On a phase--field model of damage for hybrid laminates with cohesive interface
Elena Bonetti, Cecilia Cavaterra, Francesco Freddi, Filippo Riva
aa r X i v : . [ m a t h . A P ] J u l ON A PHASE–FIELD MODEL OF DAMAGEFOR HYBRID LAMINATES WITH COHESIVE INTERFACE
ELENA BONETTI, CECILIA CAVATERRA, FRANCESCO FREDDI AND FILIPPO RIVA
Abstract.
In this paper we investigate a rate–independent model for hybrid laminates de-scribed by a damage phase–field approach on two layers coupled with a cohesive law governingthe behaviour of their interface. For the analysis we adopt the notion of energetic evolution,based on global minimisation of the involved energy. Due to the presence of the cohesive zone,as already emerged in literature, compactness mathematical issues lead to the introduction ofa fictitious variable replacing the physical one which represents the maximal opening of theinterface displacement discontinuity reached during the evolution. A new strategy which allowsto recover the equivalence between the fictitious and the real variable under general loading–unloading regimes is illustrated. The argument is based on temporal regularity of energeticevolutions. This regularity is achieved by means of a careful balance between the convexity ofthe elastic energy of the layers and the natural concavity of the cohesive energy of the interface.
Keywords:
Damage phase–field model, Cohesive interface, Energetic evolutions, Temporal regularity.
Contents
Introduction 11. Setting of the Problem 41.1. The variables 41.2. The energies 51.3. Energetic evolutions 82. Existence Result 112.1. Time-discretisation 112.2. Extraction of convergent subsequences 142.3. Existence of generalised energetic evolutions 163. PDE Form of Energetic Evolutions 214. Temporal Regularity and Equivalence between γ and δ h Introduction
Composite fibre reinforced materials are increasingly finding applications in the manufacturingindustry due to their capacity of offering high strength and stiffness with low mass density. Theironly mechanical weakness is the brittleness. Indeed, rapid failure occurs without sufficientwarning, due to the intrinsic nature of the adopted materials. A possible strategy to providea ductile failure response is to consider novel composite architectures where fibres of differentstiffness and ultimate strain values are combined through cohesive interfaces (hybridisation). Inthis case, complex rupture processes occur with diffuse crack pattern (fragmentation) and/or delamination. A deep analytical comprehension of the failure mechanisms of these kind ofmaterials is thus needed in order to predict and control the appearance and the evolution of thecracks.Among the mathematical community, the variational approach to fracture, as formulated by[12, 19], is one of the most adopted viewpoint to deal with crack problems. It is based on theGriffith’s idea [26] that the crack growth is governed by a reciprocal competition between theinternal elastic energy of the body and the energy spent to increase the crack length. In theoriginal theory the energy associated with the fracture is proportional to the measure of thefracture itself, while in the cohesive case (Baremblatt [6]), where the process is more gradual,the energy depends on the opening of the crack.Due to the complexity of the phenomenon and the technical difficulties of the related mathe-matical analysis, especially from the numerical point of view, in the last twenty years a damagephase-field approach has been developed to overcome the aforementioned issues. Nowadays itis a well established and consolidated method to approximate both brittle (see [4, 5, 24]) andcohesive fractures (see [13, 28]). It consists in the introduction of an irreversible internal variabletaking values in [0 ,
1] and representing the damage state of the material. Usually, values 0 and 1mean a completely sound and a completely broken state, respectively, while a value in betweenrepresents the case of a partial damage. The presence of a fracture is thus ideally replaced bythose parts of the body whose damage variable has reached the value 1.In this work a rigorous mathematical analysis of a one–dimensional model for hybrid laminatesintroduced and numerically investigated in [3] is proposed. Its description is given by couplingthe damage phase–field approach, which models the elastic–brittle behaviour of the layers, witha cohesive law in the interface connecting the materials. The investigation is restricted to thecase of incomplete damage in the sense that a reservoir of elastic material stiffness is alwaysmaintained, even if the damage variable reaches the maximum value 1. This situation can beconcretely justified by considering materials formed by different components from which onlya part can undergo a damage (for instance in composite materials obtained with a matrixand a reinforcement) and delamination may take place; on the other side it can be seen as amathematical approximation of the complete damage setting in which the material goes throughfull rupture. We refer for instance to [11, 30] for an analysis of complete damage between twoviscoelastic bodies, while we postpone the inspection of this model to future works, due to highmathematical difficulties.Here, the model we want to analyse describes the evolution of a unidirectional hybrid laminatein hard device: a prescribed time–dependent displacement ¯ u ( t ) is applied on one side of the bar,whereas the other is fixed. We restrict our attention to slow prescribed displacements, so thatinertial effects can be neglected and the analysis can be included in a quasi–static and rate–independent regime. For the sake of simplicity we consider a bar composed by only two layerswith thickness ρ and ρ , respectively, bonded together along the entire length by a cohesiveinterface. The thickness of the interface is very thin compared with ρ and ρ , which in turnare way smaller than the length of the laminate L >
0. Thus the model can be considered asone–dimensional.The brittle behaviour of the two elastic layers undergoes a damage phase–field approach. Itsuits with the rate–independent framework we are considering. For the reader interested insteadin dynamic and rate–dependent damage models we refer for instance to [14, 21]. The unknownsthat govern the problem are thus the displacements of the two layers, denoted by u and u ,and their irreversible damage variables α and α .In the quasi–static setting a huge variety of notions of solution can be considered, see forinstance the monograph [31]. In this paper we focus our attention on the concept of energeticevolution, based on two ingredients: at every time the solution is a global minimiser of the AMAGE IN LAMINATES WITH COHESIVE INTERFACE 3 involved total energy, and the sum of internal and dissipated energy balances the work doneby the external prescribed displacement. The same kind of evolution in an analogous cohesivefracture model between two elastic bodies is studied in [16, 18]; other notions based on stationarypoints of the energy, always in the framework of cohesive fractures, are instead analysed in[34, 35].The total energy we consider is composed by a first part taking into account elastic responsesof the layers and dissipation due to damage, and a second part reflecting the cohesive behaviourof the interface. The cohesive interface is governed by the slip between the two layers δ = | u − u | and its irreversible counterpart δ h which represents the maximal slip achieved duringthe evolution. The presence of an irreversible historical variable can be also found in differentmodels than cohesive fracture: we mention for instance the notion of fatigue, investigated in[1, 17].The expression of the energy in the model under consideration is hence given by: X i =1 ρ i Z L E i ( α i ( x ))( u ′ i ( x )) d x | {z } elastic energy of the i-th layer + 12 Z L ( α ′ i ( x )) d x | {z } internal energy ofthe i-th damage variable + Z L w i ( α i ( x )) d x | {z } energy dissipated bydamage in the i-th layer ! + Z L ϕ ( δ ( x ) , δ h ( x )) d x | {z } internal and dissipatedenergy in the interface , where the symbol prime ′ denotes the one–dimensional spatial derivative, E i : [0 , → (0 , + ∞ )is the elastic Young modulus of the i -th layer (which is strictly positive since we are in theincomplete damage framework), w i : [0 , → [0 , + ∞ ) is a dissipation density and ϕ : { ( y, z ) ∈ R | z ≥ y ≥ } → [0 , + ∞ ) is the loading-unloading density of the cohesive interface.As usual in the context of energetic evolutions, we follow a time–discretisation algorithm toshow existence of solutions. More precisely, we consider a fine partition of the time interval[0 , T ] and at each time step we select a global minimiser of the total energy; we then recover thetime-continuous evolution by sending to zero the discretisation parameter. Due to compactnessissues regarding the maximal slip δ h , the time–discretisation process leads to the introductionof a weaker notion of solution where a fictitious historical variable γ replaces the concrete one δ h . We point out that this fictitious device only appears when dealing with global minima ofthe energy, indeed it can be found in [16, 18], but not in [34, 35] where stationary points areconsidered. The issue has been partially overcome in [16, 18] with different approaches, butassuming the hypothesis of constant unloading response, namely when the loading–unloadingdensity ϕ depends only on the second variable z .Here, an original strategy based on temporal regularity properties of energetic evolutions inorder to recover the equivalence between the fictitious variable γ and the proper one δ h underreasonable assumptions on the density ϕ is illustrated. In particular, we are able to cover allthe general cases considered in [34]. Moreover, the proposed approach fits well with the modelunder consideration, but it can be also adapted to more general situations.An alternative strategy to deal with cohesive problems can be found in literature, whereadhesion is treated with the introduction of a damage variable that macroscopically defines thebond state between two solids. Detachment corresponds to full damage state. The problem hasbeen investigated theoretically in [7, 8, 9, 10] and numerically in [22, 23]The paper is organised as follows. In Section 1 we introduce in a rigorous way the variationalproblem, presenting the global and historical variables: the displacement field u i , the damagevariables α i , the slip δ and the historical slip δ h . Subsequently, details of the involved energiesare given together with a precise notion of energetic evolution and of its weak counterpart, herenamed generalised energetic evolution, including the fictitious variable γ . E. BONETTI, C. CAVATERRA, F. FREDDI AND F. RIVA
Section 2 is devoted to the proof of existence of generalised energetic evolutions under verymild assumptions on the loading–unloading cohesive density ϕ . We first introduce the time–discretisation algorithm based on global minimisation of the energy, and we provide uniformbounds on the sequence of discrete minimisers. Thanks to these bounds and by means of asuitable version of Helly’s selection theorem we are able to extract convergent subsequencesas the time step vanishes. After the introduction of the fictitious historical variable γ and byexploiting the fact that the discrete functions selected by the algorithm are global minima ofthe total energy, we finally deduce that the previously obtained limit functions actually are ageneralised energetic evolution.In Section 3 attention is focused on the equations that a generalised energetic evolutionmust satisfy; they are a byproduct of the global minimality condition together with the energybalance. It turns out that the displacements fulfil a system of equations in divergence form,see (3.1a), while the damages satisfy a Karush–Kuhn–Tucker condition, see (3.1b), assuming apriori certain regularity in time. Of course these equations have to be meant in a weak sense.The results of this third section are a first step in order to obtain the equivalence between γ andthe concrete historical variable δ h .Finally Section 4 illustrates the main result of the paper. We first adapt a convexity argumentintroduced in [32] to our setting in which a cohesive energy (concave by nature) is present,in order to gain regularity in time (absolute continuity) of generalised energetic evolutions.Once this temporal regularity is achieved, we exploit the Euler–Lagrange equations of Section 3together with the monotonicity (in time) of γ and δ h to deduce their equivalence under reasonableassumptions on ϕ . We thus obtain as a byproduct that the generalised energetic evolution foundin Section 2 is actually an energetic evolution, since γ coincides with δ h .At the end of the work we attach an Appendix in which we gather some definitions andproperties we need throughout the paper about absolutely continuous and bounded variationfunctions with values in Banach spaces.1. Setting of the Problem
In this section we present the variational formulation of the one-dimensional continuum modeldescribed in the Introduction of two layers bonded together by a cohesive interface in a harddevice setup. We list all the main assumptions we need throughout the paper. We also introducethe two notions of energetic evolution and generalised energetic evolution in our context, seeDefinitions 1.5 and 1.8.For the sake of clarity, in this work every function in the Sobolev space H ( a, b ) is always iden-tified with its continuous representative. The prime symbol ′ is used to denote spatial derivatives,while the dot symbol ˙ to denote time derivatives. In the case of a function f : [0 , T ] → H ( a, b ),which thus depends on both time and space, we write f ( t ) ′ to denote the (weak) spatial deriv-ative of f ( t ) ∈ H ( a, b ) and with a little abuse of notation we write f ′ ( t, x ) to denote its valueat a.e. x ∈ [ a, b ]. If f is sufficiently regular in time, for instance in C ([0 , T ]; H ( a, b )), for thetime derivative we instead adopt the scripts ˙ f , ˙ f ( t ) and ˙ f ( t, x ), with the obvious meanings: ˙ f isthe function from [0 , T ] to H ( a, b ), ˙ f ( t ) is its value as a function in H ( a, b ), once t ∈ [0 , T ] isfixed, and ˙ f ( t, x ) is its value (as a real number) at x ∈ [ a, b ]. By a ∨ b and a ∧ b we finally meanthe maximum and the minimum between two extended real numbers a and b in [ −∞ , + ∞ ].We fix a time T >
L >
0. We also normalise the thicknessof the two layers ρ and ρ to 1, since this does not affect the results.1.1. The variables.
To describe the evolution of the system, for i = 1 , u i : [0 , T ] × [0 , L ] → R , where u i ( t, x ) denotes the displacement at time t of the point x of the i -th layer; here u ( t, x ) represents the vector in R with components u ( t, x ) and u ( t, x ). AMAGE IN LAMINATES WITH COHESIVE INTERFACE 5
For the structure of the model itself, at every time t ∈ [0 , T ] the displacement u i ( t ) will belongto the space H (0 , L ). The function δ : [0 , T ] × [0 , L ] → [0 , + ∞ ) defined as δ ( t, x ) = δ [ u ]( t, x ) := | u ( t, x ) − u ( t, x ) | , (1.1a)instead denotes the displacement slip on the interface between the two layers. Then, we introducethe non-decreasing function δ h : [0 , T ] × [0 , L ] → [0 , + ∞ ) as δ h ( t, x ) := sup τ ∈ [0 ,t ] δ ( τ, x ) , (1.1b)namely the historical variable which records the maximal slip reached at the point x in theinterface till the time t . Internal constraints, such as unilateral conditions (see [8, 9]), are notnecessary on the kinematics as this only permits displacement slips between the two solids andinterpenetration is prevented a-priori.Finally, for i = 1 ,
2, we consider the function α i : [0 , T ] × [0 , L ] → [0 , α i ( t, x ) repre-sents the damage at time t of the point x of the i -th layer. It is non-decreasing in time withvalues in [0 , α i ( t ) will be in H (0 , L )for every t ∈ [0 , T ]. In analogy with the previous setting, α ( t, x ) denotes the vector in R withcomponents α ( t, x ) and α ( t, x ).1.2. The energies.
We now present the energies involved in our model. Given a pair ( u , α )belonging to [ H (0 , L )] × [ H (0 , L )] and representing an admissible displacement and damage,the stored elastic energy of the two layerss is given by: E [ u , α ] := X i =1 Z L E i ( α i ( x ))( u ′ i ( x )) d x, (1.2)where, for i = 1 ,
2, we assume the elastic Young moduli E i satisfy: E i ∈ C ([0 , E i ( y ) ≥ min ˜ y ∈ [0 , E i (˜ y ) =: ε i > , for every y ∈ [0 , . (1.3)We define ε := ε ∧ ε > , (1.4)which is strictly positive by (1.3). This feature reflects the fact that we are considering theincomplete damage framework, and it will be used to gain coercivity of E . This property of theenergy is indeed missing in the complete damage setting where the functions E i can vanish, anda completely different notion of solution and strategy must be adopted. We refer to [11, 30] forthe interested reader.We can now introduce for i = 1 , σ i : [0 , T ] × [0 , L ] → R , defined as σ i ( t, x ) = σ i [ u i , α i ]( t, x ) := E i ( α i ( t, x )) u ′ i ( t, x ) . (1.5)As before, by σ ( t, x ) we mean the vector with components σ ( t, x ) and σ ( t, x ).An other energy term appearing in the model is the sum of the stored and the dissipatedenergy of the phase–field variable α ∈ [ H (0 , L )] during the damaging process and expressedby D [ α ] := X i =1 (cid:18) Z L ( α ′ i ( x )) d x + Z L w i ( α i ( x )) d x (cid:19) . (1.6) E. BONETTI, C. CAVATERRA, F. FREDDI AND F. RIVA
In literature there are very different choices of dissipation functions w i (see for instance [2, 3,29, 33, 36, 38]). As a simple example we can consider w i ( y ) = y + y .In this work we permit quite general assumptions on w i as follows: w i ∈ C ([0 , w i ( y ) ≥ c i y for some c i > y ∈ [0 , . (1.7) Remark 1.1.
The dissipated damage density, usually a process dependent function (i.e. de-pending on the time derivative of the damage variable ˙ α ( t )), is here treated as a state functiondue to the underlying gradient damage model. See also [2], [3] and [30].We finally introduce the cohesive energy in the interface between the two layers: K [ δ, γ ] := Z L ϕ ( δ ( x ) , γ ( x )) d x, (1.8)where δ and γ are two non-negative functions in [0 , L ] such that γ ≥ δ and representing,respectively, the slip and the historical slip of the displacement at a given instant. The non-negative function ϕ : T → [0 , + ∞ ) , where T = { ( y, z ) ∈ R | z ≥ y ≥ } , (1.9)is the loading-unloading density of the cohesive interface; the variable y governs the unloadingregime (usually convex), while z the loading regime (usually concave).Since several assumptions on ϕ will be needed throughout the paper we prefer listing themhere. The first set of assumptions, very mild, will be used in Section 2 to prove existence of(generalised) energetic evolutions (see Definitions 1.5 and 1.8):( ϕ ϕ is lower semicontinuous;( ϕ ϕ (0 , · ) is bounded in [0 , + ∞ );( ϕ ϕ ( y, · ) is continuous and non-decreasing in [ y, + ∞ ), for every y ≥ γ (see Definition 1.8), and which in this work we are able to avoid. Wehowever include it in the list because we make use of it in Theorem 2.12, where we employ theargument of [16] in our context:( ϕ
4) there exist two functions ϕ , ϕ : [0 , + ∞ ) → [0 , + ∞ ) such that ϕ is lower semicontinu-ous, ϕ is bounded, non-decreasing and concave, and ϕ ( y, z ) = ϕ ( y ) + ϕ ( z ). Remark 1.2.
Actually, in [16, 18] the function ϕ appearing above is chosen identically 0, sothat the cohesive density ϕ depends only on the second variable z (constant unloading regime).However, their argument can be easily adapted to our case.To overcome the necessity of ( ϕ ϕ with domain T , for z ∈ [0 , + ∞ )we define ψ ( z ) := ϕ ( z, z ) , namely the restriction of ϕ on the diagonal. Moreover we introduce the constant¯ δ := inf { z > | ψ is constant in [ z, + ∞ ) } , (1.10)with the convention inf {∅} = + ∞ ; it represents the limit slip which triggers complete delam-ination. According to [2, 3], complete delamination may occour for finite or infinite slip value(see Remark 1.3).We then set T ¯ δ := { ( y, z ) ∈ T | z < ¯ δ } . We thus require:
AMAGE IN LAMINATES WITH COHESIVE INTERFACE 7 ( ϕ
5) the function ψ ∈ C ([0 , + ∞ )) is λ –convex for some λ >
0, namely for every θ ∈ [0 , z a , z b ∈ [0 , + ∞ ) it holds ψ ( θz a + (1 − θ ) z b ) ≤ θψ ( z a ) + (1 − θ ) ψ ( z b ) + λ θ (1 − θ ) | z a − z b | ;( ϕ
6) for every z ∈ (0 , + ∞ ) the map ϕ ( · , z ) ∈ C ([0 , z ]) is non-decreasing and convex;( ϕ
7) for every z ∈ (0 , + ∞ ) there hold ∂ y ϕ ( z, z ) = ψ ′ ( z ) and ∂ y ϕ (0 , z ) = 0;( ϕ
8) the partial derivative ∂ y ϕ belongs to C ( T \ (0 , T .( ϕ
9) for every y ∈ [0 , ¯ δ ) the map ϕ ( y, · ) is differentiable in [ y, ¯ δ ) and the partial derivative ∂ z ϕ is continuous and strictly positive on T ¯ δ \ { ( z, z ) ∈ R | z ≥ } .Condition ( ϕ
9) will be actually weakened in Section 4, where only a uniform strict monotonicitywith respect to z will be needed, see (4.17).We want to point out that this set of assumptions includes a huge variety of mechanicallymeaningful loading–unloading densities ϕ , as precised in the next remark. We also notice thatthese conditions are similar to the one considered in [34]. Remark 1.3 ( Main Example ) . The prototypical example of a physically meaningful loading-unloading density is obtained reasoning in the opposite way of what we presented before, namelyfirstly a function ψ is given and then the density ϕ is built from ψ . As regards ψ , which governsthe loading regime, natural assumptions are the following: ψ ∈ C ([0 , ¯ δ )) ∩ C ([0 , + ∞ )) is anon-decreasing, concave and bounded function such that ψ (0) = 0, ψ ′ > ψ ′′ is boundedfrom below in [0 , ¯ δ ). In particular ( ϕ
5) is satisfied with λ = sup z ∈ [0 , ¯ δ ) | ψ ′′ ( z ) | . For instance one canconsider: ψ ( z ) = c (1 − e − kz ) , ψ ( z ) = ( cz (2 k − z ) , if z ∈ [0 , k ) ,ck , if z ∈ [ k, + ∞ ) , for c, k > . In the first example ¯ δ = + ∞ , while in the second one ¯ δ = k < + ∞ .The function ϕ is then defined by considering a quadratic unloading regime: ϕ ( y, z ) := ( ψ ′ ( z ) z y + ψ ( z ) − zψ ′ ( z ) , if ( y, z ) ∈ T \ (0 , , , if ( y, z ) = (0 , . (1.11)By construction ϕ is continuous on T and ( ϕ ϕ
7) and ( ϕ
8) are satisfied. To verify also ( ϕ ∂ z ϕ ( y, z ) = ψ ′ ( z ) − zψ ′′ ( z )2 (cid:18) − y z (cid:19) , for every ( y, z ) ∈ T ¯ δ . Thus we deduce ∂ z ϕ is continuous in T ¯ δ \ (0 , z > y , since ψ ′ ( z ) is strictly positivein [0 , ¯ δ ), we get that ∂ z ϕ ( y, z ) > ϕ
9) is fulfilled.We finally observe that by the boundedness of ψ we also obtain ( ϕ ϕ in the case ¯ δ < + ∞ . Lemma 1.4.
Assume ϕ satisfies ( ϕ ϕ
6) and ( ϕ
7) and assume ¯ δ is finite. Then ϕ is constantin T \ T ¯ δ , and in particular: ϕ ( y, z ) = ψ (¯ δ ) , for every ( y, z ) ∈ T \ T ¯ δ . Proof.
Since ψ is C [0 , + ∞ ), then by definition of ¯ δ it holds ψ ′ ( z ) = 0 for every z ∈ [¯ δ, + ∞ ).We now fix z ∈ [¯ δ, + ∞ ); by ( ϕ
7) we deduce that ∂ y ϕ ( z, z ) = 0. Condition ( ϕ
6) thus yields ∂ y ϕ ( y, z ) = 0 for every y ∈ [0 , z ], and hence we conclude. (cid:3) E. BONETTI, C. CAVATERRA, F. FREDDI AND F. RIVA
We finally introduce the function ϕ ¯ δ , defined as: ϕ ¯ δ ( y, z ) := ϕ ( y ∧ ¯ δ, z ∧ ¯ δ ) , for every ( y, z ) ∈ T . Thanks to previous lemma, it is easy to deduce that if conditions ( ϕ ϕ
6) and ( ϕ
7) arefulfilled, then actually ϕ and ϕ ¯ δ coincide, namely it holds: ϕ ( y, z ) = ϕ ¯ δ ( y, z ) , for every ( y, z ) ∈ T . (1.12)This last equality will be widely exploited in Section 4.1.3. Energetic evolutions.
We are now in a position to introduce the notion of quasistaticsolution we want to investigate in this work. Before presenting it we need to consider the pre-scribed displacement acting on one extrema of the laminate, namely a function ¯ u ∈ AC ([0 , T ]);we also need to consider initial displacements and damages, namely functions u i , α i which mustsatisfy, for i = 1 ,
2, the following regularity and compatibility conditions: u i , α i ∈ H (0 , L ) , (1.13a) u (0) = u (0) = 0 , u ( L ) = u ( L ) = ¯ u (0) , (1.13b)0 ≤ α i ( x ) ≤ , for every x ∈ [0 , L ] . (1.13c)Once the initial displacements are given, we define the initial slip δ := | u − u | . For t ∈ [0 , T ], we denote by H , ¯ u ( t ) (0 , L ) the space of functions v ∈ H (0 , L ) attaining theboundary values v (0) = 0 and v ( L ) = ¯ u ( t ). We instead denote by H , (0 , L ) the space offunctions v ∈ H (0 , L ) such that 0 ≤ v ( x ) ≤ x ∈ [0 , L ]. Definition 1.5.
Given a prescribed displacement ¯ u ∈ AC ([0 , T ]) and initial data u , α sat-isfying (1.13) , we say that a pair ( u , α ) : [0 , T ] × [0 , L ] → R × R is an energetic evolution if: (CO) u ( t ) ∈ [ H , ¯ u ( t ) (0 , L )] , α ( t ) ∈ [ H , (0 , L )] , for every t ∈ [0 , T ] ; (ID) u (0) = u , α (0) = α ; (IR) for i = 1 , the damage function α i is non-decreasing in time, namely,for every ≤ s ≤ t ≤ T it holds: α i ( s, x ) ≤ α i ( t, x ) , for every x ∈ [0 , L ];(GS) for every t ∈ [0 , T ] , for every e u ∈ [ H , ¯ u ( t ) (0 , L )] and for every e α ∈ [ H (0 , L )] such that α i ( t ) ≤ e α i ≤ in [0 , L ] , i = 1 , , one has: E [ u ( t ) , α ( t )] + D [ α ( t )] + K [ δ ( t ) , δ h ( t )] ≤ E [ e u , e α ] + D [ e α ] + K [ e δ, δ h ( t ) ∨ e δ ]; here we mean e δ = | e u − e u | ; (EB) the function τ ˙¯ u ( τ ) L Z L X i =1 σ i ( τ, x ) d x belongs to L (0 , T ) and for every t ∈ [0 , T ] itholds: E [ u ( t ) , α ( t )] + D [ α ( t )] + K [ δ ( t ) , δ h ( t )] = E [ u , α ] + D [ α ] + K [ δ , δ ] + W [ u , α ]( t ) , where W [ u , α ]( t ) := Z t ˙¯ u ( τ ) L Z L X i =1 σ i ( τ, x ) d x dτ, (1.14) is the work done by the external prescribed displacement. AMAGE IN LAMINATES WITH COHESIVE INTERFACE 9
In the above Definition (CO) stands for compatibility, (ID) for initial data and (IR) forirreversibility (of the damage variables); the main conditions which characterise this sort ofsolution are of course the global stability (GS) and the energy balance (EB).We notice that, by (GS), a necessary condition for the existence of such an evolution is theglobal minimality of the initial data at time t = 0, namely: E [ u , α ] + D [ α ] + K [ δ , δ ] ≤ E [ e u , e α ] + D [ e α ] + K [ e δ, δ ∨ e δ ] , (1.15)for every e u ∈ [ H , ¯ u (0) (0 , L )] and for every e α ∈ [ H (0 , L )] such that α i ≤ e α i ≤ , L ], i = 1 , u and α are bounded in time with values in [ H (0 , L )] , as stated in the next Proposition.As a byproduct we also obtain both temporal and spatial regularity on the historical variable δ h ,which actually is bounded in time with values in C / ([0 , L ]), namely the space of 1 / x = 0 and x = L . Proposition 1.6.
Assume E i satisfies (1.3) , w i satisfies (1.7) , ϕ satisfies ( ϕ
2) and let ( u , α ) be an energetic evolution. Then there exists a positive constant C such that: sup t ∈ [0 ,T ] k u ( t ) k [ H (0 ,L )] ≤ C √ ε , and sup t ∈ [0 ,T ] k α ( t ) k [ H (0 ,L )] ≤ C, (1.16a) where ε > has been introduced in (1.4) . In particular, δ h ( t ) belongs to C / ([0 , L ]) , for every t ∈ [0 , T ] , and the following estimate holds true: sup t ∈ [0 ,T ] | δ h ( t, x ) − δ h ( t, y ) | ≤ C √ ε p | x − y | , for every x, y ∈ [0 , L ] . (1.16b) Proof.
Choosing as competitors in (GS) the functions e u i ( x ) = ¯ u ( t ) L x, e α i ≡ , for i = 1 , , and exploiting (1.3), (1.7) and ( ϕ
2) we deduce that C X i =1 (cid:16) ε k u i ( t ) k H (0 ,L ) + k α i ( t ) k H (0 ,L ) (cid:17) ≤ E [ u ( t ) , α ( t )] + D [ α ( t )] + K [ δ ( t ) , δ h ( t )] ≤ E [( e u , e u ) , (1 , D [(1 , K [0 , δ h ( t )] ≤ C (cid:16) k ¯ u k C ([0 ,T ]) + 1 (cid:17) , for every t ∈ [0 , T ], where C and C are suitable positive constants independent of t . Hence(1.16a) is proved.By (1.16a) and Sobolev embedding Theorems, we now know that u i ( t ) are uniformly 1/2-equiH¨older, for every t ∈ [0 , T ]. We thus fix t ∈ [0 , T ] and x, y ∈ [0 , L ]; by definition of δ h ( t, x ),for every η > τ η ∈ [0 , t ] such that δ h ( t, x ) − η ≤ | u ( τ η , x ) − u ( τ η , x ) | . Hence we can estimate: δ h ( t, x ) − η ≤| u ( τ η , x ) − u ( τ η , y ) | + | u ( τ η , y ) − u ( τ η , y ) | + | u ( τ η , y ) − u ( τ η , x ) |≤ C √ ε p | x − y | + δ h ( t, y ) , for any t ∈ [0 , T ] and x, y ∈ [0 , L ]. By the arbitrariness of η and reverting the role of x and y we deduce that δ h ( t ) is 1/2-H¨older and (1.16b) holds true. Trivially δ h ( t,
0) = δ h ( t, L ) = 0 andso we conclude. (cid:3) Remark 1.7.
In the previous proposition we stressed the dependence on ε > E i can vanish, one needs to consider the sequence of functions E i + ε , fulfilling (1.3), and then to perform an analysis of the limit ε → + , usually via Γ–convergence. We refer for instance to [11, 30] for a model of contact between two viscoelasticbodies.As we said in the Introduction, the common procedure used to prove existence of energeticevolutions (and which we will perform in Section 2) is based on a time discretisation algorithmand then on a limit passage as the time step goes to 0. Due to lack of compactness for thehistorical variable δ h , one needs to weaken the notion of energetic evolution and to introduce afictitious variable γ replacing δ h (see also [16, 18]). Thanks to Proposition 1.6 we however expectthat γ ( t ) should be at least continuous in [0 , L ]; we are thus led to the following definition: Definition 1.8.
Given a prescribed displacement ¯ u ∈ AC ([0 , T ]) and initial data u , α sat-isfying (1.13) , we say that a triple ( u , α , γ ) : [0 , T ] × [0 , L ] → R × R × R is a generalisedenergetic evolution if: (CO’) u ( t ) ∈ [ H , ¯ u ( t ) (0 , L )] , α ( t ) ∈ [ H , (0 , L )] , γ ( t ) ∈ C ([0 , L ]) , for every t ∈ [0 , T ] ; (ID’) u (0) = u , α (0) = α , γ (0) = δ ; (IR’) for i = 1 , the damage function α i and the generalised historical variable γ are non-decreasing in time, namely,for every ≤ s ≤ t ≤ T it holds: α i ( s, x ) ≤ α i ( t, x ) , for every x ∈ [0 , L ]; for every ≤ s ≤ t ≤ T it holds: γ ( s, x ) ≤ γ ( t, x ) , for every x ∈ [0 , L ];(GS’) for every t ∈ [0 , T ] one has γ ( t ) ≥ δ ( t ) in [0 , L ] and: E [ u ( t ) , α ( t )] + D [ α ( t )] + K [ δ ( t ) , γ ( t )] ≤ E [ e u , e α ] + D [ e α ] + K [ e δ, γ ( t ) ∨ e δ ] , for every e u ∈ [ H , ¯ u ( t ) (0 , L )] and for every e α ∈ [ H (0 , L )] such that α i ( t ) ≤ e α i ≤ in [0 , L ] for i = 1 , ; (EB’) the function τ ˙¯ u ( τ ) L Z L X i =1 σ i ( τ, x ) d x belongs to L (0 , T ) and for every t ∈ [0 , T ] itholds: E [ u ( t ) , α ( t )] + D [ α ( t )] + K [ δ ( t ) , γ ( t )] = E [ u , α ] + D [ α ] + K [ δ , δ ] + W [ u , α ]( t ) , where W [ u , α ]( t ) is defined as in (1.14) . Remark 1.9.
If conditions ( ϕ ϕ
6) and ( ϕ
7) are satisfied, then equality (1.12) allows us toreplace the function ϕ in the functional K (see (1.8)) by ϕ ¯ δ . This means that the functionswhich actually play a role in the cohesive energy are δ ∧ ¯ δ , δ h ∧ ¯ δ and γ ∧ ¯ δ . This observationwill be useful in Section 4.From the very definition it is easy to see that a pair ( u , α ) is an energetic evolution if andonly if the triple ( u , α , δ h ) is a generalised energetic evolution. It is also easy to see thatgiven a generalised energetic evolution ( u , α , γ ) it necessarily holds γ ( t, x ) ≥ δ h ( t, x ), for every( t, x ) ∈ [0 , T ] × [0 , L ]. Unfortunately, there are no easy arguments which ensure that γ = δ h ina general case. This will be the topic of Section 4 and the main outcome of the paper. AMAGE IN LAMINATES WITH COHESIVE INTERFACE 11
We finally notice that the same argument used to prove Proposition 1.6 leads to the bound(1.16a) also for a generalised energetic evolution. However (1.16b) only holds for δ h due to itsexplicit definition (1.1b), and nothing can be said, in general, about the generalised historicalvariable γ . 2. Existence Result
In this section we show existence of generalised energetic evolutions under very weak as-sumptions on the data, especially on the density ϕ . We indeed require (1.3), (1.7) and only( ϕ ϕ ϕ u belongs to AC ([0 , T ]). We then prove the existence of an energetic evolution assumingthe specific assumption ( ϕ ϕ
4) in Section 4, recovering the existence of energetic evolutions inmeaningful mechanical situations (namely assuming ( ϕ ϕ Time-discretisation.
We consider a sequence of partition 0 = t n < t n < · · · < t nn = T such that lim n → + ∞ max k =1 , ··· ,n ( t nk − t nk − ) = 0 , (2.1)and for k = 1 , · · · , n we perform the following implicit Euler scheme: given ( u k − , α k − , δ k − h ),we first select ( u k , α k ) by minimising the total energy among suitable natural competitors:( u k , α k ) ∈ argmin e u ∈ [ H , ¯ u ( tnk ) (0 ,L )] , e α ∈ [ H (0 ,L )] s.t. α k − i ≤ e α i ≤ n E [ e u , e α ] + D [ e α ] + K [ e δ, δ k − h ∨ e δ ] o . (2.2a)Here we want to recall that we mean e δ = | e u − e u | .We then define δ kh as: δ kh := δ k − h ∨ | u k − u k | = δ k − h ∨ δ k . (2.2b)The initial values in the minimisation algorithm are functions ( u , α ) satisfying the compati-bility conditions (1.13); moreover we set δ h := δ = | u − u | . Proposition 2.1.
Assume E i satisfies (1.3) , w i satisfies (1.7) and ϕ satisfies ( ϕ (2.2a) .Proof. We fix n ∈ N and for every k = 1 , . . . , n we prove the existence of a minimum by meansof the direct method of Calculus of Variations. For the sake of clarity we denote by F k − thefunctional we want to minimise, namely F k − [ e u , e α ] = E [ e u , e α ] + D [ e α ] + K [ e δ, δ k − h ∨ e δ ] . (2.3)Sequential coerciveness of F k − in the weak topology of [ H (0 , L )] follows by means of (1.3)and (1.7): F k − [ e u , e α ] ≥ E [ e u , e α ] + D [ e α ] ≥ C X i =1 (cid:16) ε k e u i k H (0 ,L ) + k e α i k H (0 ,L ) (cid:17) . As regards the lower semicontinuity of F k − we exploit the compact embedding H (0 , L ) ⊂⊂ C (0 , L ). By ( ϕ
1) and Fatou’s Lemma we deduce that K is sequentially lower semicontinuouswith respect to the considered topology. The same holds for D by using again Fatou’s Lemmatogether with weak lower semicontinuity of the norm. To prove finally lower semicontinuity of E it is enough to show that, given weakly convergent sequences e u ji ⇀ e u i , e α ji ⇀ e α i in H (0 , L ),we have that q E i ( e α ji )( e u ji ) ′ weakly converges to p E i ( e α i ) e u i ′ in L (0 , L ) as j → + ∞ , for i = 1 , φ ∈ L (0 , L ) and we estimate by exploiting (1.3): (cid:12)(cid:12)(cid:12)(cid:12)Z L q E i ( e α ji ( x ))( e u ji ) ′ ( x ) φ ( x ) d x − Z L p E i ( e α i ( x )) e u i ′ ( x ) φ ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:13)q E i ( e α ji ) − p E i ( e α i ) (cid:13)(cid:13)(cid:13)(cid:13) C ([0 ,L ]) k e u ji k H (0 ,L ) k φ k L (0 ,L ) + (cid:12)(cid:12)(cid:12)(cid:12)Z L ( e u ji ) ′ ( x ) p E i ( e α i ( x )) φ ( x ) d x − Z L e u i ′ ( x ) p E i ( e α i ( x )) φ ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12) . The first term goes to zero as j → + ∞ since e α ji uniformly converges to e α i as j → + ∞ and thefunction E i is continuous. The second term vanishes too as j → + ∞ since p E i ( e α i ) φ belongsto L (0 , L ) by the boundedness of E i .We conclude by noticing that, exploiting again the compactness of the embedding H (0 , L ) ⊂⊂ C (0 , L ), the set on which we minimise is sequentially closed with respect to the consideredtopology. (cid:3) To pass from discrete to continuous evolutions we now introduce the (right-continuous) piecewiseconstant interpolation ( u n , α n ) of the discrete displacement and damage, and the piecewiseconstant interpolation δ nh of the discrete historical variable, namely: ( u n ( t ) := u k , α n ( t ) := α k , δ nh ( t ) := δ kh , if t ∈ [ t nk , t nk +1 ) , u n ( T ) := u n , α n ( T ) := α n , δ nh ( T ) := δ nh . (2.4a)Of course, in the following, by the expression δ n we mean the piecewise constant slip, namely δ n ( t, x ) = | u n ( t, x ) − u n ( t, x ) | . (2.4b)Analogously, we consider a piecewise constant version ¯ u n of the prescribed displacement: ( ¯ u n ( t ) := ¯ u ( t nk ) , if t ∈ [ t nk , t nk +1 ) , ¯ u n ( T ) := ¯ u ( T ) . (2.4c)We also adopt the following notation: τ n ( t ) := max { t nk | t nk ≤ t } . (2.4d)Next proposition provides useful uniform bounds on the just introduced piecewise constantinterpolations. It is the analogue of Proposition 1.6 in this discrete setting. Proposition 2.2.
Assume E i satisfies (1.3) , w i satisfies (1.7) and ϕ satisfies ( ϕ ϕ C independent of n such that: max t ∈ [0 ,T ] k u n ( t ) k [ H (0 ,L )] ≤ C √ ε , max t ∈ [0 ,T ] k α n ( t ) k [ H (0 ,L )] ≤ C, (2.5a)max t ∈ [0 ,T ] sup x,y ∈ [0 ,L ] , x = y | δ nh ( t, x ) − δ nh ( t, y ) | p | x − y | ! ≤ C √ ε , (2.5b) where ε > has been introduced in (1.4) . AMAGE IN LAMINATES WITH COHESIVE INTERFACE 13
Proof.
The result follows by using exactly the same argument of Proposition 1.6. We only noticethat here we need to choose as competitors for ( u k , α k ) in (2.2a) the functions e u i ( x ) = ¯ u ( t nk ) L x, e α i ≡ , for i = 1 , , and then we argue in the same way. (cid:3) Since the piecewise constant interpolations are built starting from the minimisation algorithm(2.2a), they automatically fulfil the following inequality, which is related to the energy balance(EB):
Lemma 2.3 ( Discrete Energy Inequality ) . Assume E i satisfies (1.3) , w i satisfies (1.7) and ϕ satisfies ( ϕ ϕ R n suchthat for every t ∈ [0 , T ] and for every n ∈ N the following inequality holds true: E [ u n ( t ) , α n ( t )] + D [ α n ( t )] + K [ δ n ( t ) , δ nh ( t )] ≤ E [ u , α ] + D [ α ] + K [ δ , δ ] + Z t W n ( τ ) dτ + R n , where W n ( τ ) := ˙¯ u ( τ ) L X i =1 Z L E i ( α ni ( τ, x ))( u ni ) ′ ( τ, x ) d x .Proof. We fix n ∈ N and k ∈ { , · · · , n } ; for j = 1 , · · · , k we then choose as competitors for( u j , α j ) in (2.2a) the functions e u , e α , with components: e u i ( x ) = u j − i ( x ) + (¯ u ( t nj ) − ¯ u ( t nj − )) x/L, and e α i = α j − i , for i = 1 , . We thus obtain: E [ u j , α j ] + D [ α j ] + K [ δ j , δ jh ] ≤ E [ u j − + v j − , α j − ] + D [ α j − ] + K [ δ j − , δ j − h ] , where we denoted by v j − ( x ) the vector in R with both components equal to (¯ u ( t nj ) − ¯ u ( t nj − )) xL .From the above inequality we now get: E [ u j , α j ] + D [ α j ] + K [ δ j , δ jh ] − E [ u j − , α j − ] − D [ α j − ] − K [ δ j − , δ j − h ] ≤E [ u j − + v j − , α j − ] − E [ u j − , α j − ]= Z t nj t nj − ˙¯ u ( τ ) L X i =1 Z L E i ( α j − i ( x )) (cid:18) ( u j − i ) ′ ( x ) + ¯ u ( τ ) − ¯ u ( t nj − ) L (cid:19) d x d τ. Summing the obtained inequality from j = 1 to j = k we hence deduce: E [ u k , α k ] + D [ α k ] + K [ δ k , δ kh ] − E [ u , α ] − D [ α ] − K [ δ , δ ] ≤ k X j =1 Z t nj t nj − W n ( τ ) d τ + Z t nj t nj − ˙¯ u ( τ ) L ¯ u ( τ ) − ¯ u n ( τ ) L X i =1 Z L E i ( α ni ( τ, x )) d x d τ ! = Z t nk W n ( τ ) d τ + Z t nk ˙¯ u ( τ ) L ¯ u ( τ ) − ¯ u n ( τ ) L X i =1 Z L E i ( α ni ( τ, x )) d x d τ. Recalling the definition of the interpolations u n , α n and τ n , see (2.4), by the arbitrariness of k we finally obtain for every t ∈ [0 , T ]: E [ u n ( t ) , α n ( t )] + D [ α n ( t )] + K [ δ n ( t ) , δ nh ( t )] ≤ E [ u , α ] + D [ α ] + K [ δ , δ ] + Z t W n ( τ ) dτ + Z τ n ( t )0 ˙¯ u ( τ ) L ¯ u ( τ ) − ¯ u n ( τ ) L X i =1 Z L E i ( α ni ( τ, x )) d x d τ − Z tτ n ( t ) W n ( τ ) d τ. We thus conclude by defining: R n := Z T | ˙¯ u ( τ ) | L | ¯ u ( τ ) − ¯ u n ( τ ) | L X i =1 Z L E i ( α ni ( τ, x )) d x d τ + sup t ∈ [0 ,T ] Z tτ n ( t ) | W n ( τ ) | d τ. (2.6)Indeed we now show that lim n → + ∞ R n = 0. First of all by the very definition of W n and exploiting(2.5a) it is easy to see that | W n ( τ ) | ≤ C | ˙¯ u ( τ ) | , with C > n ; hence by theabsolute continuity of the integral the second term in (2.6) vanishes as n → + ∞ (we recall thatby assumption the sequence of partitions satisfies (2.1)). Then we notice that the first term isbounded by C k ˙¯ u k L (0 ,T ) sup t ∈ [0 ,T ] | ¯ u ( t ) − ¯ u n ( t ) | , which vanishes since ¯ u is absolutely continuous and the sequence of partitions satisfies (2.1). (cid:3) Extraction of convergent subsequences.
By the uniform bounds obtained in Proposi-tion 2.2 we are able to deduce the existence of convergent subsequences of the piecewise constantinterpolations u n , α n and δ nh . We first need the following Helly–type compactness result: Lemma 2.4 ( Helly ) . Let { f n } n ∈ N be a sequence of non-decreasing functions from [0 , T ] to C ([0 , L ]) , meaning that for every ≤ s ≤ t ≤ T it holds f n ( s, x ) ≤ f n ( t, x ) for all x ∈ [0 , L ] ,such that: • the families { f n (0) } n ∈ N and { f n ( T ) } n ∈ N are equibounded; • the family { f n ( t ) } n ∈ N is equicontinuous uniformly with respect to t ∈ [0 , T ] .Then there exist a subsequence (not relabelled) and a function f : [0 , T ] → C ([0 , L ]) such that f n ( t ) converges uniformly to f ( t ) as n → + ∞ for every t ∈ [0 , T ] , and f is non-decreasing intime, in the above sense.Moreover for every t ∈ [0 , T ] the right and left limits f ± ( t ) , which are well defined pointwiseby monotonicity, actually belong to C ([0 , L ]) and it holds f ± ( t ) = lim h → ± f ( t + h ) , uniformly in [0 , L ] . (2.7) Proof.
We only sketch the proof, being very similar to the one of Lemma 4.6 in [18].We consider a countable and dense set D ⊆ [0 , T ] containing 0 and T and by using Ascoli–Arzel´a theorem and a diagonal argument we can extract a subsequence (not relabelled) and afunction f from D to C ([0 , L ]) such that f n ( t ) converges uniformly to f ( t ) for every t ∈ D .Since each f n is non-decreasing, trivially f is non-decreasing on D .For every t ∈ [0 , T ] we now define f + ( t ) := inf s ≥ t, s ∈ D f ( s ) , f − ( t ) := sup s ≤ t, s ∈ D f ( s ) , (2.8)and we easily observe that(i) f − ( t ) = f ( t ) = f + ( t ) for every t ∈ D ; AMAGE IN LAMINATES WITH COHESIVE INTERFACE 15 (ii) f − ( t ) ≤ f + ( t ) for every t ∈ [0 , T ];(iii) if 0 ≤ s < t ≤ T , then f + ( s ) ≤ f − ( t ).Since the family { f n ( t ) } n ∈ N is equicontinuous uniformly with respect to time we obtain that thelimit family { f ( t ) } t ∈ D is equicontinuous. This actually ensures that (2.8) can be improved inthe following way: f + ( t ) = lim s ց t, s ∈ D f ( s ) , f − ( t ) = lim s ր t, s ∈ D f ( s ) , uniformly in [0 , L ] . (2.9)In particular for every t ∈ [0 , T ] the functions f + ( t ) and f − ( t ) are continuous in [0 , L ].We now introduce the set E := { t ∈ [0 , T ] | f + ( t ) = f − ( t ) } and for every t ∈ E we define f ( t ) := f + ( t ) = f − ( t ). Of course, by (i), the set D is contained in E and the definition of f agreeswith the one we already had on D . We now prove that for every t ∈ E we have f n ( t ) → f ( t )uniformly in [0 , L ]. We already know it is true for t ∈ D , so we assume t ∈ E \ D . We fix twopoints s ′ < t < t ′ such that s ′ , t ′ ∈ D and since the original sequence was non-decreasing in timewe easily get: k f n ( t ) − f ( t ) k C ([0 ,L ]) ≤ max (cid:8) k f n ( t ′ ) − f ( t ) k C ([0 ,L ]) , k f n ( s ′ ) − f ( t ) k C ([0 ,L ]) (cid:9) ≤ max (cid:8) k f n ( t ′ ) − f ( t ′ ) k C + k f ( t ′ ) − f ( t ) k C , k f n ( s ′ ) − f ( s ′ ) k C + k f ( s ′ ) − f ( t ) k C (cid:9) . Since s ′ and t ′ belong to D we infer:lim sup n → + ∞ k f n ( t ) − f ( t ) k C ([0 ,L ]) ≤ max (cid:8) k f ( t ′ ) − f ( t ) k C ([0 ,L ]) , k f ( s ′ ) − f ( t ) k C ([0 ,L ]) (cid:9) . Thanks to (2.9) and since t is in E , letting t ′ ց t and s ′ ր t we finally conclude that f n ( t )converges uniformly to f ( t ) for every t ∈ E .Let us now show that the set E c = [0 , T ] \ E is countable. First of all it is easy to see that E c coincides with S k ∈ N A k where for every k ∈ N we define A k = (cid:26) t ∈ [0 , T ] | Z L (cid:16) f + ( t, x ) − f − ( t, x ) (cid:17) d x ≥ k (cid:27) . We conclude if we prove that each A k is finite. So we fix t < t < · · · < t r ∈ A k and thanks to(iii) we estimate: rk ≤ r X j =1 Z L (cid:16) f + ( t j , x ) − f − ( t j , x ) (cid:17) d x ≤ Z L (cid:16) f + ( t r , x ) − f − ( t , x ) (cid:17) d x ≤ k f ( T ) − f (0) k L (0 ,L ) , thus r is bounded from above and thus A k is finite.So by using again Ascoli–Arzel´a theorem and a diagonal argument we can extract a furthersubsequence and a function f from E c to C ([0 , L ]) such that f n ( t ) converges uniformly to f ( t )for every t ∈ E c . Since in E we already obtained the result, we conclude by noticing that sucha f is non-decreasing in the whole [0 , T ] recalling that the original sequence was non-decreasingin time. Indeed (2.7) easily follows by (2.9). (cid:3) Proposition 2.5.
Assume E i satisfies (1.3) , w i satisfies (1.7) and ϕ satisfies ( ϕ ϕ u n , α n , δ nh introduced in (2.4a) . Then there exist a subsequence n j and for every t ∈ [0 , T ] a further subsequence n j ( t ) (depending on time) such that: (a) u n j ( t ) ( t ) ⇀ u ( t ) in [ H (0 , L )] as n j ( t ) → + ∞ ; (b) α n j ( t ) ( t ) ⇀ α ( t ) in [ H (0 , L )] as n j ( t ) → + ∞ ; (c) δ n j h ( t ) → γ ( t ) uniformly in [0 , L ] as n j → + ∞ .Moreover the limit functions satisfy: (1) u ( t ) ∈ [ H , ¯ u ( t ) (0 , L )] , α ( t ) ∈ [ H , (0 , L )] and γ ( t ) ∈ C / ([0 , L ]) for every t ∈ [0 , T ] ; (2) u (0) = u , α (0) = α and γ (0) = δ ; (3) α i and γ are non-decreasing in time; (4) γ ( t ) ≥ δ h ( t ) = sup τ ∈ [0 ,t ] | u ( τ ) − u ( τ ) | for every t ∈ [0 , T ] . Remark 2.6.
For the sake of clarity, in order to avoid too heavy notations, from now on weprefer not to stress the occurence of the subsequence via the subscript j ; namely we still write n instead of n j and n ( t ) instead of n j ( t ). Proof of Proposition 2.5.
The validity of (c), the 1 / γ ( t ) andthe fact that γ ( t,
0) = γ ( t, L ) = 0 are a byproduct of (2.5b) and Lemma 2.4; (a) and (b) insteadfollow by (2.5a) together with the weak sequential compactness of the unit ball in H (0 , L ).Since H (0 , L ) ⊂⊂ C ([0 , L ]) we also deduce (1), (2) and (3).We only need to prove (4). So let us assume by contradiction that there exists a pair ( t, x ) ∈ [0 , T ] × [0 , L ] such that: δ h ( t, x ) > γ ( t, x ) = lim n → + ∞ δ nh ( t, x ) . (2.10)By (2.10) and the definition of δ h , there exists a time τ t ∈ [0 , t ] for which | u ( τ t , x ) − u ( τ t , x ) | >γ ( t, x ); thus we infer: | u ( τ t , x ) − u ( τ t , x ) | > lim n → + ∞ δ nh ( t, x ) ≥ lim n → + ∞ δ nh ( τ t , x ) ≥ lim sup n → + ∞ | u n ( τ t , x ) − u n ( τ t , x ) |≥ lim n ( τ t ) → + ∞ | u n ( τ t )1 ( τ t , x ) − u n ( τ t )2 ( τ t , x ) | = | u ( τ t , x ) − u ( τ t , x ) | , which is absurd. (cid:3) Remark 2.7.
We want to point out that also the subsequence of the damage variable in (b)could be chosen independent of time, since each term of the sequence is non-decreasing in time.This follows by means of a suitable version of Helly’s selection theorem (see for instance TheoremB.5.13 in the Appendix B of [31]), and arguing as in [30], Proposition 3.2. However, for the sakeof simplicity we preferred to consider a time–dependent subsequence; indeed this will be enoughfor our purposes.The fact that the subsequence in (c) does not depend on time was instead crucial for thevalidity of (4), as the reader can check from the proof.2.3.
Existence of generalised energetic evolutions.
The aim of this subsection is provingthat the limit functions obtained in Proposition 2.5 are actually a generalised energetic evolution.We only need to show that global stability (GS’) and energy balance (EB’) hold true, being theother conditions automatically satisfied due to Lemma 2.5. This first proposition deals with theglobal stability:
Proposition 2.8.
Assume E i satisfies (1.3) , w i satisfies (1.7) , and ϕ satisfies ( ϕ ϕ u , α fulfil the stability condition (1.15) . Then the limit functions u , α , γ obtained in Proposition 2.5 satisfy (GS’).Proof. If t = 0 there is nothing to prove, so we consider t ∈ (0 , T ] and we first notice that by(4) in Proposition 2.5 we know γ ( t ) ≥ δ ( t ). Then we fix e u ∈ [ H , ¯ u ( t ) (0 , L )] and e α ∈ [ H (0 , L )] such that α i ( t ) ≤ e α i ≤ i = 1 , AMAGE IN LAMINATES WITH COHESIVE INTERFACE 17
By weak lower semicontinuity of the energy we get: E [ u ( t ) , α ( t )] + D [ α ( t )] + K [ δ ( t ) , γ ( t )] ≤ lim inf n ( t ) → + ∞ (cid:16) E [ u n ( t ) ( t ) , α n ( t ) ( t )] + D [ α n ( t ) ( t )] + K [ δ n ( t ) ( t ) , δ n ( t ) h ( t )] (cid:17) =: ( ⋆ ) . Now we can use the minimality properties of the discrete functions, considering as competitorsthe functions b u n ( t ) and b α n ( t ) whose components are b u n ( t ) i ( x ) := e u i ( x ) − (¯ u ( t ) − ¯ u ( τ n ( t ) ( t )) xL , b α n ( t ) i := min (cid:26)e α i + max [0 ,L ] (cid:12)(cid:12)(cid:12) α n ( t ) i ( t ) − α i ( t ) (cid:12)(cid:12)(cid:12) , (cid:27) . It is easy to see that they are admissible; moreover, since τ n ( t ) ( t ) → t and α n ( t ) i ( t ) → α i ( t )uniformly as n ( t ) → + ∞ , they strongly converge to e u and e α in [ H (0 , L )] . See also [30],Lemma 3.5.By minimality, going back to the previous estimate, we obtain:( ⋆ ) ≤ lim inf n ( t ) → + ∞ (cid:16) E [ b u n ( t ) , b α n ( t ) ] + D [ b α n ( t ) ] + K [ e δ, δ n ( t ) h ( t ) ∨ e δ ] (cid:17) = E [ e u , e α ] + D [ e α ] + K [ e δ, γ ( t ) ∨ e δ ] , where in the last equality we exploited the strong convergence of b u n ( t ) and b α n ( t ) towards e u and e α , plus assumption ( ϕ (cid:3) To show the validity of (EB’) we prove separately the two inequalities. The first one followsfrom the discrete energy inequality presented in Lemma 2.3:
Proposition 2.9 ( Upper Energy Estimate ) . Assume E i satisfies (1.3) , w i satisfies (1.7) ,and ϕ satisfies ( ϕ ϕ t ∈ [0 , T ] the limit functions u , α , γ obtained inProposition 2.5 satisfy the following inequality: E [ u ( t ) , α ( t )] + D [ α ( t )] + K [ δ ( t ) , γ ( t )] ≤ E [ u , α ] + D [ α ] + K [ δ , δ ] + W [ u , α ]( t ) . Proof.
We fix t ∈ [0 , T ] and by lower semicontinuity of the energy and Lemma 2.3 we deduce: E [ u ( t ) , α ( t )] + D [ α ( t )] + K [ δ ( t ) , γ ( t )] ≤ lim inf n ( t ) → + ∞ (cid:16) E [ u n ( t ) ( t ) , α n ( t ) ( t )] + D [ α n ( t ) ( t )] + K [ δ n ( t ) ( t ) , δ n ( t ) h ( t )] (cid:17) ≤E [ u , α ] + D [ α ] + K [ δ , δ ] + lim inf n ( t ) → + ∞ Z t W n ( t ) ( τ ) d τ. By means of the reverse Fatou’s Lemma we thus get:lim inf n ( t ) → + ∞ Z t W n ( t ) ( τ ) d τ ≤ Z t lim sup n ( t ) → + ∞ W n ( t ) ( τ ) d τ =: ( ∗ ) . In order to deal with ( ∗ ) we argue as follows (see also [16], Section 4). We consider the sub-sequence n obtained in Proposition 2.5 (see also Remark 2.6) and for every τ ∈ [0 , T ] we firstset W ( τ ) := lim sup n → + ∞ W n ( τ ) , (2.11)which belongs to L (0 , T ) since we recall that | W n ( τ ) | ≤ C | ˙¯ u ( τ ) | . Without loss of generality wecan assume that the time–dependent subsequences obtained in Proposition 2.5 also satisfy W ( τ ) = lim n ( τ ) → + ∞ W n ( τ ) ( τ ) , for every τ ∈ [0 , T ] . Thus exploiting (a) and (b) in Proposition 2.5 for a.e. τ ∈ [0 , T ] we obtain: W ( τ ) = lim n ( τ ) → + ∞ W n ( τ ) ( τ ) = lim n ( τ ) → + ∞ ˙¯ u ( τ ) L X i =1 Z L E i ( α n ( τ ) i ( τ, x ))( u n ( τ ) i ) ′ ( τ, x ) d x = ˙¯ u ( τ ) L X i =1 Z L E i ( α i ( τ, x ))( u i ) ′ ( τ, x ) d x. (2.12)Combining (2.11) and (2.12) we finally get( ∗ ) ≤ Z t W ( τ ) d τ = W [ u , α ]( t ) , and we conclude. (cid:3) The opposite inequality is instead a byproduct of the global stability condition we proved inProposition 2.8:
Proposition 2.10 ( Lower Energy Estimate ) . Assume E i satisfies (1.3) , w i satisfies (1.7) ,and ϕ satisfies ( ϕ ϕ u , α fulfil the stability condition (1.15) .Then for every t ∈ [0 , T ] the limit functions u , α , γ obtained in Proposition 2.5 satisfy: E [ u ( t ) , α ( t )] + D [ α ( t )] + K [ δ ( t ) , γ ( t )] ≥ E [ u , α ] + D [ α ] + K [ δ , δ ] + W [ u , α ]( t ) . Proof. If t = 0 the inequality is trivial, so we fix t ∈ (0 , T ] and we consider a sequence ofpartitions of [0 , t ] of the form 0 = t n < t n < · · · < t nn = t satisfying:(i) lim n → + ∞ max k =1 ,...,n (cid:12)(cid:12) t nk − t nk − (cid:12)(cid:12) = 0;(ii) lim n → + ∞ n X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( t nk − t nk − ) ˙¯ u ( t nk ) − Z t nk t nk − ˙¯ u ( τ ) d τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0;(iii) lim n → + ∞ n X k =1 ( t nk − t nk − ) W ( t nk ) = W [ u , α ]( t ),where W is the function introduced in (2.11) and (2.12). The existence of such a sequence ofpartitions follows from Lemma 4.5 in [20], since both ˙¯ u and W belong to L (0 , T ). In particular,by (i) and the absolute continuity of the integral, we can assume without loss of generality that:(iv) for every n ∈ N it holds Z t nk t nk − | ˙¯ u ( τ ) | d τ ≤ n for every k = 1 , . . . , n .For a given partition we fix k = 1 , . . . , n and, recalling Proposition 2.8, we choose as competitorsfor u ( t nk − ), α ( t nk − ) and γ ( t nk − ) in (GS’) the functions e u , e α , with components: e u i ( x ) = u i ( t nk , x ) + (¯ u ( t nk − ) − ¯ u ( t nk )) xL , e α i = α i ( t nk ) , for i = 1 , . Recalling that γ ( t nk − ) ∨ δ ( t nk ) ≤ γ ( t nk ), and hence K [ δ ( t nk ) , γ ( t nk − ) ∨ δ ( t nk )] ≤ K [ δ ( t nk ) , γ ( t nk )] by( ϕ E [ u ( t nk − ) , α ( t nk − )]+ D [ α ( t nk − )]+ K [ δ ( t nk − ) , γ ( t nk − )] −E [ u ( t nk ) , α ( t nk )] −D [ α ( t nk )] −K [ δ ( t nk ) , γ ( t nk )] ≤ − Z t nk t nk − ˙¯ u ( τ ) L X i =1 Z L E i ( α i ( t nk , x )) (cid:18) u ′ i ( t nk , x ) + ¯ u ( τ ) − ¯ u ( t nk ) L (cid:19) d x d τ. AMAGE IN LAMINATES WITH COHESIVE INTERFACE 19
Summing the above inequality from k = 1 to k = n we obtain: E [ u ( t ) , α ( t )] + D [ α ( t )] + K [ δ ( t ) , γ ( t )] − E [ u , α ] − D [ α ] − K [ δ , δ ] ≥ n X k =1 Z t nk t nk − ˙¯ u ( τ ) L Z L X i =1 E i ( α i ( t nk , x )) (cid:18) u ′ i ( t nk , x ) + ¯ u ( τ ) − ¯ u ( t nk ) L (cid:19) d x d τ =: J n Now we easily notice that J n can be written as: J n = n X k =1 ( t nk − t nk − ) W ( t nk )+ n X k =1 Z t nk t nk − ˙¯ u ( τ ) − ˙¯ u ( t nk ) L d τ Z L X i =1 E i ( α i ( t nk , x )) u ′ i ( t nk , x ) d x + n X k =1 Z t nk t nk − ˙¯ u ( τ ) L ¯ u ( τ ) − ¯ u ( t nk ) L d τ Z L X i =1 E i ( α i ( t nk , x )) d x =: J n + J n + J n . By (iii) we know that lim n → + ∞ J n = W [ u , α ]( t ), so we conclude if we prove that lim n → + ∞ J n =lim n → + ∞ J n = 0. With this aim we estimate: | J n | ≤ C n X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t nk t nk − ( ˙¯ u ( τ ) − ˙¯ u ( t nk )) d τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i =1 k u i ( t nk ) k H (0 ,L ) ! ≤ C n X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( t nk − t nk − ) ˙¯ u ( t nk ) − Z t nk t nk − ˙¯ u ( τ ) d τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , which goes to 0 by (ii). As regards J n , by using (iv) we get: | J n | ≤ C n X k =1 Z t nk t nk − | ˙¯ u ( τ ) || ¯ u ( τ ) − ¯ u ( t nk ) | d τ = C n X k =1 Z t nk t nk − | ˙¯ u ( τ ) | (cid:12)(cid:12)(cid:12)(cid:12)Z t nk τ ˙¯ u ( s ) d s (cid:12)(cid:12)(cid:12)(cid:12) d τ ≤ C n X k =1 Z t nk t nk − | ˙¯ u ( τ ) | d τ ! ≤ Cn n X k =1 Z t nk t nk − | ˙¯ u ( τ ) | d τ = Cn k ˙¯ u k L (0 ,t ) , and the proof is complete. (cid:3) Putting together what we obtained in this section we infer our first result of existence ofgeneralised energetic evolutions:
Theorem 2.11 ( Existence of Generalised Energetic Evolutions ) . Let the prescribed dis-placement ¯ u belong to AC ([0 , T ]) and the initial data u , α fulfil (1.13) together with the stabilitycondition (1.15) . Assume E i satisfies (1.3) , w i satisfies (1.7) , and ϕ satisfies ( ϕ ϕ u , α and γ obtained in Proposition 2.5 is a generalisedenergetic evolution. We conclude this section by showing that, assuming in addition the specific condition ( ϕ ϕ
4) there exist two functions ϕ , ϕ : [0 , + ∞ ) → [0 , + ∞ ) such that ϕ is lower semicontinu-ous, ϕ is bounded, non-decreasing and concave, and ϕ ( y, z ) = ϕ ( y ) + ϕ ( z ),the functions u and α obtained in Proposition 2.5 are actually an energetic evolution. Theapproach is exactly the same of [16]. We first notice that ( ϕ
4) implies ( ϕ ϕ
2) and ( ϕ Theorem 2.12.
Let the prescribed displacement ¯ u belong to AC ([0 , T ]) and the initial data u , α fulfil (1.13) together with the stability condition (1.15) . Assume E i satisfies (1.3) , w i satisfies (1.7) , and ϕ satisfies ( ϕ ( u , α ) obtained in Proposition 2.5 is anenergetic evolution.If in addition ϕ is strictly increasing, then the function γ obtained in Proposition 2.5 coincideswith the historical variable δ h .Proof. Thanks to Theorem 2.11 we only need to show the validity of (GS) and (EB) in Defi-nition 1.5. We first focus on (GS); so we fix t ∈ [0 , T ] and two functions e u ∈ [ H , ¯ u ( t ) (0 , L )] , e α ∈ [ H (0 , L )] such that α i ( t ) ≤ e α i ≤ , L ] for i = 1 ,
2. Since the triplet ( u , α , γ ) satisfies(GS’) we know that: E [ u ( t ) , α ( t )] + D [ α ( t )] ≤ E [ e u , e α ] + D [ e α ] + K [ e δ, γ ( t ) ∨ e δ ] − K [ δ ( t ) , γ ( t )] , thus we conclude if we prove K [ e δ, γ ( t ) ∨ e δ ] − K [ δ ( t ) , γ ( t )] ≤ K [ e δ, δ h ( t ) ∨ e δ ] − K [ δ ( t ) , δ h ( t )] . (2.13)With this aim, exploiting ( ϕ ϕ , and recallingthat γ ( t ) ≥ δ h ( t ), we get: ϕ ( γ ( t ) ∨ e δ ) = ϕ ( γ ( t ) + [ e δ − γ ( t )] + ) ≤ ϕ ( γ ( t ) + [ e δ − δ h ( t )] + ) ≤ ϕ ( γ ( t )) + ϕ ( δ h ( t ) + [ e δ − δ h ( t )] + ) − ϕ ( δ h ( t ))= ϕ ( γ ( t )) + ϕ ( δ h ( t ) ∨ e δ ) − ϕ ( δ h ( t )) . The above inequality implies: K [ e δ, γ ( t ) ∨ e δ ] − K [ e δ, δ h ( t ) ∨ e δ ] = Z L (cid:16) ϕ ( γ ( t, x ) ∨ e δ ( x )) − ϕ ( δ h ( t, x ) ∨ e δ ( x )) (cid:17) d x ≤ Z L (cid:16) ϕ ( γ ( t, x )) − ϕ ( δ h ( t, x )) (cid:17) d x = K [ δ ( t ) , γ ( t )] − K [ δ ( t ) , δ h ( t )] , which is equivalent to (2.13).We now prove (EB). Since the triplet ( u , α , γ ) satisfies (EB’), it is enough to prove K [ δ ( t ) , γ ( t )] = K [ δ ( t ) , δ h ( t )] , for every t ∈ [0 , T ] . (2.14)Since γ ( t ) ≥ δ h ( t ) we easily deduce K [ δ ( t ) , γ ( t )] ≥ K [ δ ( t ) , δ h ( t )]. To get the other inequality wefirst observe that arguing exactly as in the proof of Proposition 2.10, but replacing γ with δ h (indeed we have just proved (GS)) we get: E [ u ( t ) , α ( t )] + D [ α ( t )] + K [ δ ( t ) , δ h ( t )] ≥ E [ u , α ] + D [ α ] + K [ δ , δ ] + W [ u , α ]( t ) . Combining the above inequality with (EB’) we finally obtain K [ δ ( t ) , δ h ( t )] ≥ K [ δ ( t ) , γ ( t )] , hence (2.14) holds true.If in addition ϕ is strictly increasing, then (2.14) implies γ ( t ) = δ h ( t ) since both functionsare continuous in [0 , L ]. Thus we conclude. (cid:3) AMAGE IN LAMINATES WITH COHESIVE INTERFACE 21 PDE Form of Energetic Evolutions
In this section we compute the Euler–Lagrange equations coming from the global stabilitycondition (GS’). More precisely we prove that any generalised energetic evolution ( u , α , γ ) mustsatisfy, in a suitable weak formulation, the following system of equilibrium equations governingthe stresses σ i (see Proposition 3.3): ( − σ ( t ) ′ + ∂ y ϕ ( δ ( t ) , γ ( t )) sgn( u ( t ) − u ( t )) = 0 , in [0 , L ] , − σ ( t ) ′ − ∂ y ϕ ( δ ( t ) , γ ( t )) sgn( u ( t ) − u ( t )) = 0 , in [0 , L ] , for every t ∈ [0 , T ] , (3.1a)where sgn( · ) denotes the signum function, together with a Karush–Kuhn–Tucker condition de-scribing the evolution of the damage variables (if regular in time, see Propositions 3.4 and 3.5): ˙ α i ( t ) ≥ , in [0 , L ] , − α i ( t ) ′′ + E ′ i ( α i ( t ))( u i ( t ) ′ ) + w ′ i ( α i ( t )) ≥ , in [0 , L ] , h − α i ( t ) ′′ + E ′ i ( α i ( t ))( u i ( t ) ′ ) + w ′ i ( α i ( t )) i ˙ α i ( t ) = 0 , in [0 , L ] , for a.e. t ∈ [0 , T ] . (3.1b)The results of this section will be crucial for the achievement of our goal, namely the equivalencebetween the fictitious historical variable γ and the concrete one δ h , under meaningful assump-tions on ϕ . The argument based on temporal regularity of generalised energetic evolutions willbe developed in Section 4.We recall that, given the loading–unloading density ϕ : T → [0 , + ∞ ), we denote by ψ itsrestriction to the diagonal, namely ψ ( z ) = ϕ ( z, z ), for z ∈ [0 , + ∞ ). Throughout the section themain assumptions on ϕ (and ψ ) are:the function ψ belongs to C ([0 , + ∞ )); (3.2a)for every z ∈ (0 , + ∞ ) the map ϕ ( · , z ) belongs to C ([0 , z ]); (3.2b)for every z ∈ (0 , + ∞ ) there hold ∂ y ϕ ( z, z ) = ψ ′ ( z ) and ∂ y ϕ (0 , z ) = 0; (3.2c)the partial derivative ∂ y ϕ belongs to C ( T \ (0 , T . (3.2d)We notice that the above conditions are slightly more general than properties ( ϕ ϕ
8) listedin Section 1, since we do not require any convexity assumption (which will be instead employedin Section 4).We start the analysis with a simple but useful lemma.
Lemma 3.1.
Let f, g ∈ R such that f ≥ | g | and assume the function ϕ : T → [0 , + ∞ ) satisfies:the function z ϕ ( z, z ) =: ψ ( z ) is differentiable in [0 , + ∞ ); (3.3a) for every z ∈ (0 , + ∞ ) the map ϕ ( · , z ) is differentiable in [0 , z ]; (3.3b) for every z ∈ (0 , + ∞ ) there hold ∂ y ϕ ( z, z ) = ψ ′ ( z ) . (3.3c) Then for every v ∈ R one has: lim h → + ϕ ( | g + hv | , f ∨ | g + hv | ) − ϕ ( | g | , f ) h = ∂ y ϕ ( | g | , f ) sgn( g ) v, if f > | g | > ,ψ ′ ( | g | ) sgn( g ) v, if f = | g | > ,∂ y ϕ (0 , f ) | v | , if f > | g | = 0 ,ψ ′ (0) | v | , if f = | g | = 0 . Proof.
We denote by I the limit we want to compute and we distinguish among all the differentcases. We first assume that f > | g | , so we get: • if g = 0, then I = lim h → + ϕ ( h | v | , f ) − ϕ (0 , f ) h = ∂ y ϕ (0 , f ) | v | ; • if g >
0, then I = lim h → + ϕ ( g + hv, f ) − ϕ ( g, f ) h = ∂ y ϕ ( g, f ) v = ∂ y ϕ ( | g | , f ) sgn( g ) v ; • if g <
0, then I = lim h → + ϕ ( | g | − hv, f ) − ϕ ( | g | , f ) h = − ∂ y ϕ ( | g | , f ) v = ∂ y ϕ ( | g | , f ) sgn( g ) v .If instead f = | g | we have: • if g = 0, then I = lim h → + ϕ ( h | v | , h | v | ) − ϕ (0 , h = lim h → + ψ ( h | v | ) − ψ (0) h = ψ ′ (0) | v | ; • if g > v ≥
0, then I = lim h → + ϕ ( g + hv, g + hv ) − ϕ ( g, g ) h = ψ ′ ( g ) v = ψ ′ ( | g | ) sgn( g ) v ; • if g > v <
0, then I = lim h → + ϕ ( g + hv, g ) − ϕ ( g, g ) h = ∂ y ϕ ( g, g ) v = ψ ′ ( g ) v = ψ ′ ( | g | ) sgn( g ) v ; • if g < v ≥
0, then I = lim h → + ϕ ( | g | − hv, | g | ) − ϕ ( | g | , | g | ) h = − ∂ y ϕ ( | g | , | g | ) v = ψ ′ ( | g | ) sgn( g ) v ; • if g < v <
0, then I = lim h → + ψ ( | g | − hv ) − ψ ( | g | ) h = − ψ ′ ( | g | ) v = ψ ′ ( | g | ) sgn( g ) v .So we conclude. (cid:3) As an immediate corollary we deduce:
Corollary 3.2.
Let f, g be two measurable functions such that f ∈ L ∞ (0 , L ) and f ≥ | g | a.e. in [0 , L ] , and assume ϕ satisfies (3.2a) , (3.2b) and (3.2c) . Then for every v ∈ L ∞ (0 , L ) it holds: lim h → + K [ | g + hv | , f ∨ | g + hv | ] −K [ | g | , f ] h = Z {| g | > } ∂ y ϕ ( | g ( x ) | , f ( x )) sgn( g ( x )) v ( x ) d x + ψ ′ (0) Z { f =0 } | v ( x ) | d x. Proof.
We notice that, by the explicit expression of K given by (1.8), the limit we want tocompute can be written aslim h → + Z L ϕ ( | g ( x ) + hv ( x ) | , f ( x ) ∨ | g ( x ) + hv ( x ) | ) − ϕ ( | g ( x ) | , f ( x )) h d x. Assumptions (3.2a) and (3.2b) allow us to pass to the limit inside the integral, thus we concludeby means of Lemma 3.1 and exploiting (3.2c). (cid:3)
We are now in a position to state and prove the first result of this section, namely a weakform of the Euler–Lagrange equation for the displacement u , or better for the stress σ . Proposition 3.3.
Let E i ∈ C ([0 , and assume ϕ satisfies (3.2a) , (3.2b) and (3.2c) . Let ( u , α , γ ) satisfy (CO’) and (GS’) of Definition 1.8. Then for every t ∈ [0 , T ] and for every v ∈ [ H (0 , L )] it holds: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z L X i =1 σ i ( t ) v ′ i d x + Z { δ ( t ) > } (cid:2) ∂ y ϕ ( δ ( t ) , γ ( t )) sgn( u ( t ) − u ( t )) (cid:3) ( v − v ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ψ ′ (0) Z { γ ( t )=0 } | v − v | d x, (3.4) where the stresses σ i have been introduced in (1.5) .In particular, for every t ∈ [0 , T ] the sum of the stresses X i =1 σ i ( t ) is constant in [0 , L ] . AMAGE IN LAMINATES WITH COHESIVE INTERFACE 23
Proof.
We fix t ∈ [0 , T ] and by choosing e α = α ( t ) in (GS’) we get for every h > v ∈ [ H (0 , L )] : E [ u ( t ) , α ( t )] + K [ δ ( t ) , γ ( t )] ≤ E [ u ( t ) + h v , α ( t )] + K [ | u ( t ) − u ( t ) + h ( v − v ) | , γ ( t ) ∨ | u ( t ) − u ( t ) + h ( v − v ) | ] . Letting h → + we thus deduce0 ≤ lim h → + E [ u ( t ) + h v , α ( t )] − E [ u ( t ) , α ( t )] h + lim h → + K [ | u ( t ) − u ( t ) + h ( v − v ) | , γ ( t ) ∨ | u ( t ) − u ( t ) + h ( v − v ) | ] − K [ δ ( t ) , γ ( t )] h . The first limit is trivially equal to R L P i =1 σ i ( t ) v ′ i d x , while for the second one we employCorollary 3.2 and we finally obtain:0 ≤ Z L X i =1 σ i ( t ) v ′ i d x + Z { δ ( t ) > } (cid:2) ∂ y ϕ ( δ ( t ) , γ ( t )) sgn( u ( t ) − u ( t )) (cid:3) ( v − v ) d x + ψ ′ (0) Z { γ ( t )=0 } | v − v | d x. By following the same argument with − v , we prove (3.4).In particular if v = v =: v we deduce that Z L X i =1 σ i ( t ) ! v ′ d x = 0 , for every v ∈ H (0 , L ) , and so P i =1 σ i ( t ) is constant in [0 , L ]. (cid:3) We want to point out that if ψ ′ (0) were equal to 0 (usually false in a cohesive setting), theninequality (3.4) would actually be equivalent to the system (3.1a). The simplifications broughtby the assumption ψ ′ (0) = 0 can be also found in [3], where it has been used for numericalreasons, and in [35], where it has been exploited to perform an approximation argument.In our work, however, we do not need that additional assumption, indeed inequality (3.4) willbe enough for our purposes.Next proposition deals with the damage variable α : Proposition 3.4.
Assume E i , w i ∈ C ([0 , and let ( u , α , γ ) satisfy (CO’) and (GS’) of Defi-nition 1.8. Then, for every t ∈ [0 , T ] and for every β ∈ [ H (0 , L )] such that β i ≥ for i = 1 , ,it holds: X i =1 Z { α i ( t ) < } E ′ i ( α i ( t ))( u i ( t ) ′ ) β i d x + Z { α i ( t ) < } w ′ i ( α i ( t )) β i d x + Z { α i ( t ) < } α i ( t ) ′ β ′ i d x ! ≥ . Proof.
We fix t ∈ [0 , T ] and by choosing e u = u ( t ) in (GS’) we get E [ u ( t ) , α ( t )] + D [ α ( t )] ≤ E [ u ( t ) , e α ] + D [ e α ] , for every e α ∈ [ H (0 , L )] s.t. α i ( t ) ≤ α i ≤ . (3.5)We now fix β ∈ [ H (0 , L )] such that β i ≥ h > e α h ( t, x ) as the vector in R whose components are ( α i ( t, x ) + hβ i ( x )) ∧
1. By plugging e α h ( t ) in (3.5) as a test function and letting h → + we thus deduce:0 ≤ lim inf h → + E [ u ( t ) , e α h ( t )] − E [ u ( t ) , α ( t )] + D [ e α h ( t )] − D [ α ( t )] h = lim inf h → + X i =1 (cid:16) Z L E i ( e α hi ( t )) − E i ( α i ( t )) h ( u i ( t ) ′ ) d x + Z L w i ( e α hi ( t )) − w i ( α i ( t )) h d x + 12 Z L ( e α hi ( t ) ′ ) − ( α i ( t ) ′ ) h d x (cid:17) = lim inf h → + ( I h + II h + III h ) . (3.6)We study the limits of I h , II h , III h separately. Since E i , w i are in C ([0 , I h and II h . We also notice that given f ∈ C ([0 , a ∈ [0 , b ≥ h → + f (( a + hb ) ∧ − f ( a ) h = ( f ′ ( a ) b, if a ∈ [0 , , , if a = 1 . Thus we deduce thatlim h → + I h = X i =1 Z { α i ( t ) < } E ′ i ( α i ( t ))( u i ( t ) ′ ) β i d x, and lim h → + II h = X i =1 Z { α i ( t ) < } w ′ i ( α i ( t )) β i d x. (3.7)To deal with III h we first observe that e α hi ( t ) ′ = ( α i ( t ) ′ + hβ ′ i , a.e. in { α i ( t ) + hβ i < } , , a.e. in { α i ( t ) + hβ i ≥ } , andso III h = X i =1 Z { α i ( t )+ hβ i < } α i ( t ) ′ β ′ i d x + h Z { α i ( t )+ hβ i < } ( β ′ i ) d x − h Z { α i ( t )+ hβ i ≥ } ( α i ( t ) ′ ) d x ! ≤ X i =1 Z { α i ( t )+ hβ i < } α i ( t ) ′ β ′ i d x + h Z L ( β ′ i ) d x ! . By an easy application of dominated convergence theorem we hence obtainlim sup h → + III h ≤ X i =1 Z { α i ( t ) < } α i ( t ) ′ β ′ i d x, (3.8)and collecting (3.6), (3.7) and (3.8) we conclude. (cid:3) The last result of the section is a byproduct of the energy balance (EB’), assuming a priori thata generalised energetic evolution possesses a certain regularity in time. This kind of regularitywill be however proved in Section 4 under suitable convexity assumptions on the data, thus thisa priori requirement is not restrictive.We refer to the Appendix for the definition and the main properties of absolutely continuousfunctions in Banach spaces, concepts we use in the next proposition.
Proposition 3.5.
Assume E i , w i ∈ C ([0 , and that ϕ satisfies (3.2) and ( ϕ ( u , α , γ ) be a generalised energetic evolution such that: u , α ∈ AC ([0 , T ]; [ H (0 , L )] ) , and γ ∈ C ([0 , T ] , C ([0 , L ])) . AMAGE IN LAMINATES WITH COHESIVE INTERFACE 25
Then for a.e. t ∈ [0 , T ] one has: • Z L E ′ i ( α i ( t ))( u i ( t ) ′ ) ˙ α i ( t ) d x + Z L w ′ i ( α i ( t )) ˙ α i ( t ) d x + Z L α i ( t ) ′ ˙ α i ( t ) ′ d x = 0 , for i = 1 , • lim h → Z L ϕ ( δ ( t ) , γ ( t + h )) − ϕ ( δ ( t ) , γ ( t )) h d x = 0 . (3.9) Proof.
First of all we notice that the temporal regularity we are assuming on u and α ensuresthat the maps t
7→ E [ u ( t ) , α ( t )] and t
7→ D [ α ( t )] are absolutely continuous in [0 , T ]. Moreoverfor almost every time t ∈ [0 , T ] the following expressions for their derivatives can be easilyobtained:dd t E [ u ( t ) , α ( t )] = X i =1 (cid:18) Z L E ′ i ( α i ( t ))( u i ( t ) ′ ) ˙ α i ( t ) d x + Z L E i ( α i ( t )) u i ( t ) ′ ˙ u i ( t ) ′ d x (cid:19) ; (3.10a)dd t D [ α ( t )] = X i =1 (cid:18)Z L w ′ i ( α i ( t )) ˙ α i ( t ) d x + Z L α i ( t ) ′ ˙ α i ( t ) ′ d x (cid:19) . (3.10b)By (EB’), since the work of the prescribed displacement W [ u , α ] is absolutely continuous bydefinition, we now deduce that also the map t
7→ K [ δ ( t ) , γ ( t )] is absolutely continuous in [0 , T ].Moreover we know that δ belongs to AC ([0 , T ]; H (0 , L )), indeed both u and u are absolutelycontinuous with values in H (0 , L ) by assumption. Thus for almost every t ∈ [0 , T ] there existsthe derivative of K [ δ ( t ) , γ ( t )] and we can compute:dd t K [ δ ( t ) , γ ( t )] = lim h → Z L ϕ ( δ ( t + h ) , γ ( t + h )) − ϕ ( δ ( t ) , γ ( t )) h d x = Z L ∂ y ϕ ( δ ( t ) , γ ( t )) ˙ δ ( t ) d x + lim h → Z L ϕ ( δ ( t ) , γ ( t + h )) − ϕ ( δ ( t ) , γ ( t )) h d x, (3.11)where we exploited the continuity assumption of both ∂ y ϕ and γ .Differentiating (EB’), using (3.10) and (3.11), and recalling that the sum of the stresses σ i isconstant in [0 , L ] by Proposition 3.3, we deduce, for almost every t ∈ [0 , T ],0 = X i =1 (cid:18) Z L E ′ i ( α i ( t ))( u i ( t ) ′ ) ˙ α i ( t ) d x + Z L w ′ i ( α i ( t )) ˙ α i ( t ) d x + Z L α i ( t ) ′ ˙ α i ( t ) ′ d x (cid:19) + Z L X i =1 E i ( α i ( t )) u i ( t ) ′ ˙ u i ( t ) ′ d x + Z L ∂ y ϕ ( δ ( t ) , γ ( t )) ˙ δ ( t ) d x − ˙¯ u ( t ) X i =1 σ i ( t, h → Z L ϕ ( δ ( t ) , γ ( t + h )) − ϕ ( δ ( t ) , γ ( t )) h d x. (3.12)The term in the third line of (3.12) is nonnegative by means of ( ϕ
3) and the fact that γ isnon-decreasing (in time). We thus conclude if we show that also the sum of the terms in thesecond line and each of the two terms (for i = 1 ,
2) in the sum in the first line are nonnegative.We first focus on the first line. We notice that for i = 1 ,
2, the function ˙ α i ( t ) ∈ H (0 , L ) isnonnegative and vanishes on the set { α i ( t ) = 1 } ; indeed α i is non-decreasing in time and it isalways less or equal than 1. This means that we can use it as a test function in Proposition 3.4,getting for a.e. t ∈ [0 , T ]:12 Z L E ′ i ( α i ( t ))( u i ( t ) ′ ) ˙ α i ( t ) d x + Z L w ′ i ( α i ( t )) ˙ α i ( t ) d x + Z L α i ( t ) ′ ˙ α i ( t ) ′ d x ≥ . As regards the sum of the terms in the second line in (3.12), we actually prove it is equal to zero.To this aim we make use of Proposition 3.3 choosing as test functions v i ( x ) = ˙ u i ( t, x ) − ˙¯ u ( t ) x/L ∈ H (0 , L ), so that | v − v | = | ˙ u ( t ) − ˙ u ( t ) | . We indeed notice that | v − v | = 0 on the set { γ ( t ) = 0 } : if x belongs to that set, then u ( τ, x ) = u ( τ, x ) for every τ ∈ [0 , t ], and thus˙ u ( t, x ) = ˙ u ( t, x ). So we deduce for a.e. t ∈ [0 , T ]:0 = Z L X i =1 σ i ( t ) (cid:18) ˙ u i ( t ) ′ − ˙¯ u ( t ) L (cid:19) d x + Z { δ ( t ) > } (cid:2) ∂ y ϕ ( δ ( t ) , γ ( t )) sgn( u ( t ) − u ( t )) (cid:3) ( ˙ u ( t ) − ˙ u ( t )) d x = Z L X i =1 E i ( α i ( t )) u i ( t ) ′ ˙ u i ( t ) ′ d x + Z L ∂ y ϕ ( δ ( t ) , γ ( t )) ˙ δ ( t ) d x − ˙¯ u ( t ) X i =1 σ i ( t, . In the above equality we first used the fact that by definition˙ δ ( t ) = ( ˙ u ( t ) − ˙ u ( t )) sgn( u ( t ) − u ( t )) , in { δ ( t ) > } , and then we exploited the assumption ∂ y ϕ (0 , z ) = 0 for z > (cid:3) Temporal Regularity and Equivalence between γ and δ h In this last section we finally develop the strategy which will allow to show that the fictitioushistorical variable γ actually coincides with the concrete one δ h in some meaningful cases, seeTheorems 4.7 and 4.8. The argument, which exploits the results of Section 3, is based onthe regularity in time of generalised energetic evolutions; this feature, as noticed in [32], is apeculiarity of systems governed by convex energies. For this reason, in this section we need tostrengthen the assumptions on the data, requiring for i = 1 , E i ∈ C ([0 , E ′′ i ( y ) E i ( y ) − E ′ i ( y ) > y ∈ [0 , w i ∈ C ([0 , µ i > , namely w i ( θy a + (1 − θ ) y b ) ≤ θw i ( y a ) + (1 − θ ) w i ( y b ) − µ i θ (1 − θ ) | y a − y b | , for every θ, y a , y b ∈ [0 , . (4.2)We notice that (4.1) implies (1.3), while (4.2) is trivially satisfied for instance by the simpleexample w i ( y ) = y + y . We also define M i := max y ∈ [0 , E ′′ i ( y ) , m i := min y ∈ [0 , (cid:18) E ′′ i ( y ) E i ( y ) − E ′ i ( y ) (cid:19) , (4.3)which are strictly positive by (4.1), and we finally denote by µ the minimum between µ and µ , namely µ := µ ∧ µ > . (4.4) Remark 4.1 ( Hardening Materials ) . Condition (4.1) is a characteristic of the so calledhardening materials, namely those materials for which the compliance S ( y ) := E ( y ) − is strictlyconcave. Indeed by simple calculations one has: S ′′ ( y ) = − E ( y ) (cid:18) E ′′ ( y ) E ( y ) − E ′ ( y ) (cid:19) , from which S ′′ < S ) discontinuous evolutions are common due to snap–back phenomena (seealso the analysis of [36]). AMAGE IN LAMINATES WITH COHESIVE INTERFACE 27
Of course we also need some sort of convexity for the loading–unloading density ϕ . However,we recall that usually it originates from a concave function ψ , see Remark 1.3; thus, in order tokeep that crucial property, we only require a weak form of convexity assumption on ψ , alreadyadopted in [34]:the function ψ is λ –convex for some λ > , namely for every θ ∈ [0 ,
1] and z a , z b ∈ [0 , + ∞ ) ψ ( θz a + (1 − θ ) z b ) ≤ θψ ( z a ) + (1 − θ ) ψ ( z b ) + λ θ (1 − θ ) | z a − z b | , (4.5a)while for ϕ itself, in addition to (3.2), we assume:for every z ∈ (0 , + ∞ ) the map ϕ ( · , z ) is non-decreasing and convex. (4.5b) Remark 4.2.
Coupling (3.2) with (4.5) we have thus recovered the assumptions ( ϕ ϕ ϕ given by (1.11).A crucial condition on the involved parameters will be given by m M ∧ m M > λ L π . (4.6)It morally says that the convexity of the internal energy E , represented by m M ∧ m M , is strongerthan the concavity of K , represented by λ , and thus the overall behaviour is the one of a convexenergy. Remark 4.3.
As already observed in [32, 36], a simple example of functions satisfying (4.1) isgiven by E i ( y ) = a i (1 + y ) b i , with a i > b i ∈ (0 , . In this case indeed it holds12 E ′′ i ( y ) E i ( y ) − E ′ i ( y ) = a i b i (1 − b i )(1 + y ) b i ) ≥ a i b i (1 − b i )4 b i = m i . Moreover M i = max y ∈ [0 , E ′′ i ( y ) = a i b i (1 + b i ), so that m i M i = a i − b i b i b i . In the particular case in which a = a =: a and b = b = 1 / m M = m M = a and socondition (4.6) can be written as: aλL > π , and can be achieved by increasing the parameter a or by decreasing λ or the lenght of the bar L . For convenience, in this section we also introduce the notation of the “shifted” energy, see also[30], Remark 3.2. For t ∈ [0 , T ] and x ∈ [0 , L ] we define the function ¯ u D ( t, x ) := (cid:16) ¯ u ( t ) L x, ¯ u ( t ) L x (cid:17) and we present the shifted variable v ( t ) = u ( t ) − ¯ u D ( t ), which has zero boundary conditionsand hence it belongs to [ H (0 , L )] . We finally introduce the shifted energy: E D [ t, v ( t ) , α ( t )] := E [ v ( t ) + ¯ u D ( t ) , α ( t )] = E [ u ( t ) , α ( t )] , and we want to highlight its explicit dependence on time given by the prescribed displacementand encoded by the function ¯ u D . Written in this form, the energy allows us to recast the workof the external prescribed displacement (1.14) in the following way: W [ u , α ]( t ) = Z t ∂ t E D [ τ, v ( τ ) , α ( τ )] d τ. (4.7)Moreover, by simple computations, it is easy to see that for almost every time τ ∈ [0 , T ] and forevery t ∈ [0 , T ] the following inequality holds true: | ∂ t E D [ τ, v ( τ ) , α ( τ )] − ∂ t E D [ τ, v ( t ) , α ( t )] |≤ C | ˙¯ u ( τ ) | (cid:16) k α ( τ ) − α ( t ) k H (0 ,L )] + k v ( τ ) ′ − v ( t ) ′ k L (0 ,L )] (cid:17) , (4.8)where C > t ∈ [0 , T ] one has γ ( t ) ≥ δ ( t ) in [0 , L ] and E D [ t, v ( t ) , α ( t )] + D [ α ( t )] + K [ δ ( t ) , γ ( t )] ≤ E D [ t, e v , e α ] + D [ e α ] + K [ e δ, γ ( t ) ∨ e δ ] , for every e v ∈ [ H (0 , L )] and for every e α ∈ [ H (0 , L )] such that α i ( t ) ≤ e α i ≤ , L ]for i = 1 , Lemma 4.4.
Let ( X, k · k ) be a normed space and let f : [ a, b ] → X be a bounded measurablefunction such that: k f ( t ) − f ( s ) k ≤ Z ts k f ( t ) − f ( τ ) k g ( τ ) dτ, for every a ≤ s ≤ t ≤ b, for some nonnegative g ∈ L ( a, b ) . Then actually it holds: k f ( t ) − f ( s ) k ≤ Z ts g ( τ ) dτ, for every a ≤ s ≤ t ≤ b. We are now in a position to state and prove the first result of this section, which yields tem-poral regularity of generalised energetic evolutions under the convexity assumptions we statedbefore. The argument is based on the ideas of [32], adapted to our setting where also a cohesiveenergy (concave by nature) is taken into account.
Proposition 4.5 ( Temporal Regularity ) . Assume E i satisfies (4.1) , w i satisfies (4.2) , andassume ϕ ∈ C ( T ) satisfies ( ϕ ϕ ϕ ( u , α , γ ) be a generalised energetic evolutionrelated to the prescribed displacement ¯ u ∈ AC ([0 , T ]) . If condition (4.6) on the parameters issatisfied, then both the displacements u and the damages α belong to AC ([0 , T ]; [ H (0 , L )] ) ,and so one also has δ ∈ AC ([0 , T ]; H (0 , L )) and δ h ∈ AC ([0 , T ]; C ([0 , L ])) .If in addition the family { γ ( t ) ∧ ¯ δ } t ∈ [0 ,T ] is equicontinuous and for every y ∈ [0 , ¯ δ ) the map ϕ ( y, · ) is strictly increasing in [ y, ¯ δ ) , then the function γ ∧ ¯ δ belongs to C ([0 , T ]; C ([0 , L ]) . Remark 4.6.
We want to point out that the additional requirement of equicontinuity of thefamily { γ ( t ) ∧ ¯ δ } t ∈ [0 ,T ] , although can not be derived directly from the Definition 1.8 of generalisedenergetic evolutions, is automatically satisfied by the limit function γ obtained in Proposition 2.5thanks to the uniform estimate (2.5b). Thus it is not restrictive. AMAGE IN LAMINATES WITH COHESIVE INTERFACE 29
Proof of Proposition 4.5.
We first consider, for i = 1 ,
2, the Hessian matrix of the function[0 , × R ∋ ( α, v ) E i ( α ) v , denoted by H i ( α, v ), and its quadratic form, namely the map:( x, y )
7→ h ( x, y ) , H i ( α, v )( x, y ) i = 12 E ′′ i ( α ) v x + 2 E ′ i ( α ) vxy + E i ( α ) y . By (4.1) it must be E ′′ i ( α ) > α ∈ [0 ,
1] and so we can write: h ( x, y ) , H i ( α, v )( x, y ) i = 2 E ′′ i ( α ) "(cid:18) E ′′ i ( α ) vx + E ′ i ( α ) y (cid:19) + (cid:18) E ′′ i ( α ) E i ( α ) − E ′ i ( α ) (cid:19) y ≥ m i M i y . Thanks to this estimate on the Hessian matrix it is easy to infer that for every t ∈ [0 , T ], forevery θ ∈ [0 ,
1] and for every v a , v b ∈ [ H (0 , L )] and α a , α b ∈ [ H , (0 , L )] it holds: E D [ t, θ v a + (1 − θ ) v b , θ α a + (1 − θ ) α b ] ≤ θ E D [ t, v a , α a ] + (1 − θ ) E D [ t, v b , α b ] − m M ∧ m M θ (1 − θ ) k ( v a ) ′ − ( v b ) ′ k L (0 ,L )] . (4.9)By means of (4.2) we also deduce that for every t ∈ [0 , T ], for every θ ∈ [0 ,
1] and for every α a , α b ∈ [ H , (0 , L )] we have: D [ θ α a + (1 − θ ) α b ] ≤ θ D [ α a ] + (1 − θ ) D [ α b ] − µ ∧ θ (1 − θ ) k α a − α b k H (0 ,L )] . (4.10)Finally, by (3.2c), (4.5a) and (4.5b) (which are implied by ( ϕ ϕ z ∈ [0 , + ∞ ) the function y ϕ ( y, z ∨ y ) is λ –convex in [0 , + ∞ ); thus for every t ∈ [0 , T ], forevery θ ∈ [0 ,
1] and for every nonnegative δ a , δ b ∈ H (0 , L ) it holds: K [ θδ a + (1 − θ ) δ b , γ ( t ) ∨ ( θδ a + (1 − θ ) δ b )] ≤ θ K [ δ a , γ ( t ) ∨ δ a ] + (1 − θ ) K [ δ b , γ ( t ) ∨ δ b ] + λ θ (1 − θ ) k δ a − δ b k L (0 ,L ) . (4.11)We now fix t ∈ [0 , T ], θ ∈ (0 , e v ∈ [ H (0 , L )] , e α ∈ [ H (0 , L )] such that α i ( t ) ≤ e α i ( t ) ≤ i = 1 ,
2, and we consider as competitors in (GS’) the functions θ e v + (1 − θ ) v ( t ) and θ e α + (1 − θ ) α ( t ); by means of (4.9), (4.10) and (4.11), together with ( ϕ
3) and (4.5b), we thusget: E D [ t, v ( t ) , α ( t )] + D [ α ( t )] + K [ δ ( t ) , γ ( t )] ≤ E D [ t, θ e v + (1 − θ ) v ( t ) , θ e α + (1 − θ ) α ( t )] + D [ θ e α + (1 − θ ) α ( t )]+ K [ | θ ( e v − e v ) + (1 − θ )( v ( t ) − v ( t )) | , γ ( t ) ∨ | θ ( e v − e v ) + (1 − θ )( v ( t ) − v ( t )) | ] ≤ θ E D [ t, e v , e α ] + (1 − θ ) E D [ t, v ( t ) , α ( t )] − m M ∧ m M θ (1 − θ ) k ( e v ) ′ − ( v ( t )) ′ k L (0 ,L )] + θ D [ e α ] + (1 − θ ) D [ α ( t )] − µ ∧ θ (1 − θ ) k e α − α ( t ) k H (0 ,L )] + θ K [ e δ, γ ( t ) ∨ e δ ] + (1 − θ ) K [ δ ( t ) , γ ( t )] + λ θ (1 − θ ) k e δ − δ ( t ) k L (0 ,L ) . (4.12)We now exploit the well known sharp Poincar´e inequality: Z ba f ( x ) d x ≤ ( b − a ) π Z ba f ′ ( x ) d x, for every f ∈ H ( a, b ) , to deduce that k e δ − δ ( t ) k L (0 ,L ) ≤ L π k ( e v ) ′ − ( v ( t )) ′ k L (0 ,L )] . (4.13) By plugging (4.13) in (4.12), dividing by θ and then letting θ → + we finally deduce: (cid:18) m M ∧ m M − λ L π (cid:19) k ( e v ) ′ − ( v ( t )) ′ k L (0 ,L )] + µ ∧ k e α − α ( t ) k H (0 ,L )] + E D [ t, v ( t ) , α ( t )] + D [ α ( t )] + K [ δ ( t ) , γ ( t )] ≤ E D [ t, e v , e α ] + D [ e α ] + K [ e δ, γ ( t ) ∨ e δ ] . (4.14)For the sake of simplicity we denote by c the minimum between m M ∧ m M − λ L π and µ ∧ , andwe notice that c is strictly positive by (4.6). We now fix two times 0 ≤ s ≤ t ≤ T . Exploiting(4.14) at time s with e v = v ( t ) and e α = α ( t ), and recalling (EB’) and (4.7) we obtain: c (cid:16) k α ( t ) − α ( s ) k H (0 ,L )] + k v ( t ) ′ − v ( s ) ′ k L (0 ,L )] (cid:17) ≤ E D [ s, v ( t ) , α ( t )] + D [ α ( t )] + K [ δ ( t ) , γ ( s ) ∨ δ ( t )] − E D [ s, v ( s ) , α ( s )] − D [ α ( s )] − K [ δ ( s ) , γ ( s )] ≤ E D [ s, v ( t ) , α ( t )] − E D [ t, v ( t ) , α ( t )] + W [ u , α ]( t ) − W [ u , α ]( s ) ≤ Z ts | ∂ t E D [ τ, v ( τ ) , α ( τ )] − ∂ t E D [ τ, v ( t ) , α ( t )] | d τ. By using (4.8) we thus deduce: k α ( t ) − α ( s ) k H (0 ,L )] + k v ( t ) ′ − v ( s ) ′ k L (0 ,L )] ≤ Cc Z ts | ˙¯ u ( τ ) | (cid:16) k α ( τ ) − α ( t ) k H (0 ,L )] + k v ( τ ) ′ − v ( t ) ′ k L (0 ,L )] (cid:17) d τ. By means of (1.16a) we can apply Lemma 4.4 getting: (cid:16) k α ( t ) − α ( s ) k H (0 ,L )] + k v ( t ) ′ − v ( s ) ′ k L (0 ,L )] (cid:17) ≤ Cc Z ts | ˙¯ u ( τ ) | d τ, and so we infer that α belongs to AC ([0 , T ]; [ H (0 , L )] ) and v belongs to AC ([0 , T ]; [ H (0 , L )] ).By construction we also have k u ( t ) − u ( s ) k [ H (0 ,L )] ≤ k v ( t ) − v ( s ) k [ H (0 ,L )] + k u D ( t ) − u D ( s ) k [ H (0 ,L )] ≤ k v ( t ) − v ( s ) k [ H (0 ,L )] + C | ¯ u ( t ) − ¯ u ( s ) | , so also u belongs to AC ([0 , T ]; [ H (0 , L )] ) and as a simple byproduct we obtain that δ is AC ([0 , T ]; H (0 , L )).Since H (0 , L ) ⊆ C ([0 , L ]), in particular there exists a nonnegative function φ ∈ L (0 , T )such that k δ ( t ) − δ ( s ) k C ([0 ,L ]) ≤ Z ts φ ( τ ) d τ, for every 0 ≤ s ≤ t ≤ T. (4.15)We now show that the same inequality holds true for δ h in place of δ . We thus fix 0 ≤ s ≤ t ≤ T and x ∈ [0 , L ]. If δ h ( t, x ) = δ h ( s, x ) there is nothing to prove, so let us assume δ h ( t, x ) > δ h ( s, x ).By definition of δ h and since now we know that δ is continuous both in time and space we deducethat δ h ( t, x ) = max τ ∈ [0 ,t ] δ ( τ, x ) = δ ( t x , x ) , for some t x ∈ [ s, t ] . So we have δ h ( t, x ) − δ h ( s, x ) ≤ δ ( t x , x ) − δ ( s, x ) ≤ Z t x s φ ( τ ) d τ ≤ Z ts φ ( τ ) d τ. We have thus proved the validity of (4.15) with δ h in place of δ , and hence δ h belongs to AC ([0 , T ]; C ([0 , L ])). AMAGE IN LAMINATES WITH COHESIVE INTERFACE 31
We only need to prove that γ ∧ ¯ δ ∈ C ([0 , T ]; C ([0 , L ]) under the additional assumptions that { γ ( t ) ∧ ¯ δ } t ∈ [0 ,T ] is an equicontinuous family and ϕ ( y, · ) is strictly increasing in [ y, ¯ δ ) for any given y ∈ [0 , ¯ δ ). For the sake of clarity we prove it only in the case ¯ δ = + ∞ ; in the other situation theresult can be obtained arguing in the same way and recalling equality (1.12). To this aim weobserve that, by equicontinuity, for every t ∈ [0 , T ] the right and the left limits γ + ( t ) and γ − ( t )are continuous in [0 , L ]. By monotonicity and using classical Dini’s theorem we hence obtain γ ± ( t ) = lim h → ± γ ( t + h ) , uniformly in [0 , L ] . (4.16)So we conclude if we prove that γ + ( t ) = γ − ( t ).Arguing as in the proof of Proposition 3.5, since u and α are in AC ([0 , T ]; [ H (0 , L )] ), wededuce by (EB’) that the map t
7→ K [ δ ( t ) , γ ( t )] is continuous in [0 , T ], and thus for every t ∈ [0 , T ] we have: lim h → + K [ δ ( t + h ) , γ ( t + h )] = lim h → − K [ δ ( t + h ) , γ ( t + h )] . By using (4.16) we can pass to the limit inside the integral getting Z L ϕ ( δ ( t ) , γ + ( t )) d x = Z L ϕ ( δ ( t ) , γ − ( t )) d x. Since ϕ ( y, · ) is strictly increasing we conclude. (cid:3) Thanks to the temporal regularity obtained in the previous proposition we are able to proveour main results. The first theorem ensures the equality between γ and δ h (actually between γ ∧ ¯ δ and δ h ∧ ¯ δ , which however are the meaningful ones, see Remark 1.9) assuming a prioriequicontinuity on the family { γ ( t ) } t ∈ [0 ,T ] , which is however not restrictive due to Remark 4.6;a similar argument to the one adopted here, but in an easier setting, can be found in [37],Proposition 2.7. The second theorem states that the generalised energetic evolution obtained inSection 2 as limit of discrete minimisers is actually an energetic evolution. We thus reach ourgoal, avoiding the assumption ( ϕ ϕ ϕ ϕ
9) by the weaker (4.17)) which for instance are satisfied by the exampleprovided in Remark 1.3.
Theorem 4.7 ( Equivalence between γ and δ h ) . Let the prescribed displacement ¯ u belongto the space AC ([0 , T ]) . Assume E i satisfies (4.1) , w i satisfies (4.2) , and ϕ ∈ C ( T ) satisfies( ϕ ϕ z :for every compact set K ∈ { z > y ≥ } ∩ T ¯ δ there exists a positive constant C K > such that ϕ ( y, z ) − ϕ ( y, z ) ≥ C K ( z − z ) for every ( z , y ) , ( z , y ) ∈ K satisfying z ≥ z . (4.17) Assume also condition (4.6) on the parameters. Then, given a generalised energetic evolution ( u , α , γ ) such that the family { γ ( t ) ∧ ¯ δ } t ∈ [0 ,T ] is equicontinuous, the function γ ∧ ¯ δ coincides with δ h ∧ ¯ δ .Proof. For the sake of clarity we prove the result only in the case ¯ δ = + ∞ , being the othersituation analogous by (1.12).We know that γ ≥ δ h and that γ (0) = δ h (0) = δ and γ ( t,
0) = γ ( t, L ) = δ h ( t,
0) = δ h ( t, L ) = 0for every t ∈ [0 , T ]. Moreover by Proposition 4.5 we know that both γ and δ h are continuous on[0 , T ] × [0 , L ]. We thus assume by contradiction there exists (¯ t, ¯ x ) ∈ (0 , T ] × (0 , L ) for which γ (¯ t, ¯ x ) > δ h (¯ t, ¯ x );by continuity we thus deduce there exists η > γ ( t, x ) > δ h ( t, x ) ≥ δ ( t, x ) , for every ( t, x ) ∈ [¯ t − η, ¯ t ] × [¯ x − η, ¯ x + η ] . By assumption (4.17) we hence infer the existence of constant c η > ϕ ( δ ( s, x ) , γ ( t, x )) − ϕ ( δ ( s, x ) , γ ( s, x )) ≥ c η ( γ ( t, x ) − γ ( s, x )) , for every ¯ t − η ≤ s ≤ t ≤ ¯ t and x ∈ [¯ x − η, ¯ x + η ] . (4.18)We now recall that by Proposition 4.5 we know the map t
7→ K [ δ ( t ) , γ ( t )] is absolutely continuousin [0 , T ]. So for every 0 ≤ s ≤ t ≤ T we can estimate: Z L ( ϕ ( δ ( s ) , γ ( t )) − ϕ ( δ ( s ) , γ ( s ))) d x = K [ δ ( t ) , γ ( t )] − K [ δ ( s ) , γ ( s )] + Z L ( ϕ ( δ ( s ) , γ ( t )) − ϕ ( δ ( t ) , γ ( t ))) d x (4.19) ≤ Z ts dd t K [ δ ( τ ) , γ ( τ )] d τ + C k δ ( t ) − δ ( s ) k C ([0 ,L ]) ≤ Z ts φ ( τ ) d τ, where φ ∈ L (0 , T ) is a suitable nonnegative function.Combining (4.18) and (4.19) we now obtain: c η Z ¯ x + η ¯ x − η ( γ ( t ) − γ ( s )) d x ≤ Z ts φ ( τ ) d τ, for every ¯ t − η ≤ s ≤ t ≤ ¯ t, hence γ ∈ AC ([¯ t − η, ¯ t ]; L (¯ x − η, ¯ x + η )).By means of (3.9) we now deduce that for a.e. t ∈ [¯ t − η, ¯ t ] we have:0 = lim h → Z L ϕ ( δ ( t ) , γ ( t + h )) − ϕ ( δ ( t ) , γ ( t )) h d x ≥ c η lim sup h → Z ¯ x + η ¯ x − η γ ( t + h ) − γ ( t ) h d x ≥ , namely for almost every t ∈ [¯ t − η, ¯ t ] the function γ is strongly differentiable in L (¯ x − η, ¯ x + η )and ˙ γ ( t ) = 0. By Proposition A.3 we now obtain γ ( t ) = γ (¯ t − η )+ Z t ¯ t − η ˙ γ ( τ ) d τ = γ (¯ t − η ) , for every t ∈ [¯ t − η, ¯ t ] , as an equality in L (¯ x − η, ¯ x + η ) . In particular, since γ is continuous, we deduce that γ (¯ t, ¯ x ) = γ (¯ t − η, ¯ x ).Since δ h is non-decreasing we can iterate all the previous argument, finally getting γ (¯ t, ¯ x ) = γ (0 , ¯ x ). But this is absurd, indeed it implies: δ (¯ x ) = γ (0 , ¯ x ) = γ (¯ t, ¯ x ) > δ h (¯ t, ¯ x ) ≥ δ h (0 , ¯ x ) = δ (¯ x ) , and so we conclude. (cid:3) Theorem 4.8 ( Existence of Energetic Evolutions ) . Let the prescribed displacement ¯ u belongto the space AC ([0 , T ]) and the initial data u , α fulfil (1.13) together with the stability condition (1.15) . Assume E i satisfies (4.1) , w i satisfies (4.2) , and ϕ ∈ C ( T ) satisfies ( ϕ ϕ ϕ (4.17) . Assume also condition (4.6) on the parameters. Then the pair ( u , α ) composed bythe functions obtained in Proposition 2.5 is an energetic evolution, since it holds γ ∧ ¯ δ = δ h ∧ ¯ δ .Moreover u and α belong to AC ([0 , T ]; [ H (0 , L )] ) , and so in particular the historical slip δ h is in AC ([0 , T ]; C ([0 , L ])) .Proof. The result is a simple byproduct of Theorem 2.11 together with Proposition 4.5 andTheorem 4.7 (we also recall (1.12)). We indeed notice that the equicontinuity assumption onthe family { γ ( t ) ∧ ¯ δ } t ∈ [0 ,T ] (actually on the whole { γ ( t ) } t ∈ [0 ,T ] ) is automatically satisfied by the AMAGE IN LAMINATES WITH COHESIVE INTERFACE 33 limit function γ obtained in Proposition 2.5 thanks to the uniform bounds (2.5b) on the discreteinterpolations. (cid:3) Conclusions.
The obtained results put the basis for further investigations. First of all wemay mention the generalisation of the simple one–dimensional model here presented to higherdimensional settings, as the ones numerically investigated in [2]. A second line of explorationcould also be the analysis of the problem in case of complete damage, meant as complete lossof material stiffness.Moreover, it would be interesting to extend the proposed approach to classical problems of co-hesive fracture mechanics. In this case, dissipation combined with irreversible effects introducesdifficulties, at least when dealing with global minimisers of the energy, in considering loading-unloading cohesive laws that reflect the real behaviour of materials rather than hypothesesdictated by mere mathematical assumptions. The main difference provided by cohesive fracturemodels with respect to the considered problem of cohesive interface relies in the reduced dimen-sion of the fracture, which is a ( d − d –dimensional material. Thisfeature involves the use of weaker topologies, which can not be directly treated following ourargument, and thus requires further adaptations in order to transfer our results. Appendix A. Absolutely Continuous and BV–Vector Valued Functions
In this Appendix we briefly present the main definitions and properties of vector valuedabsolutely continuous functions and functions of bounded variation we used throughout thepaper. A deeper and more detailed analysis can be found in the Appendix of [15], to which werefer for all the proofs and examples. Here ( X, k · k ) will denote a Banach space, and by X ∗ wemean its topological dual. The duality product between w ∈ X ∗ and x ∈ X is finally denotedby h w, x i . Definition A.1.
A function f : [0 , T ] → X is said to be: • a function of bounded variation ( BV ([0 , T ]; X ) ) if V X ( f ; 0 , T ) := sup finite partitionsof [0 ,T ] X k f ( t k ) − f ( t k − ) k < + ∞ ; • absolutely continuous ( AC ([0 , T ]; X ) ) if there exists a nonnegative function φ ∈ L (0 , T ) such that k f ( t ) − f ( s ) k ≤ Z ts φ ( τ ) d τ, for every ≤ s ≤ t ≤ T ; • in the space f W ,p (0 , T ; X ) , p ∈ [1 , + ∞ ] , if there exists a nonnegative function φ ∈ L p (0 , T ) such that k f ( t ) − f ( s ) k ≤ Z ts φ ( τ ) d τ, for every ≤ s ≤ t ≤ T ; • in the Sobolev space W ,p (0 , T ; X ) , p ∈ [1 , + ∞ ] , if there exists a function g ∈ L p (0 , T ; X ) such that f ( t ) = f (0) + Z t g ( τ ) d τ, for every t ∈ [0 , T ] . As in the classical case X = R any function of bounded variation belongs to L ∞ (0 , T ; X ), itadmits right and left (strong) limits at every t ∈ [0 , T ] and the set of its discontinuity pointsis at most countable. To gain the well known property of almost everywhere differentiabilityalso in the vector valued framework it is instead crucial to require X to be reflexive (see theexamples in [15]). Proposition A.2. If X is reflexive, then any function f belonging to BV ([0 , T ]; X ) is weaklydifferentiable almost everywhere in [0 , T ] . Moreover k ˙ f ( t ) k ≤ dd t V X ( f ; 0 , t ) for a.e. t ∈ [0 , T ] and in particular ˙ f ∈ L (0 , T ; X ) . We now focus our attention on absolutely continuous and Sobolev functions. By the verydefinition it is easy to see that any absolutely continuous function is also of bounded variation;furthermore the spaces AC ([0 , T ]; X ) and f W , (0 , T ; X ) coincide, while f W , ∞ (0 , T ; X ) is thespace of Lipschitz functions from [0 , T ] to X . Moreover for every p ∈ [1 , + ∞ ] the inclusion W ,p (0 , T ; X ) ⊆ f W ,p (0 , T ; X ) always holds, but in general is strict.Next proposition states that the Sobolev space W ,p (0 , T ; X ) is actually characterised by thestrong differentiability of its elements. Proposition A.3.
Let p ∈ [1 , + ∞ ] and let f be a function from [0 , T ] to X . Then the followingare equivalent: (i) f ∈ W ,p (0 , T ; X ) ; (ii) f ∈ f W ,p (0 , T ; X ) and it is strongly differentiable for a.e. t ∈ [0 , T ] ; (iii) for every w ∈ X ∗ the map t
7→ h w, f ( t ) i is absolutely continuous in [0 , T ] , f is weaklydifferentiable for a.e. t ∈ [0 , T ] and ˙ f ∈ L p (0 , T ; X ) .If one of the above condition holds, then one has f ( t ) = f (0) + Z t ˙ f ( τ ) d τ, for every t ∈ [0 , T ] . (A.1)In the reflexive case, as in Proposition A.2, we gain differentiability of absolutely contin-uous functions and so we deduce the equivalence between the two spaces f W ,p (0 , T ; X ) and W ,p (0 , T ; X ). Proposition A.4. If X is reflexive, then for every p ∈ [1 , + ∞ ] the Sobolev space W ,p (0 , T ; X ) coincides with f W ,p (0 , T ; X ) . Acknowledgements.
E. Bonetti, C. Cavaterra and F. Riva are members of the GruppoNazionale per l’Analisi Matematica, la Probabilit e le loro Applicazioni (GNAMPA) of theIstituto Nazionale di Alta Matematica (INdAM).
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Via Cesare Saldini, 50, 20133, Milano, Italy e-mail address : [email protected] (Cecilia Cavaterra) Universit`a degli Studi di Milano, Dipartimento di Matematica “Federigo Enriques”, Via Cesare Saldini, 50, 20133, Milano, Italy e-mail address : [email protected] (Francesco Freddi) Universit`a degli Studi di Parma, Dipartimento di Ingegneria e Architettura, Parco Area delle Scienze, 181/A, 43124, Parma, Italy e-mail address : [email protected] (Filippo Riva) SISSA, Via Bonomea, 265, 34136, Trieste, Italy e-mail address ::