Parameterized Motion Driven by Global Minimization
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.
KeywordsDamage Process Recovery Sequence Convex Envelope Dimensional Hausdorff Measure Vary Boundary Condition
- 2.Braides, A., Cassano, B., Garroni, A., Sarrocco, D.: Evolution of damage in composites: the one-dimensional case. Preprint Scuola Normale Superiore, Pisa (2013). http://cvgmt.sns.it/paper/2238/