RATE-INDEPENDENT PROCESSES WITH LINEAR GROWTH ENERGIES AND TIME-DEPENDENT BOUNDARY CONDITIONS

A rate-independent evolution problem is considered for which the stored energy density depends on the gradient of the displacement. The stored energy density does not have to be quasiconvex and is assumed to exhibit linear growth at infinity; no further assumptions are made on the behaviour at infinity. We analyse an evolutionary process with positively 1-homogeneous dissipation and time-dependent Dirichlet boundary conditions.


1.
Introduction. In this contribution, we analyse a rate-independent mesoscopic process governed by time-dependent Dirichlet boundary conditions. A characteristic feature of the problem is that the stored energy has linear growth at infinity. A similar problem for fixed Dirichlet data, but a time-dependent applied force has been previously considered by the authors [9]. That formulation, however, requires a restriction on the norm of the applied force. This difficulty vanishes in the case under consideration.
The rate-independent process we consider models plastic behaviour of a solid. A sketch of the motivation is as follows (see also [9]). Crystalline materials can often be characterised via energy minimisation; for plastically deformed crystals, Ortiz and Repetto [13] provide a setting in which dislocation structures can be described by a nonconvex minimisation problem. The nature of this variational model is incremental, to reflect the irreversible nature of plastic deformations [13]. We account for these phenomena with a phenomenological dissipation functional. As discussed elsewhere [9], one is led to an energy that depends on a strain tensor and has linear growth at infinity. One important feature of the analysis is that we do not work in BV , since the time-dependent boundary data require continuity of the trace, while the variational arguments build on compactness. To get this combination, we use a fine extension developed by J. Souček, see Subsection 1.2.
The motivation for the analysis of linearly growing energies stem from appli cations in plasticity. In particular, Conti and Ortiz [2] derive an energy that is with γ j being the slip strain, s j the slip direction and m j the plane normal. If one assumes infinite latent hardening and no self-hardening, then one is led to a microscopic energy W that is linear along single-slip orbits. For the macroscopic energy, Conti and Ortiz have shown that the convex envelope in this situation has linear growth on traceless symmetric matrices, and quadratic on trace part Thus, the macroscopic energy is linear except for the trace. We focus here on this linear growth behaviour alone for the sake of clarity of the exposition; the inclusion of a quadratically growing energy of the trace is a technical issue we do want to discuss here. This article is organised as follows. Subsection 1.1 settles the notation; a short synopsis of Young measures and DiPerna-Majda measures is given in Appendix A. We refer the reader to [9] for a similar but slightly more comprehensive overview. Section 2 describes the evolutionary problem with time-dependent boundary condi tions; Section 3 states the required assumptions and Section 4 gives the (construc tive) existence proof.
1.1. Basic notation. Let X be a topological space. We denote the space of realvalued continuous functions in X by C(X). If X is a locally compact space then C 0 (X) denotes the closure of the subspace of functions with the compact support in C(X). We write (X, M, µ) for a measurable space with σ-algebra M. For simplicity, µ is omitted in the notation if X ⊂ R n is open and µ is the n-dimensional Lebesgue measure. We recall that the support of a Borel measure µ is the complement of the largest open set N with µ (N ) = 0.
If X is a locally compact Hausdorff space, we write M (X) for the set of (signed) Radon measures with finite mass supported on X; M + (X) stands for the cone of non-negative Radon measures; Prob (X) is the set of probability measures. The Jordan decomposition for signed measures µ = µ + − µ − gives rise to the total variation |µ| := µ + + µ − . The set M (X) is a Banach space when endowed with the total variation �µ� := |µ| (X) as a norm. By the Riesz Representation Theorem, the dual space to C 0 (X), C 0 (X) � , is isometrically isomorphic with M (X). The weak-� topology on M (X) is defined by this duality and weak-� convergence is denoted u k � u. Finally, if X is compact then the dual space to C(X), C(X) � , is isometrically isomorphic with M (X).
The usual Lebesgue space of p-integrable functions is denoted by L p (X, µ). Again, we suppress µ from the notation if it is the Lebesgue measure. The no tation �µ,

RATE-INDEPENDENT EVOLUTION WITH LINEARLY GROWING ENERGIES 593
domain with smooth boundary. Weak convergence respectively strong convergence is expressed as u k � u respectively u n u as usual. We follow the convention of → writing C for a generic constant, whose value may change from line to line.
1.2. Fine extensions of W 1,1 (Ω; R m ). It is well known that W 1,1 (Ω; R m ) is nonreflexive, that is, a bounded sequence does not necessarily contain a subsequence with a weak limit in W 1,1 (Ω; R m ). Hence, one often looks for an extension of W 1,1 (Ω; R m ). Instead of the usual space of functions of bounded variations, we will w ork with the so-called Souček space [15]; we denote it by W 1,µ Ω; R m . This ex tension consists of functions in L 1 (Ω; R m ) whose gradient is a measure on Ω (see [9], where a similar but more extensive summary is given). The precise formulation is as follows. Let T he weak� convergence in W 1,µ Ω; R m is defined analogously to BV (Ω; R m ); the precise formulation can be found in the literature [15,9]. Moreover, as shown in ¯[ 15, Theorem 1 (iii)], if (u, Du) ∈ W 1,µ Ω; R m , then there is a unique measure ¯T u, Du ∈ M (∂Ω; R m ) such that Ω Ω f or all ϕ ∈ C 1 Ω; R n and all 1 ≤ j ≤ m, where ν is the outward pointing normal. The measure � � � � � � ��¯¯1 ¯¯m T u, Du = T u , Du 1 , . . . , T u , Du m ¯ī s called the trace of u, Du . Here, the measure Du j denotes the jth row of the m atrix-valued measure Du. We now quote the key results which provide a math ematical justification for working in W 1,µ Ω; R m : compactness holds as for BV in the weak topology, but in addition the trace operator is continuous in suitable topologies. This enables us to impose Dirichlet boundary data, which would pose a challenge in the conventional setting of BV . While this is a mathematical justifica tion, the question whether W 1,µ Ω; R m is also the appropriate space in the sense of mechanics, giving agreement with experimental observations, is open.
. Finally, balls in W 1,µ Ω; R m are weakly� compact, which can be seen as in [15,Theorem 6]. The following Poincaré-type inequality has been proved recently [9]. Lemma 1.1. Let Ω ⊂ R n be a bounded domain, with ∂Ω belonging to class C 1 . Let Γ D ⊂ ∂Ω be relatively open and of positive (n − 1)-dimensional Lebesgue measure; suppose further that z ∈ M (Γ D ; R m ). Then there is C > 0 such that the estimate Rate-independent evolution with linearly growing energy and timedependent boundary conditions. We now have the ingredients to start the anal ysis of a rate-independent mesoscopic process governed by time-dependent Dirichlet boundary conditions. The focus is on a relaxed formulation of a problem with linear growth in the stored energy, where we want to study the influence of time-dependent boundary conditions. The analysis resembles that for temporally constant Dirichlet data with a time-dependent applied force carried out by the authors [9]. Yet, the argument is sketched to an extent that the differences become clear (for example, a restriction on the norm of the applied force required in [9] is no longer required here).
As mentioned in the Introduction, we work in a variational setting, where dislo cation structures can be described by a nonconvex minimisation problem; see the pioneering work by Ortiz and Repetto [13] for a related setting for the analysis of plastically deformed crystals. The irreversibility is described by an incremental process (as suggested by [13]); a (phenomenological) dissipation functional is intro duced to this behalf. As in the case of force-governed evolution [9], we consider an energy that depends on a strain tensor and has linear growth at infinity. The linear growth of the energy functional is necessitated by the plastic nature of the problem: it can be shown that in the setting of deformation theory of plasticity, the quasiconvex envelope of a single-slip energy has linear growth [2].
2.1. The stored energy and its relaxation. We first describe the energetic setting, casting it as a variational problem with a linear growth energy. The energy is assumed to be a continuous function W : Ω × R n×m → R such that constants β ≥ α > 0 exist with The motivation for the linear growth comes, as mentioned above, is natural in the setting of deformation theory of plasticity of single-slip systems, see [2].
The variational problem is then to In general, there is no solution to (4), because of the non-reflexivity of the under lying space and the possible non-(quasi)convexity of W (x, ). In order to capture · the limiting behaviour of minimising sequences, we state a relaxed problem, still for a fixed instance of time (which we suppress from the notation for now). The relax ation is in terms of DiPerna-Majda measures η = (ν, σ), see Appendix A. We write GDM F u D (Ω; R m×n ) for the set of DiPerna-Majda-measures generated by gradients of � For the displacement u, the appropriate function space is W 1,µ Ω; R m , the fine extension of L 1 (Ω) in the sense of J. Souček [15], see Subsection 1.2. We note that it is crucial to work in this setting here, rather than in the more familiar setting ō f the space of bounded variations: W 1,µ Ω; R m gives both weak� compactness and weak� continuity of the trace, and this combination is essential for the problem under consideration. The relaxed formulation of (4) is Ω Ω β F R m×n 1 + |s| It can be shown [9] that (5) has a solution and min Ī = inf I, with I given in (4). Moreover, minimising sequences of I generate (in the sense of (44)) minimisers of ¯Ī and every minimiser of I is generated by a minimising sequence of I.
Since microstructures can develop in the problem under consideration, it is rea sonable to introduce a concept of a phase field variable, which we denote here λ. Motivated by applications in shape memory alloys, we introduce a variable akin to one used in [11]. We give one exact formulation below, but many variants are possible.
We suppose that there is L ∈ N and a continuous bounded mapping Λ : R m×n → R L such that Λ j ∈ F for 1 ≤ j ≤ L (with F a subalgebra of the space of bounded and continuous functions, see Appendix A) such that the mesoscopic order parameter λ ā ssociated with the system configuration described by u, Du, σ, ν is given by the formula � � β F R m×n � �w hich means that λ ∈ M Ω; R L is a measure such that, for all g ∈ C Ω , Here for x ∈ Ω, ν x is a probability measure; see Appendix A for the precise defini tion. At present, it is common to augment the energy Γ by a regularising term to be able to prove existence. We follow this line of thought. We suppose that the m easure λ ∈ M Ω; R L introduced in (7) is absolutely continuous with respect to the Lebesgue measure on Ω. We identify it with its density x � → λ(x). Moreover, we will require that λ, which is by definition integrable, belongs even to W 1,2 Ω; R L ; see [11] for a similar regularisation, and a justification. Let � > 0; we then consider Though the time-dependence may not be visible at first glance, Γ � depends on time since η is time-dependent. Finally, we set +∞ otherwise with Q being the state space (defined rigorously in (10) below). Notice that (9) excludes states of the system in which λ is a measure which is not absolutely con tinuous with respect to the Lebesgue measure with fairly regular density. Existence of a minimiser for (9) follows from the existence argument given for (4) and the � weak� compactness of the set of measures η which give rise to λ ∈ W 1,2 Ω; R L .

Evolution.
We now describe the rate-independent evolution for a process with the energy (5), following the setting developed by Mielke and coworkers [12]. We consider the evolution during an arbitrary, but fixed time interval [0, T ]. The evo lution will be triggered by changes in the Dirichlet boundary data. To account for the energy that may be dissipated during the evolution, we follow Mielke and co-workers [11] in introducing a dissipation distance. As for the force-driven evolu tion [9], we define the (mesoscopic) dissipation distance between two DiPerna-Majda measures η 1 , η 2 ∈ GDM u D (Ω; R m×n ), since these measures record the microstruc-F ture.
Let Q be the set of admissible configurations. Each such configuration will be w ritten as q := u, Du, η, λ . Since the boundary data depends on time, the set Q depends on time, but we decouple the time-dependence in the following way. At a given time t, let u D ∈ W 1,µ (Ω; R m ) be the boundary data. Then, let η D be the DiPerna-Majda measure generated by a subsequence of {�u k } k∈N ⊂ L 1 (Ω; R m ) from the definition of W 1,µ (Ω; R m ). Similarly, let λ D be given by (7). We then seek a state q ∈ Q, where with Q 0 being is the set of admissible configurations with homogeneous Dirichlet data, (7), and T u 0 , Du 0 = 0 on Γ D .
Though Q depends on time, this is suppressed from the notation.
We now define the dissipation D : Q × Q → R as Since λ is derived from η, we sometimes write D (η 1 , η 2 ) instead of D (q 1 , q 2 ). We note that the time-dependent boundary conditions lead to a time-dependent DiPerna-Majda measure η and thus both λ and D vary over time. Also, as a con sequence of (9), (11) can be written as D (η 1 , η 2 ) = �λ 1 − λ 2 � L 1 (Ω;R L ) . We notice that D is symmetric, D(η 1 , η 2 ) = D(η 2 , η 1 ) for every admissible pair (η 1 , η 2 ). This condition is not essential and can be relaxed; see [1]. Also, the triangle inequality is valid for D. That is, for any three internal states η 1 , η 2 , η 3 , it holds that Finally, for a process q : [0, T ] Q and a given time interval [t 1 , t 2 ] ⊂ [0, T ], the → temporal dissipation is given by We recall the definition of rate-independent processes as developed by Mielke and co-workers [12].
Definition 2.1. Given q 0 ∈ Q, we say that the process q : [0, T ] Q is a solution → if the following conditions hold in addition to suitable regularity assumptions:
We need to define the notion of convergence in Q, and do so as follows.
The main result of this paper is the following.
Section 4 gives the proof of this theorem.
3. Assumptions. We recall the decomposition into time-dependent and homoge neous parts from (10), in particular η = η D + η 0 . Then Γ from (9) (respectively Γ ρ from (8)) can be decomposed in a contribution with time-dependent boundary data and one with homogeneous Dirichlet data, (we don't split the regularising term here; the form above is sufficient to reveal the regularity we need). For the time-dependent boundary data, we assume that F Also, we make the common assumption (see [5]) that there are constants C 0 , C 1 > 0 such that |∂ t Γ(t, q)| ≤ C 0 (C 1 + Γ(t, q)) .
As a consequence we have Further we require uniform continuity of t � → ∂ t Γ(t, q) in the sense that there is ω : [0, T ] [0, +∞) nondecreasing such that for all t 1 , We also suppose that q � → ∂ t Γ(t, q) is weakly continuous for all t ∈ [0, T ]. 4. Existence proof. The existence of an energetic solution for a suitable u D � � ∈ C 1 [0, T ]; W 1,1 (Ω; R m ) can be shown in a constructive way, using a sequence of incremental problems. We sketch the proof, which follows a now well-established argument, to highlight the incorporation of the time-dependent boundary condi tions (see [7] for a similar argument in a different context, namely that of elasto plasticity). For a given initial condition q τ 0 = q 0 and a given step size τ , it is natural to define q τ k for k = 1, . . . , N as a solution to the problem min Γ (kτ, q) + D q τ k−1 , q .
τ τ The next proposition shows that {q τ k } k∈N is well-defined; accepting this for the moment, we introduce a piecewise interpolant q τ such that q τ (t) : Proposition 4.1. The problem (22) has a solution q τ k which is stable; that is, for every q ∈ Q, Γ kτ, q τ k ≤ Γ (kτ, q) + D q τ k , q .
N Moreover, for all t 1 ≤ t 2 from the set {kτ } k=0 , the following discrete energy in equalities hold if one extends the definition of q τ (t) by setting q τ (t) : The poof is now standard and thus omitted (see, e.g., [7] for details). The next proposition gives the a priori bounds needed to pass to the limit as the step size goes to zero. Then there is κ ∈ R such that and Proof. Using (19), (21) and (24) we get the following a priori bounds for some constants C 0 , C 1 > 0: This, thanks to (3) which holds independently of N and τ . � Next, we define the set of stable states, we recall that Q depends on time even if this is suppressed from the notation. We also define The following proposition will help us to establish the stability of the limiting process.
Proposition 4.3. Let Γ be weakly sequentially lower semicontinuous (as a function of q). Suppose that for all (t * , q * ) ∈ [0, T ] × Q, for all stable sequences {(t k , q k )} k∈N with t k t and q k � q in the sense of Definition 2.2, there is a sequence Then Γ is weakly continuous as a function of t and q along stable sequences and q * ∈ S (t * ).

k→∞ k→∞
The arbitrariness of q ∈ Q shows the stability of q � * .