# 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.

## Keywords

Damage Process Recovery Sequence Convex Envelope Dimensional Hausdorff Measure Vary Boundary Condition## References

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