domain: math math

text:
Gilles A Francfort
email: gilles.francfort@univ-paris13.fr

Maria Giovanna Mora

Quasistatic evolution in non-associative plasticity revisited

Keywords: Elasto-plasticity, Functions of bounded deformation, Quasistatic evolution, Nonassociative plasticity, Radon measures

The mathematical treatment of evolutionary non-associative elasto-plasticity is still in its infancy. In particular, all existence results thus far rely on a spatially mollified stress admissibility constraint. Further, the evolution is formulated in a rescaled time from which it is very difficult to infer any useful information on the "real" time evolution. We propose a causal spatio-temporal mollification of the stress admissibility constraint that, while no more far-fetched than a purely spatial one, produces a more elegant and complete evolution for such models, and this in the "real" time variable.

Introduction

In the past few years, small strain elasto-plasticity has seen a (modest) rebirth because of a change of perspective. While in the seminal work of P.-M. Suquet [START_REF] Suquet | Sur les équations de la plasticité: existence et régularité des solutions[END_REF], elasto-plastic evolution was seen as the limit of visco-plastic evolutions as the viscosity parameter tends to 0, the evolution is now preferably viewed [START_REF] Maso | Quasistatic evolution problems for linearly elastic-perfectly plastic materials[END_REF] as an energy conserving variational evolution in the spirit of A. Mielke's general framework [START_REF] Mielke | Evolution of rate-independent systems[END_REF].

Elasto-plastic evolution of a homogeneous elasto-plastic material occupying a volume Ω ⊂ R n , with Hooke's law (elasticity tensor) A and subject to a time-dependent loading process with, say, f (t) as body loads, g(t) as surface loads on a part Γ n of ∂Ω, and w(t) as displacement loads (hard device) on the complementary part Γ d of ∂Ω results in the following system where Eu(t) denote the infinitesimal strain at t, that is, the symmetric part of the spatial gradient of the displacement field u(t) at t, σ(t) is the Cauchy stress tensor at time t, and e(t) and p(t) (a deviatoric symmetric matrix) are the elastic and plastic strain at t:

• Kinematic compatibility: Eu(t) = e(t) + p(t) in Ω and u(t) = w(t) on Γ d ;

• Equilibrium: div σ(t) + f (t) = 0 in Ω and σ(t)ν = g(t) on Γ n , where ν denotes the outer unit normal to ∂Ω; • Constitutive law: σ(t) = Ae(t) in Ω;

• Stress constraint: σ D (t) ∈ K, where σ D is the deviatoric part of the Cauchy stress σ, and K is the admissible set of stresses (a convex and compact subset of deviatoric n × n matrices); • Flow rule: ṗ(t) = 0 if σ D (t) ∈ int K, while ṗ(t) belongs to the normal cone to K at σ D (t) if σ D (t) ∈ ∂K. The corresponding variational evolution, as discussed in [START_REF] Maso | Quasistatic evolution problems for linearly elastic-perfectly plastic materials[END_REF], formally consists in the following four-pronged formulation, for t ∈ [0, T ],

• Kinematic compatibility: Eu(t) = e(t) + p(t) in Ω and u(t) = w(t) on Γ d ;

• Global stability: The triplet (u(t), e(t), p(t)) globally minimizes Unfortunately that formulation seems ill suited to deal with a slew of plastic models that share a common feature, non-associativity. The concerned material behavior ranges from that of a soil (in, e.g., Cam-Clay plasticity or Drucker-Prager plasticity) to that of a metal when nonlinear kinematic hardening is introduced (Armstrong-Frederick type models 1 ).

In any case, those non-associative models are thought to lie beyond the reach of any kind of variational formulation. However, it has been recently shown, first for Cam-Clay plasticity in [START_REF] Maso | Solombrino: Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling[END_REF][START_REF] Maso | Solombrino: Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solutions[END_REF], then for Drucker-Prager plasticity in [START_REF] Babadjian | Quasistatic evolution in non-associative plasticity -the cap model[END_REF], and finally for Armstrong-Frederick type materials in [START_REF] Francfort | Quasi-static evolution for the Armstrong-Frederick hardening-plasticity model[END_REF], that this view is not correct and that a variational formulation can be brought forth for a rather generic model of non-associative elasto-plasticity.

However that formulation is plagued by several defects. First it cannot proceed if the set of admissible stresses is not bounded in all directions in the space of symmetric matrices. But this is not so unless a cap is imposed on the admissible stresses in some directions [START_REF] Babadjian | Quasistatic evolution in non-associative plasticity -the cap model[END_REF]. Then, the variational formulation introduces a set of admissible stresses that depends on the actual stress. The analysis does not seem to handle that dependence very well because the actual stress is not continuous. The way out consists in mollifying the actual stress in the formulation [START_REF] Maso | Solombrino: Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling[END_REF].

Finally, it proves impossible to carry through the analysis in real time because of a lack of Lipschitz estimates in time (see, e.g., [START_REF] Maso | Solombrino: Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling[END_REF]). The remedy is to rescale time (essentially using the energy as the new time), so as to recover Lipschitz estimates. That idea, borrowed from [START_REF] Mielke | Modeling solutions with jumps for rate-independent systems on metric spaces[END_REF][START_REF] Mielke | BV solutions and viscosity approximations of rate-independent systems[END_REF], but whose germ is in [START_REF] Efendiev | On the rate-independent limit of systems with dry friction and small viscosity[END_REF], yields an elasto-plastic evolution in rescaled time. It is then extremely difficult to recover a real time evolution; this is the object of [START_REF] Maso | Solombrino: Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solutions[END_REF] which only partially succeeds in such an endeavor.

In this paper, we propose to demonstrate that such a rescaling is unnecessary in all models of non-associative elasto-plasticity that have been considered thus far, provided that the mollification of the actual stress is (slightly) modified. In lieu of a spatial convolution, we suggest a space-time convolution, the time part of that convolution only involving the past so that causality is preserved.

The paper is organized as follows. A first section is devoted to the generic formulation of a non-associative cap model in a framework that will be palatable to the subsequent analysis. The second section addresses the existence of a visco-plastic evolution. Viscosity type approximations for quasi-static elasto-plastic evolution problems have been used many times, starting with the pioneering work of P.-M. Suquet [START_REF] Suquet | Sur les équations de la plasticité: existence et régularité des solutions[END_REF]; more recently, approximations that combine viscosity and dynamics have also been proposed [START_REF] Maso | Scala: Quasistatic evolution in perfect plasticity as limit of dynamic processes[END_REF]. In all cases, the approximation rested on a time-incremental process whereas, in our setting, existence cannot be secured through such a process. Rather, we use a faster and, in our opinion, more elegant fixed point argument. In Section 4 we prove the main result of the paper, namely Theorem 4.1, which states an existence theorem in real time.

The obtained results conform to our expectation that one should indeed recover an evolution in real time, at least in the Drucker-Prager context, because the solution in the spatially homogeneous case has been shown in [START_REF] Laborde | A nonmonotone differential inclusion[END_REF]Theorem 5] to be continuous in time, at least for particular classes of sets of admissible stresses.

Finally, we limit the loading process to a hard device, that is, to a displacement w(t) acting on the entirety of the boundary ∂Ω of our domain. This is certainly a simplifying assumption because it alleviates in particular the need for safe load conditions on the loads f (t) and g(t) and renders duality much simpler. Those can become at times a thorny issue in plasticity. We are confident that the analysis remains the same if general loads were to be incorporated into the evolution.

Description of the model

In this section, we provide an overview of the type of models that can be addressed.

2.1. The relevant model. The context is that of small strains. Let Ω ⊂ R n be a bounded open set occupied by a homogeneous elasto-plastic material. We denote by u(t) : Ω → R n the displacement field at time t and by Eu(t) := (Du(t) + Du T (t))/2 the strain tensor. As is usual in small deformations plasticity, the strain tensor is additively decomposed as

Eu(t) = e(t) + p(t),
where e(t) and p(t) stand for the elastic and plastic strains, respectively. This is part of what will be referred to as kinematic compatibility. The constitutive equation which relates the (Cauchy) stress tensor σ to the elastic part e of the linearized strain is also assumed to be linear, i.e.,

σ(t) = Ae(t),
where A is the Hooke tensor. At equilibrium, and if no volume forces are applied to the sample, the stress satisfies div σ(t) = 0 in Ω.

It is also constrained to remain in a σ-dependent compact convex subset K(σ(t)) of the set M n×n sym of n × n symmetric matrices, that is,

σ(t) ∈ K(σ(t)).
The behavior of the plastic strain is governed by the following flow rule: denoting by ṗ the time derivative of p, ṗ(t) ∈ ∂I K(σ(t)) ,

where I K denotes the indicator function of the set K.

Finally, as announced in the introduction, the material is subject to a hard loading device; in other words, a Dirichlet boundary condition u(t, x) = w(t, x) is imposed on ∂Ω.

Our goal is to obtain a triplet (u(t, x), e(t, x), p(t, x)) such that

               Eu(t, x) = e(t, x) + p(t, x), σ(t, x) = Ae(t, x), div σ(t, x) = 0, σ(t, x) ∈ K(σ(t, x)), ṗ(t, x) ∈ ∂I K(σ(t,x)) ,
together with the Dirichlet boundary condition. Of course, we know from prior works on plasticity that we cannot expect the boundary condition to be satisfied because plastic strains may develop at the boundary, so that, as seen later, we will have to replace that condition by p(t, x) = (w(t, x) -u(t, x)) ν(x) on ∂Ω.

Throughout, the symbol stands for the symmetrized tensor product, while ν denotes the outer unit normal to ∂Ω.

When K(σ) is independent of σ, the model is associative, and the plastic strain rate obeys the usual normality flow rule (see [START_REF] Lubliner | Plasticity Theory[END_REF]) ṗ ∈ N K (σ),

where N K (σ) is the normal cone to K at σ ∈ K. However, whenever significant volume variations accompany the plastic deformation, the principle of maximum plastic work is no longer valid and thus the associative flow rule should be abandoned in favor of a non-associative model. Various works [START_REF] Babadjian | Quasistatic evolution in non-associative plasticity -the cap model[END_REF][START_REF] Maso | Solombrino: Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling[END_REF][START_REF] Maso | Solombrino: Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solutions[END_REF][START_REF] Maso | Solombrino: Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case[END_REF][START_REF] Francfort | Quasi-static evolution for the Armstrong-Frederick hardening-plasticity model[END_REF] have recently tackled the issue of non-associativity and shown that many non-associative models can actually be viewed as "regular" models of elasto-plasticity, provided that the set of admissible stresses is allowed to depend on the stress field σ.

We next define the dissipation potential H :

M n×n sym × M n×n sym → R as the support function of K(σ), that is, H(σ, p) = max τ ∈K(σ)
τ : p, and note that, for a fixed σ, it is convex, subadditive, and positively one-homogeneous in p.

By standard convex analysis, the inclusion ṗ

∈ ∂I K(σ) (σ) is equivalent to σ ∈ ∂ 2 H(σ, ṗ)
, where ∂ 2 H(σ, ṗ) denotes the subdifferential of ξ → H(σ, ξ) at ṗ. Thus, an equivalent formulation of the problem is

               Eu(t, x) = e(t, x) + p(t, x), p(t, x) = (w(t, x) -u(t, x)) ν(x) on ∂Ω, σ(t, x) = Ae(t, x), div σ(t, x) = 0, σ(t, x) ∈ ∂ 2 H(σ(t, x), ṗ(t, x)).
Note that the last condition implies the stress constraint σ(t, x) ∈ K(σ(t, x)) by the positive homogeneity of ξ → H(σ, ξ).

In what follows, we will assume that: (H0) The map ξ → H(x, ξ) is convex and positively one-homogeneous; (H1) The map H is continuous over M n×n sym × M n×n sym ; (H2) There exist 0 < α H < β H < +∞ such that

B(0, α H ) ⊂ K(σ) ⊂ B(0, β H ) for every σ ∈ M n×n sym , (2.1)
or still

α H |p| ≤ H(σ, p) ≤ β H |p| for every σ, p ∈ M n×n sym ; (2.2) (H3) There exists a constant C H > 0 such that |H(σ 1 , p) -H(σ 2 , p)| ≤ C H |p||σ 1 -σ 2 | for any σ 1 , σ 2 , p ∈ M n×n sym ; (H4) There exists a constant C H > 0 such that |P K(σ1) (τ ) -P K(σ2) (τ )| ≤ C H |σ 1 -σ 2 | for any σ 1 , σ 2 , τ ∈ M n×n
sym , where P K(σ) denotes the minimal distance projection onto the convex set K(σ).

We introduce the perturbed dissipation potential

H ε : M n×n sym × M n×n sym → [0, +∞) defined, for each ε > 0, as H ε (σ, p) := H(σ, p) + ε 2 |p| 2 . The convex conjugate H * ε : M n×n sym × M n×n sym → [0, +∞) of H ε with respect to the second variable is defined by H * ε (σ, τ ) := sup p∈M n×n sym τ : p -H ε (σ, p) .
Using standard convex analysis, see [START_REF] Rockafellar | Convex Analysis[END_REF]Theorem 16.4], one can show that

H * ε (σ, τ ) = |τ -P K(σ) (τ )| 2 2ε .
In particular, H * ε is differentiable with respect to the second variable, and its partial derivative is given by

N ε (σ, τ ) = ∂ 2 H * ε (σ, τ ) = τ -P K(σ) (τ ) ε .
Note that, since 0 ∈ K(σ) by (2.1), we have

|N ε (σ, τ )| ≤ 1 ε |τ |,
and this implies that, for any σ, τ 1 and τ 2 ∈ M n×n sym ,

|H * ε (σ, τ 1 ) -H * ε (σ, τ 2 )| ≤ 1 ε (|τ 1 | + |τ 2 |)|τ 1 -τ 2 |.
Actually, N ε is Lipschitz continuous. Indeed, we have the following result.

Lemma 2.1. Let C H := C H + 2, where C H is the constant in (H4). Then |N ε (σ 1 , τ 1 ) -N ε (σ 2 , τ 2 )| ≤ C H ε (|σ 1 -σ 2 | + |τ 1 -τ 2 |) for any σ 1 , σ 2 , τ 1 and τ 2 ∈ M n×n sym . Proof. By definition of N ε and since the projection is 1-Lipschitz continuous, |N ε (σ, τ 1 ) -N ε (σ, τ 2 )| ≤ 2 ε |τ 1 -τ 2 |.
On the other hand, by (H4) we have

|N ε (σ 1 , τ ) -N ε (σ 2 , τ )| ≤ C H ε |σ 1 -σ 2 |,
so that we obtain the thesis.

As a final note, given σ ∈ L 2 (Ω; M n×n sym ), we define the set

K(σ) := {τ ∈ L 2 (Ω; M n×n sym ) : τ (x) ∈ K(σ(x)) for a.e. x ∈ Ω}. Then, if τ ∈ L 2 (Ω; M n×n sym ), N ε (σ, τ ) 2 = dist 2 (τ, K(σ)) ε , (2.3)
where, for any closed set We use standard notation for Lebesgue and Sobolev spaces. In particular, for 1 ≤ p ≤ ∞, the L p -norms of the various quantities are denoted by • p . The space M(Ω; M n×n sym ) is that of all M n×n sym -valued finite Radon measures on Ω, and the norm in that space is denoted by • 1 . By the Riesz Representation Theorem, M(Ω; M n×n sym ) can be identified with the dual of C(Ω; M n×n sym ). Finally, BD(Ω) stands for the space of functions with bounded deformations on Ω, i.e., u ∈ BD(Ω) if u ∈ L 1 (Ω; R n ) and Eu ∈ M(Ω; M n×n sym ) (the space of all M n×n sym -valued finite Radon measures on Ω), where Eu := (Du + Du T )/2 and Du is the distributional derivative of u. We refer to [START_REF] Temam | Translation of Problèmes mathématiques en plasticité[END_REF] for general properties of that space.

C ⊂ L 2 (Ω; M n×n sym ), dist 2 (τ, C) is the L 2 -distance from τ to C.
Let A be a fourth order Hooke tensor satisfying the usual symmetry properties A ijkh = A jikh = A khij for every i, j, k, h ∈ {1, . . . , n}, and

α A |ξ| 2 ≤ Aξ : ξ ≤ β A |ξ| 2 , (2.4)
for some 0 < α A ≤ β A < +∞ and every ξ ∈ M n×n sym . We define, for any e ∈ L 2 (Ω; M n×n sym ), the elastic energy as

Q(e) :=
H(σ, p) ≤ lim inf k→+∞ H(σ k , p k ).
Indeed, hypothesis (H3) implies that

|H(σ, p k ) -H(σ k , p k )| ≤ C H σ -σ k ∞ p k 1 → 0, since {p k } is bounded in M(Ω; M n×n sym ). Hence lim inf k→+∞ H(σ k , p k ) = lim inf k→+∞ H(σ, p k ).
Since (x, ξ) → H(σ(x), ξ) is (lower semi-)continuous, while ξ → H(σ(x), ξ) is convex and positively one-homogeneous, we infer from the Reshetnyak Lower Semi-continuity Theorem (see, e.g., [1, Theorem 2.38]) that lim inf k H(σ, p k ) ≥ H(σ, p). ¶

When letting the viscosity parameter tend to 0, we will only obtain weak convergence in L 2 of the approximating σ-sequence, and convergence in the space of measures of the approximating p-sequence.

Unfortunately, the Reshetnyak Lower Semi-continuity Theorem is false when H fails to be (lower semi)-continuous, so that we are pretty much forced to restrict our analysis to continuous stresses; but continuity is not preserved under L 2 -weak convergence, which is the best we can hope for the various sequences of stresses that will enter the formulation. Consequently, the analysis will soon grind to a halt for lack of lower semi-continuity of H. This is why it was first proposed in [START_REF] Maso | Solombrino: Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling[END_REF] to perform a regularization of σ in the definition of K(σ). That was achieved by introducing a convolution kernel ρ and replacing K(σ) by K(σ * ρ) defined below.

We fix ρ ∈ C 1 c (R n ) and set, for σ ∈ L 2 (Ω; M n×n sym ),

x ∈ Ω → (σ * ρ)(x) := ˆΩ ρ(x -y)σ(y) dy.
The convolution σ * ρ defines an element in C 1 (Ω; M n×n sym ). Note that, with that definition of the convolution, if σ k σ weakly in L 2 (Ω; M n×n sym ), then, in particular,

σ k * ρ → σ * ρ uniformly on Ω.
With this kind of regularization, it proved possible to establish the following existence theorem for a quasi-static evolution in a rescaled time setting, see, e.g., [2, Theorem 4.1 and page 289]. We extend the boundary datum w ∈ H 1 (0, T ;

H 1 (Ω; R n )) to w(T ) for t ≥ T . Theorem 2.3. Let w ∈ H 1 (0, T ; H 1 (Ω; R n )) and let (u 0 , e 0 , p 0 ) ∈ BD(Ω) × L 2 (Ω; M n×n sym ) × M(Ω; M n×n sym ) be such that Eu 0 = e 0 + p 0 in Ω, p 0 = (w(0) -u 0 ) ν H n-1 on ∂Ω,
and div σ 0 = 0 in Ω, σ 0 ∈ K(σ 0 * ρ),
where σ 0 := Ae 0 . Then, there exist T > 0 and a mapping [0, T ] s → (u

• (s), e • (s), p • (s), t • (s)) such that u • : [0, T ] → BD(Ω)
is strongly continuous and a.e. weakly * differentiable;

e • : [0, T ] → L 2 (Ω; M n×n sym )
is strongly continuous and a.e. differentiable;

p • : [0, T ] → M(Ω; M n×n sym ) is 1-Lipschitz continuous; t • : [0, T ] → [0, +∞) is nondecreasing and 1-Lipschitz continuous, with t • (T ) ≥ T .
Further, setting σ • := Ae • , the following properties are satisfied:

Initial condition: (u • (0), e • (0), p • (0), t • (0)) = (u 0 , e 0 , p 0 , 0); Kinematic compatibility: For every s ∈ [0, T ], Eu • (s) = e • (s) + p • (s) in Ω, p • (s) = (w(t • (s)) -u • (s)) ν H n-1 on ∂Ω; Equilibrium condition: For every s ∈ [0, T ], div σ • (s) = 0 in Ω; Partial stress constraint: For every s ∈ [0, T ] \ U • , σ • (s) ∈ K(σ • (s) * ρ),
where U • := {s ∈ (0, T ] : t • is constant in a neighborhood of s}; Energy equality: For every S ∈ [0, T ],

Q(e • (S)) + ˆS 0 H(σ • (s) * ρ, ṗ• (s)) ds + ˆS 0 ṗ• (s) 2 dist 2 (σ • (s), K(σ • (s) * ρ)) ds = Q(e • (0)) + ˆS 0 ˆΩ σ • (s) : E ẇ• (s) dx ds.
Interpreting this evolution in "real" time is a non trivial task that was partially undertaken in [START_REF] Maso | Solombrino: Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solutions[END_REF]. But much information is lost because it may be so that there is no interval of finite length lying entirely outside U • . Also, as mentioned before, there is a certain degree of arbitrariness in the rescaling which immediately begs the question of the dependence of the jump times (the image of U • through t • ) upon said rescaling.

Our goal in what follows is to show that a slightly different regularization completely alleviates the need for rescaling in time and permits to obtain a real time evolution with well defined jump times.

We thus introduce

ρ ∈ C 1 c (R n+1 ) with 0 ≤ ρ(t, x) ≤ 1 and ´∞ 0 ´Rn ρ(t, x) dx dt = 1. Set, for σ ∈ L ∞ (-∞, T ; L 2 (Ω; M n×n sym )), (t, x) ∈ (-∞, T ] × Ω → (σ * ρ)(t, x) := ˆt -∞ ˆΩ ρ(t -s, x -y)σ(s, y) dy ds.
The convolution σ * ρ defines an element in C((-∞, T ] × Ω; M n×n sym ) ∩ L ∞ (-∞, T ; L 2 (Ω; M n×n sym )). Also remark that, with the above definition of the convolution, if

σ k σ weakly * in L ∞ (-∞, T ; L 2 (Ω; M n×n sym )), then, in particular, σ k * ρ → σ * ρ uniformly on [0, T ] × Ω.
(2.5) Indeed, we clearly have pointwise convergence. On the other hand, the functions σ k * ρ are equicontinuous on [0, T ] × Ω, so that the thesis follows from the Ascoli-Arzelà Theorem.

For now, we address in the next section the visco-plastic regularization.

The visco-plastic model

We propose in this short section to establish existence of the solution to the visco-plastic regularization. In contrast with the case of a space only regularization, it is not possible to proceed through a time incremental process because of the temporal non-local dependence of σ * ρ on σ.

Rather our proof will be based on an actually more direct and faster fixed point argument. Consider a boundary displacement w ∈ H 1 (Ω; R n ). We set

A reg (w) := (v, η, q) ∈ H 1 (Ω; R n ) × L 2 (Ω; M n×n sym ) × L 2 (Ω; M n×n sym ) : Ev = η + q a.e. in Ω, v = w H n-1 -a.e. on ∂Ω . (3.1)
The main result of this section is the following existence result for the non-associative viscoplastic evolution.

Theorem 3.1. Let w ∈ W 1,∞ (0, T ; H 1 (Ω, R n )), let (u 0 , e 0 , p 0 ) ∈ A reg (w(0)) be such that div σ 0 = 0 in Ω, where σ 0 := Ae 0 , and let ε > 0. Then, there exists a unique triplet (u ε (t), e ε (t), p ε (t)) ∈ A reg (w(t)) with

u ε ∈ W 1,∞ (0, T ; H 1 (Ω; R n )), e ε ∈ W 1,∞ (0, T ; L 2 (Ω; M n×n sym )), p ε ∈ W 1,∞ (0, T ; L 2 (Ω; M n×n sym )), such that, setting σ ε (t) :=
Ae ε (t) for 0 < t ≤ T, Ae 0 for t ≤ 0, the following conditions are satisfied: Initial condition: (u ε (0), e ε (0), p ε (0)) = (u 0 , e 0 , p 0 ); Kinematic compatibility: For every t ∈ [0, T ],

Eu ε (t) = e ε (t) + p ε (t) a.e. in Ω, u ε (t) = w(t) H n-1 -a.e.

on ∂Ω;

Equilibrium condition: For every t ∈ [0, T ], div σ ε (t) = 0 in Ω;

Regularized non-associative flow rule: For a.e. t ∈ [0, T ],

ṗε (t) = N ε ((σ ε * ρ)(t), σ ε (t)) for a.e. x ∈ Ω, or equivalently, σ ε (t) -ε ṗε (t) ∈ ∂ 2 H((σ ε * ρ)(t), ṗε (t)) for a.e. x ∈ Ω.
In particular, ε ṗε (t

) 2 = dist 2 (σ ε (t), K((σ ε * ρ)(t))).
We call such a triplet a non-associative visco-plastic solution.

Proof. Let A : L ∞ (0, T ; L 2 (Ω; M n×n sym )) → L ∞ (0, T ; L 2 (Ω; M n×n sym )) be the operator that maps p ∈ L ∞ (0, T ; L 2 (Ω; M n×n sym )) into A(p) ∈ L ∞ (0, T ; L 2 (Ω; M n×n sym )) defined as follows: for a.e. t ∈ [0, T ] A(p)(t) := Eu(t) -p(t),

where u(t) ∈ H 1 (Ω; R n ) is the solution of div A(Eu(t) -p(t)) = 0 in Ω, u(t) = w(t) on ∂Ω. Let K ε : L ∞ (0, T ; L 2 (Ω; M n×n sym )) → L ∞ (0, T ; L 2 (Ω; M n×n sym ))
be the operator defined as follows: given e ∈ L ∞ (0, T ; L 2 (Ω; M n×n sym )), we first extend e to (-∞, 0) by setting e(t) := e 0 for every t < 0; we then define K ε (e) ∈ L ∞ (0, T ; L 2 (Ω; M n×n sym )) by

K ε (e)(t) := p 0 + ˆt 0 N ε ((Ae) * ρ(s), Ae(s)) ds for a.e. t ∈ [0, T ].
In view of Lemma 2.1, (2.4), and the properties of ρ, we have that

K ε (e 1 )(t) -K ε (e 2 )(t) 2 ≤ C H C ρ ε ˆt 0 σ 1 -σ 2 L ∞ (0,t;L 2 (Ω;M n×n sym )) + σ 1 (s) -σ 2 (s) 2 ds ≤ β A C H C ρ ε t e 1 -e 2 L ∞ (0,t;L 2 (Ω;M n×n sym )) , (3.2
) where C ρ is a constant depending on ρ and we have set σ i := Ae i , i = 1, 2. By classical elliptic estimates we have

A(p 1 )(t) -A(p 2 )(t) 2 ≤ 1 + β A α A p 1 (t) -p 2 (t) 2 . (3.3)
Combining (3.2) and (3.3) we get

(A • K ε )(e 1 ) -(A • K ε )(e 2 ) L ∞ (0,t;L 2 (Ω;M n×n sym )) ≤ 1 + β A α A β A C H C ρ ε t e 1 -e 2 L ∞ (0,t;L 2 (Ω;M n×n sym )) .
Thus for t small enough, A • K ε is a strict contraction on L ∞ (0, t; L 2 (Ω; M n×n sym )). Hence there exists a unique fixed point e ε ∈ L ∞ (0, t; L 2 (Ω; M n×n sym )) and, consequently, there exists a unique triplet (u ε (s), e ε (s), p ε (s)) ∈ A reg (w(s)) satisfying for a.e. s ∈ (0, t) the kinematic compatibility, the equilibrium condition and the equality

p ε (s) = p 0 + ˆs 0 N ε ((Ae ε ) * ρ(r), Ae ε (r)) dr.
But this implies in turn that p ε ∈ W 1,∞ (0, t; L 2 (Ω; M n×n sym )), hence the additional time regularity on (u ε (s), e ε (s), p ε (s)), and that the regularized non-associative flow rule is satisfied. We deduce that the kinematic compatibility and the equilibrium condition hold for every time s ∈ [0, t], as well as the initial condition. Since the interval (0, t) on which the fixed point argument holds, does not depend upon the initial conditions on the triplet (u ε (t), e ε (t), p ε (t)), we can iterate the argument and obtain a solution over the whole interval (0, T ).

We end this subsection with two propositions stating some useful properties of visco-plastic regularized evolutions. The proof of the first proposition is very close to that of [START_REF] Maso | Solombrino: Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling[END_REF]Theorem 3.4] and is omitted here. Proposition 3.2. Under the assumptions of Theorem 3.1, let t → (u ε (t), e ε (t), p ε (t)) ∈ A reg (w(t)) with

u ε ∈ W 1,∞ (0, T ; H 1 (Ω; R n )), e ε ∈ W 1,∞ (0, T ; L 2 (Ω; M n×n sym )), p ε ∈ W 1,∞ (0, T ; L 2 (Ω; M n×n sym )
), satisfy the initial condition, the kinematic compatibility, and the equilibrium condition in Theorem 3.1. Then t → (u ε (t), e ε (t), p ε (t)) satisfies the regularized non-associative flow rule in Theorem 3.1 if and only if the following two conditions are satisfied:

1. Modified Stress Constraint:

σ ε (t) -ε ṗε (t) ∈ K((σ ε * ρ)(t)) for a.e. t ∈ [0, T ], or equivalently, since K((σ * ρ)(t)) = ∂ 2 H((σ * ρ)(t), 0), σ ε (t, x) -ε ṗε (t, x) ∈ ∂ 2 H((σ ε * ρ)(t, x), 0) for a.e. (t, x) ∈ [0, T ] × Ω;
2. Energy equality:

for every t ∈ [0, T ] Q(e ε (t)) + ˆt 0 H((σ ε * ρ)(s), ṗε (s)) ds + ε ˆt 0 ṗε (s) 2 2 ds = Q(e 0 ) + ˆt 0 ˆΩ σ ε (s) : E ẇ(s) dx ds,
or equivalently,

Q(e ε (t)) + ˆt 0 H((σ ε * ρ)(s), ṗε (s)) ds + ˆt 0 ṗε (s) 2 dist 2 (σ ε (s), K((σ ε * ρ)(s))) ds = Q(e 0 ) + ˆt 0 ˆΩ σ ε (s) : E ẇ(s) dx ds.
In view of the energy equality in Proposition 3.2 and of (2.2), (2.3), (2.4), we immediately obtain the following Proposition 3.3. Let t → (u ε (t), e ε (t), p ε (t)) be a visco-plastic regularised evolution according to Theorem 3.1. Then

sup t∈[0,T ] e ε (t) 2 ≤ C T , sup t∈[0,T ] σ ε (t) 2 ≤ C T , ˆT 0 ṗε (s) 1 ds ≤ C T , ε ˆT 0 ṗε (s) 2 2 ds ≤ C T , (3.4) hence also p ε BV ([0,T ];L 1 (Ω;M n×n sym )) ≤ C T , (3.5)
where C T is an ε-independent constant.

The evolution

In this section we propose to pass to the limit in the visco-platic evolution obtained in Theorem 3.1, as the viscosity parameter ε goes to 0. Because of the bounds in Proposition 3.3, we cannot expect to keep the L 2 -regularity of the fields Eu and p when passing to the 0-viscosity limit and we thus have to redefine A reg (w) from (3.1) as

A(w) := (v, η, q) ∈ BD(Ω) × L 2 (Ω; M n×n sym ) × M(Ω; M n×n sym ) : Ev = η + q in Ω, q = (w -v) νH n-1 on ∂Ω with w ∈ H 1 (Ω; R n ).
The interpretation of the boundary condition is that, if the displacement u does not match the prescribed boundary displacement w, then the loaded boundary can experience plastic slips. We still keep a boundary datum w ∈ W 1,∞ (0, T ; H 1 (Ω; R n )). The main result of the paper is the following existence result for the regularized quasistatic evolution model in non-associative plasticity.

Theorem 4.1. Let w ∈ W 1,∞ (0, T ; H 1 (Ω; R n )) and let (u 0 , e 0 , p 0 ) ∈ A(w(0)) be such that

div σ 0 = 0 in Ω, σ 0 ∈ K((σ 0 * ρ)(0)), (4.1)
where σ 0 (t) := Ae 0 for t ≤ 0. Then, there exists a mapping [0, T ] t → (u(t), e(t), p(t)) with p ∈ BV ([0, T ]; M(Ω; M n×n sym )) such that, setting σ(t) := Ae(t) for 0 < t ≤ T, Ae 0 for t ≤ 0, the following properties are satisfied:

Initial condition: (u(0), e(0), p(0)) = (u 0 , e 0 , p 0 ); Kinematic compatibility: For every t ∈ [0, T ],

Eu(t) = e(t) + p(t) in Ω, p(t) = (w(t)) -u(t)) ν H n-1 on ∂Ω;
Equilibrium condition: For every t ∈ [0, T ], div σ(t) = 0 in Ω;

Partial stress constraint:

For every t ∈ [0, T ] \ N , σ(t) ∈ K((σ * ρ)(t)),
where

N is the countable set of jumps of t → Var(p; 0, t) := = sup n i=1 p(t i ) -p(t i-1 ) 1 : t 0 = 0 ≤ t 1 ≤ ... ≤ t n = t, n ∈ N ; (4.2)
Energy equality: For every t ∈ [0, T ]\N ,

Q(e(t)) + ˆ[0,t]×Ω H (σ * ρ)(s, x), ṗ | ṗ| (s, x) d| ṗ| = Q(e 0 ) + ˆt 0 ˆΩ σ(s) : E ẇ(s) dx ds, (4.3)
where |p| is the variation measure associated with p viewed as a measure on [0, T ] × Ω;

Regularity: The maps t → u(t) and t → e(t) are continuous at all points of [0, T ] \ N in the strong topology of BD(Ω) and of L 2 (Ω; M n×n sym ), respectively, while | ṗ| does not charge {t} × Ω for t ∈ [0, T ] \ N ; Condition at jumps:

for every t ∈ N Q(e(t+)) -Q(e(t-)) + ˆ{t}×Ω H (σ * ρ)(s, x), ṗ | ṗ| (s, x) d| ṗ| = 0, (4.4)
where e(t-) and e(t+) denote the left and the right limit of the map t → e(t) at time t with respect to the strong topology of L 2 (Ω; M n×n sym ).

Proof. According to [4, Lemma 5.1], there exists a sequence

{u ε 0 } in H 1 (Ω; R n ) such that u ε 0 = w(0) H n-1 -a.e. on ∂Ω, u ε 0 → u 0 strongly in L 1 (Ω; R n ),
and Eu ε 0 Eu 0 weakly * in M(Ω; M n×n sym ). Setting p ε 0 := Eu ε 0 -e 0 , we have that (u ε 0 , e 0 , p ε 0 ) ∈ A reg (w(0)), u ε 0 u 0 weakly * in BD(Ω), and p ε 0 p 0 weakly * in M(Ω; M n×n sym ). By Theorem 3.1 for every ε > 0 there exists a unique non-associative visco-plastic solution (u ε , e ε , p ε ) with initial datum (u ε 0 , e 0 , p ε 0 ). By Proposition 3.3 the bounds (3.4) and (3.5) are satisfied.

Step 1 -Compactness and continuity. By application of the Helly Theorem (see [START_REF] Mainik | Existence results for energetic models for rate-independent systems[END_REF]) to (3.5), there exists an element p ∈ BV ([0, T ]; M(Ω; M n×n sym )) and a subsequence of {p ε } (still indexed by ε) such that, for every t ∈ [0, T ], p ε (t) p(t) weakly * in M(Ω; M n×n sym ). (

We extend p ε (t) to p ε 0 for t < 0 and to p ε (T ) for t > T , so that ṗε (t) = 0 for t < 0 and for t > T . In view of the third bound in (3.4), we have that

ṗε ṗ weakly * in M([0, T ] × Ω; M n×n sym ), (4.6)
that is,

ˆ[0,T ]×Ω φ : d ṗε → ˆ[0,T ]×Ω φ : d ṗ for every test function φ ∈ C([0, T ] × Ω; M n×n sym )
. By the first two bounds in (3.4) and the Korn-Poincaré Inequality in BD(Ω) we deduce that, at any given time t, there exists a time-dependent subsequence {ε t } ⊂ {ε} such that

     e εt (t) e(t) weakly in L 2 (Ω; M n×n sym ), σ εt (t) σ(t) weakly in L 2 (Ω; M n×n sym ) with div σ(t) = 0, u εt (t) u(t) weakly * in BD(Ω),
with (u(t), e(t), p(t)) ∈ A(w(t)). In fact, there is no need for subsequence extraction. Indeed, for

any ϕ ∈ H 1 0 (Ω; R n ) we infer from div σ(t) = 0 that Q(e(t)) ≤ Q(e(t) + Eϕ). (4.7)
Thus, if another subsequence {ε t } is such that e ε t e (t) weakly in L 2 (Ω; M n×n sym ) and u εt u (t) weakly * in BD(Ω) with (u (t), e (t), p(t)) ∈ A(w(t)), then u(t) -u (t) is immediately seen to be in H 1 0 (Ω; R n ) with E(u(t) -u (t)) = e(t) -e (t). We thus get from (4.7) that Q(e(t)) ≤ Q(e (t)), hence switching the roles of e and e that Q(e(t)) = Q(e (t)). Using the strict convexity of Q we conclude that e(t) = e (t), from which we also have that u(t) = u (t).

We have therefore proved that for every t ∈ [0, T ]

     e ε (t) e(t) weakly in L 2 (Ω; M n×n sym ), σ ε (t) σ(t) weakly in L 2 (Ω; M n×n sym ) with div σ(t) = 0, u ε (t) u(t) weakly * in BD(Ω). (4.8)
In particular, the initial condition is satisfied. Further, because of the L ∞ -bounds in (3.4) on e ε and σ ε and of (4.8), t → e(t) and t → σ(t) are weakly measurable (with values in L 2 (Ω; M n×n sym )), hence strongly measurable, so that

e ε e weakly * in L ∞ (0, T ; L 2 (Ω; M n×n sym )), σ ε σ weakly * in L ∞ (0, T ; L 2 (Ω; M n×n sym )).
(4.9)

Let N be the countable set of discontinuity points of t → Var(p; 0, t). for a subsequence e(t kj ) e * weakly in L 2 (Ω; M n×n sym ), with div Ae * = 0 in Ω. As t ∈ N , we have that p(t kj ) → p(t) strongly in M(Ω; M n×n sym ). Hence, u(t kj ) u * weakly * in BD(Ω) with (u * , e * , p(t)) ∈ A(w(t)). From the previous minimality argument we deduce that u * = u(t) and e * = e(t). Hence, the whole sequences {e(t k )} and {u(t k )} converge to e(t) and u(t) weakly in L 2 (Ω; M n×n sym ) and weakly * in BD(Ω), respectively.

Step 2 -Stress constraint. Recall the modified stress constraint from Proposition 3.2, namely

σ ε (t, x) -ε ṗε (t, x) ∈ ∂ 2 H((σ ε * ρ)(t, x), 0) for a.e. (t, x) ∈ [0, T ] × Ω.
Then, for any q ∈ M n×n sym , H((σ ε * ρ)(t, x), q) ≥ (σ ε (t, x) -ε ṗε (t, x)) : q for a.e. ( Now take t k → t with t k , t ∈ N . Then, in view of (4.12), σ(t k ) ∈ K((σ * ρ)(t k )), while div σ(t k ) = 0, so that, appealing to, e.g., [10, Proposition 3.9, (6.5) and (6.20)] and using (

(σ ε * ρ)(s) -(σ * ρ)(s) ∞ ˆt 0 ṗε (s) 1 ds.
Now, in view of the second convergence in (4.9) and of (2.5), for any η > 0 there exists some εη > 0 such that sup

0<ε<εη sup s∈[0,t] (σ ε * ρ)(s) -(σ * ρ)(s) ∞ ≤ η.
Consequently, using the third bound in (3.4), we have

lim inf ε→0 ˆt 0 H((σ ε * ρ)(s), ṗε (s)) ds ≥ lim inf ε→0 ˆt 0 H((σ * ρ)(s), ṗε (s)) ds -C T η.
Now, for every t ∈ [0, T ], [0, t) × Ω is a locally compact set and (4.6) implies the weak * convergence of ṗε to ṗ in M([0, t)×Ω; M n×n sym ), that is, against any test function in C 0 ([0, t)×Ω; M n×n sym ). Thus, in view of the assumptions (H0), (H1) on H, application of the Reshetnyak Theorem in the inequality above is licit; letting η 0 yields (4.14). For t = T , we can apply the Reshetnyak Theorem on [0, T ] × Ω and obtain (4.15) by virtue of the weak * convergence of ṗε to ṗ in M([0, T ] × Ω; M n×n sym ).

Step 4 -Energy equality. We first pass to the lim inf ε in the energy equality of Proposition for every t ∈ [0, T ). Similarly, using (4.15), we obtain

Q(e(T )) + ˆ[0,T ]×Ω H (σ * ρ)(s, x), ṗ | ṗ| (s, x) d| ṗ| + lim inf ε ε ˆT 0 ṗε (s) 2 2 ds ≤ Q(e 0 ) + ˆT 0 ˆΩ σ(s) : E ẇ(s) dx ds. (4.17)
Conversely, let λ be the weak * limit of | ṗε | in M([0, T ] × Ω). We introduce the set

M := N ∪ t ∈ [0, T ] : λ({t} × Ω) = 0
and we note that M is a countable set. We also remark that, since div σ(t) = 0 for every t, while (4.12) holds at t / ∈ N and also at t = 0 in view of (4.1), minimality also holds at all t ∈ N and at t = 0, that is, for any (v, η, q) ∈ A(w(t))

Q(e(t)) ≤ Q(η) + H((σ * ρ)(t), q -p(t)), (4.18)
see, e.g., [START_REF] Maso | Quasistatic evolution problems for linearly elastic-perfectly plastic materials[END_REF]Theorem 3.6].

Let t ∈ (0, T ]. Consider then a (k-indexed) sequence of partitions {s k i } i=0,...,n k of [0, t] with s k 0 = 0 and s k n k = t such that s k i ∈ M for every i = 0, n k and lim k→∞ max i (s k i -s k i-1 ) = 0. In view of (4.18), the regularity of w implies that, at

s k i , i = n k , Q(e(s k i )) ≤ Q(e(s k i+1 ) + Ew(s k i ) -Ew(s k i+1 )) + H((σ * ρ)(s k i ), p(s k i+1 ) -p(s k i )) ≤ Q(e(s k i+1 )) - ˆsk i+1 s k i ˆΩ σ(s k i+1 ) : E ẇ(s) dx ds + H((σ * ρ)(s k i ), p(s k i+1 ) -p(s k i )) + o(1) ˆsk i+1 s k i ˆΩ |E ẇ(s)| 2 dx ds.
Iterating yields, with σ k (s

) := σ(s k i+1 ) for s ∈ [s k i , s k i+1 ), Q(e 0 ) + ˆt 0 ˆΩ σ k (s) : E ẇ(s) dx ds ≤ Q(e(t)) + i H((σ * ρ)(s i ), p(s k i+1 ) -p(s k i )) + o(1) ˆt 0 ˆΩ |E ẇ(s)| 2 dx ds.
By (4.10), (4.11), and the Dominated Convergence Theorem we obtain that, as k → ∞,

ˆt 0 ˆΩ σ k (s) : E ẇ(s) dx ds → ˆt 0 ˆΩ σ(s) : E ẇ(s) dx ds.
We then get

Q(e 0 ) + ˆt 0 ˆΩ σ(s) : E ẇ(s) dx ds ≤ Q(e(t)) + Var H (p; 0, t), (4.19)
where Var H (p; 0, t) := lim inf

k→∞ i H (σ * ρ)(s k i ), p(s k i+1 ) -p(s k i ) .
We Step 5 -Proof of (4.20). Let 0 ≤ t 1 < t 2 ≤ T with t 1 , t 2 ∈ M , unless t 1 = 0 or t 2 = T . Since p ε ∈ W

2. 2 .

 2 Mathematical setting. Throughout the paper, Ω is a bounded connected open set in R n with Lipschitz boundary. The Lebesgue measure in R n and the (n -1)-dimensional Hausdorff measure are denoted by L n and H n-1 , respectively.

  (4.25) Passing to the limit as k → ∞ and using (4.23), we deduce that ˆ{t}×ΩH (σ * ρ)(s, x), ṗ | ṗ| (s, x) d| ṗ| = 0, hence | ṗ|({t} × Ω) = 0 by (2.2).

  Constitutive law: σ(t) = Ae(t) in Ω; • Energy balance:

	dE dt	(t) =	ˆΓd	(σ(t)ν) • ẇ(t) dH n-1 -ˆΩ ḟ (t) • u(t) dx -	ˆΓn	ġ(t) • u(t) dH n-1 ,
	where E(t) := 1 2 ˆΩ Ae(t) : e(t) dx +	ˆt 0	ˆΩ H( ṗ(s)) dx ds -ˆΩ f (t) • u(t) dx -	ˆΓn	g(t) • u(t) dH n-1 .

1 2 ˆΩ Aη : η dx + ˆΩ H(q -p(t)) dx among all admissible triplets (v, η, q), where H(p) is the support function of K, i.e., H(p) := sup{σ D : p : σ D ∈ K}; •

  Then the map t → p(t) is continuous in the strong topology of M(Ω; M n×n sym ) at all times in [0, T ] \ N . We now show that

	t → (e(t), σ(t)) is weakly continuous in [L 2 (Ω; M n×n sym )] 2 at all points of [0, T ] \ N	(4.10)
	and
	t → u(t) is weakly * continuous in BD(Ω) at all points of [0, T ] \ N.
	Indeed, let t k → t with t ∈ N . Since
	sup	e(s) 2 ≤ C T ,	(4.11)
	s∈[0,T ]

  t, x) ∈ [0, T ] × Ω, hence, for any measurable set E ⊂ Ω and for a.e. t, ˆE H((σ ε * ρ)(t, x), q) dx ≥ ˆE(σ ε (t, x) -ε ṗε (t, x)) : q dx. Because of the second convergence in (4.9), (σ ε * ρ)(t) → (σ * ρ)(t) uniformly in Ω for every t ∈ [0, T ]. Thanks to hypothesis (H3), (4.8) and the fourth bound in (3.4), we deduce that ˆE H((σ * ρ)(t, x), q) dx ≥ ˆE σ(t, x) : q dx for a.e. t ∈ [0, T ]. By the weak continuity (4.10) of t → σ(t) in [0, T ] \ N , we conclude that the previous relation actually holds for every t ∈ [0, T ] \ N . Thus, for t ∈ [0, T ] \ N ,

	H((σ * ρ)(t, x), q) ≥ σ(t, x) : q for a.e. x ∈ Ω,
	or, equivalently,
	σ(t) ∈ K((σ * ρ)(t)).	(4.12)

  Combining now (4.[START_REF] Mielke | Evolution of rate-independent systems[END_REF]) and (4.21) yields the following energy equality for t ∈ M : ≥ | ṗ| by[START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] Proposition 1.62], hence | ṗ|({t} × Ω) = 0 for t ∈ M . This completes the proof of the energy equality, except for those t ∈ M \ N . For such t's we consider a decreasing sequence {t k } ⊂ [0, T ] \ M that converges to t. The energy equality (4.22) holds at such t k for every k. In view of (4.13) we immediately conclude that the same equality holds at t. Moreover,t → e(t)is strongly continuous in L 2 (Ω; M n×n sym ) at all points of [0, T ] \ N (4.23) and t → u(t) is strongly continuous in BD(Ω) at all points of [0, T ] \ N. (4.24) Indeed, let t ∈ [0, T ] \ N and let {t k } be any sequence converging to t. We can always find a sequence {s k } → t such that (s k ) ⊂ [0, T ] \ N and s k < t k for every k. Applying (4.16) at t k and (4.22) at s k we obtain where the last equality is a consequence of (4.13) and of the assumption {s k } ⊂ [0, T ] \ N . This proves (4.23). Since t → p(t) is continuous in the strong topology of M(Ω; M n×n sym ) at all times in [0, T ] \ N , the Korn-Poincaré Inequality in BD(Ω) and (4.23) yield (4.24). Equality (4.22), together with (4.23), also implies that | ṗ|({t} × Ω) = 0 for t ∈ N . Indeed, let t ∈ N and let {s k } → t -and {t k } → t + , with {s k }, {t k } ⊂ [0, T ] \ N . By (4.22) at s k and at t k we have that Q(e(t k ))+ ˆ(s k ,t k ]×Ω H (σ * ρ)(s, x), ṗ | ṗ| (s, x) d| ṗ| = Q(e(s k ))+

	Var H (p; 0, t) ≤ Assuming for now that (4.20) holds true, we get from (4.19) that ˆ[0,t]×Ω H (σ * ρ)(s, x), ṗ | ṗ| (s, x) d| ṗ|. Q(e 0 ) + ˆt 0 ˆΩ σ(s) : E ẇ(s) dx ds ≤ Q(e(t)) + ˆ[0,t]×Ω H (σ * ρ)(s, x), | ṗ| ṗ ε ε ˆT 0 ṗε (s) 2 2 ds = 0. Q(e 0 ) + ˆt 0 ˆΩ σ(s) : E ẇ(s) dx ds = Q(e(t)) + ˆ[0,t]×Ω H (σ * ρ)(s, x), ṗ | ṗ| (s, x) d| ṗ|, (s, x) d| ṗ| ˆ(s k ,t k )×Ω H (σ * ρ)(s, x), ṗ | ṗ| (s, x) d| ṗ| ≤ Q(e(s k )) + ˆtk s k ˆΩ σ(s) : E ẇ(s) dx ds. (4.20) (4.21) (4.22) Since the second term on the left-handside is nonnegative, we deduce that lim sup k→∞ Q(e(t k )) ≤ lim sup k→∞ Q(e(s k )) = Q(e(t)), ˆtk where we used that λ Q(e(t k )) + s

will show below that for t ∈ M and for t = T , for t ∈ M and for t = T . Comparing (4.17) with (4.21) at t = T immediately leads to lim inf k ˆΩ σ(s) : E ẇ(s) dx ds.

  1,∞ (0, T ; L 2 (Ω; M n×n sym )), for every ψ ∈ C(Ω; M n×n sym ) we have ˆΩ (p ε (t 2 ) -p ε (t 1 )) : ψ dx = ˆt2 Proposition 1.62]). We now recall that, by[START_REF] Solombrino | Quasistatic evolution problems for nonhomogeneous elastic plastic materials[END_REF] Theorem 3.6],H((σ * ρ)(t 1 ), p(t 2 ) -p(t 1 )) = sup ˆΩ ζ : d(p(t 2 ) -p(t 1 )) : ζ ∈ C(Ω; M n×n sym ) ∩ K((σ * ρ)(t 1 )) .Therefore, for every η > 0 there existsζ ∈ C(Ω; M n×n sym ) ∩ K((σ * ρ)(t 1 )) such that H((σ * ρ)(t 1 ), p(t 2 ) -p(t 1 )) ≤ ˆΩ ζ : d(p(t 2 ) -p(t 1 )) + η.where the last inequality follows from the fact that ζ ∈ K((σ * ρ)(t 1 )) and the definition of H.Combining together the two previous inequalities and by the arbitrariness of η, we conclude that

	On the other hand, by (4.26) we have ˆΩ ζ(x) : d(p(t 2 ) -p(t 1 )) = ˆ[t1,t2]×Ω	ζ : d ṗ ≤	ˆ[t1,t2]×Ω	H (σ * ρ)(t 1 , x),	ṗ | ṗ|	(s, x) d| ṗ|(s, x),

t1

ˆΩ ṗε (s) : ψ dx ds

In view of (4.5),

(4.6)

, and the assumption on t 1 , t 2 , this implies that ˆΩ ψ : d(p(t 2 ) -p(t 1 )) = ˆ[t1,t2]×Ω ψ : d ṗ (4.26) (see, e.g., [1, H((σ * ρ)(t 1 ), p(t 2 ) -p(t 1 )) ≤ ˆ[t1,t2]×Ω H (σ * ρ)(t 1 , x), ṗ | ṗ| (s, x) d| ṗ|(s, x).

Acknowledgements

The research has been partially supported by the National Science Fundation Grant DMS-1615839, by INdAM-GNAMPA under Project 2016 "Multiscale analysis of complex systems with variational methods", and by the Università degli Studi di Pavia through the 2017 Blue Sky Research Project "Plasticity at different scales: micro to macro".

Let now t ∈ (0, T ] with t ∈ M or t = T . The inequality above immediately implies that, in the notation of the previous step, Var H (p; 0, t) ≤ lim inf

(s, x) d| ṗ|(s, x).

In turn, recalling (H3) and the uniform continuity of σ * ρ over [0, T ]×Ω, we can replace (σ * ρ)(s k i , x) by (σ * ρ)(s, x) in the right-hand side of the above inequality, since (σ * ρ)(s k i , x) -(σ * ρ)(s, x) can be chosen as small as desired, provided that s -s k i is small enough, that is, that k is small enough. But then the right-hand side identifies with

, so that the second term in the expression above is identically 0, and we get the desired inequality.

Step 6 -Proof of (4.4). Since p ∈ BV ([0, T ]; M(Ω; M n×n sym )), the map t → p(t) has a left and a right limit with respect to the strong topology of M(Ω; M n×n sym ) at every t ∈ N . By the same argument as in (4.10) we deduce that that the map t → e(t) has a left and a right limit with respect to the weak topology of L 2 (Ω; M n×n sym ) at every t ∈ N . Let now t ∈ N and {t k } → t -with {t k } ⊂ [0, T ] \ N . Arguing as in the proof of (4.13), we obtain that

) → e(t-) strongly in L 2 (Ω; M n×n sym ) for every {t k } → t -with t k ∈ [0, T ] \ N. Analogously, for every t ∈ N we have e(t k ) → e(t+) strongly in L 2 (Ω; M n×n sym ) for every {t k } → t + with t k ∈ [0, T ] \ N. Finally, arguing as in the proof of (4.23), we conclude that the map t → e(t) has a left and a right limit with respect to the strong topology of L 2 (Ω; M n×n sym ) at every t ∈ N . Let now t ∈ N . Let {s k } → t -and {t k } → t + , with {s k }, {t k } ⊂ [0, T ] \ N . The energy equality (4.25) holds and by passing to the limit we deduce (4.4).

The proof of Theorem 4.1 is complete.

Remark 4.2. Theorem 4.1 does not directly apply to the setting of the Armstrong-Frederick model described in [START_REF] Francfort | Quasi-static evolution for the Armstrong-Frederick hardening-plasticity model[END_REF]. However the interested reader will be quickly convinced that a straightforward adaptation of the theorem will yield the relevant result in that setting as well. The reader will also note that the analogue of [11, Proposition 4.14] will equally apply in "real" time. In other words a mere repetition of the proof of that proposition would yield the removal of the cap, at least for times t ∈ [0, s 1 ), where s 1 := sup{s : [0, s] ⊂ [0, T ] \ N }, with N defined in (4.2). ¶ Remark 4.3. Our main result, Theorem 4.1, produces a notion of weak solution to the regularized evolution, which is a zero-viscosity limit of some visco-plastic evolution described in Theorem 3.1.

In [START_REF] Mielke | Balanced Viscosity (BV) solutions to infinite-dimensional rate-independent systems[END_REF] a similar notion of weak solution is proposed through a limit process for a viscous approximation as the viscosity parameter vanishes. It is called a Balanced Viscosity solution by the authors and it is characterized by an energy identity at all times t ∈ [0, T ] involving a variation term, called Var f there, that encompasses the so-called Finsler cost induced by the viscous term at the points of jump for the state field under consideration (see [19, (3.9)]).

Thanks to Theorem 4.1, we are at liberty to assert that the energy equality (4.3) holds true at all times, provided that we decide to set e(t) at the value e(t+) when t ∈ N .

But then, we do conclude that the solution that we have evidenced can be viewed as a Balanced Viscosity solution, provided that ˆ[0,t]×Ω H (σ * ρ)(s, x), ṗ | ṗ| (s, x) d| ṗ| is taken to be the definition of the variation Var f (0, t). Of course, it would remain to reconcile our definition of the variation with that introduced in [START_REF] Mielke | Balanced Viscosity (BV) solutions to infinite-dimensional rate-independent systems[END_REF] which does not, in our opinion, so easily extend to the present setting. In particular, one would have to show that ˆ{t}×Ω H (σ * ρ)(s, x), ṗ | ṗ| (s, x) d| ṗ| can also be expressed as the Finsler cost defined in [19, (1.11)] at the jump points of the variation of t → Var(p; 0, t), or, in other words, that there is indeed existence of a Balanced Viscosity solution in the problem at hand. We wish to thank the referee for prompting us to look into possible connections between those two notions of weak solution and, more generally, for the various suggestions that improved our manuscript.

¶


Sylvie Coste-Marquis

Caroline Devred
email: devred@cril.univ-artois.fr

Pierre Marquis
email: marquis@cril.univ-artois.fr

Inference from controversial arguments

We present new careful semantics within Dung's theory of argumentation. Under such careful semantics, two arguments cannot belong to the same extension whenever one of them indirectly attacks a third argument while the other one indirectly defends the third. We argue that our semantics lead to a better handling of controversial arguments than Dung's ones in some settings. We compare the careful inference relations induced by our semantics w.r.t. cautiousness; we also compare them with the inference relations induced by Dung's semantics.

Introduction

Argumentation is a general approach to model defeasible reasoning, in which the two main issues are the generation of arguments and their exploitation so as to draw some conclusions based on the way arguments interact (see e.g., [START_REF] Toulmin | The Uses of Argument[END_REF][START_REF] Prakken | Logics for defeasible argumentation[END_REF][START_REF] Bondarenko | An abstract, argumentation-theoretic approach to default reasoning[END_REF][START_REF] Kakas | Computing argumentation in logic programming[END_REF]).

Among the various theories of argumentation pointed out so far (see e.g., [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF][START_REF] Pollock | How to reason defeasibly[END_REF][START_REF] Simari | A mathematical treatment of defeasible reasoning and its implementation[END_REF][START_REF] Elvang-Gøransson | Acceptability of arguments as logical uncertainty[END_REF][START_REF] Elvang-Gøransson | Dialectic reasoning with inconsistent information[END_REF][START_REF] Elvang-Gøransson | Argumentative logics: Reasoning with classically inconsistent information[END_REF][START_REF] Vreeswijk | Abstract argumentation systems[END_REF][START_REF] Besnard | A logic-based theory of deductive arguments[END_REF][START_REF] Amgoud | Inferring from inconsistency in preference-based argumentation frameworks[END_REF][START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF][START_REF] Cayrol | From non-monotonic syntax-based entailment to preference-based argumentation[END_REF][START_REF] Dimopoulos | On the computional complexity of assumption-based argumentation for default reasoning[END_REF]) is Dung's theory [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF]. Dung's theory is quite influential since it encompasses many approaches to nonmonotonic reasoning and logic programming as special cases; as such, it has been refined and extended by several authors, including [START_REF] Baroni | Extending abstract argumentation systems theory[END_REF][START_REF] Baroni | Solving semantic problems with odd-length cycles in argumentation[END_REF][START_REF] Baroni | A recursive approach to argumentation: motivation and perspectives[END_REF][START_REF] Cayrol | Minimal defence: a refinement of the preferred semantics for argumentation frameworks[END_REF][START_REF] Cayrol | Gradual handling of contradiction in argumentation frameworks[END_REF]. In Dung's approach, no assumption is made about the nature of an argument. Dung's theory of argumentation is not concerned with the generation of arguments; arguments and the way they interact w.r.t. the attack relation are considered as initial data of any argumentation framework, which can thus be viewed as a labeled digraph.

Several inference relations can be defined within Dung's theory. Usually, inference is defined at the argument level: an argument is considered derivable from an argumentation framework ¡ £¢ when it belongs to one (credulous consequence) (resp. all (skeptical consequence)) extensions of ¡ £¢ under some semantics, where an extension of ¡ £¢ is an admissible set of arguments (i.e., a conflict-free and self-defending set) that is maximal for a given criterion (made precise by the semantics under consideration). While skeptical derivability can be safely extended to the level of sets of arguments, this is not the case for credulous derivability. Indeed, it can be the case that arguments ¤ and ¥ are (individually) derivable from an argumentation framework ¡ ¦¢ while the set § ¤ ©¨¥ is not included in any extension of ¡ ¦¢ . Now, defining derivability for sets Many thanks to the anonymous reviewers for their helpful comments. The authors have been partly supported by the IUT de Lens, the Université d'Artois, the Région Nord/Pas-de-Calais through the IRCICA Consortium, and by the European Community FEDER Program.

of arguments as inclusion into some (resp. all) extensions under Dung's semantics does not always lead to expected conclusions.

Consider the following scenario: in a public meeting, a political activist presents the motivations of her policy using arguments and counter-arguments: "One should really decrease taxes (a); of course, this requires to cut staff in public services (b), but that is not so dramatic: privatizing some activities will lead to better services since free trading is good for it (c); furthermore, I am confident that we should reduce our economical exchanges with other foreign countries (d); this is antagonistic to promoting free trading, but, anyway, the productivity of our public services is definitely bad (e)". This sounds quite strange as a political speech since the speaker admits that she is in favour of conflicting arguments; a political opponent could easily point out the presence of such a conflict and concludes that such a policy is just non-sense; in order to convince the audience that ¤ , and ¡ should be accepted, a more skillful speech would be: "One should really decrease taxes (a); of course, this requires to cut staff in public services (b), but the productivity of our public services is definitely bad (e); furthermore, I'm confident that we should reduce our economical exchanges with other foreign countries (d)".

From an abstract point of view, the scenario can be encoded in Dung's setting using the following argumentation framework: If our clumsy political activist adheres to Dung's semantics, she cannot realize that her first speech must be avoided; indeed, ¡ ¦¢ ¢ has a single extension § ¤ ¨ ¨¡ whatever the semantics among Dung's ones, hence ¤ , and ¡ are considered jointly derivable, which is just what she wants.

Example 1. Let ¡ £¢ £¢ ¥¤ §¦ ¡ © with ¡ ¤ § ¤ ©¨¥ ¨ ¨¡ and ¨¤ § ¥ ¨¤ ¨ ¡ ¨¥ ¨ ¨¥ ¨ © ! . The digraph for ¡ ¦¢ £¢ is depicted on Figure 1.
One way to cope with this problem is to ask for more demanding notions of absence of conflicts than the one considered in Dung's theory. In this paper, we define and study new semantics for Dung's framework based on the idea that an admissible set 2 of arguments should not include controversies, i.e. it should not be the case that an element 3 ¢ of 2 indirectly attacks another argument 3 whenever a second element 3 54 of indirectly defends 3 . On Example 1, this prevents from deriving the set of arguments § ¤ ¨ ¨¡ as a whole; nevertheless, § and § ¤ ¨¡ remain derivable separately.

The specific case when 3 ¢ ¤ 3 4 corresponds to the notion of controversial argu- ments, as introduced by Dung. While some controversial arguments can be inferred using Dung's standard semantics, they are systematically rejected when our careful semantics are considered.

We believe that such prudent semantics can prove helpful to reason with arguments from domains like politics or justice, where a strong notion of "coherence" on the sets of arguments pointed out makes sense.

In the following, we compare the inference relations induced by our new semantics with Dung's ones and show that in many cases one obtains more cautious notions of derivability.

The rest of this paper is organized as follows. We first recall the main definitions and results pertaining to Dung's theory of argumentation. Then, we present our new, careful semantics for argumentation frameworks. In a third section, a comparison of the various notions of acceptability (including Dung's ones) is provided. A final section concludes the paper and gives a few perspectives.

Dung's Theory of Argumentation

Let us present some basic definitions at work in Dung's theory of argumentation [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF]. We restrict them to finite argumentation frameworks.

Definition 1 (finite argumentation frameworks). A finite argumentation framework

is a pair ¡ ¦¢ ¤ ¦ ¡ ¨ where ¡ is a finite set of so-called arguments and ¨is a binary relation over ¡ (a subset of ¡ ¡ ¡ ), the attacks relation.

Clearly enough, the set of finite argumentation frameworks is a proper subset of the set of Dung's finitary argumentation frameworks, where every argument must be attacked by finitely many arguments. The definition above clearly shows that a finite argumentation framework is nothing but a finite, labeled digraph.

The main issue is the inference one, i.e., charactering the sets of arguments which could be reasonably derived from a given argumentation framework. Formally, we shall note ¡ £¢ £¢ ¤ Dunne and Bench-Capon gave a sufficient condition for the unicity of preferred extensions: Proposition 1. Cor. 9 in [START_REF] Dunne | Complexity and combinatorial properties of argument systems[END_REF] Let ¡ ¦¢ ¤ ¦ ¡ ¨ be a finite argumentation framework. If ¡ £¢ has no even-length cycle, then ¡ ¦¢ has a unique preferred extension.

Example 1 (cont'ed). Let ¤ ¤ § ¤ ¨ ¨¡ . ¤ is the grounded extension of ¡ £¢ )¢ , the unique preferred extension of ¡ £¢ ¢ , the unique stable extension of ¡ ¦¢ ¢ and the unique complete extension of ¡ £¢ ¢ . Formally, complete extensions of ¡ £¢ can be characterized as the fixed points of its characteristic function ¥ §¦ ©¨:

Definition 6 (characteristic functions). The characteristic function ¥ ¦ ©¨of an argu- mentation framework ¡ £¢ ¤ ¦ ¡ © is defined as follows: ¥ ¦ ©£ ¦ ¦ ¥ ¦ ©¨ 2 ¤ § ¤ ¢ ¤ is acceptable w.r.t. 2 .

Among the complete extensions of ¡ ¦¢

, the grounded extension of ¡ £¢ is the least element w.r.t. set inclusion [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF].

Dung has shown that every argumentation framework ¡ ¦¢ has a (unique) grounded extension and at least one preferred extension, while it may have zero, one or many stable extensions. These extensions are linked up as follows: Proposition 2. Theorem 25 in [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] Let ¡ £¢ be an argumentation framework.

¡ £¢

; in the finite case, it can be stated as follows:

Definition 7 (well-foundation). Let ¡ ¦¢ ¤ ¦ ¡ ¨ be a finite argumentation frame- work. ¡ £¢ is well-founded if and only if there is no cycle in the digraph ¦ ¡ ¨ .

Proposition 3. Theorem 30 in [5]

A well-founded argumentation framework ¡ £¢ has exactly one complete extension, which is also the unique preferred extension, the unique stable extension and the groun-

ded extension of ¡ £¢ . Example 1 (cont'ed). ¡ £¢ ¢ has no cycle. Hence ¡ £¢ ¢ is well-founded.
Dung has also shown that every stable extension is preferred and every preferred extension is complete; however, none of the converse inclusions holds. When all the preferred extensions of an argumentation framework are stable ones, the framework is said to be coherent: Obviously, the absence of controversial arguments within a set is only necessary to ensure that the set is controversial-free, hence potentially c-admissible (as Example 1 shows, this is not a sufficient condition). Since every argument belonging to an oddlength cycle of ¡ ¦¢ is controversial w.r.t. any argument of the cycle [START_REF] Doutre | Autour de la sémantique préférée des systèmes d'argumentation[END_REF], no such argument can belong to a c-admissible set. In this respect, our approach departs from [START_REF] Baroni | Solving semantic problems with odd-length cycles in argumentation[END_REF][START_REF] Baroni | A recursive approach to argumentation: motivation and perspectives[END_REF] who consider that odd-length and even-length cycles in an argumentation framework should be handled in the same way.

On

¥ ¤ .

Since ¦ is c-admissible for any ¡ £¢

, we obtain as a corollary:

Corollary 1. Every finite argumentation framework ¡ ¦¢ ¤ ¦ ¡ ¨ has a preferred c- extension.
What can be found in preferred c-extensions? Though every argument which is not attacked belongs to at least one preferred c-extension of ¡ £¢ , it is not the case (in general) that it belongs to every preferred c-extension of ¡ £¢ (see Example 1). In this respect, c-preferred extensions hardly contrast with preferred extensions.

Let us now consider the notion of stable c-extension:

Definition 14 (stable c-extensions).

Let ¡ £¢ ¤ ¦ ¡ © be a finite argumentation framework. A conflict-free and contro- versial-free subset Here is a more complex example for illustrating those notions: Let us now explain how c-extensions can be characterized using some fixed point construction:

Example 2. Let ¡ £¢ 4 ¤ ¦ ¡ © with ¡ ¤ § ¤ ¨¥ ¨ ¨ § © and ¨¤ § § ¨© ¨ © ¨¤ ¨ ¥ ¨¤ ¨¤ ¨ © ! ¨ ¥ ¨ ¨ ¨¥ . The digraph for ¡ ¦¢ 4 is depicted on Figure 2. ¤ ¢ ¤ § § ¨¤ ¨ and ¤ 4 ¤ § § ¨¥

Definition 15 (c-characteristic functions).

The c-characteristic function ¥ ¥¤ ¦ ©¨o f a finite argumentation framework ¡ £¢ ¤ ¦ ¡ ©¨ is defined as follows:

¥ ¦¤ ¦ ©¨ ¦ ¦ ¥ ¤ ¦ ©¨ 2 ¤ § ¤ ¢ ¤ is acceptable w.r.t. 2
and 2 ¨ § § ¤ is conflict-free and controversial-free for ¡ £¢ .

We immediately get that: Lemma 3. Let ¡ £¢ ¤ ¦ ¡ © be a finite argumentation framework and let 2 ¥ ¡ be a conflict-free and controversial-free set for ¡ £¢

¥ ¥ ¦¤ ¦ ©¨ .

Contrariwise to the characteristic function of an argumentation framework, ¥ ©¤ ¦ ©¨i s in general nonmonotonic w.r.t. ¥ (and this is also the case for its restriction to the set of all c-admissible subsets of ¡

). Accordingly, we cannot define a notion of c-grounded extension corresponding to the grounded one.

Let us now introduce a notion of complete c-extension:

¦¥ ¨ § ¡ © ¡ § § ¡ , 3 ¤ 2 ¤ ¥ ¡ )
and is an inference principle, either credulous ( ¤ ) or skeptical (

¤ !

).

For instance,

2 ¥ ¡ is a consequence of ¡ ¦¢ w.r.t. ¢ ¤ #" $¢ % ¤ , noted ¡ £¢ ¢ ¤ &" '¢ % ¤ 2 , indicates that 2 is included in every preferred c-extension of ¡ £¢ .
We have compared all the careful inference relations induced by the different semantics w.r.t. cautiousness. We have focused on the case of finite argumentation frameworks with a stable c-extension (otherwise, both ¢ ¤ #" '¢ ( ¤ and ¢ ¤ 0) 1¢ ( ¤ trivialize). We have obtained the following results: One can note that the cautiousness picture for careful inference relations is similar to the one for the inference relations induced from Dung's semantics (assuming that the argumentation frameworks under consideration have stable extension(s)):

Proposition 6. The cautiousness relations reported in

¢ ¤ " $¢ % ¤ £ ¢ ¤ " '¢ ( ¤ £ ¢ ¤ ) ¢ ( ¤ £ ¢ ¤ ) 1¢ % ¤ ¢

Comparisons with Dung's Framework

Let us now compare our careful semantics with Dung's ones. Let us start with a comparison in terms of extensions.

Comparing extensions

First of all, we immediately obtain the following easy result: Clearly, this does not imply that every preferred c-extension is a preferred extension which is conflict-free and controversial-free since maximality w.r.t. set inclusion is required among c-admissible sets. Nevertheless, as a consequence of Proposition 7, we have: Corollary 2. Let ¡ ¦¢ §¤ ¦ ¡ ¨ be a finite argumentation framework. For every pre- ferred c-extension ¤ ¤ of ¡ £¢

, there exists at least one preferred extension ¤ of ¡ ¦¢

s.t. ¤ ¤ ¥ ¤ .
Example 2 (cont'ed). In ¡ ¦¢ 4 , ¤ £ ¤ 4 . This corollary shows in particular that when ¡ £¢ has a unique preferred extension ¤ (especially, when ¡ £¢ is well-founded or without even-length cycle, or triviali.e., when the unique preferred extension of it is empty), ¤ includes every preferred c-extension of ¡ ¦¢ . However, unlike preferred extensions, a well-founded argumentation framework

¡ £¢

can have more than one preferred c-extension (see Example 1).

It can also be the case that a preferred extension of ¡ £¢ does not include any of the preferred c-extensions of ¡ ¦¢

. Furthermore, the presence of even-length cycles in ¡ £¢

does not prevent it from having a unique preferred c-extension. Those two points are illustrated by the following example: ¤ ¢ ¤ § ¥ © and ¤ 4 ¤ § ¥ § are the preferred (and stable) extensions of ¡ ¦¢ .

Example 3. Let ¡ £¢ ¤ ¦ ¡ ¨ with ¡ ¤ § ¤ ¨¥ © ¨¡ © ¨ § and ¨¤ § ¥ ¨¡ ¨ ¥ ! ¨ ¨¡ ¥ ¨¤ ¨ ¤ ©¨ § ¨ © § ¨ § © . The digraph
¤ ¤ § © is the unique preferred c-extension of ¡ ¦¢ . We have ¤ ¥ ¤ 4 . Observe that though the digraph for ¡ ¦¢ has an even-length cycle, ¡ ¦¢ has a unique preferred c-extension.

Another easy consequence of Proposition 7 is:

Corollary 3. Let ¡ ¦¢ ¤ ¦ ¡ © be a finite argumentation framework. If ¡ £¢ is trivial, then ¡ £¢ is c-trivial, i.e.

, the unique preferred c-extension of ¡ £¢ is empty. The converse does not hold.

Let us now turn to stable c-extensions: Lemma 5. Every stable c-extension of a finite argumentation framework ¡ £¢ also is a stable extension of ¡ £¢ . The converse does not hold.

As a direct consequence, we obtain that every stable c-extension of a finite argumentation framework ¡ £¢ also is a preferred extension of ¡ £¢ . However, the converse does not hold.

While every well-founded argumentation framework has a stable extension, it is not the case that every well-founded argumentation framework has a stable c-extension; furthermore, it is also not the case that every argumentation framework which is uncontroversial has a stable c-extension. Example 1 is a counter-example for both cases.

In the same vein, a finite argumentation framework ¡ £¢ that is both well-founded and uncontroversial is not always c-coherent (i.e., such that every preferred c-extension of

¡ £¢

is a stable c-extension of ¡ ¦¢

). In particular, it is not the case that a coherent finite argumentation framework ¡ ¦¢

¥ ¨ § ¡ © ¡ § § ¡ , 3 ¤ 2 ¨ ¤ ¥ ¡ or 3 ¤ ¡ ¨ § £¢ ¥¤ © ¡ )
and is an inference principle, either credulous ( ¤ ) or skeptical ( ¤ ). Since the presence of a stable c-extension changes the picture, we have first considered this specific case, then the general case. We have obtained the following results: Proposition 11. The cautiousness relations reported in Table 2

, ¦ D E , ¦ D E , ¦ D E , ¦ 2 3 4 76 @ D E , ¦ D E , ¦ D E , ¦ D E , ¦ 2 3 A 6 8 E , D ¦ E , D ¦ D E , ¦ E , D ¦ 2 3 A 6 @ E , D ¦ E , D ¦ D E , ¦ E , D ¦ 2 3 ¨ § 6 © E , D ¦ E , D ¦ D E , ¦ E , D ¦ Table 2.
Cautiousness links between careful relations and Dung's ones for AFs with a stable c-extension.

Tables 1 and2 are summarized on Figure 4 (Each arrow can be read as "strictly more cautious than"). In the light of Figure 4, one can observe that the most cautious inference relation among those considered here is ¢ ¤ #" '¢ % ¤ and the least cautious ones are ¢ ¤ ¡) ¢ ( ¤ ¢ ¤ 0) 1¢ %

. Furthermore, strict cautiousness is a complete ordering over the set of inference relations considered in this paper.

Let us now turn to the general case, i.e., argumentation frameworks which do not have necessarily a stable c-extension, or even a stable extension. We do not put potentially trivial relations into the picture (i.e., ¢ ¤

" $¢ ( ¤ , ¢ ¤ ) ¢ ( ¤ , ¢ ¤ " '¢ ( , ¢ ¤ ) 1¢ (
are not considered hereafter): Proposition 12. The cautiousness relations reported in Tables 3 and4 hold for every finite argumentation framework. ¤ ¤ Fig. 5. Cautiousness links between inference relations.

Some Complexity Issues

Before concluding the paper, let us consider some complexity issues. Indeed, in an AI perspective, it is important to determine how hard are the new inference relations we pointed out w.r.t. the computational point of view. We assume the reader acquainted with basic notions of complexity theory, especially the complexity classes P, NP, coNP and the polynomial hierarchy (see e.g. [START_REF] Papadimitriou | Computational complexity[END_REF]). We have shown in a previous paper [START_REF] Coste-Marquis | Symmetric argumentation frameworks[END_REF] that considering sets of arguments (instead of single arguments) as input queries for the inference problem does not lead to a complexity shift when Dung's inference relations are considered (the purpose is to determine whether such sets are derivable from a given finite argumentation framework

¡ £¢

). As to the careful inference relations, the same conclusion can be drawn.

First of all, it is easy to show that, given a finite argumentation framework ¡ ¦¢ , deciding whether a given argument indirectly attacks (resp. indirectly defends) a given argument is in P, and deciding whether a set of arguments is controversial-free is in P.

Accordingly, deciding whether a given set of arguments is c-admissible for ¡ £¢ is in P. As a consequence, deciding whether a given set of arguments is a stable c-extension of ¡ £¢ is in P as well. Therefore, deciding whether a given set of arguments 2 is included in every stable c-extension of ¡ £¢ is in coNP (in order to show that the complementary problem is in NP, it is sufficient to guess a set ¤ ¥ ¡ and to check in polynomial time that ¤ is a stable c-extension of ¡ £¢ and that 2 is not included in ¤ ).

Besides, deciding whether a set of arguments 2 is a preferred c-extension of ¡ £¢ is in coNP (in order to show that the complementary problem is in NP, it is sufficient to guess a proper superset ). As a consequence, deciding whether a given set of arguments 2 is included in every preferred c-extension of ¡ ¦¢ is in ¢¡ 4 (in order to show that the complementary problem is in £ ¡ 4 , it is sufficient to guess a set ¤ ¥ ¡ and to check in polynomial time using an NP oracle that ¤ is a preferred c-extension of ¡ £¢ and that 2 is not included in ¤ ).

Finally, deciding whether a given set of arguments is included in a preferred cextension (resp. a stable c-extension) of ¡ £¢ is in NP. To be more precise: Definition 18. C-CA(AF, S) is defined as follows:

¤ Input ¡ £¢ ¤ ¦ ¡ © a finite argumentation framework, and 2 ¥ ¡ Accordingly, our careful inference relations are not computationally more complex that the corresponding ones based on Dung's semantics (see [START_REF] Dimopoulos | Graph theoretical structures in logic programs and default theories[END_REF][START_REF] Dunne | Coherence in finite argument system[END_REF]).

Conclusion

We have presented new careful semantics within Dung's theory of argumentation. Under such careful semantics, two arguments cannot belong to the same extension whenever one of them indirectly attacks a third argument, while the second one indirectly defends it. In particular, controversial arguments are always rejected. This seems to be highly desirable in domains where controversies can be interpreted as contradictions, as we exemplified it. We have also compared our careful inference relations with Dung's ones and considered some complexity issues, showing that our inference relations are not more complex than the corresponding ones based on Dung's semantics.

Our work calls for some perspectives. A first perspective consists in developing specific algorithms for computing careful extensions, based on algorithms for computing extensions like those described in [START_REF] Vreeswijk | Credulous and sceptical argument games for preferred semantics[END_REF][START_REF] Cayrol | On decision problems related to the preferred semantics for argumentation frameworks[END_REF][START_REF] Doutre | On sceptical vs credulous acceptance for abstract argument systems[END_REF]. A second perspective consists in combining the notion of safe extension introduced recently [START_REF] Cayrol | On the acceptability of arguments in bipolar argumentation framework[END_REF] with the notion of careful extension.

Fig. 1 .

 1 Fig.1. The digraph for ' )( 10 .

1 . 2 .

 12 Every preferred (resp. stable, complete) extension of ¡ £¢ contains the grounded extension of ¡ £¢ . The grounded extension of ¡ £¢ is included in the intersection of all the complete extensions of ¡ ¦¢ . The purest argumentation frameworks ¡ ¦¢ in Dung's theory are those for which all the notions of acceptability coincide. Dung has provided a sufficient condition for an argumentation framework ¡ £¢ to satisfy this requirement, called the well-foundation of

2 and 2 Lemma 1 .

 221 is conflict-free and controversial-free for ¡ ¦¢ . Example 1 (cont'ed). § , and § ¤ ©¨¡ and its subsets except § ¤ are the c-admissible sets for ¡ £¢ ¢ . From Definition 12, the next lemma follows immmediately: Let ¤ , ¥ be two arguments of a finite argumentation framework ¡ ¦¢ . If ¤ is controversial w.r.t. ¥ , then § ¤ cannot be included in a c-admissible set for ¡ ¦¢ .

Proposition 5 .

 5 admissible for ¡ ¦¢ . Example 1 (cont'ed). § ¤ ¨¡ and § are the preferred c-extensions of ¡ £¢ )¢ . We have the following easy proposition: Let ¡ ¦¢ ¤ ¦ ¡ ©¨ be a finite argumentation framework. For every c- admissible set 2 ¥ ¡ for ¡ £¢ , there exists at least one preferred c-extension ¤ ¥ ¡ of ¡ £¢ s.t.

Fig. 2 .

 2 Fig. 2. The digraph for ' )( £¢ .

.

2

  is c-admissible for ¡ £¢ if and only if

Proposition 7 .

 7 Every c-admissible set for a finite argumentation framework ¡ ¦¢ is also admissible for ¡ £¢ . The converse does not hold.

Fig. 3 .

 3 Fig. 3. The digraph for ' )( ¡ .

Fig. 4 .

 4 Fig. 4. Cautiousness links between inference relations for ' ( s with a stable c-extension.

.¤¤Proposition 14 .

 14 Quest Is 2 included in a preferred c-extension of ¡ £¢ ? Proposition 13. C-CA(AF, S) is NP-complete. Definition 19. IN-C-STAB(AF, S) is defined as follows: ¤ Input ¡ £¢ ¤ ¦ ¡ © a finite argumentation framework, and 2 ¥ ¡ . Quest Is 2 included in a stable c-extension of ¡ ¦¢ ? IN-C-STAB(AF, S) is NP-complete.

Definition 2 (acceptable sets). Let

  2where ¡ £¢ ¤ §¦ ¡ ©¨ is a finite argumentation framework and 2 ¦¥ ¡ ¡ £¢ ¤ ¦ ¡ © be a finite argumentation framework.

	A second important notion is the notion of absence of conflicts. Intuitively, two
	arguments should not be considered together whenever one of them attacks the other
	one.
	, is typically . An inference relation ¢ ¤ under ¢ ¤ based on a notion of extension, and an inference principle (credulous or skeptical), to state that 2 is a consequence of ¡ ¦¢ so that ¡ £¢ §¢ ¤ 2 holds if and only if 2 is included in all (skeptical) or at least one (credulous) extension of ¡ £¢ .
	In order to define a notion of extension, a first important notion is the notion of acceptability: an argument ¤ is acceptable w.r.t. a set of arguments whenever it is de-fended by the set, i.e., every argument which attacks ¤ is attacked by an element of the
	set.
	An argument ¤ ©¨¡ is acceptable w.r.t. a subset 2 of ¡ s.t. ¥ ¨¤ ¨¨, there exists ¨2 s.t. ¨¥ ¨¨. A set of arguments is acceptable w.r.t. if and only if for every ¥ ¨¡ 2 when each of its elements is acceptable w.r.t. 2 .

Definition 3 (conflict-free sets). Let

  ¡ ¦¢ §¤ ¦ ¡ ¨ be a finite argumentation frame-

	work. A subset ¤ ¨¥ ¡ ¨¨.	2 of ¡	is conflict-free if and only if for every ¤ ¨¥ ¨2 , we have
	Requiring the absence of conflicts and the form of autonomy captured by self-
	acceptability leads to the notion of admissible set.

Definition 4 (admissible sets). Let

  ¡ £¢ ¤ ¦ ¡ ©¨ be a finite argumentation framework. Let ¡ £¢ ¤ ¦ ¡ ©¨ be a finite argumentation framework.

	A subset	2 of ¡	is admissible if and only if	2	is conflict-free and acceptable w.r.t.	2 .
	The significance of the concept of admissible sets is reflected by the fact that ev-
	ery extension of an argumentation framework under the standard semantics introduced
	by Dung (preferred, stable, complete and grounded extensions) is an admissible set,
	satisfying some form of optimality:
	Definition 5 (extensions). -A subset 2

of ¡ is a preferred extension of ¡ £¢ if and only if it is maximal w.r.t. ¥ among the set of admissible sets for ¡ ¦¢ . -A subset 2 of ¡ is a stable extension of ¡ £¢ if and only if it is conflict-free and for every argument ¤ from ¡ £¢ 2 , there exists ¥ ¨2 s.t. ¥ ¨¤ ¨¨. -A subset 2 of ¡ is a complete extension of ¡ £¢ if and only if it is admissible and it coincides with the set of arguments acceptable w.r.t. itself. -A subset 2 of ¡ is the grounded extension of ¡ ¦¢ if and only if it is the least element w.r.t. ¥ among the complete extensions of ¡ £¢ .

Definition 8 (coherence).

  Let ¡ £¢ ¤ §¦ ¡ ©¨ be an argumentation framework. ¡ £¢ Coherence is a desirable property. Dung gave a sufficient condition for it based on the notion of controversial argument:

	coherent if and only if every preferred extension of ¡ ¦¢	is also stable.	is

Definition 9 (controversial arguments).

  Let ¤ ¨¥ ¨¡ . ¤ indirectly attacks ¥ if and only if there exists an odd-length path from ¤ to ¥ in the digraph for ¡ ¦¢ Let ¤ ©¨¥ ¨¡ . ¤ indirectly defends ¥ if and only if there exists an even-length path from ¤ to ¥ in the digraph for ¡ ¦¢ . The length of this path is not zero.-Let ¤ ¨¥ ¨¡ . ¤ is controversial w.r.t. ¥ if and only if ¤ indirectly attacks ¥ and ¤ uncontroversial if and only if there is no pair ¤ , ¥ of arguments of ¡ such that ¤ is controversial w.r.t. ¥ . -¡ £¢ is limited controversial if and only if there is no infinite sequence of arguments ¤ ¡ £¢ ¤¢ £¢ ¨¤ ¦¥ £¢ ¤¢ £¢ of ¡ s.t. ¤ ¨ § © ¢ is controversial w.r.t. ¤ ¨ § . and only if for every ¤ ¨¥ ¨2 and every ¨¡ , ¤ ©¨¥ is not super-controversial w.r.t. .

	Dung has shown the following theorem:
	Proposition 4. Theorem 33 in [5]
	Every uncontroversial or limited controversial argumentation framework is coherent.
	3 Careful Extensions
	Let us now present our new semantics for Dung's argumentation frameworks. They are
	based on the notion of super-controversial pair of arguments:
	Definition 10 (super-controversial arguments).
	Definition 12 (c-admissible sets). Let ¡ £¢ ¤ ¦ ¡ ¨ be a finite argumentation frame-work. 2 ¥ ¡ is c(areful)-admissible for ¡ ¦¢ if and only if every ¤ ¨2 is acceptable
	w.r.t.
	Let ¡ £¢ ¤ ¦ ¡ © be an argumentation framework.
	indirectly defends ¥ .

-. --¡ £¢ is Let ¡ £¢ ¤ ¦ ¡ ¨ be a finite argumentation framework and let ¤ ¨¥ © ¨¡ . ¤ ¨¥ is super-controversial w.r.t. if and only if ¤ indirectly attacks and ¥ indirectly defends . Example 1 (cont'ed). In ¡ ¦¢ , ¨¡ is super-controversial w.r.t. ¤ . Obviously enough, the notion of super-controversial pair of arguments extends the notion of controversial arguments since ¤ is controversial w.r.t. if and only if ¤ ¨¤ is super-controversial w.r.t. .

In order to address Example 1 in a more satisfying way, we need to reinforce Dung's notion of conflict-free set of arguments; we consider in addition the notion of controversial-free set of arguments: Definition 11 (controversial-free sets). Let ¡ £¢ ¤ ¦ ¡ ¨ be a finite argumentation framework.

2 ¥ ¡

is controversial-free for ¡ £¢ if

  this ground, one can define several notions of careful extensions, echoing Dung's ones. Let us start with preferred c-extensions:

	Definition 13 (preferred c-extensions). Let ¡ £¢ ¤ ¦ ¡ ¨ be a finite argumentation framework. A c-admissible set 2 ¥ ¡ for ¡ £¢ is a preferred c-extension of ¡ £¢ if and only if 2 ¢¡ ¥ ¡ s.t.

  © are the two preferred (and stable) extensions of ¡ ¦¢ 4 . ¤ ¢ is the unique stable c-extension of ¡ £¢ 4 . ¤ ¢ and ¤ ¤ § ¥ are the two preferred c-extensions of ¡ ¦¢ 4 .

Definition 16 (complete c-extensions). Let

	Definition 17 (careful inference relations). ¢ ¤ ¡ £¢ ¤ ¤ denotes the careful inference rela-tion obtained by considering a careful semantics 3 (where 3 ¤
	conflict-free and controversial-free for ¡ ¦¢	belongs to	2 .	and such that	2 § § ¤ is
	Example 1 (cont'ed). § ¤ ¨¡ is a complete c-extension of ¡ £¢ ¢ .
	From the definition, it comes immediately that:
	Lemma 4. A conflict-free and controversial-free set of arguments extension of ¡ £¢ if and only if ¥ ¤ ¦ ©¨ 2 ¤ 2 .	2 is a complete c-
	Let us now define several inference relations based on our careful semantics for
	argumentation frameworks:

¡ £¢ ¤ ¦ ¡ © be a finite argumentation framework and let 2 be a c-admissible set for ¡ ¦¢ . 2 is a complete c-extension of ¡ ¦¢ if and only if every argument ¤ which is acceptable w.r.t. 2

Table 1

 1

	hold for every finite ar-

Table 1 .

 1 Cautiousness links between careful inference relations for AFs with a stable c-extension.

  Let ¡ ¦¢ §¤ ¦ ¡ ¨ be a finite argumentation framework. If ¡ £¢ Let ¡ ¦¢ §¤ ¦ ¡ ¨ be a finite argumentation framework. If ¡ £¢ is also included in the intersection of all stable extensions of ¡ £¢ . Now, what's about controversies? We have the two following lemmata: Lemma 6. Let ¡ £¢ ¤ ¦ ¡ © be a finite argumentation framework. ¡ ¦¢ Let ¡ £¢ ¤ ¦ ¡ ©¨ be a finite argumentation framework. If ¡ £¢ Let ¡ £¢ ¤ ¦ ¡ ©¨ be a finite argumentation framework. If ¡ £¢ Let ¡ ¦¢ ¤ ¦ ¡ ¨ be a finite argumentation framework s.t. ¡ £¢ We have compared our careful inference relations with the ones induced by Dung's semantics w.r.t. cautiousness. Let ¢ ¤ ¢ ¤

	4.2 Comparing inference relations
	denote the inference relation obtained by consider-ing Dung's semantics 3 (where 3 ¤
	is always c-coherent as well (see Example 1). It turns out that argumentation frameworks ¡ £¢ with a stable c-extension are partic-ularly interesting. Indeed, whenever ¡ ¦¢ has a stable c-extension, we have:
	Proposition 8. has a stable c-extension, the grounded extension of ¡ £¢ and the intersection of all preferred extensions of ¡ £¢ coincide (i.e., ¡ £¢ is relatively grounded).
	Proposition 9. has a stable c-extension, the intersection of all preferred c-extensions of ¡ £¢ is included in the grounded extension of ¡ ¦¢
	troversial if and only if ¡ £¢	has no odd-length cycle.	is limited con-
	Lemma 7. has a stable c-extension, then ¡ £¢ is limited controversial.
	Thanks to Proposition 4, we obtain the following corollary:
	Lemma 8. has a stable c-extension, then ¡ £¢ is coherent.
	We also have:
	Proposition 10. has a stable c-extension. For every preferred c-extension ¤ ¤ of ¡ ¦¢ , there exists at least one stable extension 2 of ¡ ¦¢ . s.t. ¤ ¤ ¥ 2 Finally, we can show that the set of complete extensions of ¡ ¦¢ and the set of com-plete c-extensions of ¡ £¢ are not comparable w.r.t. ¥ .

.

Subsequently, if ¡ £¢ has a stable c-extension, the intersection of all preferred cextensions of ¡ £¢ is included in the intersection of all preferred extensions of ¡ £¢ ; hence, it

Table 3 .

 3 Cautiousness links between careful relations.

	2 3 54 76 8 9 2 3 BA 6 8 9 2 3 54 76 8 D E , ¦ D E , ¦ 2 3 BA 6 8 D E , D ¦ D E , D ¦ 2 3 § 6 © D E , D ¦ D E , D ¦

Table 4 .

 4 Cautiousness links between careful relations and Dung's ones.

	Tables 3 and 4 are summarized on Figure 5. Proposition 12 shows that the lack of a stable c-extension does not question the way ¢ ¤ 0) 1¢ % ¤ and ¢ ¤ " '¢ % ¤ are linked up (see Proposition 6). Contrastingly, when the existence of a stable c-extension is not guaranteed, the cautiousness links between our careful relations and Dung's ones are heavily modified. Compared with the results reported on Figure 4, one can observe that in the general case, it is not guaranteed that ¢ ¤ ¡ ¢ ¢ and ¢ ¤ " $¢ % coincide, that ¢ ¤ " $¢ % ¤ is more cautious than ¢ ¤ £ ¢ ¢ and that ¢ ¤ &" '¢ % is more cautious than ¢ ¤ 0) 1¢ % ¤ . 2 3 § 6 © 2 3 A 6 8 2 3 4 76 8 2 3 4 76 8 9 2 3 A 6 8 9