COHESIVE ZONE-TYPE DELAMINATION IN VISCO-ELASTICITY

. We study a model for the rate-independent evolution of cohesive zone delamination in a visco-elastic solid, also exposed to dynamics eﬀects. The main feature of this model, inspired by [32], is that the surface energy related to the crack opening depends on the history of the crack separation between the two sides of the crack path, and allows for diﬀerent responses upon loading and unloading.Duetothepresence of multivalued and unbounded operators featuring non-penetration and the ‘memory’-constraint in the strong formulation of the prob- lem, we prove existence of a weaker notion of solution, known as semistable energetic solution , pioneered in [41] and reﬁned in [38].

1. Delamination models with a process zone. In the last decade cohesive zone models have received great attention from a mathematical point of view, and different aspects have been considered. The main feature of the cohesive zone models, cf. e.g. [32, p. 1268], pioneered by Dugdale [20], Barenblatt [6], Rice [35], and others-is to regard fracture as a gradual phenomenon in which separation takes cohesive law model by replacing the cohesive surface energy density φ coh by the expression φ coh ( u n , ζ) := φ coh (ζ) 2ζ 2 u n 2 .
Hereby, the indicator term enforces the non-penetration condition [[u]] n = [[u]] · n ≥ 0, as well as the 'memory'constraint [[u]] n ≤ ζ, which is a relaxation of the characteristic feature ζ(t, x) = sup s∈[0,t] [[u(s, x)]] n of the cohesive zone internal variable. Observe that I [0,ζ] (·) imposes a convex, non-smooth constraint on the displacements, which depends on the internal variable ζ. This causes the main challenge in the analysis. In addition φ s features the indicator function I [0,ζ * ] of the fixed interval [0, ζ * ], for a fixed maximal separation amount ζ * . With this constraint we prevent that the separation can reach unphysically large values, larger than ζ * . Finally, the density G induces a regularization for ζ in terms of a Sobolev-or a Sobolev-Slobodeckij seminorm, see (24e) for the details. Relation to adhesive contact: When deciding to drop the 'memory'-aspect [[u]] n ≤ ζ, then, in view of the monotonically decreasing nature of the function φ coh (ζ)/(2ζ 2 ), see Fig. 1, column C, one may set z := φ coh (ζ)/(κζ 2 ) and introduce z as an alternative internal variable. In particular, for Smith-Ferrate's cohesive law, cf. Fig. 1, line 1, also upon renormalization, the cohesive zone surface energy density φ coh (without memory), then goes over to the adhesive contact energy φ adh from (1). An adhesive contact model involving φ adh thus can be seen as a cohesive zone model without the memory of the history of maximal separations.
In the bulk domain, with mechanical energy E bulk (t, ·) we will assume a viscoelastic material response, governed by a viscous dissipation potential V(·). More precisely, denoting by e(u) = 1 2 ∇u + ∇u ⊤ the symmetrized strain tensor, we introduce E bulk (t, u) := We refer to Sec. 2.1 for the precise assumptions on the domain, the tensors C and D, and the time-dependent loadings f (t) and h(t). In addition, we also account for dynamic effects governed by an acceleration term ̺ü . Thus, the evolution of the pair (u, ζ) is characterized by the system of equations − (Ce(u) + De(u)) · n + D u φ coh ( u n , ζ) ∈ ∂ u I [0,ζ] ( u n ) on Γ C , (7c) Scope, structure, and challenges of the paper: This contribution deals with the existence analysis for system (7). In fact, owing to its coupled ratedependent/rate-independent character we will not analyse the cohesive zone system in its strong form (7), but resort to a weaker notion of solution, which combines concepts from the theory of rate-independent systems [30] with the one of (viscous) gradient flows (̺ = 0 in (7a)), resp. hyperbolic PDEs (̺ > 0 in (7a)). Pioneered in [41] and refined in [38], this concept of semistable energetic solutions is well-suited to handle rate-independent evolution, as it replaces the subdifferential inclusion (7d) by a minimality condition for ζ combined with an energy-dissipation estimate. We introduce the notion of semistable energetic solutions at the beginning of Sec. 2, see Def. 2.1.
Observe that the transmission condition (7c) features ∂ u I [0,ζ] ([[u]] n ), the subdifferential of the combined non-penetration and 'memory'-constraint (4). Indeed, to deal with this multivalued term, we will, in a first step, replace it by its corresponding Yosida-approximation ∂ u I k ([[u]] n , ζ k ). We specify in Sec. 2.1 the analytical setup, all the functionals and function spaces involved. Our results are presented and discussed in Sec. 2.2. Our first result, Thm. 2.3, states the existence of semistable energetic solutions for the Yosida-regularized model, for both cases ̺ = 0 and ̺ > 0. Our main result, Thm. 2.4, provides the existence of semistable energetic solutions for the unregularized cohesive zone model (featuring ] n )) in the case of gradient flows, i.e. ̺ = 0 in (7a). Based on Thm. 2.3, whose proof is carried out in Sec. 3, the proof of Thm. 2.4 is obtained in Sec. 4 in terms of an evolutionary Γ-limit passage. Hereby, the main challenge will lie in the fact that I [0,ζ] ([[·]] n ) encodes the non-smooth, bilateral non-penetration & 'memory'-constraint which, on top, depends on the state ζ. Thus, for the limit passage, also its Yosida-regularization I k ([[·]] n , ζ k ) will depend on the state ζ k , the internal-variable-component of a semistable energetic solution pair (u k , ζ k ) to the regularized model. An evolutionary Γ-limit passage will therefore require the proof of Mosco-convergence for the corresponding functionals, see Prop. 3, with a careful design of recovery sequences, taylored to the constraints imposed on the states varying with k.
Comparison with other existing results: The problems attached to the approximation of a state-dependent non-smooth constraint in combination with rate-dependent effects and dynamics have already been experienced in [39,40]. Therein, the adhesive contact energy φ adh from (1) is used as a regularization of the brittle constraint z| [[u]] n | = 0 a.e. on Γ C . For the limit κ → ∞ in the energy term (1) the brittle constraint could be re-obtained, also necessitating a sophisticated design of recovery sequences involving the proof of additional fine convergence properties of the semistable internal variables z k .
The history dependence of the crack opening was addressed also in [18], where existence of globally stable quasistatic evolution (i.e. energetic solution) was obtained, in the case of a cohesive zone model featuring a general φ coh without a regularizing term G for the internal variable ζ. Hence there the main mathematical difficulty was the compactness of the approximating functions t → ζ k (t), solved by introducing a new notion of convergence, which is the counterpart of the notion of convergence of sets introduced in [15]. More recently vanishing viscosity techniques have been applied in cohesive zone models w.r.t. the internal variable, [13,3]. In particular, history dependence of the crack opening was also considered in [13] for a cohesive zone model taking into account different responses upon loading and unloading. Existence of a solution obtained by means of vanishing viscosity have been established, by replacing the internal variable by a Young measure. [12,4] study cohesive zone delamination for a visco-elastic solid without introducing an internal variable and prove existence of solutions as well as a vanishing viscosity limit. Different responses upon loading and unloading, also related to fatigue, have been recently addressed in [14], where the existence of energetic solutions is established. Fatigue effects related to cohesive fracture, more precisely in a gradient damage model coupled with plasticity were discussed in [2], and the authors of [17] show in a static setting that cohesive hypersurface energy functionals can be obtained as a limit of volume damage coupled with plasticity.
2. The mathematical model and our results. In this paper we will treat the cohesive zone model and its approximations in the framework of gradient systems (that is, ̺ = 0 a.e. on Ω \ Γ C in (7a)) and damped inertial systems (that is, ̺ > 0 a.e. on Ω \ Γ C in (7a)), which consist of: • two Hilbert spaces V and W, W identified with its dual W * , such that V ⋐ W compactly and densely, so that V ⊂ W = W * ⊂ V * (dual of V) continuously and densely, and w, u V = (w, u) W for all u ∈ V and w ∈ W; • a separable Banach space Z; • a dissipation potential R : Z → [0, ∞], with domain dom(R), lower semicontinuous, convex, positively 1-homogeneous and coercive i.e., R(λζ) = λR(ζ) for all ζ ∈ Z and λ ≥ 0, with ̺ > 0 in case of a damped inertial system (V, W, Z, V, K, R, E) and ̺ = 0 in case of a gradient system (V, Z, V, R, E) • an energy functional E : In what follows, we shall denote by ∂ u E : [0, T ] × V × Z ⇒ V * the subdifferential of the functional E(t, ·, ζ) in the sense of convex analysis. We postpone to Section 3 ahead the precise statement of the further conditions on E required in the existence result from [38] that we shall apply to deduce the existence of solutions to the cohesive zone model and its approximants. Let us only mention here that the assumptions on ζ → E(t, u, ζ) (cf. the coercivity requirement (45) ahead) also involve a second space X such that X is the dual of a separable Banach space and X ⋐ Z compactly.
If ̺ = 0 we speak of a gradient system and denote it by the characterizing quintuple (V, Z, V, R, E); for ̺ > 0 we speak of a damped inertial system, denoted by (V, W, Z, V, K, R, E). A suitable solution concept for systems that couple rate-independent processes with rate-dependent and dynamic ones goes back on the pioneering work [41]. Based on a time-discrete scheme with alternating (decoupled) minimization w.r.t. the variables u and ζ it was recently refined and developed further in Definition 2.1 (Semistable energetic solutions). Consider a gradient system (V, Z, V, R, E), resp. a damped inertial system (V, W, Z, V, K, R, E). We call a pair (u, ζ) : [0, T ] → V × Z a semistable energetic solution to the gradient system (V, Z, V, R, E), resp. the damped inertial system (V, W, Z, V, K, R, E), if u ∈ W 1,1 (0, T ; V) , and for a damped inertial system (̺ > 0) alsȯ fulfill -the subdifferential inclusion -the semistability condition -the energy-dissipation inequality for all t ∈ [0, T ]: with Var R denoting the total variation induced by the dissipation potential R, i.e., for a given subinterval [s, t] ⊂ [0, T ] it is

MARITA THOMAS AND CHIARA ZANINI
Note that for the cohesive zone models, due to the unidirectionality incorporated in the dissipation potential (5), the total variation will take the specific form 2.1. Basic assumptions, spaces, and functionals.  Assumptions on the given data: For the tensors C, D ∈ R d×d×d×d in (6) & (7) and a time-dependent external force f , we require that C, D ∈ R d×d×d×d are symmetric and positive definite, i.e., Hereby, the external force f may comprise both the volume force f from (7a) and the surface force h from (7f). Moreover, to keep notation and arguments simple, we prescribe homogeneous Dirichlet data on Γ D , as already indicated in (7e).
Assumptions on the cohesive surface energy density: In line of the works [32,34,26] we assume that is continuous, monotonically decreasing, bounded by b > 0.
In fact, it can be seen in Figure 1, columns A & C, that typical cohesive zone energy densities φ coh from engineering literature do comply with (16).

2.1.2.
Function spaces and traces. Throughout this paper the set of function spaces described in (8) will be chosen as follows: The space X will be related to the (gradient) regularization of the internal variable ζ contributing to the surface energy functionals, see Φ surf , Φ surf k in (24c) & (26b) below. Depending on its choice we will obtain existence results of different quality: The choice X = H 1/2 (Γ C ) is more natural in view of (17a), if one has in mind that the purpose of the internal variable in a cohesive zone model is to keep track of the history of the maximal jumps of the displacements across Γ C . Indeed, the choice X = H 1/2 (Γ C ) is suited to obtain existence results both in the dynamic and in the gradient-flow case if one regularizes the ζ-dependent indicator term in (24f) by its Yosida-approximation, cf. Thm. 2.3 & Sec. 3. But so far, only the enforcement X = W 1,r (Γ C ) with r > d − 1, allows it to pass from the regularized models to a model displaying the unregularized cohesive zone energy (24c), cf. Thm. 2.4 & Sec. 4, by heavily exploiting the compact embedding Note that, thanks to assumption (14d), we have for all ζ ∈ X = H 1/2 (Γ C ) : For later purpose we verify here that the operation max{·, ·} defines a bounded operator from (14). Then, the operator max{·, ·} : for any f, g ∈ H 1/2 (Γ C ) .
, which concludes the proof.
2.1.3. The functionals. We denote by T Ω±→ΓC : H 1 (Ω ± ) → H 1/2 (Γ C ; R d ) the trace operator from Ω ± to the interface Γ C . With its aid we introduce the operator indicating the jump of functions u ∈ H 1 (Ω\Γ C ; R d ) across Γ C · : as well as the operator indicating the jump of functions u ∈ H 1 (Ω\Γ C ; R d ) across Γ C in normal direction n, cf. (14d), Thus, to model cohesive zone delamination along the interface Γ C in a visco-elastic body, we introduce the Energy functional of the following form: The regularization G for the internal variable consists of the indicator function of the interval [0, ζ * ] and the seminorm of the space X, cf. (19) for X = H 1/2 (Γ C ), and ΓC |∇ζ| r dH d−1 for X = W 1,r (Γ C ). The indicator term confines the values of ζ to the interval [0, ζ * ] for some ζ * > 0. This is done in view of the properties of φ coh , cf. (3) and also (16): In order to further decrease the surface energy Φ coh ([[u]] n , ·) the model might favor to attain large values of ζ. Having in mind that the internal variable is linked to the jump of the displacements, this may be unnatural. The indicator thus prevents too high values of ζ. Since the internal variable has a rate-independent evolution, governed by the 1-homogeneous dissipation potential R, cf. (5), the evolution equation of the variable ζ can be reformulated in terms of the semistability inequality (11), which is a minimality property involving only the functionals E and R, but not their differentials. Hence, here, the non-smooth, unbounded functionals G, J ([[u]] n , ·) contributing to Φ surf , can be handled in the realm of calculus of variations. This is in contrast to the rate-dependent, dynamic evolution of the displacement variable u, which is governed by the subdifferential inclusion (10) that cannot be reformulated without the differential ∂ u E: Here, the non-smoothness and unboundedness of the functional J ([[·]] n , ζ) imposes an obstruction to the analysis as the constraint (24f) therein implemented in particular depends on the internal variable ζ. Therefore, in order to handle the non-smooth constraint (24f) in the rate-dependent setting, we will first treat a regularized problem. For this, we replace the functional J by its Yosida-approximation With E bulk from (24b), Φ coh from (24d), and G from (24e) this leads to the Yosida-regularized energy functionals for every k ∈ N : We note that the domain dom(E k ) of the functionals E k is given by Remark 1 (Interpretation of the surface functionals and their derivatives). For given ζ ∈ X, (26b) composes the cohesive functional Φ coh (·, ζ) : (28b) Then, cf. e.g. [11, Example 2.8.2 p.46], the Fréchet-derivative D v J k (·, ζ), which is the Yosida-regularization to the subdifferential ∂ v J (·, ζ) of J (·, ζ) : with Id : L 2 (Γ C ) → L 2 (Γ C ) the identity and P [0,ζ] : L 2 (Γ C ) → L 2 (Γ C ) the projection operator, i.e., Thus, the pointwise equivalent to (29) computes as and, moreover, for Revisiting (28), by the validity of the chain rule due to the continuity of the composed functionals, the differentials can be represented in L 2 (Γ C ). However, this is no longer true for the non-smooth functional J (·, ζ) : but it cannot be concluded that ξ ∈ L 2 (Γ C ). Therefore the limit passage k → ∞, i.e. from the regularized models to the cohesive zone model, has to be carried out in the topology of V, or equivalently in H 1/2 (Γ C ), which is much stronger than the L 2 (Γ C )-topology. Exactly this strengthening of the topology for the non-smooth constraint and its regularizations features the main challenge in the analysis of the limit passage.

2.2.
Our results for the Yosida-regularized model and its cohesive limit. The Yosida-regularized models (V, Z, V, R, E k ), resp. (V, W, Z, V, K, R, E k ), fall into the class of gradient systems, resp. damped inertial systems introduced at the beginning of Sec. 2. Based on abtract existence results proved in [38] we will establish the existence of semistable energetic solutions for the systems (V, Theorem 2.3 (Existence of semistable energetic solutions to the regularized systems). Let the assumptions (14)-(15) hold true. Then, for every k ∈ N fixed, for every initial datum given by (17), (26), (5), admits a semistable energetic solution (u k , ζ k ) in the sense of Def. 2.1 of regularity and such that Moreover, the energy-dissipation inequality (12) even holds as a balance In view of (31), the surface energy functional Φ surf k , cf. (26b), due to the Yosidaregularization J k , satisfies the following k-dependent (sub)gradient estimate, evaluated along solutions (u k , ζ k ) see Sec. 3 for the details of the calculation. Nevertheless, for gradient systems (V, Z, V, R, E k ), i.e. ̺ = 0 in (35), a uniform bound on ∂ u Φ surf k (u k , ζ k ) L 2 (0,T ;V * ) can be obtained by comparison in the k-momentum balance. This will be used in Sec. 4 to prove for the cohesive zone delamination model (V, Z, V, R, E ∞ ) the existence of semistable energetic solutions by performing an evolutionary Γ-limit as k → ∞ from the regularized systems (V, Z, V, R, E k ) to the cohesive limit system (V, Z, V, R, E ∞ ). The main challenge in this limit analysis lies in the fact that the non-smooth constraint incorporated in J ([[·]] n , ζ), cf. (24f), and approximated by J k ([[·]] n , ζ k ), cf. (25), depends on the internal variable itself. This will require to prove the Mosco-convergence of the functionals (J k ([[·]] n , ζ k )) k in the topology V = H 1 (Ω\Γ C ), cf. Prop. 3. We now state the main result of this work: The existence of semistable energetic solutions to the gradient cohesive zone system (V, Z, V, R, E ∞ ) obtained via evolutionary Γ-convergence: Theorem 2.4 (Evolutionary Γ-convergence of the gradient systems, ̺ = 0). Let the assumptions of Theorem 2.3 hold true. In addition assume that X = W 1,r (Γ C ) with r > d − 1 and that the initial datum complies with the following assertion: Moreover, assume that the initial data are well-prepared, i.e., be a semistable energetic solution to the regularized system (V, Z, V, R, E k ). Then, there exists a (not relabeled) subsequence (u k , ζ k ) k and a limit pair (u, ζ) of regularity such that the following convergences hold true The limit pair (u, ζ) is a semistable energetic solution of the cohesive zone system (V, Z, V, R, E ∞ ), cf. Def. 2.1. Moreover, the following statements hold true: 1. For a.e. t ∈ (0, T ) the momentum balance takes the following form For all t ∈ [0, T ] the energy dissipation inequality is satisfied: Remark 2 (Condition on the initial datum (37)). The strictly positive bound from below (37) on the initial datum means that the bonding along Γ C has already experienced an opening in normal direction of at least ζ * in the past, and at initial time of the monitoring of the loading experiment, the maximal opening ever experienced before in the point x ∈ Γ C takes the value ζ 0 (x) ≥ ζ * . Furthermore, observe that the unidirectionality of the 1-homogeneous dissipation potential R together with the initial condition ζ(0) = ζ 0 ≥ ζ * ensures that ζ(t, x) ≥ ζ 0 (x) ≥ ζ * a.e..
Remark 3 (Limit passage in case of damped inertial systems). For the systems (V, W, Z, V, K, R, E k ), a comparison argument in (10) only provides uniform bounds, independent of k ∈ N, on the sum ̺ü k (t)+D u J k ([[u k (t)]] n , ζ k (t)), but not on the two terms separately. Moreover, in view of (36) one obtains an analogous k-dependent bound on the inertial term: Since the bounds (36)&(42) blow up as k → ∞, one cannot use the notion of semistable energetic solutions from Def. 2.1 to carry out an evolutionary Γ-limit passage for the damped inertial systems (V, W, Z, V, K, R, E k ). A promising ansatz would be to follow an alternative approach established in the recent series of works [9,49,48] that successfully handle the analysis of problems combining unilateral, non-smooth constraints with the visco-elastodynamic setting. Therein, the key to circumvent the problems caused by the blow-up of the bounds (36)&(42) is to carry out the limit passage from a regularized to the original model in a notion of solution that uses a weak formulation of the momentum balance in Bochner spaces H 1 (0, T, V) instead of the pointwise-in-time formulation (10). But let us stress that their approach so far has proved successful only for non-smooth unilateral constraints that are independent of the state variables. This is in clear contrast to our model, where the constraint strongly depends on the internal variable ζ. Exactly here lies the limitation of this approach: our models combine the rate-dependent (and dynamic) evolution of the variable u with a rate-independent evolution of the internal variable. As can be seen from (9) the internal variable has much less temporal regularity than the displacement variable. Therefore, at the moment, it seems to be out of reach to exploit a weak formulation of the momentum balance in (H 1 (0, T, V)) * in order to pass to the cohesive zone limit. The problem can be phrased more precisely: Passing from a pointwise-in-time formulation of (10) (equivalently, from a formulation in L 2 (0, T ; V * )) to a formulation in (H 1 (0, T, V)) * will provide uniform bounds on (D u J k ([[u k (t)]] n , ζ k (t))) k in (H 1 (0, T, V)) * and necessitate the identification of the limit as an element of the subdifferential ∂ u This is at the price that Mosco-convergence of the functionals ( ] n , ζ k (t)) dt) k has to be established in H 1 (0, T, V). But since the internal variable ζ k is expected to be only of BV -regularity in time, the contruction of a recovery sequence for the displacements that depends on ζ k and converges strongly in H 1 (0, T, V) seems to be out of reach.
In other words, the mismatch between the temporal regularity of the diplacements and the one of the internal variable seems to inhibit the Γ-limit passage in case of damped inertial systems. This problem might be overcome if one assumes that the internal variable has a rate-dependent evolution governed by a viscous dissipation potential, which will be the scope of a follow-up work.
3. Proof of Theorem 2.3 -Existence of semistable energetic solutions (k ∈ N fixed). In this section we address the existence of semistable energetic solutions for the Yosida-regularized systems (V, Z, V, R, E k ), resp. (V, W, Z, V, K, R, E k ), with k ∈ N fixed, by resorting to the abstract existence results proved in [38], cf. [38,Thm. 4.9] for gradient systems, resp. [38, Thm. 5.6] for damped inertial systems. In what follows, for the reader's convenience we shall first revisit the prerequisites on an abstract gradient system (V, Z, V, R, E), resp. an abstract damped inertial system (V, W, Z, V, K, R, E), underlying the existence results in [38], and then verify that the systems (V, Z, V, R, E k ), resp. (V, W, Z, V, K, R, E k ), do comply with them, thus deducing the existence of semistable energetic solutions for the Yosida-regularized systems.
Let (V, Z, V, R, E) be a gradient system, resp. (V, W, Z, V, K, R, E) be a damped inertial system, complying with the basic conditions (8). In line with the direct method of the calculus of variations and with tools from rate-independent and gradient systems, [ Boundedness from below & Weak lower semicontinuity: E is bounded from below: (43a) ∀ t ∈ [0, T ] : E(t, ·, ·) is weakly sequentially lower semicontinuous on V×Z, indeed, if E is bounded from below, up to a shift we can assume that it is bounded by a positive constant.
Temporal regularity and power control: and fulfilling for all sequences t n → t, u n → u in V, ζ n → ζ in Z with sup n E(t n , u n , ζ n ) ≤ C that lim sup n→∞ ∂ t E(t n , u n , ζ n ) ≤ ∂ t E(t, u, ζ) . Coercivity: Mutual recovery sequence condition ensuring closedness of stable sets: Let (t n , u n , ζ n ) n ⊂ dom(E) for all n ∈ N satisfy semistability condition (11), Then, for everyζ ∈ Z there existsζ n ⇀ζ in Z such that lim sup n→∞ E(t n , u n ,ζ n ) As previouly mentioned, the existence results in [38] allow for a non-smooth and even non-convex (in lower order terms) dependence u → E(t, u, ζ). However, since the energies E k (t, ·, ζ) from (26) are convex and Gâteaux-differentiable, we will confine the discussion to energies with this property and denote by ∂ u E(t, ·, ζ) the Gâteaux-differential of the convex functional E(t, ·, ζ). Following [38, Thm. 4.9 & Thm. 5.6], we need to impose a suitable condition on the differentials ∂ u E in the spirit of Minty's trick: Continuity: Finally, to find a bound on the inertial term, a further requirement of [38,Thm. 4] is the following Subgradient estimate: There exists constants C 3 , C 4 , C 5 > 0 and σ ∈ [1, ∞) such that We are now in the position to recall the existence result from [38] for damped inertial systems (V, W, Z, V, K, R, E). Then for every (u 0 , u 1 , ζ 0 ) ∈ dom u × W × dom ζ fulfilling the semistability (11) at t = 0, i.e.
Proof of Theorem 2.3. As a first step we will prove the mutual recovery sequence condition (46) as a separate lemma. The proof of this lemma uses the approach developed in [51,50], but is here for the first time adapted to the character of the Sobolev-Slobodeckij seminorm. Recall from (24e) & (27) that for any triple (t, u, ζ) belonging to the domain dom(E k ), it holds in particular that ζ ∈ [0, ζ * ] a.e. on Γ C .  (46)). Let k ∈ N fixed and consider a sequence (t n , u n , ζ n ) n ⊂ dom(E k ) such that t n → t in [0, T ], u n ⇀ u in V and ζ n ⇀ ζ in X, as well as an elementζ ∈ X. Then the sequence (ζ n ) n ⊂ X, ζ n := min{ζ * , max{ζ n ,ζ + δ n }} with δ n := max ζ n − ζ 1/2 L 2 (ΓC) , 1/n , (50) serves as a mutual recovery sequence for the functionals E k (t n , u n , ·) & R, i.e., Condition (46) is satisfied. In particular, forζ ∈ X such thatζ ≥ ζ a.e. in Γ C , the following relations hold true with r = 2 for X = H 1/2 (Γ C ) and r > d − 1 for Proof. In the case X = W 1,r (Γ C ) it is straighforward to adapt the arguments of [51, Thm. 3.14] to the present situation. Let now X = H 1/2 (Γ C ). First of all, by Lemma 2.2 we convince ourselves that ζ n obtained by the superposition of the Lipschitz-functions max and min with the functions ζ n , (ζ + δ n ) ∈ X and the constant ζ * according to (50) has a bounded H 1/2 -seminorm: for n sufficiently large such that δ n ≤ 1. Moreover, byζ n ≤ζ + δ n + ζ n a.e. in Γ C , we find that also ζ n L 2 (ΓC) ≤ ζ * L 2 (ΓC) + ζ + 1 L 2 (ΓC) + ζ n L 2 (ΓC) . Hence,ζ n ∈ X for all n ∈ N.
Ad (51b): We introduce the following notation and note that B n := [ζ + δ n < ζ n ] , henceζ n = ζ n on B n , (52b) with the short-hand [a ≤ b] := {x ∈ Γ C , a(x) ≤ b(x) for a.e. x ∈ Γ C }. For the sets B n we observe that becauseζ ≥ ζ by assumption. Moreover, since ζ n ⇀ ζ in X by assumption, hence ζ n → ζ strongly in L 2 (Γ C ), we find that where we used Markoff's inequality to obtain the second estimate. The last estimate follows from the very definition of δ n := max ζ n − ζ 1/2 L 2 , 1/n , cf. (50). Thus, we have that We use that Γ C = (A n ∪ C n ) ∪ B n and in view of the definition of (ζ n ) n from (50) the term involving the Sobolev-Slobodeckij seminorms rewrites and estimates as follows: |x−y| d − 2ζ n (x)(ζ(y) + δ n ) + |ζ(y) + δ n | 2 (56d) +2ζ n (x)ζ n (y) − |ζ n (y)| 2 dx dy , Our aim is to further process the above terms in such a way that (56a) can be estimated from above by We see that the terms I n 1 , I n 2 and I n 5 -I n 9 will already nicely contribute to this estimate. But it remains to suitably estimate the mixed integrals (56d) & (56e). For this, we want to make use of the following estimate for (the integrand of) I 3 n : Verifying (58) is equivalent to showing that 0 ≤ |ζ(x)+δ n | 2 +|ζ n (y)| 2 −2(ζ(x)+δ n )(ζ(y)+δ n )+2ζ n (x)(ζ(y)+δ n )−2ζ n (x)ζ n (y) .
(59) Using that |ζ(x) + δ n | 2 + |ζ n (y)| 2 ≥ 2ζ n (y)(ζ(x) + δ n ) and keeping in mind that x ∈ B n , whereas y ∈ A n , we can further estimate the right-hand side of (59) from below as follows (ζ(x) + δ n ) ζ n (y) − (ζ(y) + δ n ) + ζ n (x) (ζ(y) + δ n ) − ζ n (y) i.e., (59), resp. (58), is verified. With the same arguments we can deduce an analogous estimate for the (integrand of) I 4 n , essentially by swapping the meaning of the variables x and y in (59), resp. (58). Putting these findings together, we see that |x − y| d dx dy and . Moreover, in view of (55), we may choose a further (not relabeled) subsequence (ζ n ) n such that Hence, the complements U N := Γ C \ ∪ ∞ n=N B n satisfy for all n ≥ N ∈ N : We may now use the sets U N to further estimate the left-hand side of (62); i.e., for N ∈ N fixed, for every n ≥ N we have , due to the weak lower semicontinuity of | · | 2 H 1/2 (UN ) and the fact that ζ n ⇀ ζ in X. For N → ∞ we then conclude (62). Thus, (51b) is proven. Ad (51c): This convergence result now follows from the continuous embedding L 2 (Γ C ) ⊂ L 1 (Γ C ). Ad (51d): In order to verify the convergence result for the cohesive surface energy Φ coh we choose further (not relabeled) subsequences (u n ) n , (ζ n ) n , (ζ n ) n such that lim ]] n , ζ n ) andζ n →ζ as well as ζ n → ζ pointwise a.e. in Γ C . With z as a placeholder forζ n and ζ n we verify for the cohesive surface density φ coh that where we made use of the assumptions (16). The term on the right-hand side is ] n 2 dx thanks to the compactness of the trace operator from H 1 (Ω\Γ C ) to L 2 (Γ C ). Thus, we are entitled to conclude (51d) with the aid of the dominated convergence theorem. Ad (51e): This estimate can be verified by arguing pointwise on the density I k of the Yosida-regularization, i.e., I k (r, z) = k 2 |(r) − | 2 + k 2 |(r−z) + | 2 . Taking into account thatζ n ≥ ζ n a.e. in Γ C according to (50), we see that ([[u n ]] n −ζ n ) + ≤ ([[u n ]] n − ζ n ) + a.e. in Γ C . Hence, (51e) follows. Ad (51f): This convergence result follows also by the dominated convergence theorem, using that the density I k of the Yosida-regularization satisfies the following estimate where the terms on the right-hand side are integrable and the integrals converge. Above, z is again a placeholder forζ n , resp. ζ n .
Ad (43): To check (43a), we calculate in view of (24e) & (26), using Korn's and Young's inequality: where we used that f (t) V * ≤ C f , the bound on ζ imposed by the indicator of the interval [0, ζ * ], cf. (24e), and we set 1 (67) The weak lower semicontinuity of E(t, ·, ·) follows by the fact that E k (t, ·, ·) is separately convex and strongly lower semicontinuous on V × Z.
Ad (44): Observe that ∂ t E(t, u, z) = − ḟ (t), u V . In view of the regularity assumption (15b) we haveḟ (t n ) →ḟ (t) in V * for t n → t in [0, T ], which immediately gives the upper semicontinuity property of the powers. In view of (67) and Young's inequality we find the following power-control estimate: Ad (45): The coercivity assumption on the sum of E k , V, and R directly follows from the the coercivity of V and R combined with the just deduced coercivity estimate (66) for E k (t, ·, ·).
Ad (48): In order to verify the subgradient estimate (48) for the energy functionals E k , cf. (42), i.e., we will check the respective estimate for each of the contributions to E k separately. For the bulk energy (24b) we verify with standard arguments relying on assumptions (15): Thanks to assumption (16) we find for the cohesive energy (24d) For the Yosida-regularization term we find To conclude the proof of Theorem 2.3 it remains to verify the validity of the energy dissipation balance (35). For this, we will make use of a general result, cf. Thm. 3.5 below, drawn from [38]. In fact, the proof of Thm. 3.5 below, cf. [38,Thm. 3.6], provides the following integral chain-rule inequality for all t ∈ [0, T ]: in the case the map u → E(t, u, ζ) is Gâteaux-differentiable, from the semistability condition (11). This is achieved by mimicking the Riemann-sum procedure from the proof of [41,Prop. 5.4], see also [42,Prop. 4.3], which in turn is based on the argument first developed in [15].
T ] E(t, u, ζ) ≤ M } the energy sublevel with M ∈ R, and that ∂ t E satisfies analogous Lipschitz estimates, i.e.
Observe that the conditions (8)  Proof. Ad (73a): We observe that the first statement of (73a) is a direct consequence of the subgradient estimate (48), which has been already verified for the functional E k along with Lemma 3.4. Moreover, for the second statement of (73a), we recall the form of D u E k (t, ·, z) from (32), and observe that the bulk contribution D u E bulk (t, ·) and the contribution arising from Φ coh (·, z), thanks to their linear character, directly comply with the second of (73a). We now check that also the term arising from D u J k (·, z) complies with the second of (73a). For this, we observe that the function ( · ) − , resp. ( · ) + , is Lipschitz-continuous, such that Because of this, the second of (73a) also holds true for the term arising from the Yosida-regularization. Thus, (73a) is verified for E k .

4.
Limit passage for gradient systems -Proof of Theorem 2.4. In this section we carry out the evolutionary Γ-limit passage from the Yosida-regularized gradient systems (V, Z, V, R, E k ) to the cohesive zone gradient system (V, Z, V, R, E ∞ ). After deducing the compactness properties (39) here below in Prop. 1, we establish the limit passage in the semistable inequality in Sec. 4.1 and the passage in the momentum & energy inequality in Sec. 4.2, respectively. (39)). Let the assumptions of Thm. 2.4 be satisfied. Then there exists a (not relabeled) subsequence (u k , ζ k ) k of semistable energetic solutions to the regularized systems (V, Z, V, R, E k ) and a limit pair (u, ζ) of regularity (38) such that convergences (39) hold true.

4.1.
Limit passage in the semistability inequality. In order to show that the limit pair (u, ζ) extracted by the convergences (39) complies with the semistability inequality for the cohesive zone system (V, Z, V, R, E ∞ ), cf. Def. 2.1, we will make use of a mutual recovery sequence condition akin to (46). More precisely, given a sequence (u k , ζ k ) k ∈ V × Z, with sup k E k (t, u k , ζ k ) ≤ C for all t ∈ [0, T ] and, for every k ∈ N (u k , ζ k ) being semistable for the functionals E k , R, and such that (u k , ζ k ) ⇀ (u, ζ) in V × Z, and for everyζ ∈ Z there exists a sequence (ζ k ) k ⊂ Z, Proposition 2 (Semistability of the system (V, Z, V, R, E ∞ )). Let the assumptions of Theorem 2.4 hold true. Then, for each t ∈ [0, T ] fixed, the sequence (u k , ζ k ) k of semistable energetic solutions to the systems (V, Z, V, R, E k ) k and the limit pair (u, ζ) extracted by convergences (39) satisfy the mutual recovery condition (76). Hence, (u, ζ) is semistable for the cohesive zone delamination system (V, Z, V, R, E ∞ ).
Proof. Taking into account the properties of E ∞ and of R from (5) and (24), we see that the right-hand side of (76) is finite if and only ifζ ∈ X such that ζ * ≥ζ ≥ ζ ≥ u n a.e. in Γ C .
If this is not the case, then (76) is trivially satisfied. Hence, assume from now on that the competitorζ complies with (77). Since X = W 1,r (Γ C ) with r > d − 1, due to the compact embedding X = W 1,r (Γ C ) ⋐ C(Γ C ), it can be verified thatζ k := min{ζ * ,ζ + ζ k − ζ C(ΓC) } is a suitable recovery sequence forζ ≥ ζ that satisfies ζ k ≤ζ k ≤ ζ * and additionally converges strongly in X. To see the strong convergenceζ k →ζ in X it has to be used that the superposition operator min{ζ * , ·} : R → R is Lipschitz-continuous. Then, [29,Sec. 2] provides that the superposition of a W 1,r (Γ C )-function with min{ζ * , ·} yields a W 1,r -function, and that the operator min{ζ * , ·} : W 1,r (Γ C ) → W 1,r (Γ C ) is continuous. Further note that, by construction, property (51e) holds true here as well. Thus, by the strong convergence of (ζ k ) k , and the lower semicontinuity of the W 1,r -seminorm it can be verified that This proves the mutual recovery condition (76).
To verify the strong convergence of the semistable sequence (ζ k (t)) k , we observe that, by the monotonicity assumption (16) and by (51e), the semistability inequality can be further reduced to the following expression Applying the construction toζ = ζ(t), i.e.,ζ k := min{ζ * , ζ(t) + ζ k (t) − ζ(t) C(ΓC) }, and inserting this in (78), allows us to verify that lim sup k→∞ |ζ k (t)| r X ≤ |ζ(t)| r X . Together with the weak convergence ζ k (t) ⇀ ζ(t) in X from (39b) and the weak lower semicontinuity of the norm this yields ζ k → ζ strongly in X.

4.2.
Limit passage in the momentum balance and in the energy-dissipation inequality. The momentum balance of the regularized systems (V, Z, V, R, E k ), integrated over (0, T ), is given by: for every v ∈ L 2 (0, T ; V). Due to the well-preparedness of the initial data it is (u k (0), ζ k (0)) = (u 0 , ζ 0 ) for every k ∈ N and thanks to convergences (39), together with (75d)-(75e), we have This will allow us in Prop. 4 below to pass to the limit k → ∞ in (79) and thus to find for the cohesive zone system (V, Z, V, R, E ∞ ) the following time-integrated momentum balance: T 0 D u E bulk (t, u(t)) + D u Φ coh ( u(t) n , ζ(t)) +ξ(t) + DuV(u(t)), v(t) V dt = 0 .
(81) Its corresponding pointwise-in-time version can then be concluded by arguing via the fundamental lemma of calculus of variations using specific test functions φ = Proof. In a first step we verify the Mosco-convergence result (86a), which will then be carried over to the time-integrated functionals in a second step.
Ad (86a): To verify the Mosco-convergence of (J k ([[·]] n , ζ k (t))) k we first show the Γ-lim inf-relation and secondly prove the Γ-lim sup-relation to hold with a recovery sequence that converges strongly in V.
Ad Γ-lim inf for (86a): Consider a sequence v k ⇀ v in V. Denoting their normal jumps as . By the compact embedding (19) it holds v k → v in L 2 (Γ C ) and, for a (not relabeled) subsequence, v k → v pointwise a.e. in Γ C and almost uniformly.
> 0, or both (the notation for the sets is the same introduced in the proof of Lemma 2.2). We consider a subsequence (v k l , ζ k l (t)) l ⊂ (v k , ζ k (t)) k , such that v k l → v almost uniformly and which additionally realizes the lim inf, i.e. lim inf k→∞ J k (v k , ζ k (t)) = lim l→∞ J k l (v k l , ζ k l (t)). Observe that the existence of an almost uniformly converging subsequence of (v k ) k is implied by the compact embedding H 1/2 (Γ C ) ⋐ L 2 (Γ C ).
Thanks to the Mosco-convergence result we are now in the position to verify the momentum balance of the limit system (82) to hold withξ(t) ∈ ∂ u J ([[u(t)]] n , ζ(t)) for a.e. t ∈ (0, T ) and to deduce enhanced convergence of solutions (u k ) k .
From the lower Γ-limit provided by the Mosco-convergence of the functionals (J k ([[·]] n , ζ k (t))) k and the uniform bound on (E k (t, u k (t), ζ k (t))) k it can be concluded that indeed 0 = J ([[u(t)]] n , ζ(t)). Also exploiting the lower semicontinuity properties of E bulk and Φ coh in combination with convergences (39) we obtain the energy-dissipation inequality (41) of the limit systems (V, Z, V, R, E ∞ ).