Introduction to the Variational Formulation in Mechanics

Part VMultiscale Modeling

Theories relating the behavior of continuum media at a macroscopic (or large) scale with the physical interactions occurring at microscopic (small) scales are termed multiscale theories. These theories originated in the second half of the last century, where the landmark contributions of Kirkwood and collaborators placed the groundwork to assemble the governing equations of transport phenomena in continuum media starting from statistical mechanics arguments within the molecular dynamics realm [148, 155–156]. Posteriorly, in the field of solid mechanics, substantial theoretical developments towards the estimation of macroscopic properties of heterogeneous materials began with the pioneer work of Hashin [137], Hill [141–144], Budiansky [43], Mandel [183], and Gurson [130], among others. A parallel stream of developments was supported by the asymptotic analysis of partial differential equations with periodic coefficients for the modeling of continuum media with periodic microstructure, which was initiated in the 1970s. Fundamental contributions in this specific field are the books by Bensoussan and collaborators [27] and by Sanchez‐Palencia [268, 269]. A common aspect in all these theories is the fact that variables at the macroscopic level, usually named homogenized variables, are invariably related to some kind of averaging process of the fields defined at the microscopic level.

Recent decades have witnessed a large increase in the appearance of multiscale ...