Rate-Independent Systems by Alexander Mielke & Tomáš Roubíček

Author:Alexander Mielke & Tomáš Roubíček
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

(4.3.78a)

(4.3.78b)

and every (u,z) obtained in this way is a local solution to the problem specified above.

Proof.

We will use the abstract Proposition 3.​4.​9(ii) proving the convergence of the semi-implicit time discretization (3.4.20). To this end, we verify the assumptions (3.​4.​25) and (3.4.27) together with the facts that for some compact , is continuous, and the family satisfies the (S+)-property.

The strict convexity of required in (3.4.27b) is now assumed in (4.3.73c); here it is important that only incomplete damage (i.e., fixed) is considered.

Further, (3.4.27c) requires that for a fixed , the functional must be weakly lower semicontinuous on , which actually easily follows by Proposition 4.1.5(i) and compactness of the embedding ; note that the -terms in and in mutually cancel.

The assumption (3.4.27d) asking for being (weak, strong)-continuous means that for in , which can be deduced from (4.3.73f) when we realize also the compact embedding . Note that we used the cancellation of the -terms, which was the motivation for (3.4.27d) having been designed in such a fine way.

The strong continuity of on its domain, as required in Proposition 3.​4.​9(ii), is obvious.

A further ingredient is the (S+)-property of the family as required in (3.4.27f). This means here that in and in together with

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