Introduction to the Variational Formulation in Mechanics

13Method of Multiscale Virtual Power

13.1 Introduction

In this chapter we will present the derivation of the proposed multiscale modeling framework. This task is accomplished by using an abstract notation which provides the theory with a general unified framework able to meet the needs for modeling a wide variety of physical problems, such as classical formulations in solid and fluid mechanics, strain localization at macro and micro scales, damage and failure mechanics, high‐order continua, multiphysics problems, and multiscaling between scales featuring different physical phenomena, among others.

For ease of reading, each time a given piece of theory has been established, the adopted abstract notation will be specified for the classical scenario encountered in solid mechanics within the infinitesimal strains regime throughout the chapter.

The multiscale modeling strategy adopted in this chapter follows the derivation/discussion of the basic foundational elements on top of which the proposed formulation is sustained. That is to say, we will sequentially address the concepts of kinematical admissibility, mathematical duality, and the Principle of Multiscale Virtual Power (PMVP). However, and since all this procedure is extensively supported by the Principle of Virtual Power (PVP), we initiate the chapter with a concise revision of the Method of Virtual Power introduced in Chapter 3. Furthermore, this methodical remembrance will serve to set the basic notation, as well as to ...