3. Rate-independent systems in Banach spaces

Premonotonicity of A means that \(\langle \xi,v\rangle \geq 0\) for all \((\xi,v)\) satisfying \(\xi \!\in \! A(v)\).

The notion of semistability was first presented in [527] and then used also, e.g., in [350, Formula (5)] in a special dynamic fracture problem.

For z fixed, the property of \(\partial _{y}\mathcal{E}(t,\cdot,z): \boldsymbol{Y } \rightarrow \boldsymbol{Y }^{{\ast}}\) being a mapping of the type (S⁺) was introduced by Browder [102, p.279].

See also [532, Remark 8.25] for a general discussion of fractional-step methods.

In fact, \(\partial _{y}\mathcal{E}(t,y(t),z_{{\ast}}^{}(t)) \ni 0\) does not need to be satisfied at t = 0, because we did not assume the full stability of the initial condition (y ₀, z ₀) but only (3.4.27h).

An example of advancing this idea might be a finer split, e.g., for the shape-memory model in Section 4.​2.​2.​1 considering \(\mathcal{E}(y,z) =\int _{\varOmega }W(\nabla y) + \frac{1} {\varepsilon } \vert z-\nabla u\vert ^{2} + \mathbb{H}\nabla ^{2}y\begin{array}{c}. \\. \\.\end{array}\,\nabla ^{2}y\) with W only rank-one convex and \(\mathcal{R}\) from (4.​2.​34).

It is worth noting that the resulting algorithm shares similar features with the backtracking scheme introduced by Bourdin [89] (see also [91, Sect. 8.3]) in the context of variational theories for fracture and crack propagation but without direct checking of the two-sided energy estimate.

The original Mosco transformation [440] was designed rather for the optimality conditions as variational inequalities, which works even for nonpotential problems.

See, e.g., [90, 467] for numerical experiments with AMA in the context of fracture mechanics.

There is no definite terminological agreement in the literature. Sometimes, “nonassociative” means that there is no unique activation threshold associated to the dissipation mechanism, i.e., \(\mathcal{R}\) depends also on q, while in other works, it means that the dissipative rate-force relation \(\xi \in A(\dot{z})\) is only premonotone but has no potential; see Sections 1.​9 and 3.2.1. For example, [257] refers to situations in which plastic behavior is more accurately modeled by prescribing a yield function, for use in determining when plastic flow takes place, and in addition a flow potential, which is used to determine the plastic strain rates.

This definition was formulated in [529, 531] for a specific delamination problem.