1 Introduction

BV solutions and viscosity approximations of rate-independent systems

Abstract.

In the nonconvex case solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential which is a viscous regularization of a given rate-independent dissipation potential.

The resulting definition of ‘BV solutions’ involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play a crucial role in the description of the associated jump trajectories.

We shall prove a general convergence result for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems.

AMS Subject Classification: 49Q20, 58E99.

A.M. has been partially supported by DFG, Research Unit FOR 797 “MicroPlast”, Mi 459/5-1. R.R. and G.S. have been partially supported by a MIUR-PRIN’06 grant for the project “Variational methods in optimal mass transportation and in geometric measure theory”.

1. Introduction

Rate-independent evolutions occur in several contexts. We refer the reader to [32] and the forthcoming monograph [39] for a survey of rate-independent modeling and analysis in a wide variety of applications, which may pertain to very different and far-apart branches of mechanics and physics. Rate-independent systems present very distinctive common features, because of their hysteretic character [54, 24]. Driven by external loadings on a time scale much slower than their internal scale, such systems respond to changes in the external actions invariantly for time-rescalings. Thus, they in fact show (almost) no intrinsic time-scale. This kind of behavior is encoded in the simplest, but still significant, example of rate-independent evolution, namely the doubly nonlinear differential inclusion

(DN${}_{0}$)

For the sake of simplicity, we will consider here the case when is a finite dimensional linear space, an energy functional ( denoting the differential of with respect to the variable ), and is a convex, nondegenerate, dissipation potential, hereafter supposed positively homogeneous of degree . Thus, ((DN${}_{0}$)) is invariant for time-rescalings, rendering the system rate independence.

Since the range of is a proper subset of , when is not strictly convex one cannot expect the existence of an absolutely continuous solution of ((DN${}_{0}$)). Over the past decade, this fact has motivated the development of suitable notions of weak solutions to ((DN${}_{0}$)). In the mainstream of [18, 35, 44], the present paper aims to contribute to this issue. Relying on the vanishing-viscosity approach, we shall propose the notion of solution to ((DN${}_{0}$)) and thoroughly analyze it.

To better motivate the use of vanishing viscosity and highlight the features of the concept of solution, in the next paragraphs we shall briefly recall the other main weak solvability notions for ((DN${}_{0}$)). For the sake of simplicity, we shall focus on the particular case

(1.1)

Energetic and local solutions.

The first attempt at a rigorous weak formulation of ((DN${}_{0}$)) goes back to [40] and the subsequent [42, 41], which advanced the notion of global energetic solution to the rate-independent system ((DN${}_{0}$)). In the simplified case (1.1), this solution concept consists of the following relations, holding for all :

($\mathrm{S}$)
($\mathrm{E}$)

The energy identity (($\mathrm{E}$)) balances at every time the dissipated energy (the latter symbol denotes the total variation of the solution on the interval ), with the stored energy , the initial energy, and the work of the external forces. On the other hand, (($\mathrm{S}$)) is a stability condition, for it asserts that the change from the current state to another state brings about a gain of potential energy smaller than the dissipated energy. Since the competitors for range in the whole space , (($\mathrm{S}$)) is in fact a global stability condition.

The global energetic formulation (($\mathrm{S}$))–(($\mathrm{E}$)) only involves the (assumedly smooth) power of the external forces , and is otherwise derivative-free. Thus, it is well suited to jumping solutions. Furthermore, as shown in [27, 32], it is amenable to analysis in very general ambient spaces, even with no underlying linear structure. Because of its flexibility, this concept has been exploited in a variety of applicative contexts, like, for instance, shape memory alloys [42, 37, 5], crack propagation [15, 14, 17], elastoplasticity [29, 30, 31, 20, 10, 11, 28], damage in brittle materials [38, 6, 52, 33], delamination [23], ferroelectricity [43], and superconductivity [50].

On the other hand, in the case of nonconvex energies condition (($\mathrm{S}$)) turns out to be a strong requirement, for it may lead the system to change instantaneously in a very drastic way, jumping into very far-apart energetic configurations (see, for instance, [30, Ex. 6.1], [21, Ex. 6.3], and [35, Ex. 1]). On the discrete level, global stability is reflected in the global minimization scheme giving raise to approximate solutions by time-discretization. Indeed, for a fixed time-step , inducing a partition of the interval , one constructs discrete solutions of (($\mathrm{S}$))–(($\mathrm{E}$)) by setting and then solving recursively the variational incremental scheme

($\mathrm{IP}_{0}$)

However, a scheme based on local minimization would be preferable, both in view of numerical analysis and from a modeling perspective, see the discussions in [30, Sec. 6] and, in the realm of crack propagation, [16, 45, 26].

As pointed out in [16], local minimization may be enforced by perturbing the variational scheme (($\mathrm{IP}_{0}$)) with a term, modulated by a viscosity parameter , which penalizes the squared distance from the previous step

($\mathrm{IP}_{\varepsilon}$)

and depends on a second norm , typically Hilbertian, on the space . In a infinite-dimensional setting, one may think of , with a domain in , , and , the and norms, respectively. Notice that, on the time-continuous level, (($\mathrm{IP}_{0}$)) corresponds to the viscous doubly nonlinear equation

(DN${}_{\varepsilon}$)

(see [9, 8] for the existence of solutions ). Then, the idea would be to consider the solutions to ((DN${}_{0}$)) arising in the passage to the limit, in the discrete scheme (($\mathrm{IP}_{\varepsilon}$)), as and tend to simultaneously, keeping . One can guess that, at least formally, this procedure should be equivalent to considering the limit of the solutions to ((DN${}_{\varepsilon}$)) as .

Vanishing viscosity has by now become an established selection criterion for mechanically feasible weak solvability notions of rate-independent evolutions. We refer the reader to [25] for rate-independent problems with convex energies and discontinuous inputs, and, in more specific applied contexts, to [12] for elasto-plasticity with softening, to [19] for general material models with nonconvex elastic energies, the recent [13] for cam-clay non-associative plasticity, and [53, 21, 22] for crack propagation. Since the energy functionals involved in such applications are usually nonsmooth and nonconvex, the passage to the limit mostly relies on lower semicontinuity arguments. Let us illustrate the latter in the prototypical case ((DN${}_{\varepsilon}$)). The key observation is that ((DN${}_{\varepsilon}$)) is equivalent (see the discussion in Section 2.4) to the -energy identity

(1.2)

for all , where the term

(1.3)

measures the distance with respect to the dual norm of from the set . The term defined in (1.3) is penalized in (1.2) by the coefficient . Thus, passing to the limit in (1.2) as , one finds

Hence,

(1.4)

which is a local version of the global stability (($\mathrm{S}$)). Furthermore, (1.2) yields, via lower-semicontinuity, the energy inequality

(1.5)

Conditions (1.4)–(1.5) give raise to the notion of local solution of the rate-independent system ((DN${}_{0}$)).

While the local stability (1.4) is more physically realistic than (($\mathrm{S}$)), its combination with the energy inequality (1.5) turns out to provide an unsatisfactory description of the solution at jumps (see the discussion in [35, Sec. 5.2] and Remark 2.8 later on). In order to capture the jump dynamics, the energetic behavior of the system in a jump regime has to be revealed. From this perspective, it seems to be crucial to recover from (1.2), as , an energy identity, rather than an energy inequality. Thus, the passage to the limit has to somehow keep track of the limit of the term

which in fact encodes the contribution of the viscous dissipation , completely missing in (1.5).

BV solutions.

Moving from these considerations, it is natural to introduce the vanishing viscosity contact potential (which is related to the bipotential discussed in [7], see Section 3) induced by , i.e. the quantity

(1.6)

Then, the -energy identity (1.2) yields the inequality

(1.7)

see Section 3.1. Passing to the limit in (1.7), in Theorem 4.10 we shall prove that, up to a subsequence, the solutions of the viscous equation ((DN${}_{\varepsilon}$)) converge, as , to a curve satisfying the local stability (1.4) and the following energy inequality

(1.8)

Without going into details (see Definition 3.4 later on), we may point out that (1.8) features a notion of (pseudo)-total variation (denoted by ) induced by the vanishing viscosity contact potential  (1.6) and the energy . The main novelty is that a -curve obeying the local stability condition (1.4) always satisfies the opposite inequality in (1.8), thus yielding the energy balance

($\mathrm{E}_{{{\mathfrak{p}}},{\mathcal{E}}}$)

In fact, provides a finer description of the dissipation of , along a jump between two values and at time : it involves not only the quantity related to the dissipation potential (1.1), but also the viscous contribution induced by the vanishing viscosity contact potential through the formula

(1.9)

By a rescaling technique, it is possible to show that, in a jump point, the system may switch to a viscous behavior, which is in fact reminiscent of the viscous approximation ((DN${}_{\varepsilon}$)). In particular, when the jump point is of viscous type, the infimum in (1.9) is attained and the states and are connected by some transition curve , fulfilling the viscous doubly nonlinear equation

(in the case the norm is Euclidean and we use its differential to identify with ). The combination of (1.4) and (1.8) yields the notion of solution to the rate-independent system . This concept was first introduced in [35], in the case the ambient space is a finite-dimensional manifold , and both the rate-independent and the viscous approximating dissipations depend on one single Finsler distance on . In this paper, while keeping to a Banach framework, we shall considerably broaden the class of rate-independent and viscous dissipation functionals, cf. Remark 2.4. Moreover, the notion of solution shall be presented here in a more compact form than in [35], amenable to a finer analysis and, hopefully, to further generalizations.

Let us now briefly comment on our main results. First of all, we are going to show in Theorems 4.3, 4.6, and 4.7 that the concept of rate-independent evolution completely encompasses the solution behavior in both a purely rate-independent, non-jumping regime, and in jump regimes, where the competition between dry-friction and viscous effects is highlighted. Indeed, from (1.4) and (1.8) it is possible to deduce suitable energy balances at jumps (cf. conditions ((J${}_{\mathrm{BV}}$)) in Theorem 4.3).

Then, in Theorem 4.10 we shall prove that, along a subsequence, the viscous approximations arising from ((DN${}_{\varepsilon}$)) converge as to a solution. Next, our second main result, Theorem 4.11, states that, up to a subsequence, also the discrete solutions constructed via the -discretization scheme (($\mathrm{IP}_{\varepsilon}$)) converge to a solution of ((DN${}_{0}$)) as and simultaneously, provided that the respective convergence rates are such that

Finally, in Section 5 we shall develop a different approach to solutions, via the rescaling technique advanced in [18] and refined in [35, 44]. The main idea is to suitably reparametrize the approximate viscous curves in order to capture, in the vanishing viscosity limit, the viscous transition paths at jumps points. This leads to performing an asymptotic analysis as of the graphs of the functions , in the extended phase space . For every the graph of can be parametrized by a couple of functions , being the (strictly increasing) rescaling function and the rescaled solution. In Theorem 5.6 we assert that, up to a subsequence, the functions converge as to a parametrized rate-independent solution. By the latter terminology we mean a curve fulfilling

(1.10a)
(1.10b)
and the energy identity
(1.10c)

As already pointed out in [18, 35], like the notion of solution, relations (1.10) as well comprise both the purely rate-independent evolution as well as the viscous transient regime at jumps. The latter regime in fact corresponds to the case : the system does not obey the local stability constraint (1.4) any longer, and switches to viscous behavior, see also Remark 5.7 later on.

As a matter of fact, Theorem 5.8 shows that parametrized rate-independent solutions may be viewed as the “continuous counterpart” to evolutions. With a suitable transformation, it is possible to associate with every parametrized rate-independent solution a one, and conversely. One advantage of the parametrized notion is that it avoids the technicalities related to functions. Hence, it is for instance more easily amenable to a stability analysis (cf. [35, Rmk. 6]). Furthermore, in [44] a highly refined vanishing viscosity analysis has been developed, with this reparametrization technique, in the infinite-dimensional -framework, where ((DN${}_{\varepsilon}$)) is replaced by a general quasilinear evolutionary PDE.

Generalizations and future developments.

So far we have focused on dissipation functionals of the type (1.1) and as in ((DN${}_{\varepsilon}$)) for expository reasons only, in order to highlight the main variational argument leading to the notion of solution. Indeed, the analysis developed in this paper is targeted to a general

positively -homogeneous, convex dissipation ,

(cf. (2.1)), and considers a fairly wide class of approximate viscous dissipation functionals , defined by conditions (($\Psi.1$))–(($\Psi.3$)) in Section 2.3. Furthermore, at the price of just technical complications, our results could be extended to the case of a Finsler-like family of dissipation functionals , depending on the state variable , and satisfying uniform bounds and Mosco-continuity with respect to , see [35, Sect. 2] and [47, Sect. 6, 8].

The extension to infinite-dimensional ambient spaces and nonsmooth energies is crucial for application of the concept of solution to the PDE systems modelling rate-independent evolutions in continuum mechanics. A first step in this direction is to generalize the known existence results for doubly nonlinear equations, driven by a viscous dissipation, to nonconvex and nonsmooth energy functionals in infinite dimensions. As shown in [48, 47], in the nonsmooth and nonconvex case one can replace the energy differential with a suitable notion of subdifferential . Accordingly, instead of continuity of , one asks for closedness of the multivalued subdifferential in the sense of graphs. These ideas shall be further advanced in the forthcoming work [36]. Therein, exploiting techniques from nonsmooth analysis, we shall also tackle energies which do not depend smoothly on time (this is relevant for rate-independent applications, see e.g. [22] and [25]).

On the other hand, the requirement that the ambient space is finite-dimensional could be replaced by suitable compactness (of the sublevels of the energy) and reflexivity assumptions on the ambient space . The latter topological requirement in fact ensures that has the so-called Radon-Nikodým property, i.e. that absolutely continuous curves with values in are almost everywhere differentiable. The vanishing viscosity analysis in spaces which do not enjoy this property requires a subtler approach, involving metric arguments (see e.g. [47, Sect. 7]), or ad-hoc stronger estimates [44]. See also [34] for some preliminary approaches to BV solutions for PDE problems.

Plan of the paper

Section 2 is devoted to an extended presentation of energetic and local solutions to rate-independent systems. In particular, after fixing the setup of the paper in Section 2.1, in Sec. 2.2 we recall the definition of global energetic solution, show its differential characterization and the related variational time-incremental scheme. We develop the vanishing-viscosity approach in Secs. 2.3 and 2.4, thus arriving at the notion of local solution (see Section 2.5), which also admits a differential characterization.

In Section 3 we introduce the concept of vanishing viscosity contact potential and thoroughly analyze its properties, as well as the induced (pseudo)-total variation. With these ingredients, in Sec. 4 we present the notion of solution. We show that rate-independent evolutions admit, too, a differential characterization, and, in Sec. 4.2, that they provide a careful description of the energetic behavior of the system. Then, in Section 4.3, we state our main results on solutions.

While Section 5 is focused on the alternative notion of parametrized rate-independent solutions, the last Sec. 6 contains some technical results which lie at the core of our theory.

2. Global energetic versus local solutions, and their viscous regularizations

In this section, we will briefly recall the notion of energetic solutions and show that their viscous regularizations give raise to local solutions.

2.1. Rate-independent setting: dissipation and energy functionals

We let

endowed with a gauge function , namely a

non-degenerate, positively -homogeneous, convex dissipation , (2.1)

i.e.  satisfies if , and

In particular, there exists a constant such that

Since is -homogeneous, its subdifferential can be characterized by

(2.2)

takes its values in the convex set , given by

(2.3)

which enjoys some useful (and well-known, see e.g. [46]) properties. For the reader’s convenience we list them here:

  1. is the proper domain of the Legendre transform of , since

    (2.4)
  2. is the support function of , since

    (2.5)

    and is the polar set of the unit ball associated with .

  3. is the unit ball of the support function of :

    (2.6)
  4. In the even case (i.e., when for all ), we have that is an equivalent norm for , is its dual norm, and are their respective unit balls.

Further, we consider a smooth energy functional

which we suppose bounded from below and with energy-bounded time derivative

(2.7)

where denotes the positive part. The rate-independent system associated with the energy functional and the dissipation potential can be formally described by the rate-independent doubly nonlinear differential inclusion

(DN${}_{0}$)

As already mentioned in the Introduction, for nonconvex energies solutions to ((DN${}_{0}$)) may exhibit discontinuities in time. The first weak solvability notion for ((DN${}_{0}$)) is the concept of (global) energetic solution to the rate-independent system ((DN${}_{0}$)) (see [42, 40, 41] and the survey [32]), which we recall in the next section.

2.2. Energetic solutions and variational incremental scheme

Definition 2.1 (Energetic solution).

A curve is an energetic solution of the rate independent system if for all the global stability (S) and the energy balance (E) holds:

(S)
(E)

functions.

Hereafter, we shall consider functions of bounded variation pointwise defined in every point , such that the pointwise total variation with respect to (any equivalent norm of can be chosen) is finite, where

Notice that a function in admits left and right limits at every

(2.8)

and its pointwise jump set is the at most countable set defined by

(2.9)

We denote by the distributional derivative of , and recall that is a Radon vector measure with finite total variation . It is well known [3] that can be decomposed into the sum of the three mutually singular measures

(2.10)

Here, is the absolutely continuous part with respect to the Lebesgue measure , whose Lebesgue density is the usual pointwise (and -a.e. defined) derivative, is a discrete measure concentrated on , and is the so-called Cantor part, still satisfying for every . Therefore is the diffuse part of the measure, which does not charge . In the following, it will be useful to use a nonnegative and diffuse reference measure on such that and are absolutely continuous w.r.t. : just to fix our ideas, we set

(2.11)

With a slight abuse of notation, for every we denote by the integral

(2.12)

Since is -homogeneous, the above integral is independent of , provided is absolutely continuous w.r.t. .

Towards a differential characterization of energetic solutions.

Let us first of all point out that ((S)) is stronger than the local stability condition

($\mathrm{S}_{\mathrm{loc}}$)

which can be formally deduced from ((DN${}_{0}$)) and (2.3). Indeed, the global stability ((S)) yields for every and

and therefore, dividing by and passing to the limit as , one gets

so that (($\mathrm{S}_{\mathrm{loc}}$)) holds. We obtain more insight into ((E)) by representing the variation in terms of the distributional derivative of . In fact, recalling (2.11) and (2.12), we have

where the jump contribution can be described, in terms of the quantities

(2.13)

by

(2.14)

Also notice that, as usual in rate-independent evolutionary problems, is pointwise everywhere defined and the jump term takes into account the value of at every time . Therefore, if is not continuous at , this part may yield a strictly bigger contribution than the total mass of the distributional jump measure (which gives rise to the so-called essential variation).

The following result provides an equivalent characterization of energetic solutions: besides the global stability condition ((S)), it involves a formulation of the differential inclusion ((DN${}_{0}$)) (cf. the subdifferential formulation of [41]) and a jump condition at any jump point of .

Proposition 2.2.

A curve satisfying the global stability condition ((S)) is an energetic solution of the rate-independent system if and only if it satisfies the differential inclusion

(DN${}_{0,\mathrm{BV}}$)

and the jump conditions

(J${}_{\text{ener}}$)

for every (recall convention (2.8) in the case ).

We shall simply sketch the proof, referring to the arguments for the forthcoming Proposition 2.7 for all details.

Proof.

By the additivity property of the total variation , ((E)) yields for every

(E’)

Arguing as in the proof of Proposition 2.7 later on, one can see that the global stability ((S)) and ((E’)) yield the differential inclusion ((DN${}_{0,\mathrm{BV}}$)) and conditions ((J${}_{\text{ener}}$)).

Conversely, repeating the arguments of Proposition 2.7 one can verify that ((DN${}_{0,\mathrm{BV}}$)) and ((J${}_{\text{ener}}$)) imply ((E)). ∎

Incremental minimization scheme

Existence of energetic solutions can be proved by solving a minimization scheme, which is also interesting as construction of an effective approximation of the solutions.

For a given time-step we consider a uniform partition (for simplicity) , , of the time interval , and an initial value . In order to find good approximations of we solve the incremental minimization scheme

(IP${}_{0}$)

Setting

(2.15)

it is possible to find a suitable vanishing sequence of step sizes (see, e.g., [41, 32] for all calculations), such that

and is an energetic solution of ((DN${}_{0}$)).

2.3. Viscous approximations of rate-independent systems

In the present paper we want to study a different approach to approximate and solve ((DN${}_{0}$)): the main idea is to replace the linearly growing dissipation potential with a suitable convex and superlinear “viscous” regularization of , depending on a “small” parameter and “converging” to in a suitable sense as . Solving the doubly nonlinear differential inclusion (we use the notation for the time derivative when is absolutely continuous)

(DN${}_{\varepsilon}$)

one can consider the sequence as a good approximation of the solution of ((DN${}_{0}$)) as .

There is also a natural discrete counterpart to ((DN${}_{\varepsilon}$)), which regularizes the incremental minimization problem ((IP${}_{0}$)). We simply substitute by in ((IP${}_{0}$)), recalling that now the time-step should explicitly appear, since is not -homogeneous any longer. The viscous incremental problem is therefore

(IP${}_{\varepsilon}$)

Setting as in (2.15)

one can study the limit of the discrete solutions when and , under some restriction on the behavior of the quotient (see Theorem 4.11 later on).

The choice of the viscosity approximation .

Here we consider the particular case when the potential can be obtained starting from a given

($\Psi.1$)

by the canonical rescaling

($\Psi.2$)

and is linked to by the relation

($\Psi.3$)
Remark 2.3.

Notice that, by convexity of and the fact that , the map is nondecreasing for all . Hence,

(2.16)

Furthermore, by the coercivity condition (($\Psi.1$)),

Here are some examples, showing that (($\Psi.2$)) still provides a great flexibility and covers several interesting cases.

Example 2.4.
  • The simplest example, still absolutely non trivial [35], is to consider

    (2.17)

    A similar regularization can be obtained by choosing a real convex and superlinear function , with , and setting

    (2.18)
  • The most interesting case involves an arbitrary norm on and considers for

    (2.19)

    where is the -duality map associated with . In particular, if is a Hilbertian norm and , then is the Riesz isomorphism and we can choose by identifying with . Hence, ((DN${}_{\varepsilon}$)) reads

    and the incremental problem ((IP${}_{\varepsilon}$)) looks for which recursively minimizes

    This is the typical situation which motivates our investigation.

  • More generally, we can choose a convex “viscous” potential satisfying

    (2.20)

    and set