Parameterized Motion Driven by Global Minimization

Abstract

Energy-driven dynamic problems are in general associated with a local minimization procedure. Nevertheless, for “slow movements” a meaningful notion of “quasi-static” motion can be defined starting from a global-minimization criterion. Loosely speaking, a quasi static motion is controlled by some parameterized forcing condition; the motion is thought to be so slow so that the solution at a fixed value of the parameter minimizes a total energy. This energy is obtained adding some “dissipation” to some “internal energy”. A further condition is that the total dissipation increases with time. An entire general theory can be developed starting from these ingredients. An important feature of these rate-independent motions is that they can be characterized as the limit of a piecewise-constant (time-)parameterized family of functions, which are defined iteratively as solutions of minimum problems. Under suitable conditions, to such a characterization the Fundamental Theorem of Γ-convergence can be applied, so that this notion can be proved to be indeed compatible with Γ-convergence. In this chapter we treat in detail the example of the homogenization of damage, and briefly introduce the theory of energetic solutions.

Appendix

Analyses of damage models linked to our presentation are contained in the work by Francfort and Marigo [5]. The higher-dimensional case is studied in a paper by Francfort and Garroni [3]. A threshold-based formulation is introduced by Garroni and Larsen [7]. The examples in Sect. 3.1.3 have been part of the course exam of B. Cassano and D. Sarrocco at Sapienza University in Rome.

An analysis of rate-independent processes is contained in the review article by Mielke [8]. The definitions given here can be traced back to the works by Mielke, Theil and Levitas [10] and [11]. The stability with respect to Γ-convergence is analyzed in the paper by Mielke, Roubiček, and Stefanelli [9]. Most of Sect. 3.2 is taken from a lecture given by Ulisse Stefanelli during the course at the University of Pavia. The homogenization examples in Sect. 3.1, framed in the theory of energetic solutions, are contained in the paper [2]

An account of the variational theory of fracture (introduced in [6]) is contained in the book by Bourdin et al. [1]. The fundamental transfer lemma is contained in the seminal paper by Francfort and Larsen [4].