FATIGUE EFFECTS IN ELASTIC MATERIALS WITH VARIATIONAL DAMAGE MODELS: A VANISHING VISCOSITY APPROACH

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1 FATIGUE EFFECTS IN ELASTIC MATERIALS WITH VARIATIONAL DAMAGE MODELS: A VANISHING VISCOSITY APPROACH ROBERTO ALESSI, VITO CRISMALE, GIANLUCA ORLANDO Abstract. We study the existence of quasistatic evolutions for a family of gradient damage models which tae into account fatigue, that is the process of weaening in a material due to repeated applied loads. The main feature of these models is the fact that damage is favoured in regions where the cumulation of the elastic strain or other relevant variables, depending on the model is higher. To prove the existence of a quasistatic evolution, we follow a vanishing viscosity approach based on two steps: we first let the time-step τ of the time-discretisation and later the viscosity parameter ε go to zero. As τ, we find ε-approximate viscous evolutions; then, as ε, we find a rescaled approximate evolution satisfying an energy-dissipation balance. Keywords: Fatigue; Gradient-damage models; Variational methods; Vanishing-viscosity approach MSC 21: 74C5, 74A45, 74R2, 35Q74, 49J45. Contents 1. Introduction 1 2. Assumptions on the model 4 3. Incremental minimum problems 8 4. Existence of viscous evolutions Vanishing viscosity limit 28 Appendix A. Auxiliary results 35 References Introduction In Material Science, fatigue refers to the process which leads to the weaening of a material due to repeated applied loads, which individually would be too small to cause the direct failure of the material itself. Macroscopic fatigue fractures appear as a consequence of the interaction of many and complicated material phenomena occurring at the micro-scale, such as, for instance, plastic slip systems and coalescence of micro-voids, 39, 35, 37. Fatigue failure is extremely dangerous, since it often occurs without forewarning resulting in devastating events, and is responsible for up to the 9% of all mechanical failures 38. The main reason is that it is very difficult, in real situations, to identify the fatigue degradation state of a material. Therefore, its prediction still represents an open challenge for modeling and simulation at the cutting edge of mechanics. 1

2 2 ROBERTO ALESSI, VITO CRISMALE, GIANLUCA ORLANDO Fatigue favours the occurrence of damage and fracture in different types of materials, both brittle and ductile. When the stress level is high enough to induce plastic deformations, the material is usually subjected to a so-called low-cycle fatigue regime; instead, high-cycle fatigue occurs if the stress is below the yield stress such that the strains are primarily elastic. Models where fatigue effects are induced by the cumulation of plastic deformations have been recently studied in 3, 4, 2, 1 and 9, 11, 12. In this paper we study a phenomenological material model where damage is the only inelastic phenomenon and the fatigue weaening of the material is a consequence of repeated cycles of elastic deformations. Our wor is inspired by the recent paper 5, where the authors propose a similar model in the one-dimensional setting and to which the reader is invited to refer to for further mechanical details. As usual, damage is expressed in terms of a scalar variable which affects the elastic response of the material and may be interpreted as the local percentage of sound interatomical bonds. In contrast to many previous damage models 2, 28, 7, 41, 4, 24, 25, 26, in this paper the dissipation depends not only on the damage variable itself, but also on the history of the evolution. Indeed, damage is favoured in regions where a suitable history variable has a higher value. This history variable is defined pointwise in the body as the cumulation in time of a given function ζ that may be the strain, or the stress, or the energy density, according to the model. As a consequence, the material may undergo a damage process even if the variable ζ remains small during the evolution. We are here interested in proving the existence of quasistatic evolutions for this model in a two-dimensional antiplane shear setting, following a vanishing viscosity approach. To present in detail our result, before expressing the strong formulation of the model in terms of differential inclusions, we introduce the time-incremental minimisation problem corresponding to a time discretisation t i := i T = iτ for the unnowns α:, 1 the damage variable and u: R the displacement assuming that the previous states α j, uj i 1 j= { α, i u i 1 argmin α α i 1 2 µα u 2 dx α 2 dx + The functional minimised above consists of three parts: the internal energy Eα, u = 1 µα u 2 dx + 1 α 2 dx, 2 2 are nown: fv i 1 α i 1 α dx + ε } α α i 1 2 L 2τ 2. given by the sum of the elastic energy and the damage regularisation term; the energy dissipated from the previous state fv i 1 α i 1 α dx = Rα α i 1 ; V i 1, with Rβ; V = fv β dx if β, + otherwise; and the viscosity term, depending on a small parameter ε. The elastic response is affected by the factor µα >, where µ in nondecreasing in α, according to the fact that α = 1 represents a sound material and α = a completely damaged one. Notice that the constraint α α i enforces the irreversibility of the damage process. The L 2 norm of α is the usual regularising term in gradient damage models see the aforementioned wors and 17, 1, 13 for coupling with plasticity. The dissipation term characterises the present model in comparison to other damage models, since the fatigue term fv i 1 weights the damage increment. For every j, the history variable V j is defined by j := ζ h ζ h 1, V j h=1

3 FATIGUE EFFECTS IN ELASTIC MATERIALS 3 where ζ h V j represents the elastic strain, or the stress, or the density of the elastic energy at time t h. Notice that = t j ζ s ds, where ζ s is the piecewise affine interpolation of ζ h. The function f is nonincreasing, so that in the minimisation it is more convenient to tae α lower where the cumulation V i 1 term prevents α i to be too far in L 2 from the previous damage state α i 1. is larger. The viscosity The approach that we follow consists of two main steps: as, e.g., in 28, 7, 41, 4, 24, 25, we let first the time-step of the discretisation τ and later the viscosity parameter ε tend to. More precisely, the starting point is to define for every the discrete-time evolution α ε, t, u ε, t as the piecewise affine interpolation of α i, u i and to derive a priori estimates cf. Proposition 3.5 which guarantee that α ε, H 1,T ;H 1, u ε, H 1,T ;W 1,p are bounded uniformly with respect to not with respect to ε and α ε, W 1,1,T ;H 1, u ε, W 1,1,T ;W 1,p are bounded uniformly with respect to and ε, for some p > 2. We exploit the a priori estimates H 1 in time to pass to the limit as + : for every ε we obtain an ε-approximate viscous evolution α εt, u εt characterised by an equilibrium condition in u εt, a unilateral stability condition in α εt Karush-Kuhn-Tucer inequality, and an energy-dissipation balance cf. Definition 4.1. This evolution may be expressed in terms of the differential inclusions cf. ev1 ε ev2 ε in Definition 4.1 and ev3 ε in Lemma 4.12 ueα εt, u εt = in H 1 D, αr α εt; V εt + ε α εt + αeα εt, u εt in H 1, for a.e. t, T, where V εt is the history variable associated to the evolution α ε, u ε, H 1 D is the dual of {v H 1 : v = on D}, and αrβ; V is the convex analysis subdifferential of R ; V, i.e. ξ αrβ; V if and only if Rβ; V + ξ, β β Rβ; V for every β H 1. For the expression of ue and αe we refer to Lemma 2.1. The a priori estimates W 1,1 in time allow us to reparametrise the ε-approximate viscous evolutions and to obtain a family of equi-lipschitz evolutions αεs, u εs in a slower time scale s. At this stage we let ε and obtain an evolution α s, u s together with a reparametrisation function t s that permits the passage from the slow to the original fast time scale t. In Theorem 5.1 we prove that α, u still satisfies an equilibrium condition in u s, a unilateral stability condition in α s Karush-Kuhn-Tucer inequality, and an energydissipation balance. However, the dissipation in the energy balance weights the rate of damage with a function f s fv s, where V s is the history variable associated to the evolution α, u. In terms of differential inclusions, this reads as cf. ev1 ev2 in Theorem 5.1 and 5.11 in Remar 5.3 ueα s, u s = in H 1 D, αr α s; f s + αeα s, u s in H 1, for a.e. s, S \ U, where R ; f s is defined as in 1.1 with f s in place of fv s. The set U corresponds to jump instants of the evolution in the fast time scale reparametrised in the slow time scale: therein the limit evolution is governed by a variational inequality of viscous type, representing a fast unstable propagation in the original time scale. An interesting issue, that we were not able to solve, is to determine whether there are explicit examples where this inequality is strict and f s is actually the correct weight to consider in the energy-dissipation balance. In the mathematical treatment of the present model some technical difficulties arise. Here we discuss the main issues in the a priori estimates and in the limits as + and ε.

4 4 ROBERTO ALESSI, VITO CRISMALE, GIANLUCA ORLANDO The proof of the a priori estimates rests upon the manipulation of the Discrete Karush-Kuhn-Tucer conditions 3.9 and 3.1 evaluated at two subsequent times t i 1 and t i, respectively, as e.g. in 33, 24, 3, 25, 11, 26. The resulting estimate 3.17 contains in the right-hand side also discrete-time derivatives at time t i 1, in contrast to the aforementioned wors, where only discrete-time derivatives at time t i appear. These additional terms are due to the presence of the fatigue weight fv i 1 in the dissipation for the i-th incremental minimisation problem and prevent the immediate application of the discrete Gronwall estimate used in the previous wors. We refine the usual technique to overcome this issue in The main difficulty in deriving the properties of the ε-approximate viscous evolutions α εt, u εt consists in passing to the limit as + in the dissipation term containing the fatigue weight fv ε, t. The a priori estimate on u ε, H 1,T ;W 1,p only guarantees that u ε, u ε wealy in L 2, T ; L p ; R 2, and this convergence is not sufficient to deduce the convergence of V ε, to V ε, even in the paradigmatic case where ζ is the elastic strain, namely when the history variable is V t = t us ds. To circumvent this problem we first let fv ε, t converge to some f εt wealy* in L for every t by an Helly-type theorem cf. Lemma 4.6, to get an evolution α εt, u εt satisfying the Karush-Kuhn-Tucer inequality, and the energy-dissipation balance with f εt in place of fv εt cf. Propositions 4.8 and 4.1. At this stage, we exploit the convergence of all the terms of the discrete-time energy-dissipation balance to the corresponding ones in the continuous-time energydissipation balance. This improves the convergence of α ε, to α ε Proposition 4.11, allowing us to deduce that u ε, u ε strongly in L 2, T ; L p ; R 2 and thus that f εt = fv εt. Eventually, we obtain the existence of an ε-approximate evolution. The scenario when ε is radically different. Indeed, here the energy-dissipation balance does not help to improve the wea convergence u ε u for the rescaled evolutions α εs, u εs, due to the rate-independence of the system as ε. As a consequence, the limit evolution is formulated with f s, the wea -L limit of the fatigue weight reparametrisations fv ε s, in place of fv s. This motivates why we pass to the limit in two steps, rather than directly taing a simultaneous limit τ /ε, +, as in the framewor developed in 31, 27 and followed in Assumptions on the model Vector-valued functions. In this paragraph we let X be a Banach space. We will often consider the Bochner integral of measurable functions v :, T X. For the definition of this notion of integral and its main properties we refer to 8, Appendix or to the textboo 19. The Lebesgue space L p, T ; X is defined accordingly. We recall that, if p 1, and X is separable, the dual of L p, T ; X is L p, T ; X, where = 1 and X p p is the dual of X. For the definition and the main properties of absolute continuous functions AC, T ; X and Sobolev functions W 1,p, T ; X, the reader is referred to 8, Appendix. We recall here the Aubin-Lions Lemma 6, 36 about the compactness property enjoyed by W 1,p, T ; X. Let Y be a Banach space compactly embedded in X, and let 1 p, q. Then the space W = {v L p, T ; Y : v L q, T ; X} is: 1 compact in L p, T ; X if p < ; 2 compact in C, T ; X if p = and q > 1. In this paper, the Banach space X will be either a Lebesgue space L q U; R m or a Sobolev space W 1,q U, where U is an open set of R n. Given an element v L p, T ; L q U; R m, p, q 1,, we identify it with the function v :, T R m defined by vt; x := vt x.

5 FATIGUE EFFECTS IN ELASTIC MATERIALS 5 The norms L p and W 1,p without any further notation will always denote the L p -norm and the W 1,p -norm with respect to the space variable x, respectively. The reference configuration. Throughout the paper, is a bounded, Lipschitz, open set in R 2 representing the cross-section of a cylindrical body in the reference configuration. The deformation v : R R taes the form vx 1, x 2, x 3 = x 1, x 2, x 3 + ux 1, x 2, where u: R is the vertical displacement. In this antiplane shear framewor, the two dimensional setting is the physical relevant one. This assumption gives the compact embedding H 1 in L p for every p 1,, which we employ in the a priori estimates. We assume that = D N, where D and N are relatively open sets in with D N = Ø and H 1 D >. A Dirichlet boundary datum will be prescribed on the set D. In order to apply the integrability result 22 to our problem see Remar 3.2 below, we assume that N is regular in the sense of 22, Definition 2. Notice that in dimension 2 this regularity assumption on N is satisfied, e.g., when the relative boundary N Γ in consists of a finite number of points. It is convenient to introduce the notation W 1,p D for the dual of the space {u W 1,p : u = on D}, where 1 p + 1 p = 1. The total energy. Following 21, the damage state of the body is represented by an internal variable α:, 1. The value α = 1 corresponds to a sound state, whereas α = corresponds to the maximum possible damage. As usual in gradient damage models 34, the system in analysis comprises a regularizing term α 2 L 2. In particular, the damage variable α belongs to the Sobolev space H 1. For every α H 1 and u H 1, the stored elastic energy is defined by 1 µα u 2 dx. 2 We mae the following assumptions on the dependence of the shear modulus µ on the damage variable α: µ: R, + is a C 1,1 R, nondecreasing function with µ >, µβ = µ for β, µβ = µ1 for β The regularity assumption on µ is needed in the proof of Proposition 3.5 see The condition 2.1 on µ forces α to tae values in, 1 in the evolution see Remar 3.1. The total energy corresponding to a damage state α and to a displacement u is Eα, u := 1 µα u 2 dx + 1 α 2 dx Notice that the constant 1 2 in the gradient damage regularisation term does not play a role in the mathematical treatment and may be replaced by any positive constant. We compute here the derivatives of the total energy. Note that an integrability strictly higher than 2 is required on u to guarantee the differentiability of the energy with respect to α. Lemma 2.1. The following statements hold true: i Let u W 1,p, with p > 2. Then the functional α H 1 Eα, u is differentiable and αeα, u, β = 1 µ α u 2 β dx + α β dx, for every α, β H 1.

6 6 ROBERTO ALESSI, VITO CRISMALE, GIANLUCA ORLANDO ii Let α H 1. Then the functional u H 1 Eα, u is differentiable and ueα, u, v = µα u v dx, for every v H 1. Proof. We only prove i, the proof of ii being trivial. The derivative of 1 2 α 2 L 2 simply gives the second integral in 2.3. As for the differentiability of µα u 2 dx, let us fix α, β H 1, and δ >. By Young s inequality we have where q = p p 2 µα + δβ µα u 2 µ L β u 2 C β q + u p, δ <. Thans to the embedding H1 L q, we can apply the Dominated Convergence Theorem to deduce that the functional α H 1 Eα, u is Gâteaux-differentiable and its Gâteaux-differential is expressed by 2.3. Moreover, since u W 1,p, with p > 2, and H 1 L r, for any r 1,, it is immediate that the functionals in i and ii are Fréchet-differentiable. Fatigue and damage dissipation. The damage dissipation is affected by the cumulation of a suitable variable of the system during the history of the evolution. This variable may be for instance the elastic strain, the stress, or the density of the elastic energy, according to the material model. In the general case, we consider a function depending on the damage variable α and on the elastic strain u: we tae, for given evolutions α AC, T ; L q ;, 1, u AC, T ; W 1,p, with p > 2, < 1, the function q p 2 ζt := gαt ut, 2.4 where g C 1,1, 1. In the following we will guarantee that the damage variable taes values in, 1, see Remar 3.1; one could also assume g C 1,1 R and constant in, and 1, as done for µ, the difference is that the terms involving g are constant in the incremental minimisation, see 3.1. For instance, if gα 1, then ζ is simply the elastic strain; if gα = µα, then ζ is the stress. By our assumption on the evolutions α, u, we have that ζ AC, T ; L 2 ; R 2, so we consider the corresponding cumulation V ζ t; x V t; x := defined as the Bochner integral in L 2. t ζs; x ds, x, 2.5 In 2.5 the notation represents the fact that we do not write in the following the dependence of the cumulated variable from ζ. We shall also use the notation V, V ε, etc. for the cumulated variable corresponding to ζ, ζ ε, etc. given by 2.4 for α and u, α ε and u ε, etc., respectively, specifying the correspondence in each case. We notice that one could consider other possible choices for the variable ζ, for which the results of this paper still hold. For instance, one could tae ζt = gα u θ, with θ 1, p, so ζ AC, T ; L p/θ see also the observations in Proposition 3.5 and Lemmas 4.3 and 4.4. This covers, e.g., the case where ζ is the density of the elastic energy, i.e., when gα = µα and θ = 2. We denote by H 1 the functions β H 1 with β a.e. in. For every measurable function V :, +, playing the role of the cumulation of ζ, and for every β H, 1 representing the damage

7 FATIGUE EFFECTS IN ELASTIC MATERIALS 7 rate, we define the corresponding dissipation potential by Rβ; V := fv β dx, 2.6 where f :, +, + is a Lipschitz, nonincreasing function with f >. The regularity assumptions on f, g are used in the proof of Proposition 3.5 see 3.14, and in Lemmas 4.3 and 4.4. fv V Figure 1. Graph of the function f appearing in the dissipation potential. The higher the value of V, the smaller the weight fv in the damage dissipation. Recall that V plays the role of the cumulation of the variable ζ. According to the general theory of Rate-Independent systems 29, R naturally induces the following dissipation between two damage states α 1, α 2 H 1 with α 1 α 2 1 a.e. in Dα 1, α 2; V := Rα 2 α 1; V. 2.7 Remar 2.2. The dissipation potential R that we choose here slightly differs from the one proposed in the model of 5. In that paper, the dissipation potential features an additional term depending on the gradient of the damage variable. More precisely, using the notation of our paper, a choice more coherent with 5 would be R α, α, α; V = fv α + α α dx. We explain here two reasons that lead us to the decision of not including the term α α in the dissipation potential. The first reason is a mathematical one. Note that a generic evolution αt may not satisfy the inequality α + α α ; the validity of this condition is however crucial for a physically consistent notion of dissipation potential. Our approach to the problem does not guarantee the a priori fulfilment of this condition. The second reason is a modelling one. The model proposed in 5 is an approach to fatigue fracture via a phasefield model. In a classical phase-field model without fatigue, the energy dissipated by a fracture is approximated by an energy of the form 1 α α 2 dx, and in that case the term 1 2 α 2 dx should be interpreted as part of the dissipation. This explains why in 5 the rate of 1 2 α 2 appears in the definition of R α, α, α; V and the fatigue weight fv also affects this term. In contrast, our aim is to study damage models, whereas the approximation of fracture via damage is not in the scope of this paper. For this reason as already done in other papers about damage models 28, 7, 41, 4, 24, 25, 26 we interpret 1 2 α 2 dx as part of the internal energy of the system. In particular, the rate of 1 2 α 2 does not appear in the definition of the dissipation potential.

8 8 ROBERTO ALESSI, VITO CRISMALE, GIANLUCA ORLANDO Boundary conditions and initial data. For every α H 1 with α 1 a.e. in and for every w H 1, the set of admissible pairs α, u with respect to the damage variable α and the boundary datum w is defined by: A α, w := {α, u H 1 H 1 : α α a.e. in, u = w on D}. The quasistatic evolution will be driven by a boundary datum satisfying w H 1, T ; W 1, p, 2.8 where p > 2 is a suitable exponent that is chosen according to Lemma 3.3. The p integrability of w is needed to control the increments of the displacement with those of the damage variable cf. Lemma 3.3. The H 1 regularity in time of the boundary datum is needed for the proof of the a priori bounds in Proposition 3.5 see We prescribe initial conditions α H 1 and u W 1, p at time t =. We assume, consistently with 2.5, that the initial cumulation V = for notation simplicity. Taing a generic initial cumulation V L 2 with V a.e. in does not entail any mathematical difficulty. Note that, in that case, definition 2.5 should be modified accordingly by adding the initial cumulation V. We require αeα, u L Notice that one could also assume that α and u are stable with V 1 =, so that the Euler conditions in Lemma 3.4 hold for i = too. The assumption 2.9 is slightly more general, since, for instance, the initial condition α =, u = is always admissible, no matter whether it is stable or not. 3. Incremental minimum problems Construction of discrete-time evolutions. We fix a sequence of subdivisions t i i= of the interval, T, where t i := i T are equispaced nodes. We denote the step of the time discretisation by τ = 1. For notational simplicity, we omit the dependence of τ on and we use the symbol τ. Moreover, we fix ε >. We define the discretisation of the boundary datum w by w i := wt i, i =,...,. Let α := α, u := u, ζ := gα u, and V := V =. Assuming that we now α i 1 and V i 1, we define α i, u i as a solution to the incremental minimisation problem cf. 2.2, 2.6, 2.7 for the definition of E and D min { Eα, u + Dα, α i 1 ; V i 1 + ε 2τ α αi 1 2 L2 : α, u A αi 1, w i } 3.1 and we set ζ i := gα i u i and V i := V i 1 + ζ i ζ i 1 = i ζ j ζj 1. The existence of a solution to 3.1 is obtained by employing the direct method of the Calculus of Variations. j=1 Remar 3.1. It is immediate to see that α i is a solution to the problem min { Eα, u i + Dα, α i 1 ; V i 1 + ε where u = u i is fixed. Notice that α i is also a solution to the problem min { Eα, u i + Dα, α i 1 2τ ; V i 1 + ε α αi 1 2 L 2 : α H1, α α i 1 1 }, 3.2 2τ α αi 1 2 L 2 : α H1, α α i 1 }, 3.3

9 FATIGUE EFFECTS IN ELASTIC MATERIALS 9 where also competitors α with negative values are taen into account. Indeed, let us fix a competitor for the problem 3.3, namely α H 1 ; R with α α i 1 α + is a competitor for 3.2, the assumption 2.1, and the fact that α i 1 Eα i, u i + Dα i, α i 1 and let us set α + := max{α, }. We employ the fact that ; V i 1 + ε 2τ αi α i 1 Eα +, u i + Dα +, α i 1 Eα, u i + Dα, α i 1 This proves the equivalence between 3.2 and L 2 to obtain ; V i 1 + ε 2τ α+ α i 1 ; V i 1 + ε 2τ 2 L 2 α αi 1 2 L 2. We define the upper and lower piecewise constant interpolations by t t := t i, α t := α i, u t := u i, ζ t := ζ i, w t := w i, t t := t i 1, α t := α i 1, u t := u i 1, ζ t := ζ i 1, w t := w i 1, and V t := V i 1 for t t i 1, t i, for i = 1,..., and α := α, u := u, V := V =, while t T := T, α T := α, u T := u. Moreover, we consider the piecewise affine interpolations defined by for i = 1,...,, where α t := α i 1 u t := u i 1 ζ t := ζ i 1 + t t i 1 α i, + t t i 1 u i, + t t i 1 ζ i, for t t i 1, t i, α i := αi αi 1, u i τ := ui ui 1, ζi τ := ζi ζi 1 τ, and define w as the affine interpolation in time of w. We set also It is not difficult to verify that Proposition A.4 yields in the sense of Bochner integral in L 2. V t := V t + t t t ζ t ζ t τ t. 3.4 V t = t ζ s ds 3.5 Note that in the above definitions we dropped the dependence on ε for notation simplicity. A priori bounds on discrete-time evolutions. We start the analysis of the discrete evolutions by deducing higher integrability properties of the strain. Following the idea of previous papers see, e.g., 24, we apply a result proved in 22, Theorem 1 see also 23, Theorem 1.1 for an extension to the case of elliptic systems with the symmetric gradient in place of u regarding the integrability of solutions to elliptic systems with measurable coefficients and with mixed boundary conditions.

10 1 ROBERTO ALESSI, VITO CRISMALE, GIANLUCA ORLANDO Remar 3.2. By 22, Theorem 1, there exist a constant C > and p > 2 depending on µ L such that the following property is satisfied: for every α H 1, for every p 2, p, and for every l W 1,p D, the wea solution v W 1,p to the problem div µα v = l in, v = on D satisfies v W 1,p C l 1,p W. D In the following lemma we apply the regularity given by Remar 3.2 to deduce higher integrability of u t and to control the increments of the displacement u with the increments of the damage variable α. Lemma 3.3 Higher integrability of the strain. There exist p > 2 depending only on µ L and a constant C > depending only on µ L, µ L, and w L,T ;W 1, p such that u t W 1, p + u t W 1, p + u t W 1, p C, for t, T 3.6a u t W 1,p C α t L q + ẇ t W 1, p, for t, T \ {t 1,..., t 1 }, 3.6b for every p 2, p, where q = p p p p. Proof. Let p > 2 be the exponent given in Remar 3.2. To prove 3.6a, let us fix t t i 1, t i for i {1,..., } notice that the inequality is trivial for t =. By 3.1, the function u i minimises Eα i, u among all u H 1 with u = w i on D. Therefore u i is a wea solution to the problem div µα u i i = in, By Remar 3.2, we have that u i = w i on D. u i w i W 1, p C div µα i w i W 1, p D C µ L w i W 1, p, which implies 3.6a recall the definition of u, u, u in terms of the family of u i. To prove 3.6b, let us fix p 2, p and t t i 1, t i for t {1,..., }. By 3.7 for i and i 1 we get that the function v := u i u i 1 where l := div µα i 1 µα u i i w i + w i 1 is a wea solution to the problem div µα i 1 v = l in, div µα i 1 v = on D, w i w i Notice that l W 1,p by 3.6a. By Remar 3.2 and by Hölder s inequality we deduce that v W 1,p C l 1,p W C µα i 1 µα u i i L p + µα i 1 w i w i 1 L p D C µ L α i α i 1 L q u i L p + µ L w i w i 1 since 1 = 1. By 3.6a and dividing by τ we conclude that q p 1 p u i W 1,p C α i L q + ẇ i W 1, p, W 1, p, D hence the thesis.

11 FATIGUE EFFECTS IN ELASTIC MATERIALS 11 We are now in a position to derive the Euler conditions satisfied by the damage variable in the discrete evolutions. These conditions are also called Discrete Karush-Kuhn-Tucer conditions, since we have a constraint of unidirectionality on the damage variable. They are a fundamental ingredient to deduce the a priori bounds in Proposition 3.5. Lemma 3.4 Euler conditions. For every t, T \ {t 1,..., t 1 } we have αeα t, u t, β + Rβ; V t + ε α t, β L for every β H 1 such that β a.e. in. Moreover αeα t, u t, α t + R α t; V t + ε α t 2 L 2 =. 3.1 Proof. Let us fix t t i 1, t i for some i {1,..., }. Let β H 1 with β a.e. in and let δ >. Since α i solves 3.3 and α i + δβ α i 1, we get Eα i + δβ, u i + Dα i + δβ, α i 1 Eα i, u i Dα i, α i 1 Dividing by δ and letting δ +, by 2.3 we get 1 µ α i u i 2 β dx + α i β dx 2 This concludes the proof of 3.9. ; V i 1 + ε 2τ αi + δβ α i 1 2 L 2+ ; V i 1 ε 2τ αi α i 1 2 L 2. fv i 1 β dx + ε α i β dx. To prove 3.1, notice that α i δ α i α i 1 for < δ < τ. Since α i solves 3.3 we get that Eα i δ α i, u i + Dα i δ α i, α i 1 Eα i, u i Dα i, α i 1 ; V i 1 + ε 2τ αi δ α i α i 1 2 L 2+ ; V i 1 ε 2τ αi α i 1 2 L 2. Dividing by δ and letting δ +, by 2.3 this implies 3.1. The following proposition ensures that the evolution of α and u is H 1 in time uniformly in for fixed ε, and AC in time uniformly in and ε, with values in the target spaces H 1 and W 1,p. Proposition 3.5 A priori bounds. Let p be as in Lemma 3.3. There exists a positive constant C independent of ε,, and t such that for every ε >, N, t, T \ {t 1,..., t 1 }, p < p, it holds that ε τ t ε α t L 2 C exp τ α s 2 H 1 ds + ε t u s 2 W 1,p ds C exp T α s H 1 ds + T C τ t ε C τ t ε, 3.11, 3.12 u s W 1,p ds C Proof. We only need to show the estimates on α t, since the estimates on u t simply follow from 3.6b. We start with computations which are common in the proofs of all the three inequalities in the statement. The starting point is to obtain an estimate on the time increments of α t by testing the Euler equations at two subsequent times of the time discretisation. To do so, we fix i {2,..., }. The case i = 1 requires slightly different arguments. By 3.1 evaluated at a time t t i 1, t i we get that αeα i, u i, α i + R α i ; V i 1 + ε α i 2 L 2 =.

12 12 ROBERTO ALESSI, VITO CRISMALE, GIANLUCA ORLANDO On the other hand, by testing 3.9 with β = α i at a time t t i 2, t i 1, we get αeα i 1, u i 1, α i + R α ; i V i 2 + ε α i 1, α i L 2. Subtracting the second inequality from the first one, we infer that αeα i, u i αeα i 1, u i 1, α i + R α ; i V i 1 R α ; i V i 2 + ε α i α i 1, α i L 2, namely, ε α i α i 1, α i L 2 + α i α i 1 α i dx 1 µ α i 1 u i 1 2 µ α i u i 2 α i dx u µ α i 1 i 1 2 u i 2 α i dx f L V i 1 V i 2 α i dx fv i µ L u i + u i u i u i 1 α i dx µ L + f L ζ i 1 ζ i 2 α i dx fv i 2 α i dx µ α i 1 µ α i u i 2 α i dx u i Cτ + u i 1 L 2 u i L p α i L q 1 + u i 2 α L i 2 p L q 1 + u i 1 L p α i 1 L q 1 α i L q 1 + u i 1 L p α i L q 1. α i α i 1 u i 2 α i dx 3.14 In the last inequality we have chosen p 2, p and q 1 2, such that 1 p + 1 q 1 the identity ζ i 1 ζ i 2 = gα i 1 gα i 2 u i 1 + gα i 2 u i 1 u i 2 = 1, and we have employed 2 that gives ζ i 1 ζ i 2 α i dx τ g L u i 1 L p α i 1 L q 1 α i L q 1 + g L u i 1 L p α i L q We remar that, taing ζ i := gα i u i θ, with θ 1, p we could also get the conclusion in 3.14 with q 1 q 1 such that θ + 1 = 1, in place of p q 1 q1. Indeed 2 ζ i 1 ζ i 2 = gα i 1 gα i 2 u i 1 θ + gα i 2 u i 1 θ u i 2 θ, and since, by the Mean Value Theorem, u i 1 θ u i 2 θ θ u i 1 + u i 2 θ 1 u i 1 u i 2, we have that ζ i 1 ζ i 2 α i dx τ g L u i 1 L p α i 1 q L 1 αi q L 1 + g L θ u i 1 L p + u i 2 L p u i 1 L p α i q L

13 FATIGUE EFFECTS IN ELASTIC MATERIALS 13 Using the fact that α i α i 1, α i L 2 α i L 2 α i L 2 α i 1 L 2 and by Lemma 3.3 we infer that ε α i L 2 Cτ c 1τ α i L 2 α i 1 L 2 + τ α i 2 L 2 α i L q 1 α i L q 2 + α i 2 L q 1 + α i L q 1 α i 1 α i 2 L r + αi L r α i 1 L r + ẇ i 2 W L q 2 + α i L q 1 ẇ i W 1, p + ẇ i 1 1, p + ẇi 1 2 W 1, p, W 1, p 3.17 where q 2 := p p p p 2, and r = max{q 1, q 2} 2,. We labelled the constant in the last inequality with c 1 in order to eep trac of it in the sequel. By the compact embedding H 1 L r notice that R 2, we have that for every δ > there exists a constant C δ > such that for every β H 1 β 2 L r δ β 2 L 2 + C δ β 2 L 1 δ β 2 L 2 + C δ β L 1 β L Adding c 1 τ α i 2 L r c1 τ αi L r α i 1 L r + τ 2 αi 2 L2 to both sides of 3.17, choosing δ suitably small in the previous inequality applied to β = α i, and multiplying by 2τ we have that ε 2 α i L 2 α i L 2 α i 1 τ L 2 + 2c 1 ε αi L r α i L r α i 1 Let us set c 2 τ ε A i := C i := ẇ i 2 W 1, p + ẇi 1 2 τ W 1, p + 2c 2 ε αi L 1 α i L 2. L r + τ ε αi 2 H 1 α i 2 L 2 + c1 τ ε αi L r τ, B i := 2ε αi H 1, c 2 τ ε ẇ i 2 W 1, p + ẇi τ W 1, p, D i := c 2 ε αi L The quantities above are actually defined for every i = 1,...,. When i = 1, we define C 1 := c 2 τ ε ẇi W 1, p. Denoting by a i := α i L 2, τ c 1 ε αi L r, we get that α i L 2 α i L 2 α i 1 τ L 2 + c 1 ε αi L r Since τ ε we have that a i a i a i 1 = A ia i A i 1. α i L r α i 1 L r = a i a i a i τ 2ε αi 2 H 1 = τ 4ε αi 2 H 1 + τ 4ε αi 2 H 1 τ 4ε αi 2 L 2 + Cτ 4ε αi 2 L r τ 4ε αi 2 L 2 + Cτ 2 α 4ε i 2 2 L r α i 2L 2 + c1 τε αi 2 L r Cτ ε = 2c 3 τ ε A2 i. τ Collecting and setting γ := c 3, we obtain that ε 3.21 for every i = 2,...,. 2A ia i A i 1 + 2γA 2 i + B 2 i C 2 i + 2A id i, 3.22 Proof of estimate Here we prove a slightly stronger inequality with an additional term on the left-hand side. Specifically, we show that ε α t 2 L 2 + c1 τ α C ε t L r C exp τ ε t By the inequalities 2A ia i A i 1 A 2 i A 2 i 1 and D i C τ Ai, from 3.22 we get in particular that ε A 2 i A 2 i 1 + B 2 i C 2 i + C τ ε A2 i, for i = 2,...,. We fix h {2,..., } and we sum the inequality above for i = 2,..., h, deducing that εa 2 h εa Bi 2 ε Ci 2 + C τa 2 i. 3.24

14 14 ROBERTO ALESSI, VITO CRISMALE, GIANLUCA ORLANDO We claim that εa 2 1 C εc1 2 + τa ε Once 3.25 is proven, summing 3.24 and 3.25, by the initial assumption on w 2.8 we conclude that εa 2 h C 1 + τ ẇ i ε 2 W i=1 1, p + ẇi 1 2 W 1, p + i=1 τa 2 i C 1 + 1ε + for every h = 1,...,. By a discrete Gronwall inequality on εa 2 h we deduce that εa 2 h C exp C th ε ε for every h = 1,...,. Multiplying by ε and taing the square root, we get εa h C exp C th ε i=1 τa 2 i and thus It remains to prove Adding and subtracting αeα, u to 3.1 evaluated at time t, t 1, we deduce that αeα 1, u 1 αeα, u, α 1 + R α 1, V + αeα, u, α 1 + ε α 1 2 L 2 =. With computations similar to those previously done in and using the assumption αeα, u L 2, we infer that ε α 1 2 L 2 + τ α1 2 L 2 Cτ u 1 + u L 2 u 1 L p α 1 L q 1 + u 1 2 α 1 L 2 p L q 1 + f L α 1 L 1 + αeα, u L 2 α 1 L 2 Cτ ẇ 1 2W 1, p + α1 2 L r + ε 2 α1 2 L 2 + C. ε Using inequality 3.18 as above, it is not difficult to see that which in turn implies ε α 1 2 L 2 + τ α1 2 H 1 Cτ ẇ 1 2 W 1, p + α1 2 L ε Proof of estimate Inequalities 3.24 and 3.25 imply in particular that, 3.29 From 3.29 we deduce that B 2 i ε Ci 2 + C τa 2 i + C εc1 2 + τa ε εb1 2 C εc1 2 + τa ε 3.31 Let us fix h {1,..., }. Summing 3.3 and 3.31, by 2.8 we obtain that ε i=1 B 2 i C 1 + τ ẇ i ε 2 W i=1 and thus, multiplying by ε and using 3.27, ε i=1 1, p + ẇi 1 2 W 1, p + i=1 τa 2 i τ α i 2 H 1 C exp C th ε. C 1 + 1ε + In the equality above we have integrated the exponential function in time and we have used the fact that τ << ε. This concludes the proof of i=1 τa 2 i

15 FATIGUE EFFECTS IN ELASTIC MATERIALS 15 Proof of estimate every h = 2,..., By the discrete Gronwall estimate proved in 24, Lemma 4.1 we deduce that for 1 + γ 2i h 1 Bi γ 2h A γ 2i h 1 Ci γ i 1 D i 21 + γ 2h A γ 2i h 1 Ci γ i 1 2 D i 21 + γ h A γ 2i h 1 Ci γ i 1 D i 3.32 Using the estimate γ 1 + γ 2i h 1 = γ1 + γ 2h γ γ 2h+2 = 1 + γ γ 2h+2 1, γ γ by the Cauchy-Schwarz inequality we estimate the left-hand side of 3.32 by γ 1 + γ 2i h 1 α i H 1 = γ1 + γ 2i h γ1 + γ 2i h 11 2 α i H 1 C 1 + γ 2i h 1 Bi 2 1 2, for h = 2,...,. Hence 3.32 reads γ 1 + γ 2i h 1 α i H 1 C γ h A 1 + γ 1 + γ 2i h 1 ẇ i 2 W 1, p + ẇi 1 2 W + γ 1, p 1 + γ i 1 α i L 1, 3.33 for h = 2,...,. We multiply both sides of 3.33 by τ and we sum over h = 2,...,. Using the expression of the partial sums of the geometric series, it is possible to show that τ α i H 1 C 1 + εa 1 + τ ẇ i 2 W 1, p + ẇi 1 2 W + 1, p τ α i L We refer to 24, Proposition 4.3 or 9, Proposition 3.8 for more details about the computations mentioned above. Multiplying 3.29 by τ, taing the square root and using the fact that τ << ε, we infer that τ α 1 H 1 Cτ ẇ 1 W 1, p + α 1 L Adding this last inequality to 3.34 we obtain that i=1 τ α i H 1 C 1 + εa 1 + τ ẇ i 2 W i=1 1, p + ẇi 1 2 W + 1, p τ α i L 1 To conclude the proof of 3.13, we observe that: εa 1 C by 3.28 evaluated for h = 1; the second sum is bounded by a constant by the initial assumption on w 2.8; the third sum is actually a telescopic sum, namely i=1. τ α i L 1 = i=1 α α dx.

16 16 ROBERTO ALESSI, VITO CRISMALE, GIANLUCA ORLANDO In order to obtain the energy dissipation balance for the evolution α, u, in Proposition 3.7, we integrate in time the energy evaluated on these affine interpolations. We are allowed to do so because they are absolutely continuous actually H 1 in time. Since we also employ the Euler equation 3.1 of Lemma 3.4, that contains also the piecewise constant interpolations, we have to estimate the difference of the piecewise affine and constant interpolations. This is done in the following remar. Remar 3.6. For every t, T α t α t H 1 = and therefore, by 3.12, t t t t α s ds α s H 1 H 1 ds τ 1 2 α H 1,T ;H 1 t t α α L,T ;H 1 C ετ a Similarly, we have α α L,T ;H 1 C ετ 1 2, 3.35b u u L,T ;W 1,p C ετ 1 2, for p 2, p, 3.35c u u L,T ;W 1,p C ετ 1 2, for p 2, p. 3.35d Discrete energy-dissipation balance. Here we obtain the energy-dissipation balance, by employing the Euler condition 3.1, correcting with the piecewise affine interpolations in place of the piecewise constant ones. Proposition 3.7 Discrete energy-dissipation balance. Eα T, u T + T = Eα, u + where R as +. R α t; V t dt + ε T T α t 2 L 2 dt µα t u t, ẇ t L 2 dt + R, 3.36 Proof. By , the piecewise affine interpolations α t and u t are absolutely continuous in t. As a consequence, t Eα t, u t is absolutely continuous and for a.e. t, where d Eα t, u t = αeα t, u t, α t + ueα t, u t, u t dt = αeα t, u t, α t + ueα t, u t, u t + η t 3.37 η t := αeα t, u t αeα t, u t, α t + ueα t, u t ueα t, u t, u t Using u t ẇ t as test function in 3.7, we deduce that ueα t, u t, u t = ueα t, u t, ẇ t = µα t u t, ẇ t L 2 Together with the Euler equation for α t 3.1 and 3.37, this gives d Eα t, u t = ε α t 2 L dt 2 R α t; V t + µα t u t, ẇ t L 2 + η t 3.39 Integrating in time the previous equality, we obtain 3.36 with R := T η t dt.

17 FATIGUE EFFECTS IN ELASTIC MATERIALS 17 Let us show that R. By Hölder s Inequality, by 3.35a, by 3.6a, and by 3.13 we deduce that T µ α t µ α t T u t 2 α t dx dt C α t α t u t 2 α t dx dt C T α t α t H 1 u t 2 W 1,p α t H 1 dt C ετ 1 2 Furthermore by Hölder s Inequality, by 3.6a, by 3.35c, and by 3.13 we infer that T µ α t u t 2 u t 2 T α t dx dt C u t + u t u t u t α t dx dt C T u t + u t W 1,p u t u t W 1,p α t H 1 dt C ετ 1 2. Finally, by 3.35a and 3.13 we get that T α t α t α t dx dt This shows that T lim + T α t α t H 1 α t H 1 dt C ετ 1 2. αeα t, u t αeα t, u t, α t dt =. With completely analogous computations it is not difficult to show that This concludes the proof. T lim + ueα t, u t ueα t, u t, u t dt =. We observe that the energy balance 3.36 holds for any couple of times t 1 < t 2, T, as one can see arguing as in Proposition 3.7 and integrating 3.39 in the time interval t 1, t Existence of viscous evolutions In this section we pass to the limit as + i.e., as the time-step goes to zero. Notice that ε > is fixed in this section. The main result is the existence of viscous evolutions, defined as follows. Given α ε AC, T ; H 1, u ε AC, T ; H 1 we define, as in 2.4, ζ ε := gα ε u ε and, as in 2.5, V εt := t ζεs ds, 4.1 as a Bochner integral in L 2. During the section we are in the constitutive assumptions of Section 2. Definition 4.1. We say that a function α ε, u ε:, T H 1 W 1,p is an ε-approximate viscous evolution if α ε H 1, T ; H 1, u ε H 1, T ; W 1,p and the following conditions are satisfied: ev ε irreversibility:, T t α εt is nonincreasing as a family of measurable functions on, ev1 ε that is α εt α εs a.e. in for all s t; equilibrium: for every t, T, u εt H 1 is a wea solution to the problem div µα εt u εt = in, u εt = wt on D. 4.2

18 18 ROBERTO ALESSI, VITO CRISMALE, GIANLUCA ORLANDO ev2 ε Karush-Kuhn-Tucer inequality: for a.e. t, T and for every β H 1 with β a.e. in we have αeα εt, u εt, β + Rβ; V εt + ε α εt, β L ev3 ε energy balance: Eα εt, u εt + T R α εt; V εt dt + ε T α εt 2 L 2 T dt = Eα, u + µα εt u εt, ẇt L 2 dt. All the section is devoted to the proof of the result below. Theorem 4.2. Let p > 2 be given by Lemma 3.3. For every ε > and p < p there exists an ε-approximate viscous evolution α ε, u ε with α ε, u ε = α, u and there is a constant C >, independent of ε, such that T α εs H 1 ds + T u εs W 1,p ds C. 4.4 The strategy of the proof consists in showing first the existence of a wea form of ε-approximate viscous evolution. This satisfies the conditions ev ε, ev1 ε, and the ev2 ε, ev3 ε with a different expression of dissipation Propositions 4.7, 4.8, and 4.1. Such a wea existence result allows us to improve, for fixed ε, the a priori convergences of the discrete-time evolutions Proposition 4.11 and to express the dissipation in terms of V εt, the cumulation of ζ ε cf. 4.1, so recovering its form in Definition 4.1, by Lemma 4.4. Compactness. We start by exploiting the a priori bounds found in Proposition 3.5 to deduce compactness of the discrete-time evolutions. By 3.12 we find a subsequence which we do not relabel such that α α ε wealy in H 1, T ; H 1, 4.5 u u ε wealy in H 1, T ; W 1,p, for p 2, p, 4.6 as +. Actually, we also extract a subsequence independent of t such that the convergence in 4.2 below holds. We do not state this here for the sae of clarity in the presentation. By the compact embeddings H 1 L q and W 1,p L p, by the Aubin-Lions lemma 6, and by 3.35 we deduce that α α ε C,T ;L q, α α ε L,T ;L q, α α ε L,T ;L q, for q 1,, u u ε C,T ;L p, u u ε L,T ;L p, u u ε L,T ;L p, for p 2, p. 4.7a 4.7b Moreover, from the inequality α L,T ;H 1 C 1+ α W 1,1,T ;H 1 and by 3.6a and 3.35a 3.35d we deduce that for every t, T we also have α t, α t, α t α εt wealy in H 1, 4.8 u t, u t, u t u εt wealy in W 1, p. 4.9 In particular, for every s t we have α εt α εs a.e. in. Moreover, for every t, T we have In view of the convergences 4.5, 4.6, by 3.13 we get T u εt W 1, p lim inf + u t W 1, p C. 4.1 α εs H 1 ds + where C is independent of ε, and then V ε is well defined as in 4.1. T u εs W 1,p ds C, 4.11

19 FATIGUE EFFECTS IN ELASTIC MATERIALS 19 Energy-dissipation balance and stability. In this subsection we pass to the limit as + in the discrete energy-dissipation balance We start by discussing the easiest terms in the energy-dissipation balance, namely the terms involving the energy, the viscous dissipation, and the wor done by the boundary forces. The dissipation involving the fatigue term requires finer techniques and will be discussed below. From the pointwise convergences and the lower semicontinuity of the energy E with respect to the wea convergence of α in H 1 and the wea convergence of u in W 1,p we deduce that Moreover, since α α ε We claim that T lim + Eα εt, u εt lim inf + Eα T, u T wealy in L 2, T ; L 2, we have that ε T α εt 2 L2 dt lim inf + µα t u t, ẇ t L 2 dt = ε T T α t 2 L 2 dt µα εt u εt, ẇt L 2 dt To show the convergence above, first of all we notice that µα t u t µα εt u εt wealy in L 2 ; R 2 for every t, T thans to In addition, 3.6a and assumption 2.8 imply µα t u t, ẇ t L 2 C T u t H 1 ẇ t H 1 dt C. Since ẇ t ẇt strongly in L 2 ; R 2 for a.e. t, T, by the Dominated Convergence Theorem the convergence in 4.14 holds true. We consider now the limit of the dissipation involving the fatigue term. We start with the following lemma, which shows that the affine interpolation of the cumulation is close to the piecewise constant interpolation. Lemma 4.3. For every N, ε > we have that fv fv L 2,T ;L 2 Cτ α H 1,T ;L 2 + u H 1,T ;H 1 C ε τ Proof. By 3.4 and 3.6a we have V t V t = t t t τ gα t gα t u t + gα t u t u t, so that V t V t τ g L α t u t + gα t u t Cτ α t + u t. Thus for any β L 2, T ; L 2 recall that f is Lipschitz T T fv t fv t βt dx dt Cτ α t + u t βt dx dt Cτ Cτ T α t L 2 + u t H 1 βt L 2 dt α H 1,T ;L 2 + u H 1,T ;H 1 β L 2,T ;L 2. Recalling 3.12, the estimate above gives We notice that we arrive at the same conclusion also with ζ defined by gα u θ, for θ 1, p, arguing similarly to what done to pass from 3.15 to 3.16 in Proposition 3.5.

20 2 ROBERTO ALESSI, VITO CRISMALE, GIANLUCA ORLANDO In the following lemma we show that a strong convergence of the discrete-time evolutions would guarantee the convergence of the dissipation term. We stress that the a priori bounds on u t found in Proposition 3.5 only guarantee the wea convergence 4.6. Therefore we are not allowed to apply Lemma 4.4 at the moment. Lemma 4.4. Assume that the following convergences for α and u hold true: α α ε strongly in W 1,1, T ; L 2, 4.16a u u ε strongly in W 1,1, T ; W 1,p, for p 2, p. 4.16b Then and fv fv ε strongly in L 2, T ; L 2, 4.17 T lim + R α t; V t dt = where the cumulations V and V ε are defined in 3.4 and 4.1, respectively. T R α εt; V εt dt, 4.18 Proof. For the proof it is convenient to introduce the function gβ + h gβ, if h, dgβ, h := h g β, if h =, for every β, h R. Observe that gβ + h = gβ + h dgβ, h and since g C 1,1 R dgβ, h g β g L h. Using the function dg, we can write for every s, T ζ s = dgα s, τ α s α s u s + gα s u s. We now estimate V V ε V t; x V εt; x dx t t by employing 3.5 and 4.1. For every t, T we have t ζ s; x ζ εs; x dx ds dgα s, τ α s α s u s + gα s u s g α εs α εs u εs gα εs u εs dx ds τ g L α s 2 u s + g L α s α εs α s u s + g L α s α εs u s + g L αεs u s u εs + g L α s α εs u s + g L u s u εs dx ds C α α ε L,T ;L q α W 1,1,T ;L q + α α ε W 1,1,T ;L 2 u L,T ;W 1,p + C τ + u u ε L,T ;W 1,p αε W 1,1,T ;L q + C α α ε L,T ;L q u W 1,1,T ;W 1,p + C u u ε W 1,1,T ;W 1,p, for q 2, such that 1 q + 1 p < 1 2.

21 FATIGUE EFFECTS IN ELASTIC MATERIALS 21 Notice that we obtain the above inequality also if ζ ε = gα ε u ε θ, with θ 1, p, up to consider q > q with 1 q + θ p < 1 2 in the estimates of α, since d u θ = θ u θ 2 u u. dt Let us now integrate in time the inequality obtained above for V V ε : using 3.6, 3.13, 3.35, 4.7, and 4.16 we deduce that and then we get 4.17, since f is bounded. V V ε L 1,T ;L 1, Moreover, by wea convergence α α ε in L 2, T ; L 2 T T T R α t; V t dt = fv t α t dx dt fv εt α εt dx dt = and, by 4.15, T T R α εt; V εt dt R α t; V t R α t; V t dt fv fv L 2,T ;L 2 α H 1,T ;L 2 C ετ, as +. This concludes the proof. Remar 4.5. Combining 4.15 and 4.17 we obtain that if 4.16b holds, then fv fv ε strongly in L 2, T ; L At the moment we do not have convergence 4.16b at our disposal, and we cannot deduce that the convergence of the functions fv t to fv εt. For this reason, in the following lemma we consider an additional variable f εt in the limit evolution, which later in the proof will turn out to be fv εt. Lemma 4.6 Compactness for the cumulated variable. For every ε > there exist a nonincreasing function t f εt L and a subsequence independent of t which we do not relabel such that for every t, T. fv t f εt wealy* in L, 4.2 Proof. To prove the lemma we apply the generalized version of the classical Helly Theorem given in 18, Helly Theorem in the space M b. For every t, T, the sequence fv t is equibounded in L, and thus is relatively compact in M b with respect to the wea* convergence. Moreover, the functions fv have uniformly bounded variation in M b. Indeed, for s t we have fv t fv s and thus, given a partition = s < < s m = T, we get m fv s j fv s j 1 dx = fv fv T dx f L. j=1 On the one hand, by 18, Helly Theorem we deduce that there exists a subsequence independent of t which we do not relabel and a function t λ t M b such that fv tl 2 λ t wealy* in M b On the other hand, for every t, T there exists a function f εt L and a subsequence jt depending on t such that fv j tt f εt wealy* in L. 4.22

22 22 ROBERTO ALESSI, VITO CRISMALE, GIANLUCA ORLANDO By 4.21 and 4.22 we conclude that λ t = f εtl 2 and the convergence in 4.22 holds on the whole subsequence where 4.21 is satisfied. Notice that f εt is nonincreasing in t. The first step is to deduce the existence of an evolution where the fatigue term fv εt is in fact replaced by the term f εt. We first prove one inequality in the energy-dissipation balance for the continuous-time evolutions. The opposite inequality will follow automatically from the differential conditions satisfied by α ε, see Proposition 4.1 below. Proposition 4.7 Energy-dissipation balance in wea form: first inequality. For every ε > we have T T T Eα εt, u εt f εt α εt dx dt + ε α εt 2 L 2 dt Eα, u + µα εt u εt, ẇt L 2 dt Proof. In order to prove 4.23, we write the dissipation with the fatigue term as a supremum of finite sums which are continuous with respect to the convergence 4.2. Specifically T { m } R α t; V t dt = sup fv s jα s j 1 α s j dx, 4.24 =s < <s m=t j=1 where the supremum is taen among all possible partitions = s < < s m = T, m N, of the interval, T. The supremum is in fact attained on the partition = t < < t = T. To chec this, let us fix a partition = s < < s m = T and let us prove that m fv s jα s j 1 α s j dx j=1 i=1 T = fv t i α t i 1 α t i dx fv t α t dx dt Note that if we refine the partition = s < < s m = T by including the nodes t,..., t, the dissipation increases, since the monotonicity of fv and of α yields the following triangular inequality: fv r 3α r 1 α r 3 dx fv r 2α r 1 α r 2 dx + fv r 3α r 2 α r 3 dx for r 1 r 2 r 3 T. Therefore we can assume without loss of generality that {t,..., t } {s,..., s m}. Let us now fix i {1,..., } and 1 h i < l i m such that t i 1 = s hi < < s li = t i. Then the sum in in the left-hand side of 4.25 can be rearranged as l i i=1 j=h i = fv s jα s j 1 α s j dx = i=1 fv i 1 ti t i 1 τ α i 1 α i dx = l i i=1 j=h i i=1 fv i 1 sj sj 1 α i 1 α i dx τ fv t i α t i 1 α t i dx. Now we pass to the limit in 4.24 as +. Let us fix a partition = s < < s m = T and let us fix j {,..., m}. By 4.8 we have in particular that α s j α εs j and α s j 1 α εs j 1 strongly in L 1 and therefore, by 4.2, we obtain that fv s jα s j 1 α s j dx f εs jα εs j 1 α εs j dx 4.26 as +.