Perfect plasticity versus damage: an unstable interaction between irreversibility and $\Gamma$-convergence through variational evolutions

This paper addresses the question of the interplay between relaxation and irreversibility through quasi-static evolutions in damage mechanics, by inquiring the following question: can the quasi-static evolution of an elastic material undergoing a rate-independent process of plastic deformation be derived as the limit model of a sequence of quasi-static brittle damage evolutions? This question is motivated by the static analysis performed by Babadjian-Iurlano-Rindler, where they have shown how the brittle damage model introduced by Francfort and Marigo can lead to a model of Hencky perfect plasticity. Problems of damage mechanics being rather described through evolution processes, it is natural to extend this analysis to quasi-static evolutions, where the inertia is neglected. We consider the case where the medium is subjected to time-dependent boundary conditions, in the one-dimensional setting. The idea is to combine the scaling law considered by Babadjian-Iurlano-Rindler with the quasi-static brittle damage evolution introduced by Francfort and Garroni, and try to understand how the irreversibility of the damage process will be expressed in the limit evolution. Surprisingly, the interplay between relaxation and irreversibility is not stable through time evolutions. Indeed, depending on the choice of the prescribed Dirichlet boundary condition, the effective quasi-static damage evolution obtained may not be of perfect plasticity type.

1. Introduction 1.1.Interplay between Γ-convergence and variational evolutions.Rate-independent systems have proved to be very useful in many problems of continuum mechanics dealing with dissipative phenomena such as elastoplasticity, damage or fracture.These models share similar energetic formulations which put in competition a stored energy and a dissipated one which does not depend on the speed of the loading (see [25] and references therein).When the model involves a scaling parameter, say ε > 0, a natural attempt consists first in studying static models before treating the related evolutions.Such considerations can be dealt within a more general setting, such as in [26] where the authors derive a sufficient condition in order for a family of parametrized time-dependent energy functionals of the form E ε + D ε to approximate the expected effective energy E 0 + D 0 in the limit, where E ε and D ε respectively stand for the stored and dissipated parts of the total energy and E 0 and D 0 are their corresponding Γ-limits.In a nutshell, for ε ∈ [0, ∞), if E ε stands for the stored energy and D ε (q, q) stands for the minimal energy dissipated as the medium changes from the states q to q, an evolution q ε is called an energetic solution during the time interval [0, T ] if it satisfies the following stability and energy balance conditions E ε (t, q ε (t)) ≤ E ε (t, q) + D ε (q ε (t), q) for all admissible state q E ε (t, q ε (t)) + Diss ε (q ε ; 0, t) = E ε (0, q ε (0)) + t 0 ∂ s E(s, q ε (s)) ds 1 for all time t ∈ [0, T ], where the cumulated dissipation Diss ε (q ε ; 0, t) is the total variation of q ε with respect to the "distance" D ε in the time interval [0, t].Given energetic solutions {q ε } ε>0 converging to some q 0 , the authors derive in [26,Theorem 3.1] a sufficient condition in order for q 0 to be an energetic solution of the limit problem associated to E 0 and D 0 .In particular, a joint condition on the interplay between the stored and dissipated energies is needed.Unfortunately, even if we have separate Γ-convergence of E ε and D ε to E 0 and D 0 respectively, the Γ-limit of the total energies E ε + D ε might differ from the sum of the Γ-limits.This particular issue is adressed in [8] where the authors consider a family of quasi-static evolutions involving internal oscillating energies E ε and dissipations D ε and show that the Γ-limit of the sum can still be additively split as the sum of a stored energy Ẽ0 and a dissipated one D0 , even though they a priori differ from E 0 and D 0 .More generally, the interaction between Γ-convergence and variational evolutions frequently involves unexpected and tedious non-commutability phenomena in various contexts.For instance, such considerations have attracted renewed interest in the derivation of lower dimensional models for thin structures in the evolutionary setting, in the context of elastoplasticity [11,22], crack propagation [6] or delamination problems [27], without being exhaustive.Another case study concerns the stability of unilateral minimality properties through variational evolutions, as in fracture mechanics [21] or periodic homogenization in multi-phase elastoplasticity [15].1.2.Motivation and results.In the static analysis led in [7,Theorem 3.1], the authors consider a family of brittle damage energies (introduced in [17,18]) within a specific scaling law, and show how an asymptotic analysis in a singular limit can lead to a model of Hencky perfect plasticity.More precisely, they introduce a small parameter ε > 0 and consider a linearly elastic material which can only exist in one of two states: a damaged one whose elastic properties are described via a symmetric fourth-order Hooke Law εA 0 and a sound one with a stronger elasticity tensor A 1 , satisfying εA 0 < A 1 in the sense of quadratic forms acting on M N ×N sym .Introducing the characteristic function of the damaged region, χ ∈ L ∞ (Ω; {0, 1}), and following the model introduced by Francfort and Marigo, the total energy associated to a displacement u ∈ H 1 (Ω; R N ) and χ is given as the sum of the elastic energy stored inside the material and a dissipative cost, taken as proportional to the volume of the damaged zone: where κ/ε > 0 is the material toughness and the symmetric gradient e(u) = ∇u + ∇u T /2 is the linearized elastic strain.As the parameter ε tends to 0, the elasticity coefficients of the weak material degenerate to zero while the diverging character of κ/ε forces the damaged region to concentrate on vanishingly small sets.It is by now well-known that for fixed ε > 0, the minimization of the above energy with respect to the couple (u, χ) is ill-posed, so that the energy must be relaxed.By doing so, the brittle character of the damage is lost as minimizing sequences tend to develop microstructures and the class of admissible solutions is extended to the set of all possible homogenized elasticities, resulting from fine mixtures of strong and weak material (see [18,14,1,3]).Given some displacement u and minimizing first pointwise with respect to χ, one can check that the asymptotic analysis of these energies is equivalent to finding the Γ-limit of the family of functionals dx when ε ց 0, or still the Γ-limit of their lower semicontinuous envelopes, given by where SQW ε is the symmetric quasiconvex envelope of W ε (see [3]).An explicit formula of SQW ε is generally unknown, as its expression is obtained through a minimization among all attainable composite materials (the G-closure set, see [1]) and makes use of the Hashin-Shtrikman bounds.Slightly adapting the proof of [7, Theorem 3.1] (see Appendix 7 for a precise statement and its proof), the authors have shown that when A 0 and A 1 are isotropic Hooke Laws defined by where λ 1 > λ 0 > 0 and µ 1 > µ 0 > 0 are the Lamé coefficients, the brittle damage energies E ε Γ-converges in L 1 (Ω; R N ) as ε ց 0 to the functional with is the support function of K, standing for the plastic dissipation potential (see [29]).In particular, for all displacement u ∈ BD(Ω), writing the Radon-Nikodým decomposition of Eu with respect to Lebesgue Eu = e(u)L N Ω + E s u and using the definition of the infimal convolution, we infer that the absolutely continuous linearized strain can be additively split as e(u) = e + p a with e and p a ∈ L 1 (Ω; M N ×N sym ) such that is indeed the energy functional corresponding to Hencky perfect plasticity, as mentionned in [24].
The objective of the present paper is to extend this work to the quasi-static case in a onedimensional setting.More specifically, we consider a linearly elastic material whose reference configuration is Ω = (0, L), a bounded open interval, with toughness κ > 0 and stiffness a 1 > 0, subjected to a prescribed time-dependent displacement on ∂Ω = {0, L}: Adapting the analysis led in [7] to the quasi-static setting, we consider a family of quasi-static brittle damage evolutions (introduced in [14]) within the same specific scaling law.More precisely, we introduce a small parameter ε > 0 and apply [14,Theorem 2] to a linearly elastic material which can only exist in a damaged state or in a sound state with respective stiffness 0 < εa 0 < a 1 , subjected to the prescribed displacement w on ∂Ω and with toughness κ/ε.Thus, we recover a triple discribing the quasi-static evolution of brittle damage undergone by the medium for a fixed ε > 0.
In other words, the state of the damaged medium (for ε > 0 fixed) at time t ∈ [0, T ] is dictated by the displacement u ε (t) while its elastic properties are given by the stiffness a ε (t) ∈ G Θε(t) (εa 0 , a 1 ) (see Section 2), where Θ ε (t) is the volume fraction of sound material a 1 (see Proposition 3.1 below).
We next wish to perform the asymptotic analysis of these evolutions when taking the limit ε ց 0, in the hope of recovering a quasi-static evolution of perfect plasticity in the limit, of which we briefly recall the fundamentals now.
In [29], Suquet proposed the (first complete) mathematical kinematical framework adapted to evolutions of perfect plasticity for dissipative materials and proves the existence of solutions in terms of the displacement field, under the assumption of small deformations.Heuristically, let Ω = (0, L) be the configuration at rest of an elastoplastic medium with stiffness a 1 , whose evolution is driven by a time-dependent boundary displacement w : [0, T ] × R → R prescribed on ∂Ω.The behaviour of the material is described via three kinematic variables (u, e, p), where the displacement u : [0, T ] × Ω → R is such that the linearized strain Du = e + p is additively decomposed in an elastic strain e : [0, T ] × Ω → R and a plastic strain p : [0, T ] × Ω → R accounting for the reversible and permanent deformations respectively.In the quasi-static setting, where inertia is neglected, the evolution satisfies the Constitutive Equations In other words, the Constitutive Equations mean that the elastic strain is proportional to the stress σ, which is constrained to lie in a given closed and convex set K ⊂ R standing for the elasticity domain and whose boundary ∂K is referred to as the yield surface.The last assertion is nothing but Hill's maximum work principle.More recently, quasi-static plastic evolutions have been revisited into a variational evolution formulation for rate-independent processes.The problem has been interpreted in an energetic form that does not require the solutions to be smooth in time nor in space, making use of modern tools of the calculus of variations instead (see [25,12,24] and references therein).Following [12, Definition 4.2], a quasi-static evolution of perfect plasticity is a triple subjected to the relaxed boundary condition p(t) ∂Ω = (w(t) − u(t)) (δ L − δ 0 ) and satisfying the additive decomposition Du(t) = e(t)L 1 Ω + p(t) Ω, such that The existence of quasi-static evolutions is (by now classically) obtained by performing a timediscretization and solving incremental minimization problems inductively, before letting the timestep tend to 0 (see [25,9,10,20,12] for instance).The purpose of the present paper is not to prove the existence of quasi-static evolutions of perfect plasticity, but to establish whether such evolutions can be derived from the quasi-static brittle damage evolutions introduced above in (1.2).By analogy with the static analysis of [7], we expect to derive the same closed convex set of plasticity K which is given by the closed interval as one can check that G(τ ) = τ 2 /a 0 for all τ ∈ R in this simplified setting.In particular, the support function of K is simply given by Therefore, following the variational framework of quasi-static perfect plasticity recalled above (see [12,25]), the dissipative cost cumulated during a time interval [s, t] ⊂ [0, T ] due to a time dependent Radon measure q : [0, T ] → M([0, L]) is defined as Diss K (q; s, t) = √ 2κa 0 V(q; s, t) where is the total variation of q during the time interval [s, t].The question inquired in the present work is then: when passing in the limit ε ց 0 (in some sense detailed in the next sections) in the above brittle damage evolutions (1.2), can we derive a quasi-static evolution of perfect plasticity (u, e, p) satisfying (1.3) and (1.4)?Contrary to the static analysis, the interplay between damage and Γ-convergence turns out to be unstable through the time evolution process.Indeed, as explained in Theorem 1.1 and Theorem 1.2, the effective quasi-static evolution derived in the subsequent sections might not be of perfect plasticity type.Instead, it can be interpreted as one of damage, characterised by means of the material's compliance as internal variable: Theorem 1.1.Let ε > 0 and (u ε , Θ ε , a ε ) be a quasi-static evolution of the homogenized brittle damage model given by Proposition 3.1.There exists a subsequence (not relabeled) and absolutely continuous functions when ε ց 0 and satisfying the following assertions for all t ∈ [0, T ]: i. Additive Decomposition: Furthermore, the effective compliance defined by is non-decreasing in time and satisfies the following assertions: vi. Constitutive Equation: As mentionned above, according to the choice of the Dirichlet condition, the medium's response to the loading might differ from a perfect plastic behaviour: For all 0 ≤ s < t ≤ T, w(t) The present study may be seen as an illustration of non-stability issues arising when dealing with problems of H-convergence in the L 1 (Ω) framework where ellipticity is lost during the time process, even when working in the simplest evolution setting and in dimension one.
1.3.Organization of the paper.In Section 2, we recall some notation and preliminary results.
In Section 3, we introduce the family of quasi-static brittle damage evolutions derived in [14, Theorem 2] associated to a linearly elastic material with toughness κ/ε and stiffness tensors εa 0 and a 1 , for ε > 0, given a prescribed boundary datum and no volume force load in the one-dimensional setting.Particularly, due to the explicit knowledge of the G-closure set of all admissible homogenized composite materials in dimension one, we collect starting information of crucial interest for the subsequent sections.In Section 4, we derive the effective quasi-static evolution when passing to the limit ε ց 0. We first give uniform bounds in Proposition 4.1 and next analyse the behaviour and regularity properties of the effective evolution.Section 5 adresses the question of the nature of the quasi-static evolution and determines in Theorem 1.2 the necessary and sufficient condition ensuring the perfect plastic behaviour of the evolution.Finally, Section 6 discusses whether the present work could be improved in order to derive a quasi-static evolution of perfect plasticity.Functional spaces.We use standard notation for Lebesgue and Sobolev spaces.If U is a bounded open subset of R N , we denote by L 0 (U ; R m ) the set of all L N -measurable functions from U to R m .We recall some properties regarding functions with values in a Banach space and refer to [16,5,12] for details and proofs on this matter.If Y is a Banach space and T > 0, we denote by AC([0, T ]; Y ) the space of absolutely continuous functions f :

Matrices
is a function of two variables, time and spatial derivatives will be respectively denoted by ḟ and f ′ .
Functions of bounded variation.Let U ⊂ R N be an open bounded set.A function u ∈ L 1 (U ; R m ) is a function of bounded variation in U , and we write u ∈ BV (U ; R m ), if its distributional derivative Du belongs to M(U ; M m×N ).We use standard notation for that space and refer to [2] for details.We just recall that if U has Lipschitz boundary, every function u ∈ BV (U ; R m ) has an inner trace on ∂U (still denoted by u and H N −1 -integrable on ∂U ) and there exists a constant C > 0 depending only on U such that according to [30, Proposition 2.4, Remark 2.5 (ii)].
Homogenization and H-convergence We refer to [1] for an exhaustive presentation of these notions and only recall minimal results.We denote, for fixed α, β > 0, the subset of fourth-order symmetric tensors Let Ω be a bounded open set of R N .We say that Convex analysis.We recall some definition and standard results from convex analysis (see [28]).Let f : R N → [0, +∞] be a proper function (i.e.not identically +∞).The convex conjugate of f is defined as which turns out to be convex and lower semicontinuous.If f is convex and finite, we define its recession function as which is convex and positively 1-homogeneous.If f, g : R N → [0, +∞] are proper convex functions, then their infimal convolution is defined as which is convex as well.The indicator function of a set C ⊂ R N is defined as I C = 0 in C and +∞ otherwise.The convex conjugate I * C of I C is called the support function of C.

Francfort-Garroni's model of Quasi-Static Brittle Damage
For all ε > 0, we consider a linearly elastic material whose reference configuration is Ω = (0, L), with toughness κ/ε and stiffness tensors εa 0 and a 1 , corresponding to its damaged and sound zones respectively.Applying Theorem 2 and Remark 5 of [14] to this linearly elastic material without volume force load and with a prescribed boundary condition w ∈ AC([0, T ]; H 1 (R)), it ensures the following existence result for a relaxed quasi-static damage evolution.Proposition 3.1.For all ε > 0, there exist a time-dependent density, a displacement and a stiffness tensor all weakly-* measurable, such that Energy Balance: for all t ∈ [0, T ], the total energy Proof.This is the direct application of [14, Theorem 2, Remark 5] together with [1, Lemma 1.3.32,Formula (1.109)] which stipulates that for all 0 < a < b and all θ ∈ L ∞ ((0, L); [0, 1]), In particular, one gets that is L 1 -measurable, as it is the difference between two non-increasing functions.For similar reasons, one also infers that Θ ) are (a priori) only weakly-* measurable.To see this, it suffices to take φ ∈ L 1 ((0, L)) instead of L 2 ((0, L)) above.
For all ε > 0, a naive first use of the One-sided Minimality (3.3) entails the following properties of the stress For all ε > 0 and all t ∈ [0, T ], the stress σ ε (t) is homogeneous in space and Proof.Indeed, for all v ∈ H 1 0 ((0, L)) and δ > 0, applying (3.3) with θ = 0 and Dividing by δ > 0 then letting δ ց 0 entails that L 0 σ ε (t)v ′ dx = 0, which implies the space homogeneity of σ ε .Formula (3.5) is a consequence of the expression of a ε and the definition of σ ε .
For all ε > 0 and t ∈ [0, T ], we define the function A second application of the One-Sided Minimality (3.3) implies that for L 1 -a.e.x ∈ (0, L) where Proof.Let ε > 0 and t ∈ [0, T ].As we are working in the scalar setting, symetric quasiconvex and convex envelopes coincide.According to [3, Lemma 3.1], we have that for all ξ ∈ R and x ∈ (0, L) ´.

The limit quasi-static evolution
As previously explained, the objective of this work is to derive an effective limit model by letting ε tend to 0. Since we expect a limit model of perfect plasticity type, one has to identify which quantities will play the role of the elastic and plastic strains at the scale ε > 0. Meanwhile, in order to pass to the limit along converging subsequences in the brittle damage evolutions described in Proposition 3.1, we rely on uniform bounds computed in Proposition 4.1 below. and Proof.Let us first prove (4.1).Note that (3.2) applied with θ = 0 and v = w(0) directly entails As for the subsequent times t ∈ [0, T ], (3.3) applied with θ = 0 and v = w(t) entails On the other hand, since w is absolutely continuous from [0, T ] into H 1 (R), we infer that is Bochner integrable.Therefore, gathering the uniform bound on the initial time energies together with Cauchy-Schwarz inequality for the scalar-product (for all time s ∈ [0, t] fixed) and the Energy Balance (3.4), we get that We next show (4.2).Let ε > 0 and t ∈ [0, T ].First, as shown in [7, Lemma 2.3, Formula (2.6)], there exists a constant c > 0 (only depending on a 0 , a 1 and κ) such that the function Remembering the definition (3.1) of a ε (t) and the fact that for all ξ ∈ R, we in particular get that Thus, using the equivalent norm in BV ((0, L)) recalled in (2.1) leads to ).Finally, the homogeneity in space of the stress σ ε (t) ∈ R implies that as well, thus concluding (4.2).
From the uniform bounds (4.2), we obtain compactness properties.
Proof.One directly deduces from (4.2) that Θ ε (t) strongly converges to 1 in L 1 ((0, L)) as ε ց 0 for all time t ∈ [0, T ].Next, using the non-decreasing character (in time) of the non-negative Radon measures In particular, we infer that , where we exchanged the infimum and the integral thanks to Aumann's criterion.
Therefore, we deduce from the Fundamental Theorem of Γ-convergence together with the bounds (4.1) and (4.2) that there exist a further subsequence (still not relabeled) and a displacement u(0) ∈ BV ((0, L)) such that when ε ց 0.
4.3.Time independence of the subsequences.We first show that, along the whole subsequence introduced in Proposition 4.2 (not relabeled and independent of t), {σ ε (t)} ε pointwise converges to some limit stress σ(t) for all time in [0, T ].
We now define what will play the role of the elastic and plastic strains at the scale ε > 0, by setting which, by (3.5), satisfy the additive decomposition u ′ ε = e ε + p ε at all time.Using Proposition 4.2 together with the homogeneity in space of σ ε and σ, we infer that for all t ∈ [0 when ε ց 0. Let us recall that The uniform bound (4.2) ensures that for all t ∈ [0, T ], there exist a further subsequence (depending on t, not relabeled) and a displacement u(t) ∈ BV ((0, L)) such that u ε (t) ⇀ u(t) weakly-* in BV ((0, L)) when ε ց 0. In particular, we deduce from (4.10c) that is independent of the subsequence defining u(t).Moreover, one can check that Indeed, extending the problem on a larger open interval [0, L] ⊂ Ω ′ and setting when ε ց Therefore, using [30, Remark 2.3 (i)], we get that which implies (4.12).In particular, we infer that the limit displacement u(t) is actually independent of the subsequence.Indeed, let u 1 (t) and u 2 (t) ∈ BV ((0, L)) be two weak limits of {u ε (t)} ε>0 in BV ((0, L)).On the one hand, (4.10c) entails that D(u On the other hand, the internal traces of u 1 (t) and u 2 (t) being prescribed on {0, L} by (4.12), we infer that . Therefore, we infer that the whole sequence converges and there exists such that u ε (t) ⇀ u(t) weakly-* in BV ((0, L)) when ε ց 0, for all t ∈ [0, T ]. (4.13) Note that u(0) ∈ BV ((0, L)) was already given by (4.4) and the static analysis led in [7], entailing the following Constitutive Equation at the initial time.Proof.Gathering (4.10c) and (4.12), we can identify the absolutely continuous and singular parts (with respect to L 1 ) in the Radon-Nikodým decompositions of Du(0) and p(0): By definition of the inf-convolution and (4.4), we get that W (D a u(0 , where we identified absolutely continuous measures with their densities.Besides, combining (4.8), (4.5) and (4.11), we also have that Therefore, we obtain that 4.4.Regularity of the evolution.Looking at the proof of Proposition 4.5 and defining, for all time t ∈ [0, T ], the function ã .
On the one hand, since we infer that E ∈ AC([0, T ]; R).On the other hand, since w ∈ AC [0, T ]; C 0 ([0, L]) and the product of absolutely continuous functions remains absolutely continuous, we infer that ∆ ∈ AC([0, T ]; R).In particular, we deduce that l, σ, e and p are continuous on the whole interval [0, T ].By non-negativity and monotonicity in time of µ, we also infer that µ is continuous from [0, T ] to M([0, L]).
Using the Energy Balance in Proposition 3.1 together with the non-decreasing character of l, (4.8) and (4.5), we can actually show that σ ∈ AC([0, T ]; R).From this, we will deduce that l, u, p and µ inherit the same regularity.This is a strong result because it is usually obtained a posteriori, once an Energy Balance of the type (4.18) is proved to be satisfied (see [12]).Remarkably, here we do not rest on such an Energy Balance in order to prove the regularity of the quasi-static evolution.This is the content of the following proposition.
At this point, we have identified good candidates for the limit evolution: (u, e, p, σ, µ) which are all absolutely continuous on [0, T ] and satisfy the following assertions for all t ∈ [0, T ]: i. Additive Decomposition: The absolute continuity of (u, σ, p, µ, l) guarantees that (u, σ) describes a quasi-static damage evolution, whose internal variable is the effective compliance (inverse effective rigidity) satisfying the Constitutive Equation and Griffith Evolution Law stated in Theorem 1.1.
Proof of Theorem 1.1: Using (4.8) and (4.5), one can check that for L 1 -a.e.t ∈ [0, T ] the following quantities are well defined and satisfy ẇ(t) a 0 and Hence Besides, using that µ is non-decreasing in time together with [12, Theorem 7.1, Formula (7.4)] and (4.11) ensures that for L 1 -a.e.t ∈ [0, T ].Thus, by non-negativity of the Radon measure μ(t), we infer that Remark 4.8.The effective limit model obtained here is a different type of damage model where the dissipative phenomena is described by means of an internal variable, the effective compliance c : [0, T ] → M([0, L]; R + ), whose non-decreasing character in time accounts for the irreversibility of damage.This is a threshold stress model, based on the conjecture that damage propagates if and only if the stress saturates the constraint.
• Formally inverting the compliance c(t), the Constitutive Equation allows us to interpret 2 as the stored (elastic) energy density of the effective damaged medium at time t ∈ [0, T ].In other words, one can interpret as the elastic energy in the body at time t.Therefore, since we can write the following energy balance where the left hand side is the sum of the elastic energy and a dissipative cost due to damage.• As explained above, the Griffith type Evolution Law states that damage can only grow when the stress σ saturates the constraint.This threshold condition generalizes the Initial Constituve Law (4.15) to the quasi-static setting.As will be explained in the Section 5, this Constitutive Law (4.15) a priori does not propagate to subsequent times through the evolution process, unless the prescribed boundary condition satisfies (1.6), corresponding to the case of perfect plasticity.In this case, one can check that 2κa 0 − σ(t) 2 l(t) = 0 for all t ∈ [0, T ].
Motivated by the static analysis led in [7] where the authors have shown how brittle damage can lead to Hencky perfect plasticity, it is natural to extend this analysis to quasi-static evolutions and inquire whether (u, e, p, σ) is of perfect plasticity type or not.Following [12,Definition 4.2], in order for this quasi-static evolution to be of perfect plasticity type, it only remains to prove the Energy Balance: for all time t ∈ [0, T ], where we used that t 0 | ṗ(s)| ([0, L]) ds = V(p; 0, t) according to [12, Theorem 7.1].As u, σ, p, µ and l are all absolutely continuous on [0, T ], we have the following Proposition: Proposition 4.9.For all t ∈ [0, T ], Proof.Indeed, for all t ∈ [0, T ], (4.8) and (4.5) entail that Using that lσ 2 = lσ 2 + 2lσ σ together with (4.15) and Theorem 1.1, we infer that Thus (4.11), Theorem 1.1 and (4.17) entail that Then, (4.8), (4.4) and (4.15) complete the proof of (4.19).
Therefore, due to the homogeneity in space of σ together with the stress constraint, (4.19) ensures that the upper bound inequality of (4.18) is always satisfied: Proposition 4.10.For all t ∈ [0, T ], Having all the previous results in mind, one would naturally be tempted to intuit the validity of the Energy Balance (4.18), hence proving that the quasi-static damage evolution (u, e, p, σ) is indeed one of perfect plasticity.Surprisingly, the interplay between relaxation and irreversibility of the damage is not stable through time evolutions.Indeed, depending on the choice of the prescribed Dirichlet boundary condition w ∈ AC([0, T ]; H 1 (R)), the effective quasi-static damage evolution may not be of perfect plasticity type, as illustrated in the example of Figure 3. Understanding on which condition the effective quasi-static evolution is of perfect plasticity type is the content of the next section.

Energy Balance
We can prove that the Energy Balance (4.18) is satisfied if and only if σ saturates the constraint once l is non-zero, until the end of the process (see Figure 2).Proof.Let us heuristically explain the argument.On the one hand, let us assume that (5.1) holds.Since µ(t) = 0 = p(t) in M [0, L] for all t ∈ [0, t 0 ] and |σ(t)| = √ 2κa 0 for all t ∈ [t 0 , T ], the Flow-Rule holds: This is immediate during the time interval [0, t 0 ], while during the time interval [t 0 , T ], by continuity we infer that σ is constant and either σ ≡ √ 2κa 0 or σ ≡ − √ 2κa 0 .Especially, since p = σ a0 µ and µ is non-decreasing in time, we infer that p is either non-decreasing or non-increasing in time on [t 0 , T ], according to the sign of σ.Hence,  is satisfied L 1 -a.e. on [0, T ].In particular, (5.1) must hold, otherwise the Flow-Rule will not be satisfied during a non L 1 -negligible set of times in [t 0 , T ].Indeed, if |σ(t)| < √ 2κa 0 for some t ∈ (t 0 , T ), considering the maximal time interval t ∈ I ⊂ (t 0 , T ] during which σ never saturates the constraint, we get by continuity of σ and Theorem 1.1 that Int(I) is a non empty interval and µ is a constant non-zero measure on I.Moreover, there exists E ⊂ I such that L 1 (E) > 0 and σ = 0 on E. If such was not the case, σ would be constant on I, which is impossible by maximality of the interval.Thus, one simultaneously has ṗ [0, L] = σ/a 0 l = 0 and |σ| < √ 2κa 0 on E, so that |σ| | σ/a 0 | l < √ 2κa 0 | σ/a 0 | l which is in contradiction with the Flow-Rule (5.2).
Yet, this sufficient and necessary condition (5.1) relies on the definition of the time t 0 .It remains to find an equivalent condition which can be expressed only in terms of the data of the setting.This is the content of Theorem 1.2, illustrated in Figure 1.
(5.9) Indeed, on the one hand, either t 0 = 0 ≤ t * 0 , or t 0 > 0. In this case, we infer that for all previous time 0 ≤ t < t 0 according to (4.5), Proposition 4.2 and the fact that l(t) = 0. Therefore, t ≤ t * 0 which leads to t 0 ≤ t * 0 when t tends to t 0 .On the other hand, assume by contradiction that t * 0 > t 0 .By definition of t 0 and t * 0 , we deduce that for all t ∈ (t 0 , t * 0 ), and l(t) > 0.

Concluding remarks
In spite of the conjecture motivated by the static analysis of [7], Theorem 1.2 determines the exact conditions on which the quasi-static evolution (u, e, p, σ) is of perfect plasticity type or not.In particular, when the prescribed boundary datum w ∈ AC([0, T ]; H 1 (R)) is such that |[w] L 0 | is decreasing and remains larger than √ 2κa 0 L a1 , the Energy Balance (4.18) is never satisfied (see Figure 3).This suggests to interpret (u, c) rather as a quasi-static evolution of damage as stated in Theorem 1.1, even when the prescribed boundary datum satisfies (1.6).In this case, the very specific nature of the plastic evolution illustrated in Figure 2 seems to confirm the interpretation of the evolution as one of damage.Indeed, the only configuration of perfect plasticity we obtain is very restrictive as the evolution remains purely elastic until a threshold time t 0 , after which the stress σ always saturates the constraint and the damage keeps on increasing, so that the elastic strain remains constant until the end of the process which is rather specific to damage than plasticity.Besides, when choosing T > 2 √ 2κa 0 /a 1 and applying a loading-unloading Dirichlet condition ,T ) (t) , the response of the limit model is indeed typical of damage, as illustrated in Figure 4. Using (5.9), one can check that t 0 = √ 2κa 0 /a 1 > 0, hence By (5.6), (1.1) and the increasing character of w L 0 during the time interval t 0 , T 2 , we infer that ò , which corresponds to the hardening phase.Then, since l is non-decreasing and w L 0 is decreasing during the unloading phase, (4.8) entails that σ decreases during the time interval T 2 , T .In particular, 0 ≤ σ(t) < √ 2κa 0 for all t ∈ T 2 , T , so that µ ≡ µ T 2 is constant during the unloading phase and In particular, at the end of the loading-unloading process, the medium goes back to its reference configuration, whereas in perfect plasticity one expects a residual plastic strain (see Figure 5).On the other hand, interpreting the evolution as one of damage also fails to be completely satisfactory, as |σ| never exceeds the damage yield threshold √ 2κa 0 , including during the hardening phase, which is specific to perfect plasticity and therefore consists in a painful lack of generality for the description of a damage evolution as well (see Figure 5).One could wonder if the effective model we lies somehow in between damage and plasticity.Without being able to answer completely this question, let us remark that the effective evolution obtained here does not fit in the large class of elastoplasticity-damage models introduced in [4,9,10] where we expect the Energy Balance to hold and the plastic yield surface to shrink as damage increases, whereas here (4.18) is not always satisfied and ∂K = {± √ 2κa 0 } is fixed.
Looking at the constructive proof of [14, Theorem 2], one could argue that passing to the limit ε ց 0 in the time-continuous quasi-static evolutions might not have been the right approach, as some incremental information (see the minimality formulae and track of the history of damage [14,Formulae (15), ( 16), (21)]) available at the stage of time discretizations is lost once the time step has been sent to 0 (see [8] for a related case study of non-commutability).Indeed, let us fix a time subdivision Following minor adaptations of the present work and passing first to the limit ε ց 0, we infer the existence of a piecewise constant in time evolution with uniformly bounded variation in N such that where Moreover, as we pass to the limit ε ց 0 in the incremental minimality [14,Formulae (15) and ( 16)] ã dx óf the discrete evolution of [14], one could hope that (u N , e N , p N ) satisfies a stronger incremental minimality as in [12, Formula (4.12)]: Assume that (6.2) holds.Then, following exactly the proof of [12,Theorem 4.5] ensures the existence of a subsequence (independent of t, not relabeled) and a quasi-static evolution of perfect plasticity (u, e, p) : [0, T ] → BV ((0, L)) × L 2 ((0, L)) × M([0, L]) satisfying (1.3) and (1.4), such that for all t ∈ [0, T ]      u N (t) ⇀ u(t) weakly-* in BV ((0, L)), e N (t) ⇀ e(t) weakly in L 2 ((0, L)), p N (t) ⇀ p(t) weakly-* in M([0, L]), when passing to the limit δ N → 0. In particular, all the above quantities in (6.1) pass to the limit as δ N → 0. Therefore, Theorem 1.2 holds true for the quasi-static evolution (u, e, p) as well.
Consequently, for any prescribed boundary datum w ∈ AC([0, T ]; H 1 (R)) not satisfying (1.6), we come to a contradiction.It unfortunately proves that commuting the limits in ε and N leads to no better statement.
Therefore, A is an admissible competitor for the One-sided minimality of [14, Theorem 2] too.
Eventually, we conclude with a general remark by noticing that this one-dimensional analysis seems to raise the question whether Hencky perfect plasticity is distinguishable from damage or not in a static setting, as mentionned in the recent survey [23, Section 1, p10].

Appendix
Lemma 7.1.Let a, b, K > 0 with a < b and f : ξ ∈ R → min K + a|ξ| 2 ; b|ξ| 2 .Then, for all ξ ∈ R, Therefore, we can assume without loss of generality that u ∈ w + C ∞ c (Ω; R N ) and conclude the proof of the upper bound arguing as in [7,Proposition 3.3].

.
If a and b ∈ R N , with N ∈ N \ {0}, we write a • b = N i=1 a i b i for the Euclidean scalar product and |a| = √ a • a for the corresponding norm.The space of symmetric N × N matrices is denoted by M N ×N sym and is endowed with the Frobenius scalar product ξ : η = tr(ξη) and the corresponding norm |ξ| = √ ξ : ξ.Measures.The Lebesgue and k-dimensional Hausdorff measures in R N are respectively denoted by L N and H k .If X is a borel subset of R N and Y is an Euclidean space, we denote by M(X; Y ) the space of Y -valued bounded Radon measures in X which, according to the Riesz Representation Theorem, can be identified to the dual of C 0 (X; Y ) (the closure of C c (X; Y ) for the sup-norm in X).The weak-* topology of M(X; Y ) is defined using this duality.The indication of the space Y is omitted when Y = R.For µ ∈ M(X; Y ), its total variation is denoted by |µ| and we denote by µ = µ a + µ s the Radon-Nikodým decomposition of µ with respect to Lebesgue, where µ a is absolutely continuous and µ s is singular with respect to the Lebesgue measure L N .

together with ( 4 . 2 ) 4 . 2 .Proposition 4 . 4 .
, one can apply the generalized version of Helly's Theorem recalled in[12,  Lemma 7.2]  for the topological duals of separable Banach spaces and find a subsequence and a limit time-dependent non-negative Radon measure µ : [0, T ] → M [0, L]; R + such that µ ε (t) ⇀ µ(t) weakly-* in M [0, L] as ε tends to 0, for all t ∈ [0, T ].Note that the monotonicity in time of µ ε is preserved by the pointwise weak-* convergence in M [0, L] .Initial time of the evolution.We begin with a corollary of the analysis led in [7, Theorem 3.1] in the static setting, taking into account a prescribed Dirichlet boundary datum.We refer to Appendix 7 for a more general statement of this proposition and its proof.The functional ẇ′ (s)(x) dxds.
Therefore, the validity of the Energy Balance (4.18) follows from (4.19) and (5.2).On the other hand, (5.1) is also a necessary condition.Let us assume that the Energy Balance (4.18) is satisfied.From (4.19), we infer that