arise from a general tendency of modern physics and applied science to account for details of the internal structure of a medium. A new branch of science, termed micromechanics, is swiftly growing up (see, e.g. Kemeny and Cook [1], Liu et al. [1], Napier and Pierce [1], Li and Wisnow [1], Tashkinov et al. [1], Dobroskok et al. [1]). Numerical simulation of doubly (in 2D) or triply (in 3D) periodic systems provides a unique opportunity to develop this science (Linkov and Koshelev [2]). It allows us to complement or substitute expensive and limited physical experiments by numerical ones. This approach has three advantages: (i) the cyclic constants of a displacement field provide rigorous definition of average (macroscopic) strains for prescribed average stresses, (ii) the ability to account for at least two hierarchical levels of structure, that of whole cells which may interact on their boundaries and that of internal elements of cells (grains, cracks, voids, etc.), (iii) calculations involve only a restricted area represented by the main cell. In 2D, the approach is strongly supported by the CV-BIE presented in this chapter.